Nonorientable hamilton cycle embeddings of complete tripartite graphs

A cyclic construction is presented for building embeddings of the complete tripartite graph Kn,n,nKn,n,n on a nonorientable surface such that the boundary of every face is a hamilton cycle. This construction works for several families of values of nn, and we extend the result to all nn with some methods of Bouchet and others. The nonorientable genus of Kt,n,n,nKt,n,n,n, for t≥2nt≥2n, is then determined using these embeddings and a surgical method called the ‘diamond sum’ technique.