Vertex-minors and the Erd\H{o}s-Hajnal conjecture

TL;DR

This paper proves an Erdős-Hajnal-type property for graphs excluding a fixed vertex-minor, showing such graphs contain large homogeneous pairs, extending the conjecture's scope to vertex-minors.

Contribution

It establishes the Erdős-Hajnal conjecture analog for vertex-minors, linking vertex-minor exclusion to large homogeneous pairs in graphs.

Findings

Graphs with no vertex-minor isomorphic to H have large homogeneous pairs.

The result applies universally to all graphs H.

It extends Erdős-Hajnal conjecture to the context of vertex-minors.

Abstract

We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erd\H{o}s-Hajnal conjecture for vertex-minors.