On total coloring the direct product of complete graphs

A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either Delta(G) + 1 or Delta(G) + 2, where Delta(G) is the maximum degree of G. We consider the direct product of complete graphs K-m, x K-n. It is known that if at least one of the numbers m or n is even, then K-m, x K-n has total chromatic number equal to Delta(K-m, x K-n) + 1, except when m = n = 2. We prove that the graph K-m x K-n has total chromatic number equal to Delta(K-m, x K-n) when both m and n are odd numbers, ensuring in this way that all graphs K-m x K-n have total chromatic number equal to Delta(K-m, x K-n) +1, except when m = n = 2. (C) 2021 The Authors. Published by Elsevier B.V.