Polynomial bounds for chromatic number VII. Disjoint holes

TL;DR

This paper proves that graphs excluding certain disjoint hole configurations have chromatic numbers bounded polynomially by their clique numbers, advancing understanding of polynomial chi-boundedness in graph theory.

Contribution

It establishes polynomial bounds on chromatic number for graphs excluding multiholes, odd holes, or holes longer than a fixed length, generalizing known results.

Findings

Graphs without k-multiholes have polynomial chi-boundedness.

Excluding all odd holes or holes of length four yields similar bounds.

Results extend to graphs with holes longer than any fixed constant.

Abstract

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any $k$, if $G$ does not contain a $k$-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 $\chi$-bounded.