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.