Grundy coloring of middle graph of wheel graph families

Abstract

A Grundy k-coloring of a graph G is a proper k-coloring of vertices in G using colors {1, 2, · · · , k} such that for any two colors i and j, i < j, any vertex colored j is adjacent to some vertex colored i. The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph G, denoted by ? (G), is the largest integer k, such that there exists a Grundy k-coloring for G. It can be easily seen that ? (G) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of G [10]. In this paper, we obtain the Grundy chromatic number of middle graph of graph G, denoted by M (G), where G be a cycle or sunlet graph or star graph or wheel graph or helm graph.