On Equitable Colorings of Windmill Graphs

Abstract

Let G be an undirected simple graph. Graph coloring is a special case of labeling, and G is said to admit a proper coloring if no two neighbouring vertices of it are given the identical color. The vertices of identical color constitute a color class. A graph is p-colorable if it has a p-coloring. The chromatic number of G, denoted by χ(G), is the minimum p such that G is p-colorable. A graph G is equitably p-colorable if it has a p-coloring and the absolute difference in size between any two distinct color classes is at most 1. The equitable chromatic number of G, denoted by χ=(G), is the minimum p such that G is equitably p-colorable. The equitable chromatic threshold of G, denoted by , is the minimum p ′ such that G is equitably p-colorable for all p ≥ p ′ . A windmill graph W_n^m consists of m copies of the complete graph K_n, with every vertex connected to a common vertex. In this paper, we give exact values of χ=(G) and  when G is a windmill graph, bistar windmill graph, cycle windmill graph, and complete windmill graph.