# Latin squares with restricted transversals

We prove that for all odd m ≥ 3 there exists a latin square of
order 3m that contains an (m-1)×m latin subrectangle consisting
of entries not in any transversal. We prove that for all even n ≥
10 there exists a latin square of order n that has a transversal but
every transversal coincides on a single entry. A corollary is a new
proof of the existence of a latin square without an orthogonal mate,
for all odd orders n ≥ 11. Finally, we report on an extensive
computational study of transversal-free entries and sets of disjoint
transversals in the latin squares of order n ≤ 9. In particular, we
count the number of species of each order that possess an orthogonal
mate.
Various data collected during the writing of this paper can be found at
http://users.monash.edu.au/~iwanless/data/transversals/.