Triangulations of orientable surfaces by complete tripartite graphs

Orientable triangular embeddings of the complete tripartite graph Kn,n,n correspond to biembeddings of Latin squares. We show that if n is prime there are at least e ln n−n(1+o(1)) nonisomorphic biembeddings of cyclic Latin squares of order n. If n = kp, where p is a large prime number, then the number of nonisomorphic biembeddings of cyclic Latin squares of order n is at least e ln p−p(1+ln . Moreover, we prove that for every n there is a unique regular triangular embedding of Kn,n,n in an orientable surface. This is a preprint of an article accepted for publication in Discrete Mathematics c ©2005 (copyright owner as specified in the journal).