Classes of graphs with small rank decompositions are chi-bounded

TL;DR

This paper proves that graph classes with small rank decompositions are chi-bounded, extending the property to classes with bounded rank-width or clique-width, which has implications for graph coloring.

Contribution

It establishes that classes of graphs admitting small rank cut decompositions inherit chi-boundedness, including those with bounded rank-width or clique-width.

Findings

Classes with small rank decompositions are chi-bounded.

Bounded rank-width or clique-width classes are chi-bounded.

The result generalizes chi-boundedness to broader graph classes.

Abstract

A class of graphs G is chi-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is chi-bounded,then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is chi-bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is chi-bounded.