On the orientable genus of the cartesian product of a complete regular tripartite graph with a 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 a even cycle. In particular, the orientable genus of $kc$ 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 patchwork method.