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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.07411 |
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| _version_ | 1866910852747100160 |
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| author | Cichacz, Sylwia |
| author_facet | Cichacz, Sylwia |
| contents | A complete mapping of a group $Γ$ is a bijection $φ\colon Γ\to Γ$ for which the mapping $x \mapsto x+φ(x)$ is a bijection. In this paper we consider the existence of a complete mapping $φ$ of $Γ$ and a partition $S_1,S_2,\ldots S_t$ of elements of $Γ$, such that $\sum_{s\in S_i}s=\sum_{s\in S_i}φ(s)=0$ for every $i$, $1 \leq i \leq t$.
A $Γ$-magic rectangle set $MRS_Γ(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $Γ$ of order $abc$, each appearing once, with all row sums in every rectangle equal to a constant $ω\in Γ$ and all column sums in every rectangle equal to a constant $δ\in Γ$.
While a complete characterization of MRS$_Γ(a,b;c)$ exists for cases where $\{a,b\}\not=\{2k+1,2^α\}$, the scenario where $\{a,b\}=\{2k+1,2^α\}$ remains unsolved for $α>1$. Using the partition of $Γ$ into zero-sum sets by complete mappings, we give some sufficient conditions that a $Γ$-magic rectangle set MRS$_Γ(2k+1, 2^α;c)$ exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07411 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partition of Abelian groups into zero-sum sets by complete mappings and its application to the existence of a magic rectangle set Cichacz, Sylwia Combinatorics A complete mapping of a group $Γ$ is a bijection $φ\colon Γ\to Γ$ for which the mapping $x \mapsto x+φ(x)$ is a bijection. In this paper we consider the existence of a complete mapping $φ$ of $Γ$ and a partition $S_1,S_2,\ldots S_t$ of elements of $Γ$, such that $\sum_{s\in S_i}s=\sum_{s\in S_i}φ(s)=0$ for every $i$, $1 \leq i \leq t$. A $Γ$-magic rectangle set $MRS_Γ(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $Γ$ of order $abc$, each appearing once, with all row sums in every rectangle equal to a constant $ω\in Γ$ and all column sums in every rectangle equal to a constant $δ\in Γ$. While a complete characterization of MRS$_Γ(a,b;c)$ exists for cases where $\{a,b\}\not=\{2k+1,2^α\}$, the scenario where $\{a,b\}=\{2k+1,2^α\}$ remains unsolved for $α>1$. Using the partition of $Γ$ into zero-sum sets by complete mappings, we give some sufficient conditions that a $Γ$-magic rectangle set MRS$_Γ(2k+1, 2^α;c)$ exists. |
| title | Partition of Abelian groups into zero-sum sets by complete mappings and its application to the existence of a magic rectangle set |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.07411 |