Finite Voronoi decompositions of infinite vertex transitive graphs

TL;DR

This paper investigates the properties of Voronoi decompositions in infinite vertex-transitive graphs, introducing the survival number s(G) to measure the number of infinite cells, and explores its behavior across different graph classes.

Contribution

It defines the survival number s(G) for infinite vertex-transitive graphs and characterizes its finiteness or infiniteness for various graph types, revealing new insights into their geometric structure.

Findings

s(G) is always at least two

Graphs with polynomial growth have finite s(G)

Graphs with infinitely many ends have infinite s(G)

Abstract

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 covering, which we use to show that s(G) is always at least two. The survival number is not a quasi-isometry invariant, but it remains open whether finiteness of the s(G) is. We show that all vertex transitive graphs with polynomial growth have a finite s(G); vertex transitive graphs with infinitely many ends have an infinite s(G); the lamplighter graph LL(Z), which has exponential growth, has a finite s(G); and the lamplighter graph LL(Z^2), which is Liouville, has an infinite s(G).