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arXiv · 2412.12466

Latin Squares whose transversals share many entries

Afsane Ghafari·Ian M. Wanless
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Abstract

We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$ transversal-free entries. We also prove that for all odd $m\geq 3$, there exists a latin square of order $n=3m$ divided into nine $m\times m$ subsquares, where every transversal hits each of these subsquares at least once.

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Afsane Ghafari, Ian M. Wanless. 2024-12-17. Latin Squares whose transversals share many entries. https://arxiv.org/abs/2412.12466

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