From χ to χp-bounded classes
Résumé
χ-bounded classes are studied here in the context of star colorings and, more generally, χ pcolorings. 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. χ 2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χ p-boundedness leads to more stability and we give structural characterizations of (strong and weak) χ p-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.
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