On distance magic labelings of Hamming graphs and folded hypercubes
Štefko Miklavič, Primož Šparl · Discussiones Mathematicae Graph Theory · 2021
Let $\Gamma = (V,E)$ be a graph of order $n$. A distance magic labeling of $\Gamma$ is a bijection $\ell \colon V \to \{1,2, \ldots, n\}$ for which there exists a positive integer $k$ such that $\sum_{x \in N(u)} \ell(x) = k$ for all vertices $u \in V$, where $N(u)$ is the neighborhood of $u$. A graph is said to be distance magic if it admits a distance magic labeling. The Hamming graph $\mathrm{H}(D,q)$, where $D, q$ are positive integers, is the graph whose vertex set consists of all words of length $D$ over an alphabet of size $q$ in which two vertices are adjacent whenever the corresponding words differ in precisely one position. The well-known hypercubes are precisely the Hamming graphs with $q = 2$. Distance magic hypercubes were classified in two papers from 2013 and 2016. In this paper we consider all Hamming graphs. We provide a sufficient condition for a Hamming graph to be distance magic and as a corollary provide an infinite number of pairs $(D, q)$ for which the corresponding Hamming graph $\mathrm{H}(D,q)$ is distance magic. A folded hypercube is a graph obtained from a hypercube by identifying pairs of vertices at maximal distance. We classify distance magic folded hypercubes by showing that the dimension-$D$ folded hypercube is distance magic if and only if $D$ is divisible by $4$.