Zero sum partition into sets of the same order and its applications

Abstract

We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-\emph{zero-sum-partition property} ($m$-\textit{ZSP-property}) if $m$ divides $n$, $m\ge 2$ and there is a partition of $\Gamma$ into pairwise disjoint subsets $A_1,A_2,\dots ,A_t$, such that $|A_i|=m$ and ${\sum }_{a\in A_i}a=g_0$ for $1\le i\le t$, where $g_0$ is the identity element of $\Gamma$. In this paper we study the $m$-ZSP property of $\Gamma$. We show that $\Gamma$ has $m$-ZSP if and only if $|\Gamma |$ is odd or $m\ge 3$ and $\Gamma$ has more than one involution. We will apply the results to the study of group distance magic graphs as well as to generalized Kotzig arrays.

References in corpus (1)

• Note on group distance magic graphs $G[C_4]$

Cited by in corpus (3)

• A magic rectangle set on Abelian groups
• Realization of digraphs in Abelian groups and its consequences
• Note on group irregularity strength of disconnected graphs