Two point concentration of maximum degree in sparse random planar graphs

TL;DR

This paper investigates the maximum degree in sparse random planar graphs, showing that with high probability, the maximum degree takes at most two different values when the edge-to-vertex ratio is less than one.

Contribution

It establishes a new probabilistic property of maximum degree concentration in sparse random planar graphs, specifically that it assumes at most two values.

Findings

Maximum degree concentrates on at most two values in sparse regimes.

High probability results for maximum degree in random planar graphs.

Applicable when the ratio of edges to vertices is less than one.

Abstract

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $\limsup_{n \to \infty} m/n<1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values.