Triangulations of orientable surfaces by complete tripartite graphs

Orientable triangular embeddings of the complete tripartite graph Kn,n,nKn,n,n correspond to biembeddings of Latin squares. We show that if n   is prime there are at least enlnn-n(1+o(1))enlnn-n(1+o(1)) nonisomorphic biembeddings of cyclic Latin squares of order n  . If n=kpn=kp, where p is a large prime number, then the number of nonisomorphic biembeddings of cyclic Latin squares of order n   is at least eplnp-p(1+lnk+o(1))eplnp-p(1+lnk+o(1)). Moreover, we prove that for every n   there is a unique regular triangular embedding of Kn,n,nKn,n,n in an orientable surface.