Longest and shortest cycles in random planar graphs

Abstract

Let P(n, m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set {1, … , n} with m = m(n) edges. We study the cycle and block structure of P(n, m) when m ∼ n∕2. More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in P(n, m) in the critical range when m = n∕2 + o(n).In addition, we describe the block structure of P(n, m) in the weakly supercritical regime when n 2∕3 ≪ m − n∕2 ≪ n.