Dominating Coloring Number of Claw-free Graphs

Let G be a graph. It is well-known that G contains a proper vertex-coloring with χ(G) colors with the property that at least one color class of the coloring is a dominating set in G. Among all such proper vertex-coloring 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 and is denoted by dχ(G). In this paper, we prove that if G is a claw-free graph with minimum degree at least two, then dχ(G)⩾2.