Total chromatic number for certain classes of product graphs

Abstract

Total coloring is a function that assigns colors to the vertices and edges of the graph such that the adjacent and the incident elements receive different colors. The minimum number of colors required for a proper total coloring of a graph [Formula: see text] is called the total chromatic number of [Formula: see text] and is denoted by [Formula: see text]. Behzad–Vizing conjecture (Total Coloring Conjecture) states that for any graph [Formula: see text], [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. In this paper, we verify the Behzad–Vizing conjecture for some product graphs.