On total chromatic number of direct product graphs
The Total Coloring Conjecture (TCC), posed independently by Behzad and Vizing, states that every simple graph G has χ′′(G) ≤ ¢(G) + 2. If χ′′(G) = ¢(G) + 1, then G is a type 1 graph; if χ′′(G) = ¢(G) + 2, then G is a type 2 graph. The TCC has been confirmed for cartesian product of graphs G and H, if the TCC holds for the graphs G and H by Zmazek, Zerovnik and for the powers of cycles Ckk by Campos, Mello .Here we confirm the TCC for direct product of a path, Pn, and a graph G, where G is type 1 graph. We further investigate the total chromatic number of direct product of a path and an arbitrary cycle.