Hamilton transversals in random Latin squares. Issue 2 (14th June 2022) - Gould, Stephen; Kelly, Tom
Abstract: Gyárfás and Sárközy conjectured that every n × n n× 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→ ∞ , all but a vanishing proportion of n × n 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 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).