On the dominated coloring in graphs

Abstract

A dominated coloring of a graph G is a proper vertex coloring where
each color class is dominated by at least one vertex of G. The dominated
chromatic number of G, denoted Xdom(G), is the minimum number of col-
ors required for a dominated coloring of G. In this paper, we provide
new bounds for Xdom(G) and characterize all graphs that achieve some of
these bounds. Also we investigate graphs G for which Xdom(G) = X(G)
where X(G) is the chromatic number of G, in particular we give a charac-
terization of cubic graphs G such that Xdom(G) = X(G).

Keywords: Coloring, Domination, Dominated coloring