Distance Magic Labeling of the Halved Folded n-Cube

Abstract

AbstractHypercube 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.KeywordsD-magic labelingDistance-regular graphHalved folded n-cubeNetworkIncomplete tournament