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.
@article{arxiv.1107.2161,
title = {Classes of graphs with small rank decompositions are chi-bounded},
author = {Zdenek Dvorak and Daniel Kral},
journal= {arXiv preprint arXiv:1107.2161},
year = {2011}
}