Triangulations of orientable surfaces by complete tripartite graphs

Grannell, M. J.; Griggs, T. S.; Knor, M. and Siran, J. (2006). Triangulations of orientable surfaces by complete tripartite graphs. Discrete Mathematics, 306(6) pp. 600–606.

DOI: https://doi.org/10.1016/j.disc.2005.10.025

Abstract

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 enlnn-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 eplnp-p(1+lnk+o(1)). Moreover, we prove that for every n there is a unique regular triangular embedding of Kn,n,n in an orientable surface.

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