On Low Rank-Width Colorings

Abstract

We introduce the concept of low rank-width colorings, generalizing the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [26]. We say that a class $\mathcal {C}$ of graphs admits low rank-width colorings if there exist functions $N:\mathbb {N}\rightarrow \mathbb {N}$ and $Q:\mathbb {N}\rightarrow \mathbb {N}$ such that for all $p\in \mathbb {N}$, every graph $G\in \mathcal {C}$ can be vertex colored with at most N(p) colors such that the union of any $i\le p$ color classes induces a subgraph of rank-width at most Q(i).

Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class $\mathcal {C}$ of bounded expansion and every positive integer r, the class $\{G^r:G\in \mathcal {C}\}$ of rth powers of graphs from $\mathcal {C}$, as well as the classes of unit interval graphs and bipartite permutation graphs admit low rank-width colorings. All of these classes have unbounded rank-width and do not admit low tree-depth colorings. We also show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. As interesting side properties, we prove that every graph class admitting low rank-width colorings has the Erdős-Hajnal property and is $\chi$-bounded.

The work of O. Kwon is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC consolidator grant DISTRUCT, agreement No. 648527). The work of M. Pilipczuk and S. Siebertz is supported by the National Science Centre of Poland via POLONEZ grant agreement UMO-2015/19/P/ST6/03998, which has received funding from the European Union’s Horizon 2020 research and innovation programme (Marie Skłodowska-Curie grant agreement No. 665778). M. Pilipczuk is supported by the Foundation for Polish Science (FNP) via the START stipend programme.