Optimal r-dynamic coloring of sparse graphs

Abstract

An r-dynamick-coloring of a graphG is a proper k-coloring such that every vertex v in V(G) has neighbors in at least $$min\{d(v),r\}$$ different classes. The r-dynamic chromatic number ofG, written $$\chi _{r}(G)$$ , is the minimum integer k such that G has such a coloring. In this paper, we investigate the r-dynamic $$(r+1)$$ -coloring (i.e. optimal r-dynamic coloring) of sparse graphs and prove that $$\chi _{r}(G)\le r+1$$ holds if G is a planar graph with $$g(G)\ge 7$$ and $$r\ge 16$$ , which is a generalization of the case $$r=\Delta $$ .