Latin Squares with Restricted Transversals

The original article to which this erratum refers was correctly published online on 1 December 2011. Due to an error at the publisher, it was then published in Journal of Combinatorial Designs 20: 124–141, 2012 without the required shading in several examples. To correct this, the article is here reprinted in full. The publisher regrets this error. We prove that for all odd m ≥ 3 there exists a latin square of order 3 m 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 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. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 344–361, 2012