D-magic labelings of the folded n-cube

Abstract

Let G be a finite undirected simple connected graph with vertex set V(G), distance function ∂ and diameter d. Let D⊆{0,1,…,d} be a set of distances. A bijection l:V(G)→{1,2,…,|V(G)|} is called a D-magic labeling of G if there exists a constant k such that ∑x∈ND(v)l(x)=k for any vertex v∈V(G), where ND(v)={x∈V(G):∂(x,v)∈D}. We say G has a D-magic labeling if G admits a D-magic labeling. In this paper, we show that the folded n-cube has a {1}-magic labeling (resp. a {0,1}-magic labeling) if and only if n≡0 (mod 4) (resp. n≡3 (mod 4)).