## Abstract

We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [Combinatorica, 1995]. For a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of ℤ ∞d , the infinite graph with vertex set ℤ d and an edge (u, v) whenever ∥u - v∥∞ = 1. The growth rate of G, denoted ρ G , is the minimum ρ such that every ball of radius r > 1 in G contains at most r ρ vertices. By simple volume arguments, dim(G) = Ω(ρ G ). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρ G ) for every graph G. Previously, it was unknown whether dim(G) could be bounded above by any function of ρ G . We show that a weaker form of Levin's conjecture holds by proving that dim(G) = O(ρ G log ρ G ) for any graph G. We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) = Ω(ρ G log ρ G ). For several special families of graphs (e.g., planar graphs), we salvage the strong form, showing that dim(G) = O(ρ G ). Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces posed by Linial and independently by Benjamini and Schramm. © 2007 Springer-Verlag.