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 ∈ N D ( v ) l ( x ) = k for any vertex v ∈ V ( G ) , where N D ( 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)).