Symmetries That Latin Squares Inherit from 1-Factorizations

A 1-factorization of a graph is a decomposition of the graph into edge disjoint perfect matchings. There is a well-known method, which we call the K-construction, for building a 1-factorization of Kn;n from a 1-factorization of Knþ1. The 1-factorization of Kn;n can be written as a latin square of order n. The K-construction has been used, among other things, to make perfect 1-factorizations, subsquare-free latin squares, and atomic latin squares. This paper studies the relationship between the factorizations involved in the K-construction. In particular, we show how symmetries (automorphisms) of the starting factorization are inherited as symmetries by the end product, either as automorphisms of the factorization or as autotopies of the latin square. Suppose that the K-construction produces a latin square L from a 1-factorization F of Knþ1. We show that the main class of L determines the isomorphism class of F, although the converse is false. We also prove a number of restrictions on the symmetries (autotopies and paratopies) which L may possess, many of which are simple consequences of the fact that L must be symmetric (in the usual matrix sense) and idempotent. In some circumstances, these restrictions are tight enough to ensure that L has trivial autotopy group. Finally, we give a cubic time algorithm for deciding whether a main class of latin squares contains any square derived from the K-construction. The algorithm also detects symmetric squares and totally symmetric squares (latin squares that equal their six conjugates). # 2005 Wiley Periodicals, Inc. J Combin Designs 13: 157–172, 2005.