diff --git "a/3dFST4oBgHgl3EQfYzh6/content/tmp_files/load_file.txt" "b/3dFST4oBgHgl3EQfYzh6/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3dFST4oBgHgl3EQfYzh6/content/tmp_files/load_file.txt" @@ -0,0 +1,992 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf,len=991 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='13789v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='CO] 31 Jan 2023 The Minimum Degree Removal Lemma Thresholds Lior Gishboliner∗ Zhihan Jin∗ Benny Sudakov∗ Abstract The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H then G contains δnv(H) copies of H for some δ = δ(ε, H) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The current proofs of the removal lemma give only very weak bounds on δ(ε, H), and it is also known that δ(ε, H) is not polynomial in ε unless H is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ(ε, H) depends polynomially or linearly on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In this paper we answer several questions of Fox and Wigderson on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 1 Introduction The graph removal lemma, first proved by Ruzsa and Szemerédi [23], is a fundamental result in extremal graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It also have important applications to additive combinatorics and property testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The lemma states that for every fixed graph H and ε > 0, if an n-vertex graph G contains εn2 edge-disjoint copies of H then G it contains δnv(H) copies of H, where δ = δ(ε, H) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Unfortunately, the current proofs of the graph removal lemma give only very weak bounds on δ = δ(ε, H) and it is a very important problem to understand the dependence of δ on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The best known result, due to Fox [11], proves that 1/δ is at most a tower of exponents of height logarithmic in 1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Ideally, one would like to have better bounds on 1/δ, where an optimal bound would be that δ is polynomial in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' However, it is known [2] that δ(ε, H) is only polynomial in ε if H is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This situation led Fox and Wigderson [12] to initiate the study of minimum degree conditions which guarantee that δ(ε, H) depends polynomially or linearly on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Formally, let δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) be the maximum δ ∈ [0, 1] such that if G is an n-vertex graph with minimum degree at least γn and with εn2 edge-disjoint copies of H, then G contains δnv(H) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The linear removal threshold of H, denoted δlin-rem(H), is the infimum γ such that δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) depends linearly on ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) ≥ µε for some µ = µ(γ) > 0 and all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The polynomial removal threshold of H, denoted δpoly-rem(H), is the infimum γ such that δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) depends polynomially on ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δ(ε, H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' γ) ≥ µε1/µ for some µ = µ(γ) > 0 and all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Trivially, δlin-rem(H) ≥ δpoly-rem(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fox and Wigderson [12] initiated the study of δlin-rem(H) and δpoly-rem(H), and proved that δlin-rem(Kr) = δpoly-rem(Kr) = 2r−5 2r−3 for every r ≥ 3, where Kr is the clique on r vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' They further asked to determine the removal lemma thresholds of odd cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Here we completely resolve this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The following theorem handles the polynomial removal threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δpoly-rem(C2k+1) = 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 also answers another question of Fox and Wigderson [12], of whether δlin-rem(H) and δpoly-rem(H) can only obtain finitely many values on r-chromatic graphs H for a given r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 shows that δpoly-rem(H) obtains infinitely many values for 3-chromatic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In contrast, δlin-rem(H) ob- tains only three possible values for 3-chromatic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, the following theorem determines δlin-rem(H) for every 3-chromatic H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' An edge xy of H is called critical if χ(H − xy) < χ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' ∗Department of Mathematics, ETH, Zürich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Research supported in part by SNSF grant 200021_196965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Email: {lior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='gishboliner, zhihan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='jin, benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='sudakov}@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a graph H with χ(H) = 3, it holds that δlin-rem(H) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 2 H has no critical edge, 1 3 H has a critical edge and contains a triangle, 1 4 H has a critical edge and odd-girth(H) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 show a separation between the polynomial and linear removal thresholds, giving a sequence of graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' C5, C7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' ) where the polynomial threshold tends to 0 while the linear threshold is constant 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The parameters δpoly-rem and δlin-rem are related to two other well-studied minimum degree thresholds: the chromatic threshold and the homomorphism threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The chromatic threshold of a graph H is the infimum γ such that every n-vertex H-free graph G with δ(G) ≥ γn has bounded cromatic number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=', there exists C = C(γ) such that χ(G) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The study of the chromatic threshold originates in the work of Erdős and Simonovits [10] from the ’70s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Following multiple works [4, 15, 16, 7, 5, 25, 26, 19, 6, 14, 20], the chromatic threshold of every graph was determined by Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moving on to the homomorphism threshold, we define it more generally for families of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The homomorphism threshold of a graph-family H, denoted δhom(H), is the infimum γ for which there exists an H-free graph F = F(γ) such that every n-vertex H-free graph G with δ(G) ≥ γn is homomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' When H = {H}, we write δhom(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This parameter was widely studied in recent years [18, 22, 17, 8, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It turns out that δhom is closely related to δpoly-rem(H), as the following theorem shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a graph H, let IH denote the set of all minimal (with respect to inclusion) graphs H′ such that H is homomorphic to H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For every graph H, δpoly-rem(H) ≤ δhom(IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that IC2k+1 = {C3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Using this, the upper bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 follows immediately by combining Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4 with the result of Ebsen and Schacht [8] that δhom({C3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}) = 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 was established in [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' for completeness, we sketch the proof in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The rest of this short paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Section 2 contains some preliminary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In Section 3 we prove the lower bounds in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Section 4 gives the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4, and Section 5 gives the proof of the upper bounds in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In the last section we discuss further related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 2 Preliminaries Throughout this paper, we always consider labeled copies of some fixed graph H and write copy of H for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We use δ(G) for the minimum degree of G, and write H → F to denote that there is a homo- morphism from H to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a graph H on [h] and integers s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh > 0, we denote by H[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh] the blow-up of H where each vertex i ∈ V (H) is replaced by a set Si of size si (and edges are replaced with complete bipartite graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The following lemma is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a fixed graph on vertex set [h] and let s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' There exists a constant c = c(H, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh) > 0 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be an n-vertex graph and V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vh ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose that G contains at least ρnh copies of H mapping i to Vi for all i ∈ [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G contains at least cρ 1 c · ns1+···+sh copies of H[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh] mapping Si to Vi for all i ∈ [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that the sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vh in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 do not have to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 works by defining an auxiliary h-uniform hypergraph G whose hyperedges correspond to the copies of H in which vertex i is mapped to Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By assumption, G has at least ρnh edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the hypergraph generalization of the Koväri-Sós-Turán theorem, see [9], G contains poly(ρ)ns1+···+sh copies of K(h) s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=',sh, the complete h-partite hypergraph with parts of size s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each copy of K(h) s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=',sh gives a copy of H[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , sh] mapping Si to Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fox and Wigderson [12, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1] proved the following useful fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If H → F and F is a subgraph of H, then δpoly-rem(H) = δpoly-rem(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 2 The following lemma is an asymmetric removal-type statement for odd cycles, which gives polynomial bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' A similar result has appeared very recently in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For 1 ≤ ℓ < k, there exists a constant c = c(k) > 0 such that if an n-vertex graph G has εn2 edge-disjoint copies of C2ℓ+1, then it has at least cε1/cn2k+1 copies of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let C be a collection of εn2 edge-disjoint copies of C2ℓ+1 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' There exists a collection C′ ⊆ C such that |C′| ≥ εn2/2 and each vertex v ∈ V (G) belongs to either 0 or at least εn/2 of the cycles in C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, to obtain C′, we repeatedly delete from C all cycles containing a vertex v which belongs to at least one but less than εn/2 of the cycles in C (without changing the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The set of cycles left at the end is C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In this process, we delete at most εn2/2 cycles altogether (because the process lasts for at most n steps);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' hence |C′| ≥ εn2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let V be the set of vertices contained in at least εn/2 cycles from C′, so |V | ≥ εn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' With a slight abuse of notation, we may replace G with G[V ], C with C′ and ε/2 with ε, and denote |V | by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, from now on, we assume that each vertex v ∈ V (G) is contained in at least εn of the cycles in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This implies that |N(v)| ≥ 2εn for every v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any v0 ∈ V (G) and let C(v0) be the set of cycles C ∈ C such that C ∩ N(v0) ̸= ∅ and v0 /∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The number of cycles C ∈ C intersecting N(v0) is at least |N(v0)| · εn/(2ℓ + 1) ≥ 2ε2n2/(2ℓ + 1), and the number of cycles containing v0 is at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, |C(v0)| ≥ 2ε2n2/(2ℓ + 1) − n ≥ ε2n2/(ℓ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Take a random partition V0, V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vℓ of V (G) \\ {v0}, where each vertex is put in one of the parts uniformly and independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For a cycle (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , x2ℓ+1) ∈ C(v0) with xℓ+1 ∈ N(v0), say that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , x2ℓ+1) is good if xℓ+1 ∈ V0 and xℓ+1−i, xℓ+1+i ∈ Vi for 1 ≤ i ≤ ℓ (so in particular x1, x2ℓ+1 ∈ Vℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The probability that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , x2ℓ+1) is good is 1/(ℓ + 1)2ℓ+1, so there is a collection of good cycles C′(v0) ⊆ C0 of size |C′(v0)| ≥ |C(v0)|/(ℓ + 1)2ℓ+1 ≥ ε2n2/(ℓ + 1)2ℓ+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put γ := ε2/(ℓ + 1)2ℓ+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the same argument as above, there is a collection C′′(v0) ⊆ C′(v0) with |C′′(v0)| ≥ γn2/2 such that each vertex is contained in either 0 or at least γn/2 cycles from C′′(v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let W be the set of vertices contained in at least γn/2 cycles from C′′(v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that W ∩ V0 ⊆ N(v0) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, each vertex in W ∩ Vℓ has at least γn/2 neighbors in W ∩ Vℓ, and for each 1 ≤ i ≤ ℓ, each vertex in W ∩ Vi has at least γn/2 neighbors in W ∩ Vi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It follows that W ∩ Vℓ contains at least 1 2|W ∩ Vℓ| · �2k−2ℓ−2 i=0 (γn/2 − i) = poly(γ)n2k−2ℓ paths of length 2k − 2ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now construct a collection of copies of C2k+1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Choose a path yℓ+1, yℓ+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k−ℓ of length 2k − 2ℓ − 1 in W ∩ Vℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For each i = ℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , 1, take a neighbor yi ∈ W ∩ Vi−1 of yi+1 and a neighbor y2k−i+1 ∈ W ∩ Vi−1 of y2k−i, such that the vertices y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k are all different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k is a path and y1, y2k ∈ W ∩ V0 ⊆ N(v0), so v0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k is a copy of C2ℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The number of choices for the path yℓ+1, yℓ+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k−ℓ is poly(γ)n2k−2ℓ and the number of choices for each vertex yi, y2k−i+1 ∈ Vi−1 (i = ℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , 1) is at least γn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, the total number of choices for y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y2k is poly(γ)n2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As there are n choices for v0, we get a total of poly(γ)n2k+1 = polyk(ε)n2k+1 copies of C2k+1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 3 Lower bounds Here we prove the lower bounds in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 was proved in [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For completeness, we include a sketch of the proof: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' δpoly-rem(C2k+1) ≥ 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix an arbitrary α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In [2] it was proved that for every ε, there exists a (2k + 1)-partite graph with parts V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , V2k+1 of size αn/(2k + 1) each, with εn2 edge-disjoint copies of C2k+1, but with only εω(1)n2k+1 copies of C2k+1 in total (where the ω(1) term may depend on α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Add sets U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , U2k+1 of size (1 − α)n/(2k + 1) each, and add the complete bipartite graphs (Ui, Vi), 1 ≤ i ≤ 2k + 1, and (Ui, Ui+1), 1 ≤ i ≤ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It is easy to see that this graph has minimum degree (1 − α)n/(2k + 1), and every copy of C2k+1 is contained in V1 ∪ · · · ∪ V2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Letting α → 0, we get that δpoly-rem(C2k+1) ≥ 1 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By combining the fact that δpoly-rem(C3) = 1 3 with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 (with F = C3), we get that δlin-rem(H) ≥ δpoly-rem(H) = 1 3 for every 3-chromatic graph H containing a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This proves the lower bound in the second case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we prove the lower bounds in the other two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We prove a more general statement for r-chromatic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 3 V2 V3 V4 V5 V1 U2 U3 U4 U5 U1 Figure 1: Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 for C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Heavy edges indicate complete bipartite graphs while dashed edges form the Ruzsa–Szemerédi construction for C5 (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph with χ(H) = r ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, 3r−8 3r−5 ≤ δlin-rem(H) ≤ r−2 r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moreover, δlin-rem(H) = r−2 r−1 if H contains no critical edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote h = |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The bound δlin-rem(H) ≤ r−2 r−1 holds for every r-chromatic graph H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' this follows from the Erdős-Simonovits supersaturation theorem, see by [12, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose now that H contains no critical edge, and let us show that δlin-rem(H) ≥ r−2 r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' To this end, we construct, for every small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ r−2 r−1n, such that G has at most O(ε2nh) copies of H, but Ω(εn2) edges must be deleted to turn G into an H-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let T (n, r − 1) be the Turán graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' the complete (r − 1)-partite graph with balanced parts V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Add an εn-regular graph inside V1 and let the resulting graph be G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We first claim that G contains O(ε2nh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As H contains no critical edge and χ(H) = r, every copy of H in G contains two edges e and e′ inside V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If e and e′ are disjoint, then there are at most n2(εn)2 = ε2n4 choices for e and e′ and then at most nh−4 choices for the other h− 4 vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Therefore, there are at most ε2nh such H-copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if e and e′ intersect, then there are at most n(εn)2 = ε2n3 choices for e and e′ and then at most nh−3 choices for the remaining vertices, again giving at most ε2nh such H-copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So G indeed has O(ε2nh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' On the other hand, we claim that one must delete Ω(εn2) edges to destroy all H-copies in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Observe that G has at least 1 2 |V1|·εn·|V2|·· · ··|Vr−1| = Ωr(εnr) copies of Kr, and every edge participates in at most nr−2 of these copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, deleting cεn2 edges can destroy at most cεnr copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If c is a small enough constant (depending on r), then after deleting any cεn2 edges, there are still Ω(εnr) copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, the remaining graph contains Kr[h], the h-blowup of Kr, and hence H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof that δlin-rem(H) ≥ r−2 r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now prove that δlin-rem(H) ≥ 3r−8 3r−5 for every r-chromatic graph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It suffices to construct, for every small enough ε and infinitely many n, an n-vertex graph G with δ(G) ≥ 3r−8 3r−5n, such that G has at most O(ε2nh) copies of H but at least Ω(εn2) edges must be deleted to turn G into an H-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The vertex set of G consists of r + 1 disjoint sets V0, V1, V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr, where |Vi| = n 3r−5 for i = 0, 1, 2, 3 and |Vi| = 3n 3r−5 for i = 4, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=', r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put complete bipartite graphs between V0 and V1, between V0 ∪ V1 and V4 ∪ · · · ∪ Vr, and between Vi to Vj for all 2 ≤ i < j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put εn-regular bipartite graphs between V1 and V2, and between V1 and V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The resulting graph is G (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It is easy check that δ(G) ≥ 3r−8 3r−5n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, let 0 ≤ i ≤ r and v ∈ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If 4 ≤ i ≤ r then v is connected to all vertices except for Vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' if i ∈ {2, 3} then v is connected to all vertices except V0 ∪ V1 ∪ Vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' and if i ∈ {0, 1} then v is connected to all vertices except V2 ∪ V3 ∪ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In any case, the neighborhood of v misses at most 3n 3r−5 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that G has at most O(ε2nh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, observe that if we delete all edges between V1 and V2 then the remaining graph is (r − 1)-colorable with coloring V1 ∪ V2, V0 ∪ V3, V4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, every copy of H must contain an edge e between V1 and V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Similarly, every copy of H must contain an edge e′ between V1 and V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If e, e′ are disjoint then there are at most n2(εn)2 = ε2n4 ways to choose e, e′ and then at most nh−4 ways to choose the remaining vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if e and e′ intersect then there are at most n(εn)2 = ε2n3 ways to choose e, e′ and at most nh−3 for the remaining h − 3 vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In both cases, the number of H-copies is at most ε2nh, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we show that one must delete Ω(εn2) edges to destroy all copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Observe that G has |V1| · (εn)2 · |V4| · · · · · |Vr| = Ω(ε2nr) copies of Kr between the sets V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that every edge f 4 V1 V2 V3 V0 V1 V2 V3 V4 V0 Figure 2: Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, r = 3 (left) and r = 4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Heavy edges indicate complete bipartite graphs while dashed edges indicate εn-regular bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' participates in at most εnr−2 of these r-cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, by the same argument as above, every copy of Kr containing f must contain an edge e from E(V1, V2) and an edge e′ from E(V1, V3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose without loss of generality that e ̸= f (the case e′ ̸= f is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In the case f ∩ e = ∅, there are at most n · εn = εn2 choices for e and at most nr−4 choices for the remaining vertices of Kr, giving at most εnr−2 copies of Kr containing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if f, e intersect, then there are at most εn choices for e and at most nr−3 for the remaining r − 3 vertices, giving again εnr−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We see that deleting cεn2 edges of G can destroy at most cε2nr copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, if c is a small enough constant, then after deleting any cεn2 edges there are still Ω(ε2nr) copies of Kr left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, the remaining graph contains a copy of Kr[h] and hence H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 4 Polynomial removal thresholds: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4 We say that an n-vertex graph G is ε-far from a graph property P (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' being H-free for a given graph H, or being homomorphic to a given graph F) if one must delete at least εn2 edges to make G satisfy P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Trivially, if G has εn2 edge-disjoint copies of H, then it is ε-far from being H-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We need the following result from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For every graph F on f vertices and for every ε > 0, there is q = qF (ε) = poly(f/ε), such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If a graph G is ε-far from being homomorphic to F, then for a sample of q vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq ∈ V (G), taken uniformly with repetitions, it holds that G[{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq}] is not homomorphic to F with probability at least 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 is proved in Section 2 of [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In fact, [21] proves a more general result on property testing of the so-called 0/1-partition properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Such a property is given by an integer f and a function d : [f]2 → {0, 1, ⊥}, and a graph G satisfies the property if it has a partition V (G) = V1 ∪ · · · ∪ Vf such that for every 1 ≤ i, j ≤ f (possibly i = j), it holds that (Vi, Vj) is complete if d(i, j) = 1 and (Vi, Vj) is empty if d(i, j) = 0 (if d(i, j) =⊥ then there are no restrictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' One can express the property of having a homomorphism into F in this language, simply by setting d(i, j) = 0 for i = j and ij /∈ E(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In [21], the class of these partition properties is denoted GPP0,1, and every such property is shown to be testable by sampling poly(f/ε) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This implies Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that IH is the set of minimal graphs H′ (with respect to inclusion) such that H is homomorphic to H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For convenience, put δ := δhom(IH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Our goal is to show that δpoly-rem(H) ≤ δ+α for every α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So fix α > 0 and let G be a graph with minimum degree δ(G) ≥ (δ + α)n and with εn2 edge-disjoint copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of the homomorphism threshold, there is an IH-free graph F (depending only on IH and α) such that if a graph G0 is IH-free and has minimum degree at least (δ + α 2 ) · |V (G0)|, then G0 is homomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Observe that if a graph G0 is homomorphic to F then G0 is H-free, because F is free of any homomorphic image of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It follows that G is ε-far from being homomorphic to F, because G is ε-far from being H-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let q = qF (ε) be given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We assume that q ≫ log(1/α) α2 and n ≫ q2 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Sample q vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq ∈ V (G) with repetition and let X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, G[X] is not homomorphic to F with probability at least 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As n ≫ q2, the vertices x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xq are pairwise-distinct with probability at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, for every i ∈ [q], the number of indices j ∈ [q] \\ {i} with xixj ∈ E(G) dominates a binomial 5 distribution B(q − 1, δ(G) n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the Chernoff bound (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' [3, Appendix A]) and as δ(G) ≥ (δ + α)n, the number of such indices is at least (δ + α 2 )q with probability 1 − e−Ω(qα2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Taking the union bound over i ∈ [q], we get that δ(G[X]) ≥ (δ + α 2 )|X| with probability at least 1 − qe−Ω(qα2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='9, as q ≫ log(1/α) α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, with probability at least 1 2 it holds that δ(G[X]) ≥ (δ + α 2 )|X| and G[X] is not homomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If this happens, then G[X] is not IH-free (by the choice of F), hence G[X] contains a copy of some H′ ∈ IH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By averaging, there is H′ ∈ IH such that G[X] contains a copy of H′ with probability at least 1 2|IH|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put k = |V (H′)| and let M be the number of copies of H′ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The probability that G[X] contains a copy of H′ is at most M( q n)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Using the fact that q = polyH,α( 1 ε), we conclude that M ≥ 1 2|IH| · ( n q )k ≥ polyH,α(ε)nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As H → H′, there exists H′′, a blow-up of H′, such that H′′ have the same number of vertices as H, and that H ⊂ H′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 for H′ with Vi = V (G) for all i, there exist polyH,α(ε)nv(H′′) copies of H′′ in G, and thus polyH,α(ε)nv(H) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 5 Linear removal thresholds: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 Here we prove the upper bounds in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' the lower bounds were proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The first case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, so it remains to prove the other two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We begin with some preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For disjoint sets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Am, we write � i∈[m] Ai × Ai+1 to denote all pairs of vertices which have one endpoint in Ai and one in Ai+1 for some 1 ≤ i ≤ m, with subscripts always taken modulo m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So a graph G has a homomorphism to the cycle Cm if and only if there is a partition V (G) = A1 ∪ · · · ∪ Am with E(G) ⊆ � i∈[m] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose H is a graph such that χ(H) = 3, H contains a critical edge xy, and odd-girth(H) ≥ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, There is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 = {x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪ (� i∈[3] Ai × Ai+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' if k ≥ 2, there is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that A1 = {x}, A2 = {y} and E(H) ⊆ � i∈[2k+1] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In particular, H is homomorphic to C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write H′ = H − xy, so H′ is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let V (H) = V (H′) = L ·∪ R be a bipartition of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As χ(H) = 3, x and y must both lie in the same side of the bipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Without loss of generality, assume that x, y ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For the first item, set A1 = {x}, A2 = {y}, A3 = R and B = L\\{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then every edge of G goes between B and A3 or between two of the sets A1, A2, A3, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose now that k ≥ 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' odd-girth(H) = 2k + 1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For 1 ≤ i ≤ k, let Xi be the set of vertices at distance (i − 1) from x in H′, and let Yi be the set of vertices at distance (i − 1) from y in H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that X1 = {x} and Y1 = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, Xi, Yi lie in L if i is odd and in R if i is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write L′ := L\\ k� i=1 (Xi ∪ Yi), R′ := R\\ k� i=1 (Xi ∪ Yi), We first claim that {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Xk, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk, L′, R′} forms a partition of V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The sets X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Xk are clearly pairwise-disjoint, and so are Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, all of these sets are disjoint from L′, R′ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So we only need to check Xi and Yj are disjoint for every pair 1 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose for contradiction that there exists u ∈ Xi ∩ Yj for some 1 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then i ≡ j (mod 2), because otherwise Xi, Yj are contained in different parts of the bipartition L, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of Xi and Yj, H′ has a path x = x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xi = u and a path y = y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , yj = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, x = x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , xi = u = yj, yj−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , y1, y, x forms a closed walk of length i+j −1, which is odd as i ≡ j (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, odd-girth(H) ≤ 2k−1, contradicting our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By definition, there are no edges between Xi and Xj for j − i ≥ 2, and similarly for Yi, Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, there are no edges between L′ ∪ R′ and �k−1 i=1 (Xi ∪ Yi) because the vertices in L′ ∪ R′ are at distance more than k to x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moreover, if k is even then there are no edges between Xk ∪ Yk and R′, and if k is odd then there are no edges between Xk ∪ Yk and L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Next, we show that there are no edges between Xi and Yj for any 1 ≤ i, j ≤ k except (i, j) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, if i = j then e(Xi, Yj) = 0 because Xi, Yj are on the same side 6 x y X2 Y2 L′ R′ x y X2 Y2 X3 Y3 R′ L′ Figure 3: Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, k = 2 (left) and k = 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Edges indicate bipartite graphs where edges can be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' of the bipartition L, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So suppose that i ̸= j, say i < j, and assume by contradiction that there is an edge uv with u ∈ Xi, v ∈ Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then v is at distance at most i + 1 ≤ k from x, implying that Yj intersects X1 ∪ · · · ∪ Xi+1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Finally, we define the partition A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , A2k+1 that satisfies the assertion of the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If k is even then take A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , A2k+1 to be X1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk−1, Yk∪R′, L′, Xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , X2, and if k is odd then take A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , A2k+1 to be X1, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Yk−1, Yk ∪ L′, R′, Xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the above, in both cases it holds that E(H) ⊆ � i∈[2k+1] Ai × Ai+1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For vertex u ∈ V (G), denote by NG(u) the neighborhood of u and let degG(u) = |NG(u)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For vertices u, v ∈ V (G), denote by NG(u, v) the common neighborhood of u, v and let degG(u, v) = |NG(u, v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph on h vertices such that χ(H) = 3 and H contains a critical edge xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a graph on n vertices with δ(G) ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let ab ∈ E(G) such that degG(a, b) ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, there are at least poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the first item of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, there is a partition V (H) = A1 ·∪ A2 ·∪ A3 ·∪ B such that A1 = {x}, A2 = {y} and E(H) ⊆ (A3 × B) ∪ � i∈[3] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let s = |A3| and t = |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each u ∈ NG(a, b) has at least αn − 2 ≥ αn 2 neighbors not equal to a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, there are at least 1 2 · |NG(a, b)| · αn 2 ≥ α2n2 4 edges uv with u ∈ NG(a, b) and v /∈ {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 with H = K2, V1 = NG(a, b) and V2 = V (G)\\{a, b}, we see that there are poly(α)ns+t pairs of disjoint sets (S, T ) such that |S| = s, |T | = t, S ⊆ NG(a, b), a, b /∈ T , and S, T form a complete bipartite graph in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Given any such pair, it is safe to map x to a, y to b, A3 to S and B to T to obtain an H-copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, G contains at least poly(α)ns+t = poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph on h vertices such that χ(H) = 3, H contains a critical edge xy, and odd-girth(H) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a graph on n vertices, let ab ∈ E(G), and suppose that there exists A ⊂ NG(a) and B ⊂ NG(b) such that |A| , |B| ≥ αn and |NG(a′, b′)| ≥ αn for all distinct a′ ∈ A and b′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then there are at least poly(α)nh−2 copies of H in G mapping xy ∈ E(H) to ab ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 (using odd-girth(H) ≥ 5), there exists a partition V (H) = A1 ·∪ · · · ·∪ A5 such that A1 = {x}, A2 = {y}, and E(H) ⊆ � i∈[5] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put si = |Ai| for i ∈ [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' There are at least (|A||B| − |A|)/2 ≥ α2n2/3 pairs {a′, b′} of distinct vertices with a′ ∈ A, b′ ∈ B (the factor of 2 is due to the fact that each pair in A ∩ B is counted twice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each such pair a′, b′ has at least αn − 2 ≥ αn/2 common neighbors c′ /∈ {a, b}, by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Therefore, there are at least α2n2 3 αn 2 = α3n3 6 triples (a′, b′, c′) such that a′ ∈ A, b′ ∈ B, and c′ ̸= a, b is a common neighbor of a′, b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 with H = K2,1 and V1 = A, V2 = B, V3 = V (G)\\{a, b}, there are at least poly(α)ns3+s4+s5 corresponding copies of K2,1[s3, s5, s4], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=', triples of disjoint sets (R, S, T ) such that R ⊆ A, S ⊆ B, a, b /∈ T , |R| = s5, |S| = s3, |T | = s4, and (R, T ) and (S, T ) form complete bipartite graphs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Given any such 7 triple, we can safely map A1 = {x} to a, A2 = {y} to b, A5 to R, A3 to S, and A4 to T to obtain a copy of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, there are at least poly(α)ns3+s4+s5 = poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In the following theorem we prove the upper bound in the second case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph such that χ(H) = 3, H has a critical edge xy, and H contains a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, δlin-rem(H) ≤ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write h = v(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix an arbitrary α > 0, and let G be an n-vertex graph with minimum degree δ(G) ≥ ( 1 3 + α)n and with a collection C = {H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Hm} of m := εn2 edge-disjoint copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , m, there exist u, v, w ∈ V (Hi) forming a triangle (because H contains a triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As degG(u) + degG(v) + degG(w) ≥ 3δ(G) ≥ (1 + 3α)n, two of u, v, w have at least αn common neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We denote these two vertices by ai and bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, G has at least poly(α)nh−2 copies of H which map xy to aibi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The edges a1b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , ambm are distinct because H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , Hm are edge-disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, summing over all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , m, we see that G contains at least εn2 · poly(α)nh−2 = poly(α)εnh copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This proves that δlin-rem(H) ≤ 1 3 + α, and taking α → 0 gives δlin-rem(H) ≤ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In what follows, we need the following very well-known observation, originating in the work of Andrásfai, Erdős and Sós, see [4, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If δ(G) > 2 2k+1n and odd-girth(G) ≥ 2k + 1 for k ≥ 2, then G is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose by contradiction that G is not bipartite and take a shortest odd cycle C in G, so |C| ≥ 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As � x∈C deg(x) ≥ (2k+1)δ(G) > 2n, there exists a vertex v /∈ C with at least 3 neighbors on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then there are two neighbors x, y ∈ C of v such that the distance of x, y along C is not equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then by taking the odd path between x, y along C and adding the edges vx, vy, we get a shorter odd cycle, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We will also use the following result of Letzter and Snyder, see [17, Corollary 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='6 ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a {C3, C5}-free graph on n vertices with δ(G) > n 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G is homomorphic to C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We can now finally prove the upper bound in the last case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let H be a graph such that χ(H) = 3, H contains critical edge xy, and odd-girth(H) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then δlin-rem(H) ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote h = |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Write odd-girth(G) = 2k + 1 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the second item of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1, there is a partition V (H) = A1 ·∪ A2 ·∪ · · · ·∪ A2k+1 such that |A1| = |A2| = 1, and E(H) ⊆ � i∈[2k+1] Ai × Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote si = |Ai| for each i ∈ [2k + 1], so H is a subgraph of the blow-up C2k+1[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , s2k+1] of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let c1 = c1(C2k+1, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , s2k+1) > 0 and c2 = c2(k) > 0 be the constants given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' According to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, δpoly-rem(C2k+1) = 1 2k+1 < 1 4, and hence there exists a constant c3 = c3(k) > 0 such that if G is a graph on n vertices with δ(G) ≥ n 4 and at least εn2 edge-disjoint C2k+1-copies, then G contains at least c3ε 1 c3 n2k+1 copies of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Set c := c1 · min(c2, c3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let α > 0 and ε be small enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' it suffices to assume that ε < � α2 200k(k+2) �1/c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G be a graph on n vertices with δ(G) ≥ ( 1 4 + α)n which contains at least εn2 edge-disjoint copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Our goal is to show that G contains ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose first that G contains at least εcn2 edge-disjoint copies of C2ℓ+1 for some 1 ≤ ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If ℓ < k, then G contains Ωk(εc/c2n2k+1) = Ωk(εc1n2k+1) copies of C2k+1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 and the choice of c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' And if ℓ = k, then G contains Ωk(εc/c3n2k+1) = Ωk(εc1n2k+1) copies of C2k+1 by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2 and the choice of c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In either case, G contains Ωk(εc1n2k+1) copies of C2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' But then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='1 (with V1 = · · · = V2k+1 = V (G)), G contains at least ΩH(εc1/c1nh) = ΩH(εnh) copies of C2k+1[s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , s2k+1], and hence ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This concludes the proof of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' From now on, assume that G contains at most εcn2 edge-disjoint C2ℓ+1-copies for every ℓ ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let Cℓ be a maximal collection of edge-disjoint C2ℓ+1-copies in G, so |Cℓ| ≤ εcn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let Ec be the set of edges which 8 are contained in one of the cycles in C1 ∪ · · · ∪ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let S be the set of vertices which are incident with at least αn 10 edges from Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then |Ec| ≤ k � ℓ=1 (2ℓ + 1)εcn2 = k(k + 2)εcn2 and |S| ≤ 2 |Ec| αn/10 ≤ 20k(k + 2)εc α n < αn 10 , (1) where the last inequality holds by our assumed bound on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G′ be the subgraph of G obtained by deleting the edges in Ec and the vertices in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that G′ ⊆ G − Ec is {C3, C5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}-free because for every 1 ≤ ℓ ≤ k, we removed all edges from a maximal collection of edge-disjoint C2ℓ+1-copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' |V (G′)| > (1 − α 10)n and δ(G′) > ( 1 4 + 4α 5 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The first inequality follows from (1) as |V (G′)| = n − |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each v ∈ V (G)\\S has at most αn 10 incident edges from Ec, and at most |S| < αn 10 neighbors in S, thereby degG′(v) > degG(v) − αn 5 ≥ ( 1 4 + 4α 5 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, δ(G′) > ( 1 4 + 4α 5 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' G′ is homomorphic to C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Moreover, G′ is bipartite unless k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that G′ is {C3, C5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , C2k+1}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As k ≥ 2, G′ is {C3, C5}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, δ(G′) > n 4 ≥ |V (G′)| 4 by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So G′ is homomorphic to C7 by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If k ≥ 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' odd-girth(H) ≥ 7, then odd-girth(G′) ≥ 2k + 3 ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As δ(G′) > n 4 , G′ is bipartite by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The rest of the proof is divided into two cases based whether or not G′ is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' These cases are handled by Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose that G′ is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G has ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let (L′, R′) be a bipartition of G′, so V (G) = L′ ·∪ R′ ·∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let L1 ⊆ S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' R1 ⊆ S) be the set of vertices of S having at most αn 5 neighbors in L′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G′′ be the bipartite subgraph of G induced by the bipartition (L′′, R′′) := (L′ ·∪ L1, R′ ·∪ R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let S′′ = V (G)\\(L′′ ·∪ R′′), so V (G) = L′′ ·∪ R′′ ·∪ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' First, as G′ is a subgraph of G′′, we have degG′′(v) > ( 1 4 + 4α 5 )n for each v ∈ V (G′) ⊆ V (G′′) by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now we consider vertices in V (G′′) \\ V (G′) = L1 ∪ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Each v ∈ L1 has at most |S| ≤ αn 10 neighbors in S and at most αn 5 neighbors in L′, by the definition of L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, v has at least degG(v) − 3α 10 n ≥ ( 1 4 + α 2 )n neighbors in R′ ⊆ V (G′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the symmetric argument for vertices v ∈ R1, we get that δ(G′′) ≥ ( 1 4 + α 2 )n, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For an edge uv ∈ E(G)\\E(G′′), we say uv is of type I if u, v ∈ L′′ or u, v ∈ R′′, and we say that uv is of type II if u ∈ S′′ or v ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Every edge in E(G)\\E(G′′) is of type I or II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Since χ(H) = 3 and G′′ is bipartite, each copy of H in G must contain an edge of type I or an edge of type II (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As G has εn2 edge-disjoint H-copies, G contains at least εn2 2 edges of type I or at least εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now consider these two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that xy ∈ E(H) denotes a critical edge of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 1: G contains εn2 2 edges of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any edge ab ∈ E(G) of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Without loss of generality, assume a, b ∈ L′′ (the case a, b ∈ R′′ is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that G has poly(α)nh−2 copies of H mapping xy ∈ E(H) to ab ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If degG(a, b) ≥ αn 2 then this holds by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Otherwise, degG(a, b) < αn 2 , and thus |R′′| ≥ |NG′′(a) ∪ NG′′(b)| ≥ degG′′(a) + degG′′(b) − degG(a, b) > 2δ(G′′) − αn 2 > n 2 , using that δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, |L′′| < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This implies that for all a′ ∈ NG′′(a), b′ ∈ NG′′(b), degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3 (with A = NG′′(a) and B = NG′′(b)), there are poly(α)nh−2 copies of H mapping xy to ab, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Summing over all edges ab of type I, we get εn2 2 · poly(α)nh−2 = poly(α)εnh copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 9 L′′ R′′ a b L′′ R′′ a b a′ b′ L′′ R′′ S′′ a b a′ b′ Figure 4: Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10: Case 1 with degG(a, b) ≥ αn 2 (left), Case 1 with degG(a, b) < αn 2 (middle) and Case 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The red part is the common neighborhood of a and b (or a′ and b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 2: G contains εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Note that the number of edges of type II is trivially at most |S′′| n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Thus, |S′′| ≥ εn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix some a ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of L′′, R′′ and S′′, v has at least αn 5 neighbors in L′ ⊆ L′′ and at least αn 5 neighbors in R′ ⊆ R′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Without loss of generality, assume |L′′| ≤ |R′′|, thereby |L′′| ≤ n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now fix any b ∈ L′′ adjacent to a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' there are at least αn 5 choices for b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have |NG(a) ∩ R′′| ≥ αn 5 and |NG′′(b)| ≥ δ(G′′) > n 4 , and for all a′ ∈ NG(a) ∩ R′′, b′ ∈ NG′′(b) ⊆ R′′ it holds that degG′′(a′, b′) ≥ 2δ(G′′) − |L′′| ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Therefore, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, G has poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Enumerating over all a ∈ S′′ and b ∈ NG(a) ∩ L′′, we again get ΩH,α(εnh) copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose G′ is non-bipartite but homomorphic to C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then G has ΩH,α(εnh) copies of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='9 we must have k = 2 , so odd-girth(H) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The proof is similar to that of Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10, but instead of a bipartition of G′, we use a partition corresponding to a homomorphism into C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let V (G)\\S = V (G′) = V ′ 1 ·∪ V ′ 2 ·∪ · · · ·∪ V ′ 7 be a partition of V (G′) such that E(G′) ⊆ � i∈[7] V ′ i × V ′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Here and later, all subscripts are modulo 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have V ′ i ̸= ∅ for all i ∈ [7], because otherwise G′ would be bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For i ∈ [7], let Si be the set of vertices in S having at most 2αn 5 neighbors in V (G′)\\ (V ′ i−1 ∪V ′ i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In case v lies in multiple Si’s, we put v arbitrarily in one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Set V ′′ i := V ′ i ∪ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Let G′′ be the 7-partite subgraph of G with parts V ′′ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , V ′′ 7 and with all edges of G between V ′′ i and V ′′ i+1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' , 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By definition, G′ is a subgraph of G′′, and G′′ is homomorphic to C7 via the homomorphism V ′′ i �→ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Put S′′ := V (G)\\V (G′′) = S \\ �7 i=1 Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now collect the following useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The following holds: (i) δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (ii) For every i ∈ [7] and for every u, v ∈ V ′′ i or u ∈ V ′′ i , v ∈ V ′′ i+2, it holds that degG′′(u, v) ≥ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iii) For every i ∈ [7], every v ∈ V ′′ i has at least αn neighbors in V ′′ i−1 and at least αn neighbors in V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iv) For every a ∈ S′′, there are i, j with j − i ≡ 1, 3 (mod 7) and |NG(a) ∩ V ′′ i | , ��NG(a) ∩ V ′′ j �� > 2αn 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' fds (i) Let i ∈ [7] and v ∈ V ′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If v ∈ V (G′), then degG′′(v) ≥ degG′(v) ≥ δ(G′) > ( 1 4 + α 2 )n, using Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Otherwise, v ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By definition, v has at most 2αn 5 neighbours in V (G′)\\(V ′ i−1 ∪V ′ i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, v has at most |S| ≤ αn 10 neighbours in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' It follows that v has at least degG(v)− 2αn 5 − αn 10 ≥ ( 1 4 + α 2 )n neighbors in V ′′ i−1 ∪ V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, degG′′(v) > ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (ii) First, observe that |V ′′ i | + ��V ′′ i+2 �� ≥ �1 4 + α 2 � n (2) 10 for all i ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, V ′′ i+1 is non-empty, and fixing any v ∈ V ′′ i+1, we have |V ′′ i | + ��V ′′ i+2 �� ≥ degG′′(v) ≥ δ(G′′) ≥ ( 1 4 + α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By applying (2) to the pairs (i + 2, i + 4) and (i − 2, i), we get ��V ′′ i−1 �� + ��V ′′ i+1 �� + ��V ′′ i+3 �� ≤ n − ( ��V ′′ i+2 �� + ��V ′′ i+4 ��) − ( ��V ′′ i−2 �� + |V ′′ i |) ≤ n − 2 �1 4 + α 2 � n < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (3) Now let i ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For u, v ∈ V ′′ i we have NG′′(u) ∪ NG′′(v) ⊆ V ′′ i−1 ∪ V ′′ i+1, and for u ∈ V ′′ i , v ∈ V ′′ i+2 we have NG′′(u) ∪ NG′′(v) ⊆ V ′′ i−1 ∪ V ′′ i+1 ∪ V ′′ i+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' In both cases, |NG′′(u) ∪ NG′′(v)| < n 2 by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As degG′′(u) + degG′′(v) ≥ 2δ(G′′) ≥ ( 1 2 + α)n, we have degG′′(u, v) > αn, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iii) We first argue that |V ′′ i | ≤ ( 1 4 − 3α 2 )n for each i ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Indeed, by applying (2) to the pairs (i − 1, i + 1), (i + 2, i + 4), (i + 3, i + 5), we get |V ′′ i | ≤ n − ( ��V ′′ i−1 �� + ��V ′′ i+1 ��) − ( ��V ′′ i+2 �� + ��V ′′ i+4 ��) − ( ��V ′′ i+3 �� + ��V ′′ i+5 ��) ≤ n − 3 �1 4 + α 2 � n = �1 4 − 3α 2 � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, for every v ∈ V ′′ i , we have NG′′(v) ⊆ V ′′ i−1 ∪ V ′′ i+1 and ��V ′′ i−1 �� , ��V ′′ i+1 �� < ( 1 4 − 3α 2 )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, v has at least degG′′(v) − ( 1 4 − 3α 2 )n ≥ αn neighbors in each of V ′′ i−1, V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' (iv) Let I be the set of i with |NG(a) ∩ V ′′ i | ≥ 2αn 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' If I is empty, then a has less than 5 · 2αn 25 = 2αn 5 neighbors in every V (G′)\\(V ′ i−1 ∪V ′ i+1) and therefore can not be in S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Suppose for contradiction that there exist no i, j ∈ I with j − i ≡ 1, 3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We claim that there is j ∈ [7] such that I ⊆ {j, j + 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix an arbitrary i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, i ± 1, i ± 3 /∈ I by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, at most one of i + 2, i − 2 is in I, because (i − 2) − (i + 2) ≡ 3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So I ⊆ {i, i + 2} or I ⊆ {i − 2, i}, proving our claim that I ⊆ {j, j + 2} for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By the definition of I, a has at most 5 · 2αn 25 = 2αn 5 neighbors in V (G′)\\(V ′ j ∪ V ′ j+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Hence, a ∈ Sj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This contradicts the fact that a ∈ S′′, as S′′ ∩ Si+1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We continue with the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that the edges in E(G) \\ E(G′′) are precisely the edges of G not belonging to � i∈[7] V ′′ i × V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' For an edge ab ∈ E(G)\\E(G′′), we say ab is of type I if a, b ∈ V (G′′), and of type II if a ∈ S′′ or b ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Clearly, every edge in E(G)\\E(G′′) is either of type I or of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Since odd-girth(H) = 5 and C5 is not homomorphic to C7, every H-copy in G must contain some edge of type I or of type II (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As G has εn2 edge-disjoint H-copies, G must have at least εn2 2 edges of type I or at least εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We consider these two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 5 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Recall that xy ∈ E(H) denotes a critical edge of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 1: G contains εn2 2 edges of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any edge ab of type I, where a ∈ V ′′ i and b ∈ V ′′ j for i, j ∈ [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We now show that G has poly(α)nh−2 copies of H mapping xy ∈ E(H) to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' As ab /∈ E(G′′), we have i−j ≡ 0, ±2, ±3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' When j−i ≡ 0, ±2 (mod 7), we have degG(a, b) ≥ degG′′(a, b) > αn by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='2, G has poly(α)nh−2 copies of H mapping xy to ab, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now suppose that j−i ≡ ±3 (mod 7), say j ≡ i+3 (mod 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Denote A := NG(a)∩V ′′ i−1 and B := NG(b)∩V ′′ j+1 = NG(b)∩V ′′ i−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have that |A| , |B| ≥ αn by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (iii), and |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, G has poly(α)nh−2 copies of H mapping xy to ab, proving our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Summing over all edges ab of type I, we get εn2 2 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G, finishing this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Case 2: G contains εn2 2 edges of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Notice that the number edges incident to S′′ is at most |S′′| n, meaning that |S′′| ≥ εn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any a ∈ S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' By Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (iv), there exist i, j ∈ [7] with j − i ≡ 1, 3 (mod 7) and |NG(a) ∩ V ′′ i | , ��NG(a) ∩ V ′′ j �� > 2αn 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Fix any b ∈ NG(a) ∩ V ′′ i (there are at least 2αn 25 choices for b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Take A = NG(a)∩V ′′ j and B = NG(b)∩V ′′ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We have that |A| ≥ 2αn 25 , and |B| ≥ αn by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Further, as j − (i + 1) ≡ 0, 2 (mod 7), Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='12 (ii) implies that |NG(a′, b′)| > αn for all a′ ∈ A, b′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Now, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='3, G has poly(α)nh−2 copies of H mapping xy to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Summing over all choices of a ∈ S′′ and b ∈ V ′′ i , we acquire |S′′| · 2αn 25 · poly(α)nh−2 = ΩH,α(εnh) copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' This completes the proof of Case 2, and hence the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='10 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11 imply the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 11 V ′′ 1 V ′′ 2 V ′′ 3 V ′′ 4 V ′′ 5 V ′′ 6 V ′′ 7 a b V ′′ 1 V ′′ 2 V ′′ 3 V ′′ 4 V ′′ 5 V ′′ 6 V ′′ 7 a b a′ b′ V ′′ 1 V ′′ 2 V ′′ 3 V ′′ 4 V ′′ 5 V ′′ 6 V ′′ 7 S′′ a b a′ b′ Figure 5: Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='11: Case 1 for j = i + 2 (left), Case 1 for j = i + 3 (middle) and Case 2 for j = i + 3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' The red part is the common neighborhood of a and b (or a′ and b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' 6 Concluding remarks and open questions It would be interesting to determine the possible values of δpoly-rem(H) for 3-chromatic graphs H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' So far we know that 1 2k+1 is a value for each k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Is there a graph H with 1 5 < δpoly-rem(H) < 1 3?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Also, is it true that δpoly-rem(H) > 1 5 if H is not homomorphic to C5?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Another question is whether the inequality in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='4 is always tight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' is it always true that δpoly-rem(H) = δhom(IH)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Finally, we wonder whether the parameters δpoly-rem(H) and δlin-rem(H) are monotone, in the sense that they do not increase when passing to a subgraph of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' We are not aware of a way of proving this without finding δpoly-rem(H), δlin-rem(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Allen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFST4oBgHgl3EQfYzh6/content/2301.13789v1.pdf'} +page_content=' Böttcher, S.' 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