Hamilton transversals in random Latin squares

Gyárfás and Sárközy conjectured that every n×n$$ n\times n $$ Latin square has a “cycle‐free” partial transversal of size n−2$$ n-2 $$. We confirm this conjecture in a strong sense for almost all Latin squares, by showing that as n→∞$$ n\to \infty $$, all but a vanishing proportion of n×n$$ n\times n $$ 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 bound on the number of full transversals. This result strengthens a result of Kwan (which in turn implies that almost all Latin squares also satisfy the famous Ryser–Brualdi–Stein conjecture).