On Low Rank-Width Colorings

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 [25]. 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 {G : G ∈ C} of rth powers of graphs from 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 χ-bounded.