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.

  1. For n=3 there is 1 main class and 1 biembedding.
  2. For n=4 there are 2 main classes and 1 biembedding.
  3. For n=5 there is 1 main class and 3 biembeddings.
  4. For n=6 there are 12 main classes and 29 biembeddings.
  5. For n=7 there are 147 main classes and 23,664 biembeddings.