Growth and Isoperimetric Profile of Planar Graphs

Light Structures in Innnite Planar Graphs without the Strong Isoperimetric Property

It is shown that the discharging method can be successfully applied on innnite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality. The general outline of the method is presented and the following applications are given: Planar graphs with only nitely many ver-tices of degree 5 and with subexponential growth contain arbitr...

Light Structures in Infinite Planar Graphs without the Strong Isoperimetric Property

It is shown that the discharging method can be successfully applied on infinite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality. The general outline of the method is presented and the following applications are given: Planar graphs with only finitely many vertices of degree ≤ 5 and with subexponential growth contain ar...

Isoperimetric Constants of (d,f )-Regular Planar Graphs

Edge Separators for Graphs of Bounded Genus with Applications

We prove that every n-vertex graph of genus 9 and maximal degree k has an edge separator of size O( ..;gTffl). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separatorto the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.

Asymptotic Isoperimetry of Balls in Metric Measure Spaces

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. Let A be a family of subsets of a metric measure space (X, d, μ), with finite, unbounded volume. For t > 0, we define: I ↓ A(t) = inf A∈A,μ(A)≥t μ(∂A). We say that A is...