Two point concentration of maximum degree in sparse 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 $\left\{1,\dots ,n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when ${\mathrm{limsup}}_{n\to \infty }m/n<1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values.