Concentration of maximum degree in random planar graphs

TL;DR

This paper investigates the maximum degree concentration in random planar graphs, revealing a phase transition: sparse graphs have a highly concentrated maximum degree, while dense graphs do not.

Contribution

It establishes the contrasting behaviors of maximum degree concentration in sparse versus dense random planar graphs.

Findings

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

In dense graphs, maximum degree is not concentrated on any small subset.

Phase transition occurs at the ratio m/n=1.

Abstract

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $[n]:=\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $m/n\leq 1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values. In contrast, this is not true anymore in the dense regime, when $m/n>1$, where the maximum degree of $P(n,m)$ is not concentrated on any subset of $[n]$ with bounded size.