On Metric Dimensions of Hypercubes

TL;DR

This paper investigates the metric, edge metric, and mixed metric dimensions of hypercube graphs, revealing surprising relationships and proposing a conjecture about their equality for large dimensions.

Contribution

It establishes that the metric and edge metric dimensions differ by one, and that the metric and mixed metric dimensions are equal for hypercubes, with a conjecture for all three being equal at large dimensions.

Findings

Metric and edge metric dimensions differ by one for hypercubes.

Metric and mixed metric dimensions are equal for all hypercubes with dimension ≥ 3.

Conjecture that all three metric dimensions are equal for large d.

Abstract

The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of $Q_d$ differ by only one for every integer $d$. In particular, if $d$ is odd, then the metric and edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube $Q_d$ are equal for every $d \ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when $d$ is large enough.