Random planar graphs with bounds on the maximum and minimum degrees

TL;DR

This paper studies the properties of random planar graphs with specified degree bounds, analyzing the likelihood of containing certain subgraphs and components as the number of vertices grows large.

Contribution

It provides a detailed probabilistic analysis of degree-bounded random planar graphs, establishing conditions for the presence of specific subgraphs.

Findings

Identifies thresholds for the presence of particular components.

Determines when certain subgraphs appear with high probability.

Provides bounds on the probability of subgraph containment.

Abstract

Let P_{n,d,D} denote the graph taken uniformly at random from the set of all labelled planar graphs on {1,2,...,n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability that P_{n,d,D} wll contain given components and subgraphs, showing exactly when this is bounded away from 0 and 1 as n tends to infinity.