Hamilton transversals in random Latin squares

Understand

Gy\'{a}rf\'{a}s and S\'{a}rk\"{o}zy conjectured that every $n\times n$ 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 $n \rightarrow \infty$, all but a vanishing proportion of $n\times n$ Latin squares have a Hamilton transversal, i.e.
• 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.