On the diameter of random planar graphs
arXiv:1203.3079 · doi:10.1017/S0963548314000467
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-ε},n^{1/4+ε}) is greater than 1-\exp(-n^{cε}) for ε small enough and n>n_0(ε). We prove similar statements for 2-connected and 3-connected planar graphs and maps.
24 pages, 7 figures