On a conjecture by Sylwia Cichacz and Tomasz Hinc, and a related problem
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| Format: | Preprint |
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2024
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| _version_ | 1866914906294452224 |
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| author | Morini, Fiorenza Pellegrini, Marco Antonio Sora, Stefania |
| author_facet | Morini, Fiorenza Pellegrini, Marco Antonio Sora, Stefania |
| contents | A $Γ$-magic rectangle set $\mathrm{MRS}_Γ(a, b; c)$ is a collection of $c$ arrays of size $a\times b$ whose entries are the elements of an abelian group $Γ$ of order $abc$, each one appearing once and in a unique array in such a way that the sum of the elements of each row is equal to a constant $ω\in Γ$ and the sum of the elements of each column is equal to a constant $δ\in Γ$. In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an $\mathrm{MRS}_Γ(a,b;c)$. We also generalize this problem, describing constructions of $Γ$-magic rectangle sets, whose elements are partially filled arrays. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04897 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture by Sylwia Cichacz and Tomasz Hinc, and a related problem Morini, Fiorenza Pellegrini, Marco Antonio Sora, Stefania Combinatorics 05B15, 05C78, 05B30 A $Γ$-magic rectangle set $\mathrm{MRS}_Γ(a, b; c)$ is a collection of $c$ arrays of size $a\times b$ whose entries are the elements of an abelian group $Γ$ of order $abc$, each one appearing once and in a unique array in such a way that the sum of the elements of each row is equal to a constant $ω\in Γ$ and the sum of the elements of each column is equal to a constant $δ\in Γ$. In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an $\mathrm{MRS}_Γ(a,b;c)$. We also generalize this problem, describing constructions of $Γ$-magic rectangle sets, whose elements are partially filled arrays. |
| title | On a conjecture by Sylwia Cichacz and Tomasz Hinc, and a related problem |
| topic | Combinatorics 05B15, 05C78, 05B30 |
| url | https://arxiv.org/abs/2408.04897 |