About the Erdös-Hajnal conjecture for seven-vertex tournaments

Soukaina, Zayat,; Ghazal, Salman

doi:10.48550/arxiv.2010.12331

Cited by 4 publications

(6 citation statements)

References 3 publications

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“…Corollary 2.9 (Zayat and Ghazal [8]). Let S be a tournament, let w be a {0, 1}-vector, and let ∈ m λ *, 0 < < 1 1 2 be constants.…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 97%

“…Lemma 2.10 [ Zayat and Ghazal,8]. Let ≤ λ γ 0 < < 1, 0 < 1 be constants and let w be a {0, 1} −vector.…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 99%

“…-A 2 is the 12-vertex tournament, with {(11, 4), (11, 5), (5, 2), (3, 1), (12, 1), (8,6), (10, 6), (9, 7)} is the set of backward arcs of A 2 under the ordering γ = (1, …, 12) 2 of its vertices. Under γ A , 2 2 is the disjoint union of the left β 2 -asteroid {1, …, 5, 11, 12} and the two left stars {6, 8, 10} and {7, 9}.…”

Section: Definitions and Toolsmentioning

confidence: 99%

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Erdös–Hajnal conjecture for new infinite families of tournaments

Soukaina

Ghazal

2022

Journal of Graph Theory

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Erdös-Hajnal conjecture states that for every undirected graph H there exists H ϵ( ) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n H ϵ( ) . This conjecture has a directed equivalent version stating that for every tournament H there exists H ϵ( ) > 0 such that every H -free n-vertex tournament T contains a transitive subtournament of order at least n H ϵ( ) . This conjecture is known to hold for a few infinite families of tournaments. In this article we construct two new infinite families of tournaments-the family of socalled galaxies with spiders and the family of so-called asterisms, and we prove the correctness of the conjecture for these two families.

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“…Corollary 2.9 (Zayat and Ghazal [8]). Let S be a tournament, let w be a {0, 1}-vector, and let ∈ m λ *, 0 < < 1 1 2 be constants.…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 97%

“…Lemma 2.10 [ Zayat and Ghazal,8]. Let ≤ λ γ 0 < < 1, 0 < 1 be constants and let w be a {0, 1} −vector.…”

Section: Definitions and Preliminary Lemmasmentioning

confidence: 99%

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Erdös–Hajnal conjecture for new infinite families of tournaments

Soukaina

Ghazal

2022

Journal of Graph Theory

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“…There are other classes of tournaments that have been shown to have the EH-property: see for instance [2,5,17].…”

Section: Introductionmentioning

confidence: 99%

Pure pairs. X. Tournaments and the strong Erdos-Hajnal property

Chudnovsky¹,

Scott²,

Seymour³

et al. 2022

Preprint

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A pure pair in a tournament G is an ordered pair (A, B) of disjoint subsets of V (G) such that every vertex in B is adjacent from every vertex in A. Which tournaments H have the property that if G is a tournament not containing H as a subtournament, and |G| > 1, there is a pure pair (A, B) in G with |A|, |B| ≥ c|G|, where c > 0 is a constant independent of G? Let us say that such a tournament H has the strong EH-property.As far as we know, it might be that a tournament H has this property if and only if its vertex set has a linear ordering in which its backedges form a forest. Certainly this condition is necessary, but we are far from proving sufficiency. We make a small step in this direction, showing that if a tournament can be ordered with at most three backedges then it has the strong EH-property (except for one case, that we could not decide). In particular, every tournament with at most six vertices has the property, except for three that we could not decide. We also give a seven-vertex tournament that does not have the strong EH-property. This is related to the Erdős-Hajnal conjecture, which in one form says that for every tournament H there exists τ > 0 such that every tournament G not containing H as a subtournament has a transitive subtournament of cardinality at least |G| τ . Let us say that a tournament H satisfying this has the EH-property. It is known that every tournament with the strong EH-property also has the EH-property; so our result extends work by Berger, Choromanski and Chudnovsky, who proved that every tournament with at most six vertices has the EH-property, except for one that they did not decide.

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“…In [2] Berger et al proved that every galaxy has the EH-property, and in [5,8,9] Conjecture 2 was proved for more general classes of tournaments.…”

Section: Introductionmentioning

confidence: 99%

Forests and the Strong Erdos-Hajnal Property

Soukaina¹

2022

Preprint

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An equivalent directed version of the celebrated unresolved conjecture of Erdös and Hajnal proposed by Alon et al. states that for every tournament H there exists ǫ(H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n ǫ(H) . A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T | > 1, there exist disjoint vertex subsets A and B, each of cardinality at least c|T | and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-vertex tournament denoted by C 5 , where every vertex has two inneighbors and two outneighbors has the strong EH-property. It is known that every tournament with the strong EH-property also has the EH-property. In this paper we construct an infinite class of tournaments − the so-called spiral galaxies and we prove that every spiral galaxy has the strong EH-property.

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About the Erdös-Hajnal conjecture for seven-vertex tournaments

Cited by 4 publications

References 3 publications

Erdös–Hajnal conjecture for new infinite families of tournaments

Erdös–Hajnal conjecture for new infinite families of tournaments

Pure pairs. X. Tournaments and the strong Erdos-Hajnal property

Forests and the Strong Erdos-Hajnal Property

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