D-magic labelings of the halved n-cube

Abstract

Let G=(V,E) be a graph with path-length distance function ∂ and diameter d. Let D⊆{0,1,…,d} be a set of distances in G and let ND(x)={y|∂(x,y)∈D} for a fixed vertex x∈V. A bijection φ:V→{1,2,…,|V|} is called a D-magic labeling of G if there exists a constant k such that ∑y∈ND(x)f(y)=k for any x∈V. In this paper, we will study D-magic labelings of the halved n-cube (n≥2) that is on all binary strings of length n with even number of 1s as vertices and edges between any two strings of Hamming distance 2. We prove that the halved n-cube is {1}-magic if and only if n=m2 where m≥2 and m≢0(mod4), and is {0,1}-magic if and only if n=m2+2 where m≥0 and m≢2(mod4).