### 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 ∈ Ai}

*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*(ZSP(

*t*sets*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.

#### Keywords

DOI: https://doi.org/10.26493/1855-3974.1054.fcd

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications