Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications

In an earlier paper the authors constructed a hamilton cycle embedding of Kn,n,n in a nonorientable surface for all n ≥ 1 and then used these embeddings to determine the genus of some large families of graphs. In this two-part series, we extend those results to orientable surfaces for all n 6= 2. In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph Kn,n,n on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all n = 2p such that p is prime, completing the proof started by Part I (which covers the case n 6= 2p) that there exists an orientable hamilton cycle embedding of Kn,n,n for all n ≥ 1, n 6= 2. These embeddings are then used to determine the genus of several families of graphs, notably Kt,n,n,n for t ≥ 2n and, in some cases, Km +Kn for m ≥ n− 1.