diff --git "a/2dA0T4oBgHgl3EQfM_90/content/tmp_files/load_file.txt" "b/2dA0T4oBgHgl3EQfM_90/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2dA0T4oBgHgl3EQfM_90/content/tmp_files/load_file.txt" @@ -0,0 +1,1384 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf,len=1383 +page_content='The average degree of edge chromatic critical graphs with maximum degree seven Yan Cao Scdool of Mathematical Sciences, Dalian University of Technology Dalian, Liaoning, 116024, China Email: ycao@dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='cn Rong Luo∗ Department of Mathematics, West Virginia University Morgantown, WV 26505 Email: rluo@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='wvu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='edu Zhengke Miao† School of Mathematics and Statistics, Jiangsu Normal University Xuzhou, Jiangsu, 221116, China Email: zkmiao@jsnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='cn Yue Zhao Department of Mathematics, University of Central Florida Orlando, FL 32816-1364 Email: Yue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='Zhao@ucf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='edu Abstract In this paper, by developing several new adjacency lemmas about a path on 4 or 5 ver- tices, we show that the average degree of 7-critical graphs is at least 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' It implies Vizing’s planar graph conjecture for planar graphs with maximum degree 7 and its extension to graphs embeddable in a surface with nonnegative Euler characteristic due to Sanders and Zhao (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' B 83 (2001) 201-212 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' B 87 (2003) 254-263) and Zhang (Graphs and Combinatorics 16 (2000) 467-495).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Keywords:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Edge coloring, critical graphs, Euler’s formula, planar graphs 1 Introduction An edge coloring of a graph is a function assigning values (colors) to the edges of the graph in such a way that any two adjacent edges receive different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A graph is edge k-colorable if there is an edge coloring of the graph with colors from C = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A finite simple graph ∗Partially supported by a grant from Simons Foundation (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 839830) †Partially supported by NSFC under grant numbers 12031018 and 11971205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='02140v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='CO] 5 Jan 2023 G of maximum degree ∆ is class one if it is edge ∆-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise, G is said to be class two, in which case Vizing’s Theorem [20] guarantees that it is edge (∆ + 1)-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' G is said to be edge chromatic critical (or critical for short) if it is connected, class two and χ′(G − e) < χ′(G) for every edge e ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A critical graph G of maximum degree ∆ is called a ∆-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Vizing proposed the following conjecture in 1968 [21] on the average degree of ∆-critical graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 Let G be a ∆-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(G) ≥ ∆ − 1 + 3 |V (G)|, where d(G) is the average degree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' There are direct consequences of a progress towards solving this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For example, if there is a better bound for the size of ∆-critical graphs, then one can obtain better bounds for ∆(S), where S is a surface and ∆(S) = max{∆(G)|G is a class two connected graph that can be embedded in S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' It is well known that if Vizing’s conjecture is true for ∆ = 7, then ∆(S) ≤ 6 where S is a surface of Euler characteristic at least 1, which was proved in [17] by other means in 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If this average degree conjecture is true, for a ∆-critical graph G, by applying the inequality α ≤ n− m ∆, where n = |V (G)|, m = |E(G)|, and α is the independence number of G, one can easily obtain α ≤ n 2 as ∆ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This provides a strong evidence for the independence number conjecture proposed by Vizing in 1968 [21], which claims that if G is a critical graph, then α ≤ n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 was verified for ∆ = 3 by Jakobsen [12], for ∆ = 4 by Fiorini and Wilson [10], for ∆ = 5 by Kayathri [13], and for ∆ = 6 by Luo, Miao and Zhao [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' As for the lower bound of d(G), Woodall [22] proved that if G is a ∆-critical graph, then d(G) ≥ 2(∆+1) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Cao and Chen [5] further improved to 3∆ 4 − 8 and they [5, 6] also showed that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 is asymptotically true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this paper, we will prove that if G is a 7-critical graph, then d(G) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This result implies Vizing’s planar graph conjecture for ∆ = 7 claiming that every planar graph with maximum degree at least 7 is class one, which was verified independently by Sanders and Zhao [17] and Zhang [23] and its extension to graphs embeddable in a surface with nonnegative Euler characteristic due to Sanders and Zhao in [17] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Before proceeding, we introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Throughout this paper, let G = (V, E) be a simple graph with n vertices, m edges, and maximum degree ∆(G) (or ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A k-vertex, k+-vertex, or k−-vertex is a vertex of degree k, at least k, or at most k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We use d(x), dk(x), dk+(x), dk−(x) to denote the degree of a vertex x, the number of k-vertices adjacent to x, the number of k+-vertices adjacent to x, and the number of k−-vertices adjacent to x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For a vertex v ∈ V , let N(x) = {v|xv ∈ E} be the neighborhood of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A k-neighbor of a vertex v is a neighbor of v that is a k-vertex in G, a k+-neighbor or k−-neighbor of a vertex v is a neighbor of v that is a k+-vertex or k−-vertex in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For two disjoint vertex sets U and U ′, denote by [U, U′] the set of edges with one end in U and the other in U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For a vertex set A of V (G), denote by N(A) = ∪x∈AN(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 2 2 Lemmas In this section, we present some old lemmas and develop some new lemmas needed in the proofs of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 Old lemmas Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 (Vizing’s Adjacency Lemma [20]) Let G be a ∆-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(u) + d(v) ≥ ∆ + 2 for any two adjacent vertices u and v, and d∆(x) ≥ max{2, ∆ − k + 1} if x has a k-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 (Luo, Miao, and Zhao [14]) Let G be a ∆-critical graph with ∆ ≥ 5 and x be a 3-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then x has at least two ∆-neighbors which are not adjacent to any (∆−2)−-vertices except x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3 (Luo, Miao, and Zhao [16]) Let G be a ∆-critical graph with ∆ ≥ 6 and x be a 3-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then x has a ∆-neighbor which is adjacent to at least ∆ − 4 − ⌊ ∆−1 3 ⌋ vertices z with d(z) = ∆ and d(∆−3)−(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4 (Sanders and Zhao [17] and Zhang [23]) Let G be ∆-critical graph and xyrs be a path with d(x) + d(y) = ∆ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(r) = ∆ and d(s) ≥ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Moreover if d(x), d(y) < ∆, then d(s) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='5 (Luo, Miao, and Zhao [14]) Let G be a ∆-critical graph with ∆ ≥ 6 and x be a 4-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (1) If x is adjacent to a (∆ − 2)-vertex, say y, then N(N(x)) \\ {x, y} ⊆ V∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (2) If x is not adjacent to any (∆ − 2)-vertex and if one of the neighbors y of x is adjacent to d(y) − (∆ − 3) vertices of degree at most ∆ − 2, then each of the other three neighbors of x is adjacent to only one (∆ − 2)−-vertex, which is x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (3) If x is adjacent to two (∆ − 1)-vertices, then each of the neighbors of x is adjacent to exactly one (∆ − 2)−-vertex, which is x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The following lemma is a special case of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4 in [17] due to Sanders and Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='6 Let G be a 7-critical graph and xyz be a path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If 3 ≤ d(x) ≤ 4, d(y) = 7 and d(x) + d(z) ≤ 8, then y and z have at most d(x) − 3 common neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 New lemmas The following lemmas will be proved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let G be a ∆-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For each vertex v, denote N∆∼2(v) = {z ∈ N(v) : z has a neighbor of degree 2} Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='7 Let G be a ∆-critical graph with ∆ ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then |N∆∼2(v)| ≤ 5 for every v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='8 Let G be a ∆-critical graph and xyrst be a path with d(x) + d(y) = ∆ + 2 and max{d(x), d(y)} < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(t) ≥ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='9 Let G be a ∆-critical graph and xyrst be a path with d(x) = 3 and d(y) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that y has a neighbor z ̸∈ {x, r, s} with d(z) ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(s) ≥ ∆ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' and d(z) + d(t) ≥ ∆ + 1 if d(t) ≤ ∆ − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 3 So far all adjacency lemmas are about a path on at most four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='9 is the first lemma that deals with a path with five vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='8, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='9, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='10 Let G be a 7-critical graph and xyrst be a path with d(x) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have the following: (1) if d(y) = 6, then d(r) = d(s) = 7 and d(t) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (2) if d(y) = 7 and y has another 4−-neighbor other than x, then d(s) ≥ 6 and d(t) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (3) if d(y) = 7 and y has a 5-neighbor, then either d(s) = 6 and d(t) ≥ 4 or d(s) = 7 and d(t) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='11 Let G be a ∆-critical graph and xy be an edge with d(x) + d(y) = ∆ + 3 and max{d(x), d(y)} < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then x has d(x)−2 neighbors of degree ∆ having no (∆−2)−-neighbors other than x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='12 Let G be a 7-critical graph and x be a 5-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (1) if x has three 6-neighbors, then each 7-neighbor of x has exactly one 5−-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (2) if x has two 6-neighbors, then x has two 7-neighbors, each of which has at most two 5−-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (3) if x has exactly four 7-neighbors, then x has two 7-neighbors, each of which has at most three 5−-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 3 The average degree of 7-critical graphs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 Main result In this section we will prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 d(G) ≥ 6 for every 7-critical graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let G be a 7-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We define the following subsets of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A = {u|d(u) = 7 and u is adjacent to a 2-vertex}, B = {u|d(u) = 6 and u is adjacent to a 3-vertex}, C = {u|d(u) = 7 and u is adjacent to a 3-vertex and a 5−-vertex}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The following proposition is straightforward from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='7 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 Let x be a 7-vertex which is not adjacent to a 5−-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then at most one of the three sets N(x)∩A, N(x)∩B, and N(x)∩C is a nonempty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Moreover |N(x)∩A| ≤ 5 and |N(x) ∩ B| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For each vertex x, denote by M(x) = d(x) − 6 to be the initial charge of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' R1 Let u be a 7-vertex not adjacent to a 5−-vertex but adjacent to a vertex in A ∪ B ∪ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then u sends 1 |N(x)∩A|+|N(x)∩B|+|N(x)∩C| to each neighbor in A ∪ B ∪ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' R2 Let u be a 7-vertex adjacent to a 5−-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then u sends 1 d5−(u) to each neighbor with degree 4 or 5, 1 to each 3-neighbor, and 2 to each 2-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' R3 Every 6-vertex sends 1 to each 3-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 4 R4 If a 5-vertex u is adjacent to a 7-vertex v ∈ C, then u sends 1 8 to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' R5 If a 4-vertex is adjacent to a 5-vertex, then the 4-vertex receives 1 2 from its 5-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote by M′(x) to be the new charge of the vertex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We have the following estimation for M′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (I) Let u be a vertex with degree 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then M′(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By (R2), each 2-vertex receives 2 from each neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, each 3-vertex is not adjacent to a 5−-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by (R2) and (R3), each 3-vertex receives 1 from each neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore M′(u) = 0 if d(u) = 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (II) Let uv be an edge with d(u) + d(v) = ∆ + 2 = 9 and 3 ≤ d(u) ≤ d(v) < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then M′(u) ≥ 0 and M′(v) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let w ∈ N(u)∪N(v) and w ̸∈ {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If w ∈ N(u)∩N(v), then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4, d(w) = 7, and w has only two 6−-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by (R2), w sends 1 2 to each of u and v if d(u) = 4 and d(v) = 5 and sends 1 to u, 0 to v if d(u) = 3 and d(v) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If w ̸∈ N(v) ∩ N(u), then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4, d(w) = 7 and w has only one 6−-neighbor, which is either u or v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If w ∈ N(u), then by (R2), w sends 1 to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Assume w ∈ N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If d(v) = 6, then v ∈ B, and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, w sends 1 to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If d(v) = 5, then N(w) ∩ (A ∪ B ∪ C) = ∅ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='8 and thus w sends 1 to v by (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore in any case w sends 1 to either u or v if w ̸∈ N(v) ∩ N(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If d(u) = 4 and d(v) = 5, then u receives 1 2 from v by (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus M′(u) ≥ 4−6+4× 1 2 = 0 and M′(v) = 5 − 6 + 1 2|N(u) ∩ N(v)| + |N(v) \\ N(u)| − 1 2 ≥ 5 − 6 + 3 2 + 1 − 1 2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If d(u) = 3 and d(v) = 6, then M′(u) = 0 by (I) and v sends 1 to u by (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus M′(v) = 6 − 6 + |N(v) \\ N(u)| − 1 ≥ 6 − 6 + 3 − 1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (III) Let u be a 4-vertex with four 6+-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then M′(u) > 0 unless u has either four 7-neighbors or has two 6-neighbors and two 7-neighbors, in which case M′(u) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, u is adjacent to at least two 7-vertices and each 7-neighbor of u is adjacent to at most three 5−-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If u has a 7-neighbor v adjacent to three 5−-vertices, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='5, u is adjacent to four 7-vertices and except v, each 7-neighbor is adjacent to only one 5−-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore by (R2), M′(u) ≥ 4 − 6 + 3 × 1 + 1 3 = 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now assume that each 7-neighbor is adjacent to at most two 5−-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then u receives at least 1 2 from each 7-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If u has four 7-neighbors, then M′(u) ≥ 4 − 6 + 4 × 1 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If u has a 6-neighbor, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='11, there are two 7-neighbors of u having only one 5−-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus M′(u) ≥ −2 + 2 + 1 2(d7(u) − 2) ≥ 0 with equality when u has exactly two 6-neighbors and two 7-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 5 (IV) M′(u) > 0 for each 5-vertex u with five 5+-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, u is adjacent to at least two 7-vertices and each 7-neighbor of u is adjacent to at most four 6−-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If v is a 7-neighbor of u and v is adjacent to a 3-vertex, then v sends 1 2 to u by (R2) and u sends 1 8 to v by (R4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore the total net charge u receives from v is 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus in general, u receives at least min{ 3 8, 1 4} from each 7-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If u has at least four 7-neighbors, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='12(3), M′(u) ≥ −1+2× 1 4 +2× 1 3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now assume that u is adjacent to at most three 7-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If u is adjacent to a 5-vertex, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='11, u has three 7-neighbors, each of which could be adjacent to at most two 5−-vertex (u and the 5-neighbor of u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus M′(u) ≥ −1 + 3 × 1 2 = 1 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Finally, we may assume that u is adjacent to at least two 6-vertices and at most three 7-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='12(1) and (2), M′(u) ≥ −1 + min{1 4 + 2 × 1 2, 1 + 1} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (V) Let u be a 6-vertex adjacent to six 4+-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by the discharging rules, M′(u) = M(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (VI) M′(u) ≥ 0 if d(u) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let u be a 7-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then u ̸∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By (R1) and (R2), we have M′(u) ≥ 0 if u ̸∈ A ∪ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (a) Assume u ∈ A (that is u has a 2-neighbor v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let w be the other neighbor of v and x ∈ N(u) \\ {v, w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4, d(x) = 7 and x is not adjacent to a 5−-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since u ∈ A, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, x is adjacent to at most five vertices in A∪C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by (R1), x sends at least 1 5 to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since |N(u)\\{v, w}| ≥ 5, we have M′(u) ≥ 7 − 6 − 2 + 5 × 1 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (b) Assume u ∈ C (that is u is adjacent to a 3-vertex x and another 5−-vertex z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, x and z are not adjacent and u has five 7-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='6, u and z have no common neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus u has at least three 7-neighbors which are not adjacent to x or z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let w be such a 7-neighbor of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, N(w)∩(A∪B) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If d(z) ≤ 4, then 3 ≤ d(z) ≤ 4 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, and thus u sends at most 1 to each of x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='10(2), u is the only vertex in C adjacent to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' So w sends 1 to u by (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus M′(u) ≥ 7 − 6 − 1 − 1 + 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If d(z) = 5, then w is adjacent to at most seven vertices in C and thus sends at least 1 7 to u by (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By (R2), u sends 1 to x and 1 2 to z and by (R4), z sends 1 8 to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore M′(u) ≥ 7 − 6 − 1 − 1 2 + 1 8 + 3 7 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of (VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By (I)-(VI), M′(x) ≥ 0 for each vertex x and thus 0 ≤ � x∈V M′(x) = � x∈V M(x) = (d(G) − 6)|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore d(G) ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 Concluding remarks One may wonder why our result does not include the term 3 |V | in the lower bound for the average degree as Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The reason is that we can construct some infinite families of graphs with maximum degree 7 and average degree 6 which satisfy all currently known adjacency lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For example, for any positive integer t, consider a graph G with degree sequence (4t, 72t) such that each 4-vertex is adjacent to four 7-vertices and each 7-vertex is adjacent two 4-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' One can easily check that G satisfies all adjacent lemmas that we currently have and d(G) = 7 − 1 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The above example can be generalized for arbitrary maximum degree ∆ = 2k + 1 ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For each t ≥ 1, let G be a graph with degree sequence (kt, ∆kt) such that each k-vertex is adjacent to k vertices of degree ∆ and each ∆-vertex is adjacent to exactly one k-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(G) = ∆ − 1 = 2k and G satisfies all adjacency lemmas that we know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The above examples and several other examples not only present a challenge but also indicate the necessity to develop new adjacency lemmas to attack Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 and other edge coloring problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In particular, so far all adjacency lemmas are about a path on at most four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='9 is indeed a lemma that deals with a path with five vertices and it is the key lemma in the proof of our main result, but it is only for degree 3-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' To completely solve the case of 7-critical graphs and beyond, more general adjacency lemmas concerning paths on five vertices are needed although it is very challenging to develop such lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' It would be practical and very useful to use computer program to complete the remaining cases for 7-critical graphs and to develop some forbidden structures for critical graphs in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 4 Applications to graphs embedded on surfaces with nonneg- ative Euler characteristics Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 clearly implies that every planar graph with maximum degree 7 is class one which was conjectured by Vizing and independently proved by Sanders and Zhao [17], and Zhang [23] and its extension to projective planar graphs [18] since every graph which can be embedded in a plane or a projective plane has average degree strictly less than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Our result also implies the following result due to Sanders and Zhao [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 (Sanders and Zhao [18]) Let G be a graph with maximum degree 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If G can be embedded in the torus or Klein bottle, then G is class one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Prove by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that G is not class one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we may assume that G is 7-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Euler’s formula, d(G) ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, we have d(G) = 6 and d(f) = 3 for each face f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since G is simple, we further have δ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote by M′(x) the new charge of the vertex x and A, B, C the sets defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then � x∈V (G) M′(x) = � x∈V (G)(d(G) − 6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus M′(x) = 0 for every vertex x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since δ(G) ≥ 3, we have A = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By (II) and (IV) in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, d(u)+d(v) ≥ ∆ + 3 and there are no 5-vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since every face is a 3-face and G is 2-connected, every two adjacent vertices share at least two common neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 7 Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 δ(G) = 4 and every 4-vertex is adjacent to exactly two 7-vertices and two 6- vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let y be a 7-vertex with a neighbor x where 3 ≤ d(x) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since any two adjacent vertices share at least two neighbors, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='6, y is adjacent to only one 4−-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since there are no 5-vertices in G, y is adjacent to exactly one 5−-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This implies C = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore A = B = C = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Hence every 7-vertex is adjacent to a 4−-vertex otherwise M′(x) = M(x) = 1 > 0 if x is a 7-vertex without a 4−-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore every 7-vertex has exactly one 4−-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If there is a 3-vertex, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3, there is one 7-vertex x that has no 4−-neighbors, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore δ = 4 and every 7-vertex is adjacent to exactly one 4-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By (III), every 4-vertex is adjacent to exactly two 7-vertices and two 6-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote by Vi the set of i-vertices and ni = |Vi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, n4 = 2n7 and n4 ≤ 2n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since every 7-vertex is adjacent to a 4-vertex, every 7-vertex is adjacent to at least 4 vertices in V7 and every vertex has at least two neighbors in V7 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus 2n6 + 2n4 ≤ |[V7, V6 ∪ V4]| ≤ 3n7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This implies 6n7 = 3n4 ≤ 3n7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This contradiction completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 5 Proofs of new lemmas Before giving the proofs, we first introduce some notations and lemmas that are needed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The set of all k-edge-colorings of a graph G is denoted by Ck(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ ∈ Ck(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For any color α, let Eα = {e ∈ E : ϕ(e) = α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For any two distinct colors α and β, denote by Gϕ(α, β) the subgraph of G induced by Eα ∪ Eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The components of Gϕ(α, β) are called (α, β)-chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly, each (α, β)-chain is either a path or a cycle of edges alternately colored with α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For each (α, β)-chain P, let ϕ/P denote the k-edge-coloring obtained from ϕ by exchanging colors α and β on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For any v ∈ V , let Pv(α, β, ϕ) denote the unique (α, β)-chain containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Notice that, for any two vertices u, v ∈ V , either Pu(α, β, ϕ) = Pv(α, β, ϕ) or Pu(α, β, ϕ) is vertex-disjoint from Pv(α, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This fact will be used very often without mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For convenience, we define Pv(α, β, ϕ) = v and ϕ/Pv(��, β, ϕ) = ϕ when α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For any v ∈ V , let ϕ(v) = {ϕ(e) : e ∈ E(v)} denote the set of colors presented at v and ¯ϕ(v) = C \\ ϕ(v) the set of colors not assigned to any edge incident to v, which are called missing colors at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' For a vertex set X ⊆ V (G), we call X elementary (with respect to ϕ) if all missing color sets ¯ϕ(x) (x ∈ X) are mutually disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A multi-fan at x with respect to the edge e = xy ∈ E(G) and the coloring ϕ ∈ C∆(G − e) is a sequence F = (x, e1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , ep, yp) with p ≥ 1 consisting of edges e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , ep and vertices x, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , yp satisfying the following two conditions: The edges e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , ep are distinct, e1 = e and ei = xyi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 8 For every edge ei with 2 ≤ i ≤ p, there is a vertex yj with 1 ≤ j < i such that ϕ(ei) ∈ ¯ϕ(yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that a multi-fan is slightly more general than a Vizing-fan which requires j = i − 1 in the second condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 (Stiebitz, Scheide, Toft and Favrholdt [19]) Let G be a ∆-critical graph, xy1 = e ∈ E(G) and ϕ ∈ C∆(G − e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If F = (x, e1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , ep, yp) is a multi-fan at x with respect to e and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then the following statements hold: (a) {x, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , yp} is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (b) If α ∈ ¯ϕ(x) and β ∈ ¯ϕ(yi) for some i, then Px(α, β, ϕ) = Pyi(α, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The following lemma is a direct corollary of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 Let G be a ∆-critical graph, xy = e ∈ E(G) and ϕ ∈ C∆(G − e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let xyz be a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (1) If d(z) ≤ 2∆ − (d(x) + d(y)) + 1, then α = ϕ(yz) ∈ ϕ(x) ∩ ϕ(y) and for any color β ∈ ¯ϕ(z) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)), Pz(α, β, ϕ) ends at x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (2) If ϕ(yz) ∈ ¯ϕ(x), then ¯ϕ(x) ∪ ¯ϕ(y) ⊆ ϕ(z) and thus d(z) ≥ 2∆ − (d(x) + d(y)) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A Kierstead path with respect to e = y0y1 and ϕ ∈ C∆(G − e) is a path K = y0y1 · · · yp with p ≥ 1 such that for every edge yiyi+1 with 1 ≤ i ≤ p − 1, there is a vertex yj with 0 ≤ j < i such that ϕ(yiyi+1) ∈ ¯ϕ(yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly a Kierstead path with 3 vertices is a multi-fan with center y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The next two lemmas are elementary properties of a Kierstead path with 4 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3 (Kostochka and Stiebitz [19], Luo and Zhao [15]) Let G be a ∆-critical graph, y0y1 = e ∈ E(G) and ϕ ∈ C∆(G − e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let K = y0y1y2y3 be a Kierstead path with respect to e and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then V (K) is elementary unless d(y1) = d(y2) = ∆(G), in which case, all colors in ¯ϕ(y0), ¯ϕ(y1), ¯ϕ(y2) and ¯ϕ(y3) are distinct except one possible common missing color in ¯ϕ(y3) ∩ ( ¯ϕ(y0) ∪ ¯ϕ(y1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4 Let G be a ∆-critical graph, y0y1 = e ∈ E(G) and ϕ ∈ C∆(G − e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that K = y0y1y2y3 is a Kierstead path with respect to e and ϕ, min{d(y1), d(y2)} < ∆, α ∈ ¯ϕ(y3) and β ∈ ¯ϕ(yi) for some i ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β /∈ {ϕ(y1y2), ϕ(y2y3)}, then Py3(α, β, ϕ) ends at yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since K is a Kierstead path and {y0, y1, y2, y3} is elementary by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3, we have α /∈ {ϕ(y1y2), ϕ(y2y3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that Py3(α, β, ϕ) does not end at yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then after interchanging α, β on this path, K is still a Kierstead path, but β is missing at both yi and y3, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A ϕ-broom (Figure 1 (a)) with respect to y0y1 and ϕ ∈ C∆(G − y0y1) is a sequence B = (y0, e1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , ep, yp) with p ≥ 3 such that e1 = y0y1, e2 = y1y2, ϕ(e2) ∈ ¯ϕ(y0) and for all i ≥ 3, ei = y2yi and ϕ(ei) ∈ ¯ϕ(yj) for some j < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='5 (Cao, Chen, Jing, Stiebitz and Toft [7]) Let G be a ∆-critical graph, y0y1 = e1 ∈ E(G) and ϕ ∈ C∆(G−e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If B = (y0, e1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' , ep, yp) is a ϕ-broom and min{d(y1), d(y2)} < ∆, then the vertex set of B is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 9 … 0 y 1 y 2 y 3 y 4 y p y \uf061 \uf061 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' some for ) ( ) ( Broom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' ) ( 2 i j y y y a j i \uf03c \uf0ce\uf06a \uf06a 2 \uf067 u b a 1t 2t 1s 2s c Kite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' ) (b \uf061 \uf061 1 \uf062 2 \uf062 2 \uf067 u b a 1t 2t 1s 2s 1 \uf067 1 \uf067 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=') ( ) ( , , , Fork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' ) ( 2 1 2 1 b a c \uf06a \uf06a \uf067 \uf067 \uf062 \uf062 \uf0c8 \uf0ce Figure 1: Brooms, kites and forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A kite H (Figure 1 (b)) is a graph with V (H) = {a, b, c, u, s1, s2, t1, t2} and E(H) = {ab, ac, bu, cu, us1, us2, s1t1, s2t2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The lemma below reveals some properties of a kite with specified colors on its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='6 (Cao, Chen and Shan [8]) Let G be a ∆-critical graph, H ⊆ G be a kite with V (H) = {a, b, c, u, s1, s2, t1, t2}, and let ϕ ∈ C∆(G − ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that both K = abus1t1 and K∗ = bacus2t2 are Kierstead paths with respect to ab and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ϕ(s1t1) = ϕ(s2t2), then | ¯ϕ(t1) ∩ ¯ϕ(t2) ∩ ( ¯ϕ(a) ∪ ¯ϕ(b))| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let G be a ∆-critical graph, ab ∈ E(G), and ϕ ∈ C∆(G−ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' A fork H (Figure 1 (c)) with respect to ϕ is a graph with V (H) = {a, b, u, s1, s2, t1, t2} and E(H) = {ab, bu, us1, us2, s1t1, s2t2} such that ϕ(bu) ∈ ¯ϕ(a), ϕ(us1), ϕ(us2) ∈ ¯ϕ(a) ∪ ¯ϕ(b), and ϕ(s1t1) ∈ ( ¯ϕ(a) ∪ ¯ϕ(b)) ∩ ¯ϕ(t2) and ϕ(s2t2) ∈ ( ¯ϕ(a) ∪ ¯ϕ(b)) ∩ ¯ϕ(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Forks may not exist in a ∆-critical graph if the degree sum of a, t1 and t2 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='7 (Cao and Chen [6]) Let G be a ∆-critical graph, ab ∈ E(G), and {u, s1, s2, t1, t2} ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ∆ ≥ dG(a) + dG(t1) + dG(t2) + 1, then for any ϕ ∈ C∆(G − ab), G does not contain a fork on {a, b, u, s1, s2, t1, t2} with respect to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='7 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='7 Let G be a ∆-critical graph with ∆ ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then |N∆∼2(v)| ≤ 5 for every v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that there is a ∆-vertex v with |N∆∼2(v)| ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4, v has no 2-neighbors and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, each vertex z ∈ N∆∼2(v) has exactly one 2-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let N2(v) = N(N(v))\\N[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since |N∆∼2(v)| ≥ 6, there are at least three 2-vertices in N2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let x be a 2-vertex in N2(v) and y be a vertex in N(x) ∩ N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly y ∈ N∆∼2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ ∈ C∆(G − xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ¯ϕ(x) ∪ ¯ϕ(y) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We first point out one fact that will be used very often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let t1, t2 be two 2-vertices in N2(v)\\{x}, s1 ∈ N(v) ∩ N(t1) and s2 ∈ N(v) ∩ N(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (a) If |N(x) ∩ N(v)| = 2 and ϕ(s1t1) = ϕ(s2t2), then ϕ(t1) ̸= ϕ(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (b) If ϕ(s1t1) ̸= ϕ(s2t2), then either ϕ(s1t1) ∈ ϕ(t2) or ϕ(s2t2) ∈ ϕ(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (a) Denote N(x) ∩ N(v) = {y, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that ϕ(t1) = ϕ(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then | ¯ϕ(t1)∩ ¯ϕ(t2)| ≥ 5 since ∆ ≥ 7, and {x, y, z, v, s1, s2, t1, t2} form a kite with ϕ(s1t1) = ϕ(s2t2), a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (b) Suppose to the contrary that ϕ(s1t1) ∈ ¯ϕ(t2) and ϕ(s2t2) ∈ ¯ϕ(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then {x, y, v, s1, s2, t1, t2} form a fork with ∆ ≥ 7 = d(x) + d(t1) + d(t2) + 1, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We consider two cases in the following: there are three 2-vertices in N2(v), or there are at least four 2-vertices in N2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 1: There are exactly three 2-vertices in N2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let t1, t2 be the two 2-vertices in N2(v)\\{x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since N∆∼2(v) ≥ 6, we have |N(ti)∩N(v)| = 2 for each i = 1, 2 and |N(x) ∩ N(v)| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let N(ti) ∩ N(v) = {si, s′ i} for each i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By the symmetry between si and s′ i, we may assume that ϕ(s1t1) ̸= ϕ(s2t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(b), we may assume ϕ(s′ 1t1) = ϕ(s2t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Applying Fact 1(a) on s′ 1, t1, s2, t2, we have ϕ(t1) ̸= ϕ(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus ϕ(s′ 2t2) ̸= ϕ(s1t1), ϕ(s′ 2t2) ̸∈ ϕ(t1) and ϕ(s1t1) /∈ ϕ(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This gives a contradiction to Fact 1(b) on s1, t1, s′ 2, t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 2: There are at least four 2-vertices in N2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let t1, t2, t3 be three 2-vertices in N2(v)\\{x}, si be a vertex in N(ti) ∩ N(v), and s′ i be the other neighbor of ti for each i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' ϕ(siti) ̸= ϕ(sjtj) for any 1 ≤ i < j ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Prove by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since ∆ ≥ 7 > d(t1)+d(t2)+d(t3), let η ∈ ¯ϕ(t1)∩ ¯ϕ(t2)∩ ¯ϕ(t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By symmetry, we only need to consider the following two cases: ϕ(s1t1) = ϕ(s2t2) = ϕ(s3t3) = α, or ϕ(s1t1) = ϕ(s2t2) = α and ϕ(s3t3) = β ̸= α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that ϕ(s1t1) = ϕ(s2t2) = ϕ(s3t3) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by symmetry, we may assume that Pt1(α, η, ϕ) does not pass through t2, t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ′ = ϕ/Pt1(α, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then s1, t1, s2, t2 give a contradiction to Fact 1(b) under ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that ϕ(s1t1) = ϕ(s2t2) = α and ϕ(s3t3) = β ̸= α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pt1(α, η, ϕ) does not end at t2, let ϕ′ = ϕ/Pt1(α, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then s1, t1, s2, t2 give a contradiction to Fact 1 (b) under ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Pt1(α, η, ϕ) ends at t2, so Pt3(α, η, ϕ) does not pass through t1, t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt3(α, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now α ∈ ¯ϕ1(t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by Fact 1(b), we have ϕ1(t1) = ϕ1(t2) = {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η′ ∈ ¯ϕ1(t1) ∩ ¯ϕ1(t2) ∩ ¯ϕ(t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By symmetry, we may assume that Pt3(β, η′, ϕ1) does not pass 11 through t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ1/Pt3(β, η′, ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then s1, t1, s3, t3 give a contradiction to Fact 1(b) under ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This proves Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ(s1t1) = α, ϕ(s2t2) = β, ϕ(s3t3) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' {ϕ(s′ 1t1), ϕ(s′ 2t2), ϕ(s′ 3t3)} = {ϕ(s1t1), ϕ(s2t2), ϕ(s3t3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Claim A, α, β, γ are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that ϕ(t1) = {α, η} where η /∈ {β, γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(b), we have ϕ(t2) = {β, α} and ϕ(t3) = {γ, α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then s2, t2, s3, t3 give a contradiction to Fact 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by symmetry, we may assume that ϕ(t1) = {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now by applying Fact 1(b) on s1, t1, s3, t3, we have ϕ(t3) = {α, γ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By applying Fact 1(b) on s2, t2, s3, t3, we have ϕ(t2) = {β, γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This proves Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' The final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Without loss of generality, assume ϕ(t1) = {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since |N∆∼2| ≥ 6, let s4 ∈ N∆∼2\\{s1, s2, s3} and t4 be the 2-neighbor of s4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If t4 ∈ {t1, t2, t3}, then by symmetry, we may assume that t4 = t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ(s4t1) = β and s4, t1, s3, t3 give a contradiction to Fact 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If t4 ̸∈ {t1, t2, t3}, then by Claim A, ϕ(s4t4) ̸= ϕ(siti) for each i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus {s1, s2, s4, t1, t2, t4} does not satisfy Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case 2 and thus of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='8 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='8 Let G be a ∆-critical graph and xyrst be a path with d(x) + d(y) = ∆ + 2 and max{d(x), d(y)} < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(t) ≥ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ ∈ C∆(G − xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(x) + d(y) = ∆ + 2, we have ¯ϕ(x) ∪ ¯ϕ(y) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ(yr) = α, ϕ(rs) = β, ϕ(st) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then α ∈ ¯ϕ(x) and β, γ ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(x) < ∆ and d(y) < ∆, we have | ¯ϕ(x)| ≥ 2 and | ¯ϕ(y)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that d(t) ≤ ∆ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then | ¯ϕ(t)| ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' There is a coloring in C∆(G − xy) such that yr and st are colored differently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=', we may assume α ̸= γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that α = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆ − 3, let η ∈ ¯ϕ(t) \\ {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η ∈ ¯ϕ(y), then Px(α, η, ϕ) = Py(α, η, ϕ) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 and thus is disjoint from Pt(α, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(α, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ1(yr) ̸= ϕ1(st), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since | ¯ϕ(y)| ≥ 2, let δ ∈ ¯ϕ(y) \\ {β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly δ ̸∈ {ϕ(yr), ϕ(rs), ϕ(st)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(δ, η, ϕ) and we are back to the case when η ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This proves Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' From now on, we assume that α ̸= γ in the following proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may further assume that α, β ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We consider two cases: β ∈ ¯ϕ(t) and β /∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: β ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may assume α ∈ ϕ(t) otherwise we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η ∈ ¯ϕ(t) \\ {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly η ̸= γ since ϕ(st) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η ∈ ¯ϕ(y), let ϕ1 = ϕ/Pt(α, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have α, β ∈ ¯ϕ1(t), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 12 If η ∈ ¯ϕ(x), let δ ∈ ¯ϕ(y) \\ {β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, regardless of whether δ = γ or not, Px(δ, η, ϕ) does not contain yr, rs or st since η ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(δ, η, ϕ) and we are back to the case when η ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: β /∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: α ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β ∈ ¯ϕ(y), then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, Px(α, β, ϕ) is disjoint from Pt(α, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Pt(α, β, ϕ) does not contain yr or rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(α, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β ∈ ¯ϕ1(t) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now assume β ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If there is a color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t), let ϕ1 = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise, let δ ∈ ¯ϕ(y) and η ∈ ¯ϕ(t)\\{α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Pt(η, δ, ϕ) does not pass through x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(η, δ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then δ ∈ ¯ϕ1(y) ∩ ¯ϕ1(t) and β ∈ ¯ϕ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Px(δ, β, ϕ1) and Pt(δ, β, ϕ1) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pt(δ, β, ϕ1) does not contain rs, let φ2 = ϕ1/Pt(δ, β, ϕ1) and then ϕ2 is a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Px(δ, β, ϕ1) does not contain rs, let φ2 = ϕ1/Px(δ, β, ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β ∈ ¯ϕ2(y) and we are back to the case when β ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This proves Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: α /∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If there is a color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t), let ϕ1 = ϕ/Pt(α, δ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then α ∈ ¯ϕ1(t) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose ¯ϕ(y)∩ ¯ϕ(t) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η ∈ ¯ϕ(t) and δ ∈ ¯ϕ(y)\\{β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then δ ∈ ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, regardless of whether δ = γ or not, Px(δ, η, ϕ) does not contain yr, rs or st since η ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(δ, η, ϕ), we are back to the case when ¯ϕ(y) ∩ ¯ϕ(t) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 and thus the proof of Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Claim B, we assume that α, β ∈ ¯ϕ(t) in the following proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may further assume that β, γ ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We consider two cases: β ∈ ¯ϕ(y) and β /∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: β ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may assume γ ∈ ϕ(y) otherwise we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η ∈ ¯ϕ(t) \\ {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similar to the argument in Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, we may assume that there is a color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t) and δ ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Pt(δ, γ, ϕ) and Px(δ, γ, ϕ) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(δ, γ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have β, γ ∈ ¯ϕ1(y), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: β /∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If γ ∈ ¯ϕ(y), then Pt(γ, β, ϕ) and Px(γ, β, ϕ) are disjoint by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that rs and st are contained in Pt(γ, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(γ, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β ∈ ¯ϕ1(y) and we are back to Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose γ ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similar to the argument in Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, we can assume that there is a color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then δ ̸∈ {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Px(η, γ, ϕ) is disjoint from Pt(η, γ, ϕ), so it does not contain st since η ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(η, γ, ϕ) and we are back to the case when γ ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, and thus Claim C holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now by Claims A, B and C, we assume that ϕ ∈ C∆(G − xy) satisfies the following properties: 13 ϕ(yr) = α, ϕ(rs) = β, ϕ(st) = γ, α ̸= γ, α, β ∈ ¯ϕ(t) and β, γ ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(α, γ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Under the coloring ϕ1, Py(β, α, ϕ1) = yrst ends at t but not x, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3 Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='9 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='9 Let G be a ∆-critical graph and xyrst be a path with d(x) = 3 and d(y) = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose that y has a neighbor z ̸∈ {x, r, s} with d(z) ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then d(s) ≥ ∆ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' and d(z) + d(t) ≥ ∆ + 1 if d(t) ≤ ∆ − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ be a coloring in C∆(G − xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z) ≤ ∆ − 2, d(x) = 3 and d(y) = ∆, we have |ϕ(x) ∩ ϕ(y)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, without loss of generality, assume ϕ(x) = {1, 2}, ϕ(yz) = 2, ϕ(yr) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote ϕ(rs) = β and ϕ(st) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that ¯ϕ(y) = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (1) We first show d(s) ≥ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary d(s) ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We first consider the case when ϕ(rs) = β ̸= 2 = ϕ(yz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then K = xyrs is a Kierstead path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3, | ¯ϕ(s) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus d(s) ≥ 2∆ − (d(x) + d(y) + 1 = ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(s) ≤ ∆ − 2, we have d(s) = ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that d(s) = ∆ − 2 only if 2 ∈ ¯ϕ(s) and | ¯ϕ(s) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y))| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote ¯ϕ(s) = {2, α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ¯ϕ(z) \\ {α, β} ̸= ∅, then η ∈ ¯ϕ(z) \\ {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, Pz(η, 2, ϕ) ends at x or y and thus it does not pass through s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pz(η, 2, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then xyrs remains a Kierstead path with respect to ϕ1 and xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' However, ¯ϕ1(s) = {2, α} ⊆ ¯ϕ(x) ∪ ¯ϕ(y), a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore ¯ϕ(z) \\ {α, β} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z) ≤ ∆ − 2, we have ¯ϕ(z) = {α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β ̸= 1, then we may assume α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise both Pz(1, α, ϕ) and Ps(1, α, ϕ) are disjoint from Px(1, α, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ/(Pz(1, α, ϕ) ∪ Ps(1, α, ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then 1 is missing at both z and s and 3, β ∈ ¯ϕ1(x) ∪ ¯ϕ1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since 1 ∈ ¯ϕ(z) ∩ ¯ϕ(s), both Pz(1, 3, ϕ) and Ps(1, 3, ϕ) are disjoint from Px(1, 3, ϕ) and thus neither passes through x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ/(Pz(1, 3, ϕ) ∪ Ps(1, 3, ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then 3 ∈ ¯ϕ2(z) ∩ ¯ϕ2(s) and 2 ∈ ϕ2(x) ∩ ϕ2(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, Pz(2, β, ϕ2) ends at either x or y and thus is disjoint from Ps(2, β, ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ3 = ϕ2/Ps(2, β, ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Pz(2, β, ϕ3) = zyrs which does not end at x or y, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now assume β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ¯ϕ(z) = {1, α} and thus Ps(1, α, ϕ) does not pass through x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Interchange colors on Ps(1, α, ϕ) and we are back to the case when β ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore this completes the proof when β ̸= ϕ(yz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now we consider the case when β = ϕ(yz) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η be a color in ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly η ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η = 1, then by recoloring yz with 1, we are back to the case when β ̸= ϕ(yz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Px(η, 1, ϕ) = Py(η, 1, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by interchanging η and 1 on Px(η, 1, ϕ) and then recoloring yz with η, we are back to the case when β ̸= ϕ(yz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof that d(s) ≥ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (2) Now we assume d(t) ≤ ∆ − 4 and show d(z) + d(t) ≥ ∆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 14 Suppose to the contrary that d(z) + d(t) ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' There is a coloring in C∆(G − xy) such that yr and st receive distinct colors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=', we may assume that γ ̸= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that γ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η ∈ ¯ϕ(t) \\ {2, 3, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ∈ ¯ϕ(x) ∩ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η = 1, then Px(3, η, ϕ) = Py(3, η, ϕ) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, so Pt(3, η, ϕ) is disjoint from Px(3, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(3, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We have that ϕ1(yr) ̸= ϕ1(st) now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η ̸= 1, then Pt(1, η, ϕ) does not contain x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(1, η, ϕ) and we are back to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This proves Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' From now on, we assume that ϕ(yr) ̸= ϕ(st) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' γ ̸= 3) in the following proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may further assume that 3, β ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We split the proof into two cases: β ∈ ¯ϕ(t) and β /∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: ϕ(rs) = β ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: β ̸∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If 1 ∈ ¯ϕ(t), then Pt(1, 3, ϕ) is disjoint from Px(1, 3, ϕ) = Py(1, 3, ϕ) and yr, rs ̸∈ Pt(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ1(yr) = 3, ϕ1(rs) = β, ϕ1(st) = γ and 3, β ∈ ¯ϕ1(t), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now assume 1 ̸∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t) \\ {2, 3, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Pt(1, η, ϕ) does not pass through x or y and does not contain yr, rs, or st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(1, η, ϕ) and we are back to the case when 1 ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: β ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If γ ̸= 2, then γ ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus, Pt(1, γ, ϕ) is disjoint from Px(1, γ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(1, γ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ1(rs) = γ ∈ ¯ϕ1(x) ∩ ¯ϕ1(t) and γ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now assume ϕ(st) = γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z) + d(t) ≤ ∆ and 2 ∈ ϕ(z) ∩ ϕ(t), there is a color η ∈ ¯ϕ(z) ∩ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since η ̸= γ, we have η ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η ̸= 1, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, Pz(2, η, ϕ) ends at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Pt(2, η, ϕ) does not pass through x or y and does not contain the edge rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(2, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ1(st) = η ∈ ¯ϕ1(x) ∪ ¯ϕ1(y) and we are back to the previous case If η = 1, then Pz(1, 2, ϕ) = yz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pz(1, 2, ϕ) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: ϕ(rs) = β /∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: ϕ(yr) = 3 ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: β ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' That is β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Px(3, β, ϕ) ends at y by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 and it contains both yr and rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Px(3, β, ϕ) and Pt(3, β, ϕ) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(3, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β ∈ ¯ϕ1(t) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: β = ϕ(yz) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If 1 ∈ ¯ϕ(z), recolor yz with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Assume 1 ̸∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z) ≤ ∆ − 2 and 2 ∈ ϕ(z), let η ∈ ¯ϕ(z)\\{3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly η ̸= 1, 2 and η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Pz(1, η, ϕ) does not pass through x or y and does not contain the edge 15 rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pz(1, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then 1 ∈ ¯ϕ1(z) and we are back to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3: β ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may further assume 1 ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise, since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t) \\ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ̸∈ {1, 2, 3, β}, and Px(1, η, ϕ) and Pt(1, η, ϕ) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(1, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then 1 ∈ ¯ϕ1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note Px(β, 1, ϕ1) and Pt(β, 1, ϕ1) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Px(��, 1, ϕ1) does not contain the edge rs, let ϕ2 = ϕ1/Px(β, 1, ϕ1) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pt(β, 1, ϕ1) does not contain the edge rs, let ϕ2 = ϕ1/Pt(β, 1, ϕ1) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: 3 /∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t) \\ {2, 3, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η = 1, then Px(3, 1, ϕ) and Pt(3, 1, ϕ) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ1(yr) = 3 ∈ ¯ϕ1(t) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore η ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β ̸= 1, then Px(1, η, ϕ) does not contain yr, rs or st since η ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(1, η, ϕ) and we are back to the case when η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β = 1, then Px(η, 1, ϕ) and Pt(η, 1, ϕ) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Px(η, 1, ϕ) does not pass through rs, let ϕ1 = ϕ/Px(η, 1, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η is missing at y1 now and we are back to the case when η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pt(η, 1, ϕ) does not contain rs, let ϕ1 = ϕ/Pt(η, 1, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β ∈ ¯ϕ1(t) and we are back to Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, and so Claim B holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Claims A and B, we assume that ϕ satisfies the following properties: ϕ(yr) = 3 ∈ ¯ϕ(t), ϕ(rs) = β ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' ϕ(st) = γ ̸= 3 Claim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may further assume β = ϕ(yz) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary β ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: γ ̸= ϕ(yz) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' γ ̸= 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: 1 ∈ {γ, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β = 1, then Pt(γ, 1, ϕ) does not pass through x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(γ, 1, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ1(st) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we assume γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β ∈ ¯ϕ(z), let ϕ1 = ϕ/Px(β, 1, ϕ) and then recolor yz with β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ1 is a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If 3 ∈ ¯ϕ(z), let ϕ1 = ϕ/Px(β, 1, ϕ) and ϕ2 = ϕ1/Pz(3, β, ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Notice that the second Kempe exchange will not effect yr or rs since they are on Px(3, β, ϕ1) = Py(3, β, ϕ1) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we obtain a desired coloring by recoloring yz with β under ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now we assume 3, β ̸∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ¯ϕ(z) ∩ ¯ϕ(t) ̸= ∅, let η ∈ ¯ϕ(z) ∩ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ̸∈ {1, 2, 3, β} and η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Px(1, η, ϕ) = Py(1, η, ϕ) does not contain st since η ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(1, η, ϕ) and then η ∈ ¯ϕ1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ1/Pz(η, 3, ϕ1) and then 3 ∈ ¯ϕ2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Pz(η, 3, ϕ1) does not contain yr or t since yr is on Px(η, 3, ϕ1) = Py(η, 3, ϕ1) and 3, η ∈ ¯ϕ1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Finally let ϕ3 = ϕ2/Px(η, 1, ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We are back to the case when ϕ(yr) ∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 16 Now assume ¯ϕ(z)∩ ¯ϕ(t) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z)+d(t) ≤ ∆, ϕ(z) and ϕ(t) form a partition of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Consequently, we have 1 ∈ ¯ϕ(z) and 2 ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z) ≤ ∆ − 2, let η ∈ ¯ϕ(z) \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Clearly η ∈ ¯ϕ(x) and η /∈ {1, 2, 3, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 be the coloring obtained from ϕ by recoloring yz with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then 2 ∈ ¯ϕ1(y)∩ ¯ϕ1(z) and Px(2, η, ϕ1) = Py(2, η, ϕ1) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ1/Px(2, η, ϕ1) and ϕ3 be the coloring obtained from ϕ2 by recoloring yz with η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now we have γ = 1 ∈ ¯ϕ3(y), ϕ3(yz) = η ̸= β and 2 ∈ ¯ϕ3(z) ∩ ¯ϕ3(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we are back to the case when ¯ϕ(z) ∩ ¯ϕ(t) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: 1 /∈ {γ, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t)\\{2, 3, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may assume η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise, η ∈ ¯ϕ(x) since ¯ϕ(x) = C\\{1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by interchanging colors on Pt(1, η, ϕ), 1 is missing at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since γ ∈ ¯ϕ(x), we have Px(γ, 1, ϕ) = Py(γ, 1, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since 1 ∈ ¯ϕ(t), Px(γ, 1, ϕ) does not contain st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore, by interchanging γ and 1 on Px(γ, 1, ϕ), we are back to Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: γ = ϕ(yz) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this case, ϕ(yz) = ϕ(st) = 2 ∈ ϕ(z) ∩ ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If 1 ∈ ¯ϕ(z), recolor yz with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we are back to Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 if β ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise, we have a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus in the following we assume 1 ∈ ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: {3, β} ∩ ¯ϕ(z) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β ∈ ¯ϕ(z), then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, Pz(2, β, ϕ) ends at x since β ∈ ¯ϕ(x) and it is disjoint from Pt(2, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus ϕ1 = ϕ/Pz(2, β, ϕ) is a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Assume 3 ∈ ¯ϕ(z) and β ∈ ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β = 1, then Py(1, 3, ϕ) contains the edges yr and rs and is disjoint from Pz(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that 1, β ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pz(1, 3, ϕ) and we are back to the case when 1 ∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Assume β ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z) ≤ ∆ − 2, let η ∈ ¯ϕ(z) \\ {3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ̸∈ {1, 2, 3, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Pz(1, η, ϕ) does not contain the vertices x, y or the edges rs, st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pz(1, η, ϕ) and we are back to the case when 1 ∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: {3, β} ∩ ¯ϕ(z) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since 2 ∈ ϕ(z) ∩ ϕ(t) and d(z) + d(t) ≤ ∆, let η ∈ ¯ϕ(t) ∩ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β ̸= 1, by interchanging colors on Px(η, 1, ϕ) and then recoloring yz with η, we are back to Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Px(η, 1, ϕ) and Pz(η, 1, ϕ) are disjoint and either Px(η, 1, ϕ) or Pz(η, 1, ϕ) does not contain rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In the former case, by interchanging η and 1 on Px(η, 1, ϕ) and then recoloring yz with η, we are back to Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In the later case by interchanging η and 1 on Pz(η, 1, ϕ) and then recoloring yz with 1, we have a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, and so Claim C holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Claim C, we further assume ϕ(yz) = ϕ(rs) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that ϕ(x) ∩ ϕ(y) = {2} and ¯ϕ(x) ∪ ¯ϕ(y) = C\\{2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may further assume that ¯ϕ(y) ∩ ¯ϕ(z) ̸= ∅ and γ ∈ ¯ϕ(y) ∩ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' That is γ = 1 ∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We split the proof into the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 17 Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: ϕ(yr) = 3 ∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this case Px(1, 3, ϕ) is disjoint from Pz(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pz(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pz(1, 3, ϕ) does not end at t, then ϕ1 is a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pz(1, 3, ϕ) ends at t, let ϕ2 be the coloring obtained from ϕ1 by recoloring yz with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In the coloring ϕ2, 2 is missing at y, 3 is missing at x, and Py(3, 2, ϕ2) = yrst, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This proves Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: γ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then γ ̸∈ {1, 2, 3} and γ ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If 1 ∈ ¯ϕ(t), then Px(1, γ, ϕ) ends at y and thus does not contain the edge st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by interchanging 1 and γ on Px(1, γ, ϕ), we are back to Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Assume 1 ̸∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t)\\{2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ̸∈ {1, 2, 3, γ} and η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By interchanging the colors on Pt(η, 1, ϕ), we are back to the case when 1 ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This proves Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: ϕ(yr) = 3 /∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(z) + d(t) ≤ ∆, either ϕ(z) and ϕ(t) form a partition of C or there exists a color η ∈ ¯ϕ(z) ∩ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: There exists a color η ∈ ¯ϕ(z) ∩ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this case we have η /∈ {2, 3, γ} and η ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η = 1, then Pz(1, 3, ϕ) does not pass through x, y or t since both α and η are missing at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We are back to Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 by interchanging 1 and 3 on Pz(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If η ̸= 1, then η ∈ ¯ϕ(x) and Px(η, 1, ϕ) does not pass through t since η ∈ ¯ϕ(t) ∩ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by interchanging η and 1 on Px(η, 1, ϕ), we are back to the case when η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: ϕ(z) and ϕ(t) form a partition of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this case γ ∈ ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If γ = 1, then ϕ is a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore we assume γ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus γ ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η ∈ ¯ϕ(t)\\{2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, Px(1, η, ϕ) does not pass through z or t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that if 1 = η, then Px(1, η, ϕ) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Px(1, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Px(η, γ, ϕ1) = Py(η, γ, ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Px(η, γ, ϕ1) does not contain t since η ∈ ¯ϕ1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ1/Px(η, γ, ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have γ ∈ ¯ϕ2(y) ∩ ¯ϕ2(z) and thus ϕ1 is a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, and so Claim D holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In summary, by Claims A, B, C, and D, we assume that ϕ satisfies the following properties: ϕ(x) = {1, 2} and 1 ∈ ¯ϕ(y) ∩ ¯ϕ(z) ϕ(yr) = 3, ϕ(yz) = ϕ(rs) = 2, and ϕ(st) = 1 2, 3 ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Px(1, 3, ϕ) ends at y and is disjoint from Pt(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pt(1, 3, ϕ) does not end at z, let ϕ1 be the coloring obtained from ϕ by interchanging colors on Pt(1, 3, ϕ) and recoloring yz with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then 3 ∈ ¯ϕ1(x), 2 ∈ ¯ϕ1(y) and Py(3, 2, ϕ1) = yrst not ending at x, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Pt(1, 3, ϕ) ends at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ/Pt(1, 3, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Pz(2, 3, ϕ2) = zyrst which does not end at x, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4 Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='11 Let G be a ∆-critical graph and xy be an edge with d(x) + d(y) = ∆ + 3 and max{d(x), d(y)} < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then x has d(x)−2 neighbors of degree ∆ having no (∆−2)−-neighbors other than x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ ∈ C∆(G − xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since G is ∆-critical and d(x) + d(y) = ∆ + 3, we have |ϕ(x) ∩ ϕ(y)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let δ be the color in ϕ(x) ∩ ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ¯ϕ(x) ∪ ¯ϕ(y) = C\\{δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, x has at least d(x) − 2 neighbors of degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus including y, x has at most two neighbors of degree less than ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2, we have the following fact which will be applied frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let yxzt be a path with ϕ(xz) ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (1) ¯ϕ(z) ⊆ {δ} and thus d(z) ≥ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If δ ∈ ¯ϕ(z), then for any color η ∈ ϕ(z) \\ {ϕ(xz)}, Pz(δ, η, ϕ) ends at x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (2) If yxzt is a Kierstead path, then ¯ϕ(t) ⊆ {δ} and thus d(t) ≥ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We consider two cases in the following according to the number of ∆-neighbors of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' x has a neighbor z0 ̸= y with d(z0) < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' It is sufficient to show that for any path yxzt with z ̸= z0, we have d(t) ≥ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that there is a path yxzt such that z ̸= z0 but d(t) ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We consider two cases according to ϕ(xz0) = δ or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1: α = ϕ(xz0) ̸= δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(1), ¯ϕ(z0) = {δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' First assume ϕ(xz) ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by Fact 1(2), ϕ(zt) = δ otherwise yxzt is a Kierstead path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆ − 2 and δ ∈ ϕ(t), let η ∈ ¯ϕ(t) \\ {α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(1), Pz0(δ, η, ϕ) ends at x or y and thus is disjoint from Pt(δ, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(δ, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then yxzt is a Kierstead path in ϕ1 and thus d(t) ≥ ∆ − 1 by Fact 1(2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now assume ϕ(xz) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote β = ϕ(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then β ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may assume that β ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise if there is a color η ∈ ¯ϕ(t) ∩ ¯ϕ(x), interchange colors on the path Pt(η, β, ϕ) which does not contain x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If no such η exists, let η ∈ ¯ϕ(x) and γ ∈ ¯ϕ(t) \\ {δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(η, γ, ϕ) and then let ϕ2 = ϕ1/Pt(η, β, ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(1), Pz0(δ, β, ϕ) ends at x and thus contains xzt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This implies δ ∈ ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus | ¯ϕ(t) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y))| ≥ 2 since d(t) ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η ∈ ¯ϕ(t) \\ {α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(1) again, Pz0(δ, η, ϕ) ends at x or y and thus is disjoint from Pt(δ, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(δ, η, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then in ϕ1, Px(δ, β, ϕ1) = xzt which is disjoint from Pz0(δ, β, ϕ1), a contradiction to Fact 1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2: ϕ(xz0) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ(xz) ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆−2, by Fact 1(2), ϕ(zt) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η ∈ ¯ϕ(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similar to the argument in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, we assume η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Recolor xz0 with η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then yxzt is a Kierstead path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(2), d(t) ≥ ∆ − 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' All vertices in N(x)\\{y} are ∆-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 19 Since | ¯ϕ(y)∩ϕ(x)| = d(x)−2, we are done if d(t) ≥ ∆−1 for every path yxzt with ϕ(xz) ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus assume that there is a path yxz0t0 such that ϕ(xz0) ∈ ¯ϕ(y) and d(t) ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Fact 1(2), ϕ(z0t0) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote α = ϕ(xz0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then α ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' With a similar argument as before, we may assume α ∈ ¯ϕ(t0) and there is a color η ∈ ¯ϕ(t0) ∩ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η ̸= α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now it is sufficient to show that for any path yxzt with z ̸= z0, we have d(t) ≥ ∆ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' ϕ(xz) = β ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by Fact 1(2), ϕ(zt) = δ, so t ̸= t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆ − 2, there is a color η1 ∈ ¯ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then η1 ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similarly we may assume η, η1 ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that d(t) ≤ ∆ − 2 and d(t0) ≤ ∆ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus η ̸= η1 since otherwise both Pt0(δ, η, ϕ) and Pt(δ, η, ϕ) end at x by Fact 1(2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now let ϕ1 be the coloring obtained from ϕ by coloring xy with α, leaving xz0 uncolored and recoloring z0t0 with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then Px(η1, δ, ϕ1) = Pz0(η1, δ, ϕ1) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ2 = ϕ1/Pt(η1, δ, ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ2(zt) = η1 ∈ ¯ϕ2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that the last Kempe exchange may affect the colors of the edges incident to t0, so δ may not be missing at t0 under ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' But we still have η ∈ ¯ϕ2(x) ∩ ¯ϕ2(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If δ ∈ ϕ2(t0), let ϕ3 = ϕ2/Pt0(η, δ, ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise let ϕ3 = ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have δ ∈ ¯ϕ3(z0)∩ ¯ϕ3(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Finally let ϕ4 be the coloring obtained from ϕ3 by recoloring z0t0 with δ, coloring xz0 with α and leaving xy uncolored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then yxzt is a Kierstead path under ϕ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' However d(t) ≤ ∆ − 2, a contradiction to Fact 1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 ϕ(xz) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote ϕ(zt) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' With similar arguments as before we may assume that there is a color η′ ∈ ¯ϕ(t) ∩ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may then assume that β ∈ ¯ϕ(x) since otherwise we can interchange β and η′ on Px(β, η′, ϕ) to get a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ ∆−2, let η1 ∈ ¯ϕ(t)\\{α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We then show that we may assume η1 ∈ ¯ϕ(x)∪{δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose otherwise η1 ∈ ¯ϕ(y)\\{α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(x) ≤ ∆ − 1, we have | ¯ϕ(x)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let α′ be a color in ¯ϕ(x)\\{ϕ(zt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By interchanging η1 and α′ on Px(η1, α′, ϕ), we obtain a coloring as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 be the coloring obtained from ϕ by coloring xy with α, leaving xz0 uncolored and recoloring z0t0 with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then under ϕ1, z0xzt is a Kierstead path with η1 ∈ ( ¯ϕ1(x) ∪ ¯ϕ1(z0)) ∩ ¯ϕ1(t), a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='5 Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='12 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='12 Let G be a 7-critical graph and x be a 5-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (1) if x has three 6-neighbors, then each 7-neighbor of x has exactly one 5−-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (2) if x has two 6-neighbors, then x has two 7-neighbors, each of which has at most two 5−-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (3) if x has exactly four 7-neighbors, then x has two 7-neighbors, each of which has at most three 5−-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If x has a 5-neighbor, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, x has at least three 7-neighbors and thus has at most one 6-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' To show the lemma in this case, we only need to consider the case 20 when x has four 7-neighbors and one 5-neighbor which is (3), and it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In the rest of the proof, we assume that x has no 5-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By the assumption of the lemma, x has a 6-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let y be a 6-neighbor of x, ϕ ∈ C∆(G − xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Without loss of generality we assume that ¯ϕ(y) = {1, 2}, ¯ϕ(x) = {3, 4, 5}, and ϕ(x) ∩ ϕ(y) = {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, x has at least two 7-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' (1) Denote the two 6-vertices in N(x)\\{y} by z1, z2, the two 7-vertices in N(x) by v1, v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We need to show that for any path yxvt with v ∈ {v1, v2}, d(t) ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We consider three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 x, y, z1, z2 form the vertex set of a multi-fan with respect to xy and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this case, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, we have ¯ϕ(z1) ∪ ¯ϕ(z2) = {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Assume without loss of generality that ¯ϕ(z1) = {6} and ¯ϕ(z2) = {7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then for each α ∈ ¯ϕ(x)∪ ¯ϕ(y), both Pz1(6, α, ϕ) and Pz2(7, α, ϕ) end at x if α ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let yxvt be a path where d(v) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η be a color in ¯ϕ(t) and β = ϕ(vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may assume that η ∈ ¯ϕ(x) since otherwise η ∈ {1, 2, 6, 7}, and we can interchange η and 3 on Pt(η, 3, ϕ), which doesn’t pass through x or y by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, to obtain a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we assume η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may further assume that β = ϕ(v1t) ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise β ∈ {1, 2, 6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Pt(β, η, ϕ) does not end at x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let α ∈ ¯ϕ(x) \\ {η}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Interchange η and ϕ(vt) = β on Pt(β, η, ϕ) first and then interchange β, α on the (β, α)-chain starting at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We obtain a desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we assume that β ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now let ϕ1 = ϕ/Pt(η, ϕ(xv), ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ(xv) ∈ ¯ϕ1(t) and Px(ϕ(xv), ϕ(vt), ϕ1) = xvt does not end at y, z1, or z2, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 x, y, z1, z2 do not form the vertex set of a multi-fan with respect to xy and ϕ, and |{ϕ(xz1), ϕ(xz2)} ∩ {1, 2}| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By symmetry, assume that ϕ(xz1) = 1, ¯ϕ(z1) = {6}, ϕ(xz2) = 7, ϕ(xv1) = 2, and ϕ(xv2) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then for each color η ∈ {2, 3, 4, 5}, Pz1(η, 6, ϕ) ends at x or y depending on whether η ∈ ¯ϕ(x) or η ∈ ¯ϕ(y) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similar to the argument in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, we may further assume 3 ∈ ¯ϕ(z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let yxvt be a path where d(v) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ(xv) ∈ {2, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We first assume ϕ(xv) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y), then yxvt is a Kierstead path with d(x) < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus ¯ϕ(t1) = {6, 7} by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η be a color in ¯ϕ(x) \\ {ϕ(vt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4, Pt(η, 6, ϕ) ends at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' However, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, Pz1(η, 6, ϕ) ends at x, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ϕ(vt) = 7, recolor xz2 with 3 and we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ϕ(vt) = 6, let η ∈ ¯ϕ(t) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may assume η ∈ ¯ϕ(x) since otherwise we can pick a color β ∈ ¯ϕ(x) and interchange colors on Pt(η, β, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since Pz1(η, 6, ϕ) ends at x, Pt(η, 6, ϕ) and Pz1(η, 6, ϕ) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Interchange colors on Pt(η, 6, ϕ) and we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now we assume ϕ(xv) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote ϕ(vt) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If β = 7, then recolor the edge xz2 with 3 and then 7 is missing at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we may assume β ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If ϕ(vt) = β ∈ ¯ϕ(x), then Px(6, β, ϕ) ends at z1 and thus 6 ∈ ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since d(t) ≤ 5, let 21 α ∈ ¯ϕ(t) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similarly as before we may further assume that α ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Pz1(α, 6, ϕ) and Pt(α, 6, ϕ) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ1 = ϕ/Pt(α, 6, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then 6 is missing at t and thus Px(6, β, ϕ1) = xvt does not end at z1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose ϕ(vt) = β ∈ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let α′ be a color in ¯ϕ(t)\\{7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then similarly, we can assume that α′ ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By interchanging α′ and β on Pt(α′, β, ϕ), we are back to the case when ϕ(vt) ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3 {ϕ(xz1), ϕ(xz2)} = {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let yxvt be a path where d(v) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Without loss of generality, assume ϕ(xz1) = 6, ϕ(xz2) = 7, and ϕ(xv) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote ϕ(vt) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We first assume β ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then yxvt is a Kierstead path with d(x) < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus ¯ϕ(t) = {6, 7} by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let α be a color in ¯ϕ(x)\\{β} and η be a color in ¯ϕ(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Px(α, 7, ϕ) ends at t by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we may assume that η ∈ ¯ϕ(x) since otherwise η ∈ {1, 2, 7} and we can interchange η, α on Pz1(η, α, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' So we assume η ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We then claim that we may further assume that η ∈ ¯ϕ(x)\\{ϕ(vt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise η = ϕ(vt) ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Interchange η, 1 on Pz1(η, 1, ϕ) first and then interchange 1, α on the (1, α)-chain starting at z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we assume that η ∈ ¯ϕ(x)\\{ϕ(vt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now Px(η, 6, ϕ) ends at z1 but not t, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now we further assume β ∈ {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Without loss of generality assume ϕ(vt) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η′ ∈ ¯ϕ(t)\\{7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' With a similar argument as before, we assume η′ ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η1 be the color missing at z1 and η2 be the color missing at z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We first claim η1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since otherwise, we have η1 ∈ {1, 2, 3, 4, 5} and by interchanging η1, 3 on Pz1(η1, 3, ϕ) if necessary, we may assume that η1 ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then by recoloring xz1 with η1, we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We then claim η2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since otherwise, η2 ∈ {1, 2, 3, 4, 5} and by interchanging η2, 3 on Pz2(η2, 3, ϕ) if necessary, we may assume that η2 ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By recoloring xz2 with η2 and then recoloring xz1 with 7, we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus η2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that the above argument also implies that Pz2(6, η′, ϕ) ends at x, since otherwise by interchanging 6, η′ on this path, we are back to the case when η2 ̸= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now let ϕ1 = ϕ/Pt(η′, 6, ϕ), we have ϕ1(vt) = η′ ∈ ¯ϕ1(x), and thus we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' □ (2) Since x has no 5−-neighbors, by (1) x has two 6-neighbors and three 7-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Denote by v1, v2, v3 the three 7-vertices and z the 6-neighbor of x distinct from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then ϕ(xz) ∈ {1, 2} or ϕ(xz) ∈ {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1 ϕ(xz) ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this case, x, y, z form the vertex set of a multi-fan with respect to xy and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1, we have ¯ϕ(z) ∈ {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Assume without loss of generality that ϕ(xz) = 1, ¯ϕ(z) = {6}, ϕ(xv1) = 2 and ϕ(xv2) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that if each of v1 and v2 has at most two 5−-neighbors, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If v1 has three 5−-neighbors, then there exists t1 ∈ N(v1)\\{x} such that d(t1) ≤ 5 and 22 ϕ(v1t1) ̸= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η1 be a color in ¯ϕ(t1)\\{7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' With similar arguments as before we may assume that η1 ∈ ¯ϕ(x) and ϕ(v1t1) ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now yxv1t1 is a Kierstead path with respect to xy and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' But η1 ∈ ¯ϕ(x) ∩ ¯ϕ(t1), a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If v2 has three 5−-neighbors, then there exists t2 ∈ N(v2)\\{x} such that d(t1) ≤ 5 and ϕ(v2t2) ̸= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η2 be a color in ¯ϕ(t2)\\{7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similar to the argument before, we may assume that η2 and ϕ(v2t2) are in ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let ϕ′ = ϕ/Pt2(η2, 6, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have 6 ∈ ¯ϕ′(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus Px(6, ϕ′(v2t2), ϕ′) = xv2t2 does not end at z, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='2 ϕ(xz) ∈ {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' In this case, we may assume without loss of generality that ϕ(xz) = 6, ϕ(xv1) = 1 and ϕ(xv2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that if each of v1 and v2 has at most two 5−-neighbors, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus by the symmetry, assume that v1 has three 5−-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then there exist two vertices t, t′ ∈ N(v1)\\{x} such that d(t) ≤ 5 and d(t′) ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim 1 {ϕ(v1t), ϕ(v1t′)} = {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise, without loss of generality, assume ϕ(v1t) ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then y, x, v1, t form the vertex set of a Kierstead path with d(x) < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus ¯ϕ(t) = {6, 7} by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let α be a color in ¯ϕ(x)\\{ϕ(v1t)} and η be the color in ¯ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Note that Px(α, 7, ϕ) ends at t by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus we may assume that η ∈ ¯ϕ(x) since otherwise η ∈ {1, 2, 7} and we can interchange η, α on Pz1(η, α, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Furthermore, we may assume that η ∈ ¯ϕ(x)\\{ϕ(v1t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise η = ϕ(v1t) ∈ ¯ϕ(x), and we can interchange η, 1 on Pz(η, 1, ϕ) first and then interchange 1, α on the (1, α)-chain starting at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now the (6, η)-chain starting at x ends at z but not t, a contradiction to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Therefore {ϕ(v1t), ϕ(v1t′)} = {6, 7} and without loss of generality, we assume that ϕ(v1t) = 6 and ϕ(v1t′) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Claim 2 ¯ϕ1(z) ̸= {7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Let η be the color missing at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Otherwise η ∈ ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' We may assume that η ∈ ¯ϕ(x) since otherwise we can interchange η and 3 on Pz(η, 3, ϕ) to get the desired coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now by recoloring xz with η, we have {ϕ(v1t), ϕ(v1t′)} ̸= {6, 7}, a contradiction to Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus ¯ϕ(z) = {7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Now let η′ be a color in ¯ϕ(t′)\\{6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Similarly as before, we may assume that η′ ∈ ¯ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pt′(η′, 7, ϕ) does not end at x, let ϕ1 = ϕ/Pt′(η′, 7, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have {ϕ1(v1t), ϕ2(v1t′)} ̸= {6, 7}, a contradiction to Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' If Pt′(η′, 7, ϕ) ends at x, let ϕ1 = ϕ/Pz(η′, 7, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then we have ¯ϕ1(z) ̸= {7}, a contradiction to Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' This completes the proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' □ (3) Since y is the only 6-neighbor of x and |ϕ(x) ∩ ϕ(y)| = 2, there are two 7-neighbors of x, say v1, v2, such that {ϕ(xv1), ϕ(xv2)} ⊆ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' It is sufficient to show that each v1 and v2 has at most three 5−-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Suppose to the contrary that v1 has three 5−-neighbors other than x, say t1, t2, t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Since | ¯ϕ(x)| = 3, | ¯ϕ(y)| ≥ 2 and | ¯ϕ(ti)| ≥ 2 for each i = 1, 2, 3, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='5, at most one of ϕ(v1t1), ϕ(v1t2), ϕ(v1t3) is in ¯ϕ(x) ∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Without loss of generality, assume ϕ(v1t1) ∈ 23 ¯ϕ(x)∪ ¯ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Then {ϕ(v1t2), ϕ(v1t3)} = {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='3, we have ¯ϕ(t1) = ϕ(x)∩ϕ(y) = {6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Thus 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=', 87 (2003) 254-263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Stiebitz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Scheide, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Toft, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Favrholdt, Graph Edge Coloring: Vizing’s Theorem and Goldberg’s Conjecture, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 75, Wiley, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Vizing, Critical graphs with a given chromatic class (Russian), Diskret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Analiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 5 (1965) 9-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' [21] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Vizing, Some unsolved problems in graph theory, Uspekhi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Nauk 23 (1968) 117-134, Russian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Surveys 23 (1968) 125-142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Woodall, The average degree of an edge-chromatic critical graph II, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Graph Theory, 42 (2007) 194-218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' Zhang, Every planar graph with maximum degree 7 is of class 1, Graph Theory and Combinatorics, 16 (2000) 467-495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf'}