Growth and isoperimetric profile of planar graphs

TL;DR

This paper investigates the relationship between growth conditions and isoperimetric profiles in planar graphs, establishing bounds on boundary sizes of certain domains based on volume growth constraints.

Contribution

It introduces a method to find domains with controlled boundary sizes in planar graphs satisfying specific volume growth conditions.

Findings

Boundaries of certain domains are at most proportional to n

Volume growth condition implies existence of domains with small boundary

Results connect graph growth with isoperimetric properties

Abstract

Let G be a planar graph such that the volume function of G satisfies V(2n)< CV(n) for some constant C > 0. Then for every vertex v of G and integer n, there is a domain \Omega such that B(v,n) \subset \Omega, \Omega \subset B(v, 6n) and the size of the boundary of \Omega is at most order n.