Excluding Hooks and their Complements

Abstract

The celebrated Erdős-Hajnal conjecture states that for every n-vertex undirected graph H there exists ε(H) > 0 such that every graph G that does not contain H as an induced subgraph contains a clique or an independent set of size at least n ε(H) . A weaker version of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both H and its complement H c . We show that the weaker conjecture holds if H is any path with a pendant edge at its third vertex; thus we give a new infinite family of graphs for which the conjecture holds.

A survey of χ‐boundedness

“…This was extended by Choromanski, Falik, Liebenau, Patel, and Pilipczuk [23], who proved the same for trees formed from a path by adding a leaf adjacent to its third vertex. Which other graphs can we take here?…”

Caterpillars in Erdős–Hajnal

“…Let us say H is a hook if H is a tree obtained from a path by adding a vertex adjacent to the third vertex of the path. Two of us, with Choromanski, Falik, and Patel [2], extended 1.2, proving:…”

“…It follows from a theorem of Rödl [13] (and see [8] for a version with much better constants) that every graph with the sparse property has the symmetric property; and Erdős's construction [5] of a graph with large girth and large chromatic number also shows that every graph with the sparse property is a forest, and every graph with the symmetric property is either a forest or the complement of one. (We omit all these proofs, which are easy; see [2] for more details.) We conjecture the converses, that is:…”

Towards Erdős-Hajnal for Graphs with No 5-Hole

“…It is easy to prove that if H has the strong Erdős-Hajnal property then it has the Erdős-Hajnal property (see [1,15]). This approach has been used in a number of papers to prove the Erdős-Hajnal property for various sets H (see, for example, [3,4,5,8,10,17]).…”

“…There has been some recent progress on small sets of graphs with the Erdős-Hajnal property. After partial results by a number of authors (see [4,5,17]), the following result was shown in [8]:…”