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Mathematics > Combinatorics
Title: Zero-sum partitions of Abelian groups and their applications to magic-type labelings
(Submitted on 17 Mar 2022 (v1), last revised 24 Apr 2024 (this version, v3))
Abstract: The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, 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}.
$|I(\Gamma)|\neq 1$, where $I(\Gamma)$ is the set of involutions in $\Gamma$, 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-type labelings of graphs.
Submission history
From: Karol Suchan [view email][v1] Thu, 17 Mar 2022 15:38:50 GMT (17kb,D)
[v2] Tue, 28 Feb 2023 13:01:12 GMT (19kb,D)
[v3] Wed, 24 Apr 2024 21:42:33 GMT (20kb,D)
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