Distance magic labeling of the halved folded n-cube

Abstract

Hypercube is an important structure for computer networks. The distance plays an important role in its applications. In this paper, we study a magic labeling of the halved folded n-cube which is a variation of the n-cube. This labeling is determined by the distance. Let G be a finite undirected simple connected graph with vertex set V(G), distance function $\partial$ and diameter d. Let $D\subseteq \{0,1,\dots ,d\}$ be a set of distances. A bijection $l:V(G)\rightarrow \{1,2,\dots ,|V(G)|\}$ is called a D-magic labeling of G whenever $\sum \limits _{x\in G_D(v)}l(x)$ is a constant for any vertex $v\in V(G)$, where $G_D(v)=\{x\in V(G): \partial (x,v)\in D\}$. A $\{1\}$-magic labeling is also called a distance magic labeling. We show that the halved folded n-cube has a distance magic labeling (resp. a $\{0,1\}$-magic labeling) if and only if $n=16q^2$(resp. $n=16q^2+16q+6$), where q is a positive integer.