The Genus of Repeated Cartesian Products of Bipartite Graphs

With the aid of techniques developed by Edmonds, Ringel, and Youngs, it is shown that the genus of the cartesian product of the complete bipartite graph with itself is . Furthermore, let be the graph and recursively define the cartesian product for . The genus of is shown to be , for all n, and s even; or for 1$">, and or. The graph is the 1-skeleton of the n-cube, and the formula for this case gives a result familiar in the literature. Analogous results are developed for repeated cartesian products of paths and of even cycles.