The r-Dynamic Chromatic Number is Bounded in the Strong 2-Coloring Number

TL;DR

This paper proves that graphs with bounded strong 2-coloring number have bounded r-dynamic chromatic number, providing linear bounds for classes like bounded expansion and planar graphs.

Contribution

It establishes a bound on the r-dynamic chromatic number based on the strong 2-coloring number, extending to classes like bounded expansion and planar graphs.

Findings

r-dynamic chromatic number is bounded by a linear function of r

Graphs with bounded strong 2-coloring number admit bounded r-dynamic colorings

Concrete bounds provided for graphs of bounded row-treewidth, including planar graphs

Abstract

A proper vertex-coloring of a graph is $r$-dynamic if the neighbors of each vertex $v$ receive at least $\min(r, \mathrm{deg}(v))$ different colors. In this note, we prove that if $G$ has a strong $2$-coloring number at most $k$, then $G$ admits an $r$-dynamic coloring with no more than $(k-1)r+1$ colors. As a consequence, for every class of graphs of bounded expansion, the $r$-dynamic chromatic number is bounded by a linear function in $r$. We give a concrete upper bound for graphs of bounded row-treewidth, which includes for example all planar graphs.