Dominator Colorings of Products of Graphs

Abstract

A dominator coloring $\mathcal {C}$ of a graph G is a proper coloring of G such that closed neighborhood of each vertex of G contains a color class of $\mathcal {C}.$ The minimum number of colors required for a dominator coloring of G is called the dominator chromatic number of G,  denoted by $\chi _d(G)$. In this paper we obtain the exact value of $\chi _d$ for some classes of graphs, such as $K_m\times K_n, \ K_{m(n)}\times K_{r(s)}, \ (K_r\circ K_1)\times K_s, \ K_n\Box Q_{r+2},$ where $\times , \Box$ and $\circ$ denote the tensor product, Cartesian product and corona of graphs, respectively, and $K_{m(n)}$ denotes the complete m-partite graph in which each partite set has n vertices. Also we present an upper bound for $\chi _d(G\times K_m)$ in terms of $\chi _d(G)$ and $\gamma _t(G)$, where $\gamma _t(G)$ is the total domination number of G.