Orientable hamilton cycle embeddings of complete tripartite graphs II: Voltage graph constructions and applications

Abstract

In an earlier article 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≠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≠2p) that there exists an orientable hamilton cycle embedding of Kn,n,n for all n≥1, n≠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, Km̄±Kn for m≥n-1. © 2014 Wiley Periodicals, Inc.