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arXiv · 2202.09118

Polynomial bounds for chromatic number VII. Disjoint holes

Maria Chudnovsky·Alex Scott·Paul Seymour·Sophie Spirkl
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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.

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Maria Chudnovsky, Alex Scott, Paul Seymour, Sophie Spirkl. 2022-02-18. Polynomial bounds for chromatic number VII. Disjoint holes. https://arxiv.org/abs/2202.09118

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