Zero Sum Partition of Abelian Groups into Sets of the Same Order and its Applications

Abstract

We will say that an Abelian group Γ of order n has the m-zerosum-partition property (m-ZSP-property) if m divides n, m ≥ 2 and there is a partition of Γ into pairwise disjoint subsets A 1 , A 2 , . . . , A t , such that |A i | = m and a∈A i a = g 0 for 1 ≤ i ≤ t, where g 0 is the identity element of Γ.In this paper we study the m-ZSP property of Γ. We show that Γ has m-ZSP if and only if |Γ| is odd or m ≥ 3 and Γ 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.

Zero-sum partitions of Abelian groups of order $2^n$

“…We will need the following lemma. Lemma 3.1 ( [12,31]). Let Γ be a finite Abelian group such that |I(Γ)| = 1.…”

“…Given any graph G, the lexicographic product G • K n [30], for some positive integer n, is the graph obtained from G by replacing each vertex v by an independent set of n vertices adjacent to all vertices in N G (v). If G is a graph of order t, then the lexicographic product G • K n has a Γ-distance magic labeling for any integer n, n ≥ 3, and any Γ of order nt such that |I(Γ)| = 1 [12]. Finally, some general results relating group distance magic and antimagic labelings of Cayley graphs, and Cartesian, direct and strong products of graphs can be found in [13] 2 Realizable triples in…”

“…Note that, in general, even the weaker version of Conjecture 1.5 posed in [12] is still open. Recently Cichacz and Tuza [14] proved that, if |Γ| is large enough and |I(Γ)| > 1, then Γ has 4-Zero-Sum Partition Property.…”

“…For m and t even, we only know the following. Theorem 5.2 ( [10,12]). Let G = K n 1 ,n 2 ,...,nt be a complete t-partite graph of order m such that 1 ≤ n 1 ≤ n 2 ≤ .…”

Zero-sum Partitions of Abelian Groups

“…Note that, in general, even the weaker version of Conjecture 1.7 posed in [4] is still open. Conjecture 1.9 ([4]).…”