In this paper, we introduce and study a new coloring prob- lem of a graph called the Dominated coloring (short Domcoloring). A Domcoloring of a graph G is a vertex proper coloring of G such that each color class is dominated by at least one vertex of G. The minimum number of colors among all Domcolorings is called dominated chromatic (short Domchromatic) number, denoted by χdom(G). In this paper we prove the NP-completeness of the problem and give some bounds and exact values for the Domchromatic number for some graph classes.