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 \(m \geq 1\) and for all \(n \geq 3\) 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.