Group distance magic labeling of direct product of graphs - Anholcer, Marcin (Ars mathematica contemporanea, 2015)
Let ?$G = (V,E)$? be a graph and ?$Γ$? an Abelian group, both of order ?$n$?. A group distance magic labeling of ?$G$? is a bijection ?$\ell \colon V → Γ$? for which there exists ?$μ \in Γ$? such that ?$\sum_{x \in N(v)}\ell (x) = μ $? for all ?$v \in V$?, where ?$N(v)$? is the neighborhood of ?$v$?. In this paper we consider group distance magic labelings of direct product of graphs. We show that if ?$G$? is an ?$r$?-regular graph of order ?$n$? and ?$m = 4$? or ?$m = 8$? and ?$r$? is even, then the direct product ?$C_{m} × G$? ?$Γ$?-distance magic for every Abelian group of order ?$mn$?. We also prove that ?$C_{m} × C_{n}$? is ?$\mathbb{Z}_{mn}$?-distance magic if and only if ?$m \in \{4,8\}$? or ?$n \in \{4,8\}$? or ?$m,n \equiv 0 \pmod 4$?. It is also shown that if ?$m,n \not\equiv 0 \pmod 4$? then ?$C_{m} × C_{n}$? is not ?$Γ$?-distance magic for any Abelian group ?$Γ$? of order ?$mn$?.
Keywords: Abelian group; Abelova grupa; direct product of graphs; direktni produkt grafov; distance magic labeling; group labeling; grupno označevanje; razdaljno magično označevanje