Affiliation:
1. Department of Mathematics Princeton University Princeton New Jersey USA
2. Mathematical Institute University of Oxford Oxford UK
3. Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario Canada
Abstract
AbstractA hole in a graph is an induced cycle of length at least four, and a ‐multihole in is the union of pairwise disjoint and nonneighbouring 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 integer , 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.
Funder
Engineering and Physical Sciences Research Council
Air Force Office of Scientific Research
Natural Sciences and Engineering Research Council of Canada
National Science Foundation
Subject
Geometry and Topology,Discrete Mathematics and Combinatorics