The Erdős–Hajnal conjecture for paths and antipaths

Abstract

International audienceWe prove that for every k, there exists c k > 0 such that every graph G on n vertices with no induced path P k or its complement P k contains a clique or a stable set of size n c k. An n-graph is a graph on n vertices. For every vertex x, N (x) denotes the neighborhood of x, that is the set of vertices y such that xy is an edge. The degree deg(x) is the size of N (x). In this note, we only consider classes of graphs that are closed under induced subgraphs. Moreover a class C is strict if it does not contain all graphs. It is said to have the (weak) Erdős-Hajnal property if there exists some c > 0 such that every graph of C contains a clique or a stable set of size n c where n is the size of G. The Erdős-Hajnal conjecture [8] asserts that every strict class of graphs has the Erdős-Hajnal property; see [3] for a survey. This fascinating question is open even for graphs not inducing a cycle of length five. When excluding a single graph H, Alon, Pach and Solymosi showed in [2] that it suffices to consider prime H, namely graphs without nontrivial modules (a module is a subset V of vertices such that for every x, y ∈ V , N (x) \ V = N (y) \ V). A natural approach is then to study classes of graphs with intermediate difficulty, hoping to get a proof scheme which could be extended. A natural prime candidate to forbid is certainly the path. Unfortunately, even excluding the path on five vertices seems already hard. Chudnovsky and Zwols studied the class C k of graphs not inducing the path P k on k vertices or its complement P k. They proved the Erdős-Hajnal property for P 5 and P 6-free graphs [7]. This was extended for P 5 and P 7-free graphs by Chudnovsky and Seymour [6]. Moreover structural results have been provided for C 5 [4, 5]. We show in this note that for every fixed k, the class C k has the Erdős-Hajnal property. An n-graph is an ε-stable set if it has at most ε n

Ordered graphs and large bi-cliques in intersection graphs of curves

“…Proof of Theorem 4, part (1). Let V be an n-element set and let V 1 , V 2 , V 3 , V 4 be a partition of V into four sets of size roughly n/4.…”

The Erdös--Hajnal Conjecture for Long Holes and Antiholes

“…Recently, a new approach for tackling this conjecture has been introduced: forbidding both a graph and its complement. This approach provides a large amount of results for paths (see [4,9,7,8], for instance). In particular, Bousquet, Lagoutte, and Thomassé proved that for every k, the class of graphs with no P k nor its complement satisfies the Erdős-Hajnal property [4].…”

“…This approach provides a large amount of results for paths (see [4,9,7,8], for instance). In particular, Bousquet, Lagoutte, and Thomassé proved that for every k, the class of graphs with no P k nor its complement satisfies the Erdős-Hajnal property [4]. A recent survey of Chudnovsky [6] overviews most of the results known about this conjecture and its equivalent directed version.…”

Excluding pairs of tournaments

“…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.…”

“…These results were further refined in . Surprisingly, a much more general result holds: excluding an arbitrary path and its complement implies the existence of a polynomial‐size clique or stable set (). Several interesting results regarding excluding pairs of graphs are included in .…”