From χ- to χp-bounded classes

Abstract

χ-bounded classes are studied here in the context of star colorings and, more generally, χp-colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χp-boundedness leads to more stability and we give structural characterizations of (strong and weak) χp-bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g>3 even hole-free graphs G contain at most φ(g,ω(G))|G| holes of length g.