Equitable colorings of Cartesian products of square of cycles and paths with complete bipartite graphs

Abstract

A graph G is said to be equitably k-colorable if the vertex set of G can be divided into k independent sets for which any two sets differ in size at most one. The equitable chromatic number of G, $$\chi _{=}(G)$$ź=(G), is the minimum k for which G is equitably k-colorable. The equitable chromatic threshold of G, $$\chi _{=}^{*}(G)$$ź=ź(G), is the minimum k for which G is equitably $$k'$$kź-colorable for all $$k'\ge k$$kźźk. In this paper, the exact values of $$\chi _{=}^{*}(G\Box H)$$ź=ź(GźH) and $$\chi _{=}(G\Box H)$$ź=(GźH) are obtained when G is the square of a cycle or a path and H is a complete bipartite graph.