Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs I: Latin Square Constructions

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≠2. In part I, we explore a connection between orthogonal latin squares and embeddings. A product construction is presented for building pairs of orthogonal latin squares such that one member of the pair has a certain hamiltonian property. These hamiltonian squares are then used to construct 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≥1 such that n≠2 and n≠2p for every prime p. Moreover, it is shown that the latin square construction utilized to get hamilton cycle embeddings of Kn,n,n can also be used to obtain triangulations of Kn,n,n. Part II of this series covers the case n=2p for every prime p and applies these embeddings to obtain some genus results.