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

Abstract

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 vertices, consider the edges of the complete graph 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 for “large" . In this paper we extend their results by determining the asymptotic number of edges added up to step in the early evolution of the process when . We also show that this result holds for a much more general class of graphs, including outerplanar graphs, planar graphs, and graphs on surfaces.