Latin squares with restricted transversals

We prove that for all odd m >= 3 there exists a latin square of order 3 m that contains an (m - 1) x 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 in which there is at least one transversal, but all transversals coincide 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. (C) 2011 Wiley Periodicals, Inc. J Combin Designs 20:124-141, 2012