Zero-sum partitions of Abelian groups of order $2^n$
The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero element...
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Discrete Mathematics & Theoretical Computer Science
2023-03-01
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| Online Access: | http://dmtcs.episciences.org/9914/pdf |
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| author | Sylwia Cichacz Karol Suchan |
| author_facet | Sylwia Cichacz Karol Suchan |
| author_sort | Sylwia Cichacz |
| collection | DOAJ |
| description | The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and $|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was proved that the condition $m_i\geq 2$ is sufficient if and only if $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$ and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with $\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings. |
| format | Article |
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| institution | OA Journals |
| issn | 1365-8050 |
| language | English |
| publishDate | 2023-03-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj-art-0d14bcb3e0814918890f0cbb80e25b692025-08-20T01:49:32ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502023-03-01vol. 25:1Combinatorics10.46298/dmtcs.99149914Zero-sum partitions of Abelian groups of order $2^n$Sylwia Cichaczhttps://orcid.org/0000-0002-7738-0924Karol Suchanhttps://orcid.org/0000-0003-0793-0924The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and $|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was proved that the condition $m_i\geq 2$ is sufficient if and only if $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$ and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with $\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.http://dmtcs.episciences.org/9914/pdfmathematics - combinatoricsmathematics - group theory05e16, 20k01, 05c25, 05c78g.2.1g.2.2 |
| spellingShingle | Sylwia Cichacz Karol Suchan Zero-sum partitions of Abelian groups of order $2^n$ Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics mathematics - group theory 05e16, 20k01, 05c25, 05c78 g.2.1 g.2.2 |
| title | Zero-sum partitions of Abelian groups of order $2^n$ |
| title_full | Zero-sum partitions of Abelian groups of order $2^n$ |
| title_fullStr | Zero-sum partitions of Abelian groups of order $2^n$ |
| title_full_unstemmed | Zero-sum partitions of Abelian groups of order $2^n$ |
| title_short | Zero-sum partitions of Abelian groups of order $2^n$ |
| title_sort | zero sum partitions of abelian groups of order 2 n |
| topic | mathematics - combinatorics mathematics - group theory 05e16, 20k01, 05c25, 05c78 g.2.1 g.2.2 |
| url | http://dmtcs.episciences.org/9914/pdf |
| work_keys_str_mv | AT sylwiacichacz zerosumpartitionsofabeliangroupsoforder2n AT karolsuchan zerosumpartitionsofabeliangroupsoforder2n |