The Existence of Latin Squares without Orthogonal Mates

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.

Click here to download the whole paper.

Last modified: Mon May 19 16:45:27 EST 2006