Caterpillars in Erd\H{o}s-Hajnal

TL;DR

This paper proves a structural property of graphs excluding certain caterpillar subdivisions, leading to large homogeneous sets or specific bipartite structures, extending previous results for paths and hooks.

Contribution

It establishes a new Erdős-Hajnal type theorem for caterpillar subdivisions, generalizing prior results for paths and hooks.

Findings

Existence of a vertex with many neighbors or a large bipartite structure in graphs excluding a caterpillar subdivision.

Graphs excluding a caterpillar subdivision or its complement contain large cliques or stable sets.

Extension of previous Erdős-Hajnal theorems to a broader class of trees.

Abstract

Let $T$ be a tree such that all its vertices of degree more than two lie on one path, that is, $T$ is a caterpillar subdivision. We prove that there exists $\epsilon>0$ such that for every graph $G$ with $|V(G)|\ge 2$ not containing $T$ as an induced subgraph, either some vertex has at least $\epsilon|V(G)|$ neighbours, or there are two disjoint sets of vertices $A,B$, both of cardinality at least $\epsilon|V(G)|$, where there is no edge joining $A$ and $B$. A consequence is: for every caterpillar subdivision $T$, there exists $c>0$ such that for every graph $G$ containing neither of $T$ and its complement as an induced subgraph, $G$ has a clique or stable set with at least $|V(G)|^c$ vertices. This extends a theorem of Bousquet, Lagoutte and Thomass\'e [JCTB 2015], who proved the same when $T$ is a path, and a recent theorem of Choromanski, Falik, Liebenau, Patel and Pilipczuk [Electron. J. Combin. 2018], who proved it when $T$ is a "hook".