Equitable coloring of Kronecker products of complete multipartite graphs and complete graphs

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic number of a graph G, denoted by χ=(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by χ=∗(G), is the minimum t such that G is equitably k-colorable for k≥t. In this paper, we give the exact values of χ=(Km1,…,mr×Kn) and χ=∗(Km1,…,mr×Kn) for ∑i=1rmi≤n.