On Distance Magic Labelings of Hamming Graphs and Folded Hypercubes

Let Gamma = (V, E) be a graph of order n. A distance magic labeling of Gamma is a bijection l: V -> {1, 2, . . ., n} for which there exists a positive integer k such that sigma(x)(is an element of)(N)(()(u)()) l(x) = k for all vertices u is an element of 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 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 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.