Cycle Switches in Latin Squares

Abstract

Cycle switches are the simplest changes which can be used to alter latin squares, and as such have found many applications in the generation of latin squares. They also provide the simplest examples of latin interchanges or trades in latin square designs. In this paper we construct graphs in which the vertices are classes of latin squares. Edges arise from switching cycles to move from one class to another. Such graphs are constructed on sets of isotopy or main classes of latin squares for orders up to and including eight. Variants considered are when (i) only intercalates may be switched, (ii) any row cycle may be switched and (iii) all cycles may be switched. The structure of these graphs reveals special roles played by N2, pan-Hamiltonian, atomic, semi-symmetric and totally symmetric latin squares. In some of the graphs parity is important because, for example, the odd latin squares may be disconnected from the even latin squares. An application of our results to the compact storage of large catalogues of latin squares is discussed. We also prove lower bounds on the number of cycles in latin squares of both even and odd orders and show these bounds are sharp for infinitely many orders.