On the diameter of random planar graphs

TL;DR

This paper establishes that the diameter of a random labeled connected planar graph with n vertices is typically on the order of n^{1/4}, with high probability bounds, extending results to 2-connected and 3-connected planar graphs and maps.

Contribution

It provides probabilistic bounds on the diameter of random planar graphs, including connected, 2-connected, and 3-connected cases, with precise asymptotic behavior.

Findings

Diameter of random planar graphs is approximately n^{1/4} with high probability.

Similar diameter bounds hold for 2-connected and 3-connected planar graphs and maps.

Probability bounds show the diameter concentrates around n^{1/4}.

Abstract

We show that the diameter D(G_n) of a random labelled connected planar graph with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there exists a constant c>0 such that the probability that D(G_n) lies in the interval (n^{1/4-\epsilon},n^{1/4+\epsilon}) is greater than 1-\exp(-n^{c\epsilon}) for {\epsilon} small enough and n>n_0(\epsilon). We prove similar statements for 2-connected and 3-connected planar graphs and maps.