Group Distance Magic Labeling of $C_n^r$

Abstract

Let $G=(V,E)$ be a graph and $\varGamma$ be an abelian group both of order n. For $D \subset \{0,1,\ldots , diam(G)\}$, the D -distance neighbourhood of a vertex v in G is defined to be the set $N_D(v)=\{x \in V \ | \ d(x,v) \in D\}$. A bijection $f: V \rightarrow \varGamma$ is called a $(\varGamma , D)$ -distance magic labeling of G if there exists an $\alpha \in \varGamma$ such that $\sum _{x \in N_D(v)} f(x)=\alpha$ for every $v \in V$. In this paper we study $(\varGamma , D)$-distance magic labeling of the graph $C_n^r$ for $D=\{d\}$. We obtain $(\varGamma , \{d\})$-distance magic labelings of $C_n^r$ with respect to certain classes of abelian groups. We also obtain necessary conditions for existence of such labelings.