On the b-Chromatic Sum of Mycielskian of Km,n, Kn and Cn

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by phi(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by phi'(G) is defined as the minimum of sum of colors c(v) of v for all v is an element of V in a b-coloring of G using phi(G) colors. The Mycielskian or Mycielski, mu(H) of a graph H with vertex set {v(1), v(2), ..., v(n)} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u(1), u(2), ..., u(n)} joining u to each vertex u(i)(1 <= i <= n) and joining u(i) to each neighbour of v(i) in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.