On the metric dimension of Cartesian powers of a graph

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≥2 vertices, and let M be the distance matrix of G. We prove that if there exists w∈Zq such that ∑iwi=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/logq⁡n. In the special case that G is a complete graph, our results close the gap between the lower bound attributed to Erdős and Rényi and the upper bounds developed subsequently by Lindström, Chvátal, Kabatianski, Lebedev and Thorpe.