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

The following problem has been known since the 80s. Let Γ be an Abelian group of order m (denoted |Γ| = m), and let t and (Formula presented), be positive integers such that (Formula presented). Determine when Γ∗ = Γ {0}, the set of non-zero elements of Γ, can be partitioned into disjoint subsets (Formula presented) such that |Si| = mi and (Formula presented) for every 1 ≤ i ≤ t. Such a subset partition is called a zero-sum partition. |I(Γ)| ̸= 1, where I(Γ) is the set of involutions in Γ, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of mi ≥ 4 for every 1 ≤ i ≤ t, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs. © 2024 by the author(s)