Pure pairs. II. Excluding all subdivisions of a graph
Combinatorics
2020-06-03 v2
Abstract
We prove for every graph H there exists a>0 such that, for every graph G with at least two vertices, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least a|G| neighbours, or there are two disjoint sets A,B of at least a|G| vertices such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or its complement is a subdivision of H, then G has a clique or stable set of cardinality at least |G|^c. This is related to the Erdos-Hajnal conjecture.
Cite
@article{arxiv.1804.01060,
title = {Pure pairs. II. Excluding all subdivisions of a graph},
author = {Maria Chudnovsky and Alex Scott and Paul Seymour and Sophie Spirkl},
journal= {arXiv preprint arXiv:1804.01060},
year = {2020}
}