# New Properties of Numbers of Plane Graphs

## TL;DR

This paper investigates the enumeration of crossing-free straight-edge graphs on planar point sets, deriving bounds and techniques that could enhance understanding of plane graph counts with specific edge constraints.

## Contribution

It introduces new bounds on the ratio of plane graphs with successive edge counts and employs a cross-graph charging scheme for sets with few extreme points.

## Key findings

- Derived lower bounds on the ratio of plane graphs with m+1 and m edges.
- Showed how small improvements in bounds could impact existing enumeration results.
- Applied a cross-graph charging scheme to sets with few extreme points.

## Abstract

We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the number of graphs with $m$ edges. We show how a relatively small improvement of this bound would improve existing bounds concerning numbers of plane graphs with a prescribed number of edges. Furthermore, we use a cross-graph charging scheme to derive lower bounds for the number of such graphs when the point set has few extreme points.

---
Source: https://tomesphere.com/paper/1704.03066