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  =  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.