The maximum degree of random planar graphs

Abstract

McDiarmid and Reed [‘On the maximum degree of a random planar graph’, Combin. Probab. Comput. 17 (2008) 591–601] showed that the maximum degree Δ n of a random labeled planar graph with n vertices satisfies with high probability (w.h.p.) c 1 log n < Δ n < c 2 log n , for suitable constants . In this paper, we determine the precise asymptotics, showing in particular that w.h.p. | Δ n − c log n | = O ( log log n ) , for a constant c ≈ 2 . 52946 that we determine explicitly. The proof combines tools from analytic combinatorics and Boltzmann sampling techniques.