The random planar graph process

We consider the following variant of the classical random graph process introduced by Erdos and Renyi. Starting with an empty graph on n vertices, choose the next edge uniformly at random among all edges not yet considered, but only insert it if the graph remains planar. We show that for all epsilon > 0, with high probability, theta(n(2)) edges have to be tested before the number of edges in the graph reaches (1 + epsilon)n. At this point, the graph is connected with high probability and contains a linear number of induced copies of any fixed connected planar graph, the first property being in contrast and the second one in accordance with the uniform random planar graph model. (c) 2007 Wiley Periodicals, Inc.