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

A \textit{$k$-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 $K_m \times K_n$.  It is known that if at least one of the numbers $m$ or $n$ is even, then  $K_m \times K_n$ is Type~1, except for $K_2 \timesK_2$.  We prove that the graph $K_m \times K_n$ is Type~1 when both $m$ and $n$ are odd numbers, by using that the conformable condition is sufficient for the graph $K_m \times K_n$ 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 $C_m \times K_n$ of a cycle with a complete graph. Interestingly, we are able to find a Type 2 infinite family $C_m \times K_n$, when $m$ is not a multiple of 3 and $n = 2$. We provide evidence to conjecture that all other $C_m \times K_n$ are Type 1.