Latin squares with no transversals

Abstract

Ak-plex in a latin square of ordernis a selection of kn entries that includes k representatives from each row and column and k occurrences of each symbol. A 1-plex is also known as a transversal. It is well known that if n is even then Bₙ, the addition table for the integers modulo n, possesses no transversals. We show that there are a great many latin squares that are similar to Bₙ and have no transversal. As a consequence, the number of species of transversal-free latin squares is shown to be at least nⁿ³⁼²⁽¹⁼²ᵒ⁽¹⁾ for even n→∞. We also produce various constructions for latin squares that have no transversal but do have a k-plex for some odd k >1. We prove a 2002 conjecture of the second author that for all even orders n >4 there is a latin square of order n that contains a 3-plex but no transversal. We also show that for odd k and m≥2, there exists a latin square of order 2km with a k-plex but no k'-plex for odd k'< k.