Doost Ali Mojdeh

The dominated coloring of a graph 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 dom(G) . In this paper, dominated coloring of graphs is compared with (open) packing number of G and it is shown that if G is a graph of order n with diam(G)  3, then dom(G)  n 􀀀 (G) and if 0(G) = 2n/3, then dom(G) = 0(G) , and if (G) = n/2, then dom(G) = (G) . The dominated chromatic numbers of the corona of two graphs are investigated and it is shown that if (G) is the Mycielsky graph of G, then we have dom((G)) = dom(G) + 1. It is also proved that the Vizing-type conjecture holds for dominated colorings of the direct product of two graphs. Finally we obtain some Nordhaus–Gaddum-type results for the dominated chromatic number dom(G) .