The evolution of uniform random planar graphs

TL;DR

This paper investigates the asymptotic properties of uniform random planar graphs with a fixed number of edges, revealing how their component and subgraph probabilities change with the edge-to-vertex ratio.

Contribution

It provides a detailed analysis of the probability of specific components and subgraphs in uniform random planar graphs based on edge density.

Findings

Different asymptotic behaviours depending on m/n ratio.

Probabilities of certain subgraphs vary with edge-to-vertex ratio.

Counting arguments reveal structural properties of random planar graphs.

Abstract

Let P_{n,m} denote the graph taken uniformly at random from the set of all planar graphs on {1,2,..., n} with exactly m(n) edges. We use counting arguments to investigate the probability that P_{n,m} will contain given components and subgraphs, finding that there is different asymptotic behaviour depending on the ratio m/n.