Equitable colorings of Cartesian products of square of paths and cycles with square of paths and cycles

Abstract

Let [p] = {1, 2, 3, . . . , p} and 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 an identical color. The vertices of an identical color constitute a color class. G is p - colorable if it admits proper p - coloring. The chromatic number, χ(G) = min {p : G is proper p - colorable} and G is equitably p – colorable if it admits proper p - coloring and the absolute difference in size between any distinct pairwise color class is at most 1. The equitable chromatic number, χ=(G) = min {p : G is equitably p - colorable}. The equitable chromatic threshold, χ∗=(G) = min {p′: G is equitably p - colorable ∀ p ≥ p′}. In this paper, we obtain exact values or bounds of χ∗=(G1□G2) and χ=(G1□G2), where G1 = P2m or C2m and G2 = P2n or C2n.