$D$-Magic and Antimagic Labelings of Hypercubes

Abstract

For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where $N_D(x)=\{y|d(y,x)=j, j\in D\}$. In this paper we shall find sets of distances $D$s, such that the hypercube is $D-$magic. We shall utilise well-known properties of (bipartite) distance-regular graphs to construct the $D-$magic labelings.