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arXiv · 2501.05438

Almost every Latin square has a decomposition into transversals

Candida Bowtell·Richard Montgomery
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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.

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Candida Bowtell, Richard Montgomery. 2025-01-09. Almost every Latin square has a decomposition into transversals. https://arxiv.org/abs/2501.05438

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