Abstract
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.
Original language | English |
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Pages (from-to) | 124-141 |
Number of pages | 18 |
Journal | Journal of Combinatorial Designs |
Volume | 20 |
Issue number | 2 |
DOIs | |
Publication status | Published - Feb 2012 |
Externally published | Yes |