Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929565356523520 |
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| author | Cichacz, Sylwia Suchan, Karol |
| author_facet | Cichacz, Sylwia Suchan, Karol |
| contents | The following problem has been known since the 80s. Let $Γ$ be an Abelian group of order $m$ (denoted $|Γ|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $Γ^*=Γ\setminus\{0\}$, the set of non-zero elements of $Γ$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(Γ)|\neq 1$, where $I(Γ)$ is the set of involutions in $Γ$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_09395 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings Cichacz, Sylwia Suchan, Karol Combinatorics Group Theory 05E16, 20K01, 05C25, 05C78 G.2.1; G.2.2 The following problem has been known since the 80s. Let $Γ$ be an Abelian group of order $m$ (denoted $|Γ|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $Γ^*=Γ\setminus\{0\}$, the set of non-zero elements of $Γ$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(Γ)|\neq 1$, where $I(Γ)$ is the set of involutions in $Γ$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs. |
| title | Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings |
| topic | Combinatorics Group Theory 05E16, 20K01, 05C25, 05C78 G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2203.09395 |