Total coloring conjecture on certain classes of product graphs

Kanagasabapathi Somasundaram (Department of Mathematics Amrita School of Engineering Amrita Vishwa Vidyapeetham University Coimbatore, India)
Jayabalan Geetha (Department of Mathematics Amrita School of Engineering Amrita Vishwa Vidyapeetham University Coimbatore, India)
Radhakrishnan Vignesh (School of Computer Science and Engineering Presidency University Bengaluru India)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 11, No 1 (2023): Electronic Journal of Graph Theory and Applications

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.