Distance magic labeling of the halved folded n-cube

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 $$D\subseteq \{0,1,\dots ,d\}$$ D ⊆ { 0 , 1 , ⋯ , d } be a set of distances. A bijection $$l:V(G)\rightarrow \{1,2,\dots ,|V(G)|\}$$ l : V ( G ) → { 1 , 2 , ⋯ , | V ( G ) | } is called a D-magic labeling of G whenever $$\sum \limits _{x\in G_D(v)}l(x)$$ ∑ x ∈ G D ( v ) l ( x ) is a constant for any vertex $$v\in V(G)$$ v ∈ V ( G ) , where $$G_D(v)=\{x\in V(G): \partial (x,v)\in D\}$$ G D ( v ) = { x ∈ V ( G ) : ∂ ( x , v ) ∈ D } . A $$\{1\}$$ { 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 $$\{0,1\}$$ { 0 , 1 } -magic labeling) if and only if $$n=16q^2$$ n = 16 q 2 (resp. $$n=16q^2+16q+6$$ n = 16 q 2 + 16 q + 6 ), where q is a positive integer.