## On the Connectivity of Random Graphs from Addable Classes

##### Abstract

A class \mathcal G of graphs is called weakly addable if for any G\in \mathcal G and any two distinct components C_1 and C_2 in G, any graph that can be obtained by adding an edge between C_1 and C_2 is also in \mathcal G. McDiarmid, Steger and Welsh conjectured [Random graphs from planar and other addable classes. In Topics in discrete mathematics, volume 26 of Algorithms Combin., pages 231-246. Springer, 2006] that a random graph on n vertices from \mathcal G is connected with probability at least e^{-1/2}+ o(1), as n\to \infty. We prove this conjecture under the additional assumption that \mathcal G is monotone, i.e., if for any G\in \mathcal G and any edge e\in G, the graph G without e is also in \mathcal G. This is joint work with Konstantinos Panagiotou.