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

Sylwia Cichacz¹

¹Faculty of Applied Mathematics AGH University of Science and Technology Krakow, Poland

The electronic journal of combinatorics, Tome 25 (2018) no. 1

Résumé

We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-zero-sum-partition property ($m$-ZSP-property) if $m$ divides $n$, $m\geq 2$ and there is a partition of $\Gamma$ into pairwise disjoint subsets $A_1, A_2,\ldots , A_t$, such that $|A_i| = m$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$, 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 the $m$-ZSP property if and only if $m\geq 3$ and $|\Gamma|$ is odd or $\Gamma$ 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.

Détail

DOI : 10.37236/6977

Classification : 05C25, 05C78
Mots-clés : abelian group, zero sum partition, group distance magic labelling, Kotzig arrays