On the dominated chromatic number of certain graphs

Abstract

Let $G$ be a simple graph. The dominated coloring of $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated chromatic number of $G$, denoted by $\chi_{dom}(G)$. Stability (bondage number) of dominated chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the dominated chromatic number of $G$. In this paper, we study the dominated chromatic number, dominated stability and dominated bondage number of certain graphs.