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

• Published in: Journal of graph theory Vol. 77; no. 3; pp. 219 - 236
• Main Authors: Ellingham, M. N., Schroeder, Justin Z.
• Format: Journal Article
• Language: English
• Published: Hoboken Blackwell Publishing Ltd 01.11.2014; Wiley Subscription Services, Inc
• Subjects: complete tripartite graph; genus; graph embedding; Graphs; hamilton cycle; Primary 05C10; voltage graph
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.21783

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, Km¯+Kn for m≥n−1.