On low rank-width colorings

We introduce the concept of low rank-width colorings, generalizing the notion of low tree-depth colorings introduced by Nesetřil 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).