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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 pairdoi:10.1002/jcd.21375 fatcat:rfxf6fla7rfgdpoeawmatfzt74