Maximal independent sets in minimum colorings

Every graph G contains a minimum vertex-coloring with the property that at least one color class of the coloring is a maximal independent set (equivalently, a dominating set) in G . Among all such minimum vertex-colorings of the vertices of G , a coloring with the maximum number of color classes that are dominating sets in G is called a dominating- χ -coloring of G . The number of color classes that are dominating sets in a dominating- χ -coloring of G is defined to be the dominating- χ -color number of G . In this paper, we continue to investigate the dominating- χ -color number of a graph first defined and studied in [1].