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.

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