# On low rank-width colorings

## TL;DR

This paper introduces low rank-width colorings, a generalization of low tree-depth colorings, and demonstrates their applicability to various graph classes, including powers of bounded expansion graphs, unit interval graphs, and bipartite permutation graphs.

## Contribution

It defines low rank-width colorings and proves their existence for several important graph classes, extending the scope of graph coloring techniques beyond bounded rank-width classes.

## Key findings

- Graph classes of bounded expansion and their powers admit low rank-width colorings.
- Unit interval graphs and bipartite permutation graphs admit low rank-width colorings.
- Interval graphs and permutation graphs do not admit low rank-width colorings.

## Abstract

We introduce the concept of low rank-width colorings, generalising the notion of low tree-depth colorings introduced by Ne\v{s}et\v{r}il and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC, 2008]. We say that a class $\mathcal{C}$ of graphs admits low rank-width colourings if there exist functions $N\colon \mathbb{N}\rightarrow\mathbb{N}$ and $Q\colon \mathbb{N}\rightarrow\mathbb{N}$ 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 $i\leq p$ 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 $\{G^r\colon G\in \mathcal{C}\}$ of $r$th 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 Erd\H{o}s-Hajnal property and is $\chi$-bounded.