On zero sum-partition of Abelian groups into three sets and group distance magic labeling - Cichacz, Sylwia (Ars mathematica contemporanea, 2017)
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 ?$ν \in Γ$?, ?$\sum_{a \in A_i} a = ν$ 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 ?$\ell$? from ?$V$? to an abelian group ?$Γ$? of order ?$n$? such that the weight ?$w(x) = \sum_{y\in N(x)}\ell(y)$? of every vertex ?$x \in V$? is equal to the same element ?$μ \in Γ$?, 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: abelska grupa; grupe; grupno razdaljno magično označevanje; razčlenitev s konstantno vsoto