On metric dimensions of hypercubes

Abstract

In this note we show two unexpected results concerning the metric, the edge metric and the mixed metric dimensions of hypercube graphs. First, we show that the metric and the edge metric dimensions of Q_d differ by at most one for every integer d. In particular, if d is odd, then the metric and the edge metric dimensions of Q_d are equal. Second, we prove that the metric and the mixed metric dimensions of the hypercube Q_d are equal for every d ≥ 3. We conclude the paper by conjecturing that all these three types of metric dimensions of Q_d are equal when d is large enough.