The Erdös--Hajnal Conjecture for Long Holes and Antiholes

Abstract

International audienceErdös and Hajnal conjectured that for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain an induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$ there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size at least $n^{c_k}$

Excluding pairs of tournaments

“…It can be noted that the family of left nebulae contains infinitely many tournaments with prime subtournaments that are neither galaxies nor constellations (see: [10] for a definition of the family of constellations). For example, take a tournament of 12 vertices {1, … , 12} and with the set of backward edges under ordering (1, … , 12) of the form: {(5, 1), (9, 1), (8,6), (11,6), (4, 2), (10, 3), (12, 7)}. Using that ordering, one can note that is a subtournament of a tournament from the class of left nebulae (see Figure 1).…”

“…Finally, the family of central nebulae contains infinitely many tournaments with prime subtournaments that are neither galaxies nor constellations. For example, take a tournament of twelve vertices {1, … , 12} and with the set of backward edges under ordering (1, … , 12) of the form: {(4, 1), (8,4), (5,3), (9,5), (6,2), (11,6), (10,7), (12,10)}. Using that ordering one can note that is a central nebula.…”

“…This is a (2,1)-triple since an edge {̂,̂,̂} is white. The algorithm then looks for two disjoint subsets: ⊆ ( ) and ⊆ ( ), such that one of these subsets is complete to the other one and furthermore | |, | | ≥ min(|̂|,|̂|,|̂|) 6 . If such subsets do exist, then the algorithm reaches state 2 and outputs and .…”

“…We have = = 6. Rectangular shapes correspond to the sets: 1 , … , 6 . Marked vertices are the embeddings of the vertices of the stars of that were found so far.…”

“…Marked vertices are the embeddings of the vertices of the stars of that were found so far. Vertex , is an embedding of the th vertex of (under ordering (1,2,3,4,5,6)) that is in the th star of and is a part of the th triple. Different vertices of the embedding are found in different sets and sets  are six-element subsets.…”