Abstract
A dominator coloring of a graph G is a proper coloring of the vertices of G in which each vertex of the graph dominates every vertex of some color class, where a vertex dominates itself and all vertices adjacent to it. The dominator chromatic number of G is the minimum number of colors among all dominator coloring of G. A total dominator coloring of a graph G is a proper coloring of the vertices of G in which each vertex of the graph dominates every vertex of some color class other than its own. The total dominator chromatic number of G is the minimum number of colors among all total dominator coloring of G. In this paper, we present bounds on the dominator chromatic number and total dominator chromatic number of a planar graphs with small diameter. In particular we show the dominator chromatic number of a planar graph of diameter 2 is at most 5. We also present results for the special cases of outerplanar graphs and bipartite planar graphs.
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