Abstract:Gyárfás and Sárközy conjectured that every n × n Latin square has a "cycle-free" partial transversal of size n−2. We confirm this conjecture in a strong sense for almost all Latin squares, by showing that as n → ∞, all but a vanishing proportion of n × n Latin squares have a Hamilton transversal, that is, a full transversal for which any proper subset is cycle-free. In fact, we prove a counting result that in almost all Latin squares, the number of Hamilton transversals is essentially that of Taranenko's upper… Show more
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