Orientable $\mathbb{Z}{}_{n}$-distance magic regular graphs

Abstract

Hefetz, M\"{u}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper we support the analogous question for distance magic labeling. Let $\Gamma$ be an Abelian group of order $n$. A \textit{directed $\Gamma$-distance magic labeling} of an oriented graph $\vec{G} = (V,A)$ of order $n$ is a bijection $\vec{l}:V \rightarrow \Gamma$ with the property that there is a \textit{magic constant} $\mu \in \Gamma$ such that for every $x \in V(G)$ $ w(x) = \sum_{y \in N^{+}(x)}\vec{l}(y) - \sum_{y \in N^{-}(x)} \vec{l}(y) = \mu. $ In this paper we provide an infinite family of odd regular graphs possessing an orientable $\mathbb{Z}_{n}$-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable $\mathbb{Z}_{n}$-distance magic.