On zero sum-partition of Abelian groups into three sets and group distance magic labeling

We say that a finite Abelian group Γ has the constant-sum-partition property into t sets (CSP(t)-property) if for every partition n = r1 +r2 + . . .+rt of n, with ri ≥ 2 for 2 ≤ i ≤ t, there is a partition of Γ into pairwise disjoint subsets A1, A2, . . . , At, such that |Ai| = ri and for some ν ∈ Γ, ∑ a∈Ai a = ν for 1 ≤ i ≤ t. For ν = g0 (where g0 is the identity element of Γ) we say that Γ has zero-sum-partition property into t sets (ZSP(t)-property). A Γ-distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ` from V to an Abelian group Γ of order n such that the weight w(x) = ∑ y∈N(x) `(y) of every vertex x ∈ V is equal to the same element μ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V (G)|. In this paper we study the CSP(3)-property of Γ, and apply the results to the study of group distance magic complete tripartite graphs.