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

Sylwia Cichacz

Ars Mathematica Contemporanea, Tome 13 (2017) no. 2, pp. 417-425.

Résumé

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 ∈ 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.

Détail

DOI : 10.26493/1855-3974.1054.fcd

Keywords: Abelian group, constant sum partition, group distance magic labeling