Dominated chromatic number of central graphs

Abstract

Let [Formula: see text] be a graph with no isolated vertex. A dominated coloring of [Formula: see text] is a proper coloring of [Formula: see text] such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of [Formula: see text] is called the dominated chromatic number of [Formula: see text], denoted by [Formula: see text]. 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 [Formula: see text] in terms of some invariants of the graph [Formula: see text]. 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.