Excluding hooks and their complements

TL;DR

This paper proves a new case of the Erdos-Hajnal conjecture, showing that excluding certain path graphs with pendant edges guarantees large cliques or independent sets in the remaining graph.

Contribution

It establishes the weaker Erdos-Hajnal conjecture for a new infinite family of graphs, specifically paths with a pendant edge at the third vertex.

Findings

The weaker conjecture holds for paths with a pendant edge at the third vertex.

Provides a new infinite family of graphs satisfying the conjecture.

Advances understanding of graph classes related to the Erdos-Hajnal property.

Abstract

The celebrated Erdos-Hajnal conjecture states that for every $n$-vertex undirected graph $H$ there exists $\eps(H)>0$ 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 $n^{\eps(H)}$. A weaker version of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both $H$ and its complement $H^{\compl}$. 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.