Fall coloring and b-coloring of graphs

Abstract

The b-chromatic number of G, denoted by φ(G), is the maximum k for which G has a b-coloring by k colors. A b-coloring of G by k colors is a proper k-coloring of the vertices of G such that in each color class i there exists a vertex xi having neighbors in all the other k – 1 color classes. Such a vertex xi is called a b-dominating vertex, and the set of vertices {x1, x2 … xk} is called a b-dominating system. A fall coloring of a graph G is a proper coloring, where every vertex of G is a b-dominating vertex. The fall chromatic number and the fall achromatic number of G are, respectively, the minimum and maximum integers χf(G), ψf(G) for which G has a fall coloring. In this paper, we obtain the results that, all these three coloring parameters are equal for the Complement graph of flower graph and sunflower graph and b-coloring for some girth graphs.