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Dynamic coloring parameters for graphs with given genus. (English) Zbl 1375.05098
Summary: 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 $$g\leq 2$$ and $$r$$-dynamically $$((r+1)(2g+2)+3)$$-colorable when $$g\geq 3$$. Furthermore, each of these upper bounds for $$r$$-dynamic $$k$$-colorability also holds for $$r$$-dynamic $$k$$-choosability and for $$r$$-dynamic $$k$$-paintability.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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##### References:
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