On the diameter of random planar graphs

Abstract

We show that the diameter D(Gn) of a random labelled connected planar graph with n vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant c > 0 such that $P(D(G_n) \in{} (n^{1/4-\in{}} , n^{1/4+\in{}}))$ ≥ 1 − exp(−n^{ce})for $\in{}$ small enough and n large enough (n ≥ $n_0$($\in{}$)). We prove similar statements for rooted 2-connected and 3-connected maps and planar graphs.