On the Orientable Genus of the Cartesian Product of a Complete Regular Tripartite Graph with an Even Cycle

Abstract

We apply the technique of patchwork embeddings to find orientable genus embeddings of the Cartesian product of a complete regular tripartite graph with an even cycle. In particular, the orientable genus of $K_{m,m,m}\times C_{2n}$ is determined for $m\ge 1$ and for all $n\ge 3$ and $n=1$. For $n=2$ both lower and upper bounds are given.

We see that the resulting embeddings may have a mixture of triangular and quadrilateral faces, in contrast to previous applications of the patchwork method.