On the metric dimension of Cartesian powers of a graph

TL;DR

This paper investigates the metric dimension of Cartesian powers of a graph, providing asymptotic formulas under certain algebraic conditions, and refines bounds for complete graphs.

Contribution

It establishes a new asymptotic formula for the metric dimension of Cartesian powers of graphs satisfying specific algebraic conditions, improving understanding of resolving sets.

Findings

Metric dimension of Cartesian powers is approximately (2+o(1))n/ log_q n under certain conditions.

Results close the gap between known bounds for complete graphs.

Provides conditions involving the distance matrix for determining metric dimension asymptotics.

Abstract

A set of vertices $S$ resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected graph $G$ on $q \ge 2$ vertices, and let $M$ be the distance matrix of $G$. We prove that if there exists $w \in \mathbb{Z}^q$ such that $\sum_i w_i = 0$ and the vector $Mw$, after sorting its coordinates, is an arithmetic progression with nonzero common difference, then the metric dimension of the Cartesian product of $n$ copies of $G$ is $(2+o(1))n/\log_q n$. In the special case that $G$ is a complete graph, our results close the gap between the lower bound attributed to Erd\H{o}s and R\'enyi and the upper bounds developed subsequently by Lindstr\"om, Chv\'atal, Kabatianski, Lebedev and Thorpe.