Orientable [formula omitted]-distance magic regular graphs

Abstract

Hefetz, Mütze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation (Hefetz et al., 2010). In this paper we support the analogous question for distance magic labeling. Let Γ be an Abelian group of order n. A directedΓ-distance magic labeling of an oriented graph G→=(V,A) of order n is a bijection l→:V→Γ with the property that there is a magic constantμ∈Γ such that for every x∈V(G)w(x)=∑y∈N+(x)l→(y)−∑y∈N−(x)l→(y)=μ. In this paper we provide an infinite family of odd regular graphs possessing an orientable Zn-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable Zn-distance magic.