Almost every Latin square has a decomposition into transversals

TL;DR

This paper demonstrates that most Latin squares of large order have a decomposition into transversals, countering earlier conjectures and showing such decompositions are very common.

Contribution

The paper proves that a randomly chosen Latin square of large order almost certainly has a decomposition into transversals, indicating such structures are prevalent.

Findings

Most Latin squares of large order have transversals

Counterexamples to Euler's conjecture are very rare

Random Latin squares typically decompose into transversals

Abstract

In 1782, Euler conjectured that no Latin square of order $n\equiv 2\; \textrm{mod}\; 4$ has a decomposition into transversals. While confirmed for $n=6$ by Tarry in 1900, Bose, Parker, and Shrikhande constructed counterexamples in 1960 for each $n\equiv 2\; \textrm{mod}\; 4$ with $n\geq 10$. We show that, in fact, counterexamples are extremely common, by showing that if a Latin square of order $n$ is chosen uniformly at random then with high probability it has a decomposition into transversals.