Hamilton transversals in random Latin squares

TL;DR

This paper proves that almost all large Latin squares contain a Hamilton transversal, a cycle-free full transversal, confirming a conjecture for most cases and showing the abundance of such structures.

Contribution

It establishes that nearly all Latin squares have a Hamilton transversal, confirming a conjecture and providing a counting result that aligns with known upper bounds.

Findings

Almost all Latin squares have a Hamilton transversal.

The number of Hamilton transversals matches the upper bound.

Supports the Ryser-Brualdi-Stein conjecture for most Latin squares.

Abstract

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. This result strengthens a result of Kwan (which in turn implies that almost all Latin squares also satisfy the famous Ryser-Brualdi-Stein conjecture).