Dynamic coloring parameters for graphs with given genus

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.

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