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.

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.