Ryser's Theorem for Symmetric $ρ$-latin Squares
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916947865632768 |
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| author | Bahmanian, Amin Hilton, A. J. W. |
| author_facet | Bahmanian, Amin Hilton, A. J. W. |
| contents | Let $L$ be an $n\times n$ array whose top left $r\times r$ subarray is filled with $k$ different symbols, each occurring at most once in each row and at most once in each column. We establish necessary and sufficient conditions that ensure the remaining cells of $L$ can be filled such that each symbol occurs at most once in each row and at most once in each column, $L$ is symmetric with respect to the main diagonal, and each symbol occurs a prescribed number of times in $L$. The case where the prescribed number of times each symbol occurs is $n$ was solved by Cruse (J. Combin. Theory Ser. A 16 (1974), 18--22), and the case where the top left subarray is $r\times n$ and the symmetry is not required, was settled by Goldwasser et al. (J. Combin. Theory Ser. A 130 (2015), 26--41). Our result allows the entries of the main diagonal to be specified as well, which leads to an extension of the Andersen-Hoffman Theorem (Annals of Disc. Math. 15 (1982) 9--26, European J. Combin. 4 (1983) 33--35). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06401 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ryser's Theorem for Symmetric $ρ$-latin Squares Bahmanian, Amin Hilton, A. J. W. Combinatorics Discrete Mathematics 05B15, 05C70, 05C15 Let $L$ be an $n\times n$ array whose top left $r\times r$ subarray is filled with $k$ different symbols, each occurring at most once in each row and at most once in each column. We establish necessary and sufficient conditions that ensure the remaining cells of $L$ can be filled such that each symbol occurs at most once in each row and at most once in each column, $L$ is symmetric with respect to the main diagonal, and each symbol occurs a prescribed number of times in $L$. The case where the prescribed number of times each symbol occurs is $n$ was solved by Cruse (J. Combin. Theory Ser. A 16 (1974), 18--22), and the case where the top left subarray is $r\times n$ and the symmetry is not required, was settled by Goldwasser et al. (J. Combin. Theory Ser. A 130 (2015), 26--41). Our result allows the entries of the main diagonal to be specified as well, which leads to an extension of the Andersen-Hoffman Theorem (Annals of Disc. Math. 15 (1982) 9--26, European J. Combin. 4 (1983) 33--35). |
| title | Ryser's Theorem for Symmetric $ρ$-latin Squares |
| topic | Combinatorics Discrete Mathematics 05B15, 05C70, 05C15 |
| url | https://arxiv.org/abs/2209.06401 |