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 (2008). We say that a class C of graphs admits low rank-width colorings if there exist functions N:N→N and Q:N→N such that for all p∈N, every graph G∈C can be vertex colored with at most N(p) colors such that the union of any i≤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 C of bounded expansion and every positive integer r, the class {Gr:G∈C} of rth powers of graphs from C admits low rank-width colorings. On the negative side, we 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 hereditary graph class admitting low rank-width colorings has the Erdős–Hajnal property and is χ-bounded.