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 $q \ge 2$ 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 Erdős and Rényi and the upper bounds developed subsequently by Lindström, Chvátal, Kabatianski, Lebedev and Thorpe.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1712.02723
 Bibcode:
 2017arXiv171202723J
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 Computer Science  Information Theory;
 05C12;
 05C76;
 06A07
 EPrint:
 12 pages, 1 figure, 1 table, accepted to J. Comb. Theory A, corrections suggested by the referees have been incorporated