Abstract. Orientable triangular embeddings of the complete tripartite graphs Kn,n,n correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for n=3, 4, 5, 6 and we summarize the corresponding results for n=7. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs K*n,n,n, and we also give computational results for these in the cases n=3, 4, 5, 6.
In the following table we list the number of main classes of Latin squares of order n and the total number of nonisomorphic biembeddings.