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The Erd˝Os-Hajnal Property for Graphs with No Fixed Cycle As a Pivot-Minor

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The Erd˝Os-Hajnal Property for Graphs with No Fixed Cycle As a Pivot-Minor The Erd}os-Hajnal property for graphs with no fixed cycle as a pivot-minor Jaehoon Kim∗ Sang-il Oum† Department of Mathematical Sciences Discrete Mathematics Group KAIST Institute for Basic Science (IBS) Daejeon, Korea Daejeon, Korea [email protected] Department of Mathematical Sciences KAIST Daejeon, Korea [email protected] Submitted: Apr 23, 2020; Accepted: Mar 22, 2021; Published: Apr 9, 2021 © The authors. Released under the CC BY-ND license (International 4.0). Abstract We prove that for every integer k, there exists " > 0 such that for every n-vertex graph G with no pivot-minor isomorphic to Ck, there exist disjoint sets A; B ⊆ V (G) such that jAj; jBj > "n, and A is either complete or anticomplete to B. This proves the analog of the Erd}os-Hajnal conjecture for the class of graphs with no pivot-minor isomorphic to Ck. Mathematics Subject Classifications: 05CC55, 05C75 1 Introduction In this paper all graphs are simple, having no loops and no parallel edges. For a graph G, let !(G) be the maximum size of a clique, that is a set of pairwise adjacent vertices and let α(G) be the maximum size of an independent set, that is a set of pairwise non-adjacent vertices. Erd}osand Hajnal [9] proposed the following conjecture in 1989. Conjecture 1 (Erd}osand Hajnal [9]). For every graph H, there is " > 0 such that all graphs G with no induced subgraph isomorphic to H satisfies " max(α(G);!(G)) > jV (G)j : ∗Supported by the POSCO Science Fellowship of POSCO TJ Park Foundation and by the KAIX Challenge program of KAIST Advanced Institute for Science-X. †Supported by the Institute for Basic Science (IBS-R029-C1). the electronic journal of combinatorics 28(2) (2021), #P2.9 https://doi.org/10.37236/9536 This conjecture still remains open. See [5] for a survey on this conjecture. We can ask the same question for weaker containment relations. Recently Chudnovsky and Oum [6] proved that this conjecture holds if we replace \induced subgraphs" with \vertex-minors" as follows. This is weaker in the sense that every induced subgraph G is a vertex-minor of G but not every vertex-minor of G is an induced subgraph of G. Theorem 2 (Chudnovsky and Oum [6]). For every graph H, there exists " > 0 such that every graph with no vertex-minors isomorphic to H satisfies " max(α(G);!(G)) > jV (G)j : We ask whether Conjecture1 holds if we replace \induced subgraphs" with \pivot- minors" as follows. Conjecture 3. For every graph H, there exists " > 0 such that every graph G with no pivot-minor isomorphic to H satisfies " max(α(G);!(G)) > jV (G)j : The detailed definition of pivot-minors will be presented in Section3. For now, note that the analog for vertex-minors is weakest, the analog for pivot-minors is weaker than that for induced subgraphs but stronger than that for vertex-minors. This is because every induced subgraph of G is a pivot-minor of G, and every pivot-minor of G is a vertex-minor of G. In other words, Conjecture1 implies Conjecture3 and Conjecture3 implies Theorem2. We verify Conjecture3 for H = Ck, the cycle graph on k vertices as follows. Theorem 4. For every k > 3, there exists " > 0 such that every graph with no pivot-minor isomorphic to Ck satisfies " max(α(G);!(G)) > jV (G)j : We actually prove a stronger property, as Chudnovsky and Oum [6] did. Before stating this property, let us first state a few terminologies. A class G of graphs closed under taking induced subgraphs is said to have the Erd}os-Hajnalproperty if there exists " > 0 such that every graph G in G satisfies " max(α(G);!(G)) > jV (G)j : A class G of graphs closed under taking induced subgraphs is said to have the strong Erd}os-Hajnalproperty if there exists " > 0 such that every n-vertex graph in G with n > 1 has disjoint sets A, B of vertices such that jAj; jBj > "n and A is either complete or anti-complete to B. It is an easy exercise to show that the strong Erd}os-Hajnal property implies the Erd}os-Hajnal property, see [1, 10]. Chudnovsky and Oum [6] proved that the class of graphs with no vertex-minors iso- morphic to H for a fixed graph H has the strong Erd}os-Hajnal property, implying Theo- rem2. We propose its analog for pivot-minors as a conjecture, which implies the theorem the electronic journal of combinatorics 28(2) (2021), #P2.9 2 of Chudnovsky and Oum [6]. Note that this conjecture is not true if we replace the pivot- minor with induced graphs. For example, the class of triangle-free graphs does not have the strong Erd}os-Hajnal property [10]. Conjecture 5. For every graph H, there exists " > 0 such that for all n > 1, every n-vertex graph with no pivot-minor isomorphic to H has two disjoint sets A, B of vertices such that jAj; jBj > "n and A is complete or anti-complete to B. We prove that this conjecture holds if H = Ck. In other words, the class of graphs with no pivot-minor isomorphic to Ck has the strong Erd}os-Hajnal property as follows. This implies Theorem4. Theorem 6. For every integer k > 3, there exists " > 0 such that for all n > 1, every n-vertex graph with no pivot-minor isomorphic to Ck has two disjoint sets A, B of vertices such that jAj; jBj > "n and A is complete or anti-complete to B. This paper is organized as follows. In Section2, we will introduce basic definitions and review necessary theorems of R¨odl[20] and Bonamy, Bousquet, and Thomass´e[2]. In Section3, we will present several tools to find a pivot-minor isomorphic to Ck. In particular, it proves that a long anti-hole contains Ck as a pivot-minor. In Section4, we will present the proof of the main theorem, Theorem6. In Section5, we will relate our theorem to the problem on χ-boundedness, and discuss known results and open problems related to polynomial χ-boundedness and the Erd}os-Hajnal property. 2 Preliminaries Let N be the set of positive integers and for each n 2 N, we write [n] := f1; 2; : : : ; ng. For V a graph G = (V; E), let G = (V; 2 − E) be the complement of G. We write ∆(G) and δ(G) to denote the maximum degree of G and the minimum degree of G respectively. Let T be a tree rooted at a specified node vr, called the root. If the path from vr to a node y in T contains x 2 V (T ) − fyg, we say that x is an ancestor of y, and y is a descendant of x. If one of x and y is an ancestor of the other, we say that x, y are related. We say that two disjoint sets X and Y of nodes of T are unrelated if no pairs of x 2 X and y 2 Y are related. For disjoint vertex sets X and Y , we say X is complete to Y if every vertex of X is adjacent to all vertices of Y . We say X is anti-complete to Y if every vertex of X is non-adjacent to Y .A pure pair of a graph G is a pair (A; B) of disjoint subsets of V (G) such that A is complete or anticomplete to B. For a vertex u, let NG(u) denote the set of neighbors of u in G. For each U ⊆ V (G), we write [ NG(U) := NG(u) − U: u2U The following lemma is proved in Section 2 of [2]. the electronic journal of combinatorics 28(2) (2021), #P2.9 3 Lemma 7 (Bonamy, Bousquet, and Thomass´e[2]). For every connected graph G and a vertex vr 2 V (G), there exist an induced subtree T of G rooted at vr and a function r : V (G) ! V (T ) satisfying the following. (T1) r(vr) = vr and for each u 2 V (G) − fvrg, the vertex r(u) is a neighbor of u. In particular, T is a dominating tree of G. (T2) If r(x) and r(y) are not related, then xy2 = E(G). R¨odl[20] proved the following theorem. Its weaker version was later proved by Fox and Sudakov [11] without using the regularity lemma. A set U of vertices of G is an jUj "-stable set of a graph G if G[U] has at most " 2 edges. Similarly, U is an "-clique of a jUj graph G if G[U] has at least (1 − ") 2 edges. Theorem 8 (R¨odl[20]). For all " > 0 and a graph H, there exists δ > 0 such that every n-vertex graph G with no induced subgraph isomorphic to H has an "-stable set or an "-clique of size at least δn. We will use the following simple lemma. We present its proof for completeness. Lemma 9. Let G be a graph. Every "-stable set U of G has a subset U 0 of size at least 0 0 jUj=2 with ∆(G[U ]) 6 4"jU j. Proof. Let U 0 be the set of vertices of degree at most 2"jUj in G[U]. Because X 2 degG[U](v) < "jUj ; v2U 0 0 we have jU j > jUj=2. Moreover, for each vertex v 2 U , we have degG[U 0](v) 6 2"jUj 6 4"jU 0j.
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