## 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 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.

Original language | English |
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Pages (from-to) | 146-162 |

Number of pages | 17 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 37 |

Issue number | 1 |

DOIs | |

Publication status | Published - 31 Mar 2023 |

## Keywords

- random graph process
- random graphs
- random planar graph process
- random planar graphs

## ASJC Scopus subject areas

- General Mathematics

## Fields of Expertise

- Information, Communication & Computing