The early evolution of the random graph process in planar graphs and related classes

We study the random planar graph process introduced by Gerke, Schlatter, Steger, and Taraz [The random planar graph process, Random Structures Algorithms 32 (2008), no. 2, 236--261; MR2387559]: Begin with an empty graph on $n$ vertices, consider the edges of the complete graph $K_n$ one by one in a random ordering, and at each step add an edge to a current graph only if the graph remains planar. They studied the number of edges added up to step $t$ for 'large' $t=\omega (n)$. In this paper we extend their results by determining the asymptotic number of edges added up to step $t$ in the early evolution of the process when $t=O(n)$. We also show that this result holds for a much more general class of graphs, including outerplanar graphs, planar graphs, and graphs on surfaces.