# Drmota, Michael; Matematični kolokvij december 2012

## The maximum degree of planar graphs

Michael Drmota

Technische Universität Wien

13. december 2012

McDiarmid and Reed showed in 2008 that the maximum degree Δn of a random labeled planar graph with n vertices satisfies with high probability c1logn < Δn < c2logn for suitable constants 0 < c1 < c2. The purpose of this talk is to make this statement more accurate by showing that the precise limiting behavior of Δn is (with high probability) > | Δnclogn | = O(loglogn) for a constant $c \approx 2.52946$ that can be determined explicitly. The proof combines tools from analytic combinatorics and Boltzmann sampling techniques.

This is joint work with Omer Gimenez, Marc Noy, Konstantinos Panagiotou, and Angelika Steger.