The Existence of Latin Squares without Orthogonal Mates

Authors:
Ian M. Wanless;Bridget S. Webb
Affiliations:
School of Mathematical Sciences, Monash University, Australia Vic 3800;Department of Mathematics, The Open University, UK MK7 6AA
Venue:
Designs, Codes and Cryptography
Year:
2006

Citing 4
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Abstract

A latin square is a bachelor square if it does not possess an orthogonal mate; equivalently, it does not have a decomposition into disjoint transversals. We define a latin square to be a confirmed bachelor square if it contains an entry through which there is no transversal. We prove the existence of confirmed bachelor squares for all orders greater than three. This resolves the existence question for bachelor squares.