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 = r 1 + r 2 + … + r t of n , with r i ≥ 2 for 2 ≤ i ≤ t , there is a partition of Γ into pairwise disjoint subsets A 1 , A 2 , …, A t , such that ∣ A i ∣ = r i and for some ν ∈ Γ , ∑ a ∈ A i a = ν for 1 ≤ i ≤ t . For ν = g 0 (where g 0 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.