Examples of distance magic labelings of the $6$-dimensional hypercube

TL;DR

This paper presents examples of distance magic labelings of the 6-dimensional hypercube, including non-neighbor-balanced cases, obtained through computational methods using a SAT solver.

Contribution

It provides the first known examples of non-neighbor-balanced distance magic labelings for the 6-dimensional hypercube, expanding understanding of such labelings.

Findings

Existence of non-neighbor-balanced labelings demonstrated

Examples obtained via SAT solver techniques

Contributes to the classification of hypercube labelings

Abstract

A distance magic labeling of an $n$-dimensional hypercube is a labeling of its vertices by natural numbers from $\{0, \ldots, 2^n-1\}$, such that for all vertices $v$ the sum of the labels of the neighbors of $v$ is the same. Such a labeling is called neighbor-balanced, if, moreover, for each vertex $v$ and an index $i=0,\ldots,n-1$, exactly half of the neighbors of $v$ have digit $1$ at $i$-th position of the binary representation of their label. We demonstrate examples of non-neighbor-balanced distance magic labelings of $6$-dimensional hypercube obtained by a SAT solver.