Dominated Coloring of Some Classes of Perfect and Unicyclic Graphs

Abstract

The dominated coloring of a graph $G$ is a proper vertex coloring such that each color class is dominated by a vertex. The minimum number of colors required for a dominated coloring of a graph $G$ is called the dominated chromatic number of the graph $G$ and is denoted by ${\chi }_{dom}(G)$. A graph $G$ is said to be chromatic dominated if $\chi (G)={\chi }_{dom}(G)$. In this paper, we characterized the perfect graphs, which are chromatic dominated. Also, we studied the dominated chromatic number of some classes of unicyclic graphs.