D-magic labelings of the halved n-cube.

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 N D (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 ∈ N D (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 = m 2 where m ≥ 2 and m ≢ 0 (mod 4) , and is { 0 , 1 } -magic if and only if n = m 2 + 2 where m ≥ 0 and m ≢ 2 (mod 4). [ABSTRACT FROM AUTHOR]