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.

Edge metric dimension of some Cartesian product of graphs

“…In this section, we prove that the edge metric dimension of the Cartesian product of a cycle and a path (cylinder graph) is a constant, regardless of the order of the cycle and the path. We will use the following theorems on the metric dimension of some Cartesian products [1] and an interesting relationship between metric dimension and edge metric dimension of a connected bipartite graph [8]. We compute the edge metric representation of each edge of…”

“…[8] Let G be a connected bipartite graph. Then, every metric generator for G is also an edge metric generator, i.e., edim(G) ≤ dim(G).…”