On the total chromatic number of the direct product of cycles and complete graphs

A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either Delta(G) + 1 (called Type 1) or Delta(G) + 2 (called Type 2), where Delta(G) is the maximum degree of G. We consider the direct product of complete graphs Km x Kn. It is known that if at least one of the numbers m or n is even, then Km x Kn is Type 1, except for K2 x K2. We prove that the graph Km x Kn is Type 1 when both m and n are odd numbers, by using that the conformable condition is sufficient for the graph Km x Kn to be Type 1 when both m and n are large enough, and by constructing the target total colorings by using Hamiltonian decompositions and a specific color class, called guiding color. We additionally apply our technique to the direct product Cm x Kn of a cycle with a complete graph. Interestingly, we are able to find a Type 2 infinite family Cm x Kn, when m is not a multiple of 3 and n = 2. We provide evidence to conjecture that all other Cm x Kn are Type 1.