Dominated Colorings of Graphs
Résumé
In this paper, we introduce and study a new coloring problem of a graph called the dominated coloring. A dominated coloring of a graph $G$ is a proper vertex coloring of $G$ such that each color class is dominated by at least one vertex of $G$. The minimum number of colors among all dominated colorings is called the dominated chromatic number, denoted by $\chi_{dom}(G)$. In this paper, we establish the close relationship between the dominated chromatic number $\chi_{dom}(G)$ and the total domination number $\gamma_t(G)$; and the equivalence for triangle-free graphs. We study the complexity of the problem by proving its NP-completeness for arbitrary graphs having $\chi_{dom}(G) \ge 4$ and by giving a polynomial time algorithm for recognizing graphs having $\chi_{dom}(G) \le 3$. We also give some bounds for planar and star-free graphs and exact values for split graphs.