Latin Squares whose transversals share many entries
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915067140767744 |
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| author | Ghafari, Afsane Wanless, Ian M. |
| author_facet | Ghafari, Afsane Wanless, Ian M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Latin Squares whose transversals share many entries Ghafari, Afsane Wanless, Ian M. Combinatorics 05B15 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. |
| title | Latin Squares whose transversals share many entries |
| topic | Combinatorics 05B15 |
| url | https://arxiv.org/abs/2412.12466 |