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

Abstract

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 + … + r_t of n, with r_i ≥ 2 for 2 ≤ i ≤ t, there is a partition of Γ  into pairwise disjoint subsets A₁, A₂, …, A_t, such that ∣A_i∣ = r_i and for some ν ∈ Γ , ∑ _a ∈ Aia = 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 ℓ from V to an Abelian group Γ  of order n such that the weight w(x) = ∑ _{y ∈ N(x)}ℓ(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.