The Early Evolution of the Random Graph Process in Planar Graphs and Related Classes

We study the random planar graph process introduced by Gerke et al. [Random Structures Algorithms, 32 (2008), pp. 236-261]: Begin with an empty graph on n vertices, consider the edges of the complete graph Kn 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 = ω (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.