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

TL;DR

This paper analyzes the early stages of a random process that builds planar and related graphs by adding edges one by one, providing asymptotic counts for the number of edges added when the process is in its initial phase.

Contribution

It extends previous work by determining the asymptotic number of edges in the early evolution of the process for planar and other surface-embedded graphs, covering a broader class of graphs.

Findings

Asymptotic number of edges in early process stage determined

Results apply to outerplanar, planar, and surface graphs

Extends prior work from large to early process stages

Abstract

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.