Polynomial bounds for chromatic number VII. Disjoint holes
Combinatorics
2022-02-21 v1
Abstract
A hole in a graph is an induced cycle of length at least four, and a -multihole in is a set of pairwise disjoint and nonadjacent holes. It is well known that if does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any , if does not contain a -multihole, then its chromatic number is at most a polynomial function of its clique number. We show that the same result holds if we ask for all the holes to be odd or of length four; and if we ask for the holes to be longer than any fixed constant or of length four. This is part of a broader study of graph classes that are polynomially -bounded.
Cite
@article{arxiv.2202.09118,
title = {Polynomial bounds for chromatic number VII. Disjoint holes},
author = {Maria Chudnovsky and Alex Scott and Paul Seymour and Sophie Spirkl},
journal= {arXiv preprint arXiv:2202.09118},
year = {2022}
}