Dynamic coloring parameters for graphs with given genus

A proper vertex coloring of a graph G is r-dynamic if for each v∈V(G), at least min{r,d(v)} colors appear in NG(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≤2 and r-dynamically ((r+1)(2g+2)+3)-colorable when g≥3. Furthermore, each of these upper bounds for r-dynamic k-colorability also holds for r-dynamic k-choosability and for r-dynamic k-paintability.