From $\chi$- to $\chi_p$-bounded classes

TL;DR

This paper explores extensions of $ ext{chi}$-bounded classes through $ ext{chi}_p$-colorings, providing structural characterizations, disproving a conjecture, and applying results to graph hole structures.

Contribution

It introduces $ ext{chi}_p$-bounded classes, disproves a related conjecture, and offers structural characterizations and applications to graph hole analysis.

Findings

Disproved a conjecture on star coloring boundedness.

Provided structural characterizations of $ ext{chi}_p$-bounded classes.

Established bounds on holes in even hole-free graphs.

Abstract

$\chi$-bounded classes are studied here in the context of star colorings and more generally $\chi_p$-colorings. This leads to natural extensions of the notion of bounded expansion class and to structural characterization of these. In this paper we solve two conjectures related to star coloring boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. We give structural characterizations of (strong and weak) $\chi_p$-bounded classes. On the way, we 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 $\varphi(g,\omega(G))\,|G|$ holes of length $g$.