Third-regular bi-embeddings of Latin squares

Donovan, D. M.; Grannell, M. J. and Griggs, T. S. (2010). Third-regular bi-embeddings of Latin squares. Glasgow Mathematical Journal, 52(3) pp. 497–503.

DOI: https://doi.org/10.1017/S0017089510000376

Abstract

For each positive integer $n\ge 2$ , there is a well-known regular orientable Hamiltonian embedding of $K_{n,n}$ , and this generates a regular face 2-colourable triangular embedding of $K_{n,n,n}$ . In the case $n\equiv 0$ (mod 8), and only in this case, there is a second regular orientable Hamiltonian embedding of $K_{n,n}$ . The current paper presents an analysis of the face 2-colourable triangular embedding of $K_{n,n,n}$ that results from this. The corresponding Latin squares of side $n$ are determined, together with the full automorphism group of the embedding.