Orientable Zn-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 $\Gamma$ be an Abelian group of order $n$. A directed $\Gamma$-distance magic labeling of an oriented graph $\overrightarrow{G}=(V,A)$ of order $n$ is a bijection $\overrightarrow{l}:V\to \Gamma$ with the property that there is a magic constant $\mu \in \Gamma$ such that for every $x\in V\left(G\right)$ $w\left(x\right)={\sum }_{y\in N^{+}\left(x\right)}\overrightarrow{l}\left(y\right)-{\sum }_{y\in N^{-}\left(x\right)}\overrightarrow{l}\left(y\right)=\mu .$ 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.

Keywords:

• Orientable group
• Distance magic
• Labeling

Acknowledgments

We would like to thank Sylwia Cichacz for her support, encouragement, assistance in proofreading this researchand delivering valuable tips and resources. We could not have imagined having a better advisor and mentor. We are also very grateful to Dominika Datoń, Kinga Patera, Natalia Pondel, Maciej Gabryś and Przemysław Zietek from “Snark” Research Student Association for their help and involvement in initial phase of our analysis.