Let $G$ be a graph with no isolated vertex. A 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 $\chi_{dom}(G)$. In this paper, we study the dominated chromatic number of central graphs. We obtain some tight bounds for the dominated chromatic number of a central graph $C(G)$ in terms of some invariants of the graph $G$. Also we characterize the dominated chromatic number of the central graph of some families of graphs such as star graphs, path graphs, spider graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the dominated chromatic number of central graphs. % %by giving some tight bounds for the total dominator chromatic number of the middle of a graph, join of a graph with an empty graph and Nordhaus-Gaddum-like relations. Also we will calculate the total dominator chromatic number of the middle of a path, a cycle, a wheel, a complete graph and a complete multipartite graph.
