Concentration of maximum degree 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 [n] := {1,... ,n} with m = m(n) edges. We show that in the sparse regime, when m/n ≤ 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.