Total coloring conjecture on certain classes of product graphs

Abstract

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G, denoted by χ ′′ (G), is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, ∆(G) + 1 ≤ χ ′′ (G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G. In this paper, we prove the Behzad and Vizing conjecture for Indu -Bala product graph, Skew and Converse Skew product graph, Cover product graph, Clique cover product graph and Comb product graph.