Dominated Colorings of Graphs

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 χdom(G). In this paper, we establish the close relationship between the dominated chromatic number χdom(G) and the total domination number γt(G); and the equivalence for triangle-free graphs. We study the complexity of the problem by proving its NP-completeness for arbitrary graphs having χdom(G) ≥ 4 and by giving a polynomial time algorithm for recognizing graphs having χdom(G) ≤ 3. We also give some bounds for planar and star-free graphs and exact values for split graphs.