All direct product $C5 \times Kn$ graphs are Type~1

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 $G$.  
In this paper, we establish that all the direct product $C_5 \times K_n$ graphs are Type~1, when $n$ is odd and not a multiple of 5, providing evidence for the conjecture that all $C_m \times K_n$ graphs are Type 1.