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

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 ${\Gamma }^{\ast }=\Gamma \setminus \{0\}$, 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 $1\le i\le t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(\Gamma )|\ne 1$, 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 $m_i\ge 4$ for every $1\le i\le t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.

arXiv admin note: text overlap with arXiv:2111.05394