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 , Δ ( G )+1 ≤ χ ″( G )≤ Δ ( G )+2 , where Δ ( 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.
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