Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings

TL;DR

This paper establishes sufficient conditions for zero-sum partitions of Abelian groups and explores their applications to magic- and antimagic-type graph labelings, extending known results from the 1980s.

Contribution

It proves that, under certain conditions, zero-sum partitions of Abelian groups always exist and applies these results to graph labeling problems.

Findings

Zero-sum partitions exist when all subset sizes are at least 4 and the group has no involutions.

Necessary condition: the group must have more than one involution.

Applications to magic- and antimagic-type labelings of graphs are demonstrated.

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- and antimagic-type labelings of graphs.