On the orientable genus of the cartesian product of a complete regular tripartite graph with an even cycle

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 Km,m,m x C-2n is determined for m greater than or equal to 1 and for all n greater than or equal to 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.