In this paper, we consider the Voronoi decompositions of an arbitrary infinite vertex-transitive graph G. In particular, we are interested in the following question: what is the largest number of Voronoi cells that must be infinite, given sufficiently (but finitely) many Voronoi sites which are sufficiently far from each other? We call this number the survival number s(G). The survival number of a graph has an alternative characterization in terms of the number of balls of radius r-1 required to cover a sphere of radius r. The survival number is not a quasi-isometry invariant, but it remains open whether finiteness of s(G) is. We show that all vertex-transitive graphs with polynomial growth have finite s(G); vertex-transitive graphs with infinitely many ends have infinite s(G); the lamplighter graph LL(Z), which has exponential growth, has finite s(G); and the lamplighter graph LL(Z2), which is Liouville, has infinite s(G).
A Finitary Structure Theorem for Vertex-Transitive Graphs of Polynomial Growth