Dynamic coloring parameters for graphs with given genus

TL;DR

This paper explores the properties of $r$-dynamic coloring in graphs with given genus, establishing bounds on the number of colors needed and extending results to list coloring and paintability.

Contribution

It introduces new bounds for $r$-dynamic coloring, list coloring, and paintability of graphs based on genus, and develops a method for proving reducibility of configurations.

Findings

Planar and toroidal graphs are 3-dynamically 10-colorable.

Bounds on colors needed depend on the genus and parameter $r$.

Results extend to list coloring and paintability.

Abstract

A proper vertex coloring of a graph $G$ is $r$-dynamic if for each $v\in V(G)$, at least $\min\{r,d(v)\}$ colors appear in $N_G(v)$. In this paper we investigate $r$-dynamic versions of coloring, list coloring, and paintability. We prove that planar and toroidal graphs are 3-dynamically 10-colorable, and this bound is sharp for toroidal graphs. We also give bounds on the minimum number of colors needed for any $r$ in terms of the genus of the graph: for sufficiently large $r$, every graph with genus $g$ is $r$-dynamically $((r+1)(g+5)+3)$-colorable when $g\leq2$ and $r$-dynamically $((r+1)(2g+2)+3)$-colorable when $g\geq3$. Furthermore, each of these upper bounds for $r$-dynamic $k$-colorability also holds for $r$-dynamic $k$-choosability and for $r$-dynamic $k$-paintability. We develop a method to prove that certain configurations are reducible for each of the corresponding $r$-dynamic parameters.