A constraint on the biembedding of Latin squares

A biembedding of two latin squares of order n is equivalent to a face 2-colourable triangulation of Kn,n,n. We consider the following question: Given two latin squares A and B of order n, is there any relabelling of A and B for which there exists a biembedding? Grannell, Griggs and Knor answered this question computationally for n ≤ 7. A main class of latin squares is a set of latin squares which are equivalent under some relabelling. The number of main classes and distinct biembeddings increases rapidly with n, so that there is relatively little information for n ≤ 6, while the problem is not computationally feasible for n ≥ 8. For n = 7 a pattern emerged without parallel in the equivalent results for Steiner triple systems; the 147 main classes partition into 16 sets, such that a biembedding exists for most pairs of main classes from the same set, but there is no biembedding between any two latin squares which are not in the same set. Using a argument based on permutation parity, we give a necessary condition explaining this pattern, and briefly explore some other implications of this result. Based on joint work with Diane Donovan, Mike Grannell and Terry Griggs.