Nonorientable hamilton cycle embeddings of complete tripartite graphs

A cyclic construction is presented for building embeddings of the complete tripartite graph K n , 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 n , and we extend the result to all n with some methods of Bouchet and others. The nonorientable genus of K t , n , n , n , for t ≥ 2 n , is then determined using these embeddings and a surgical method called the ‘diamond sum’ technique.