Maximal Planar Subgraphs of Fixed Girth in Random Graphs

TL;DR

This paper investigates the threshold probabilities for random graphs to contain various types of maximal planar subgraphs, including bipartite and fixed girth cases, extending previous results from 1991.

Contribution

It provides new threshold ranges for the appearance of maximal bipartite and fixed girth planar subgraphs in random graphs, broadening understanding of their probabilistic properties.

Findings

Threshold for bipartite maximal planar subgraphs identified

Threshold for fixed girth maximal planar subgraphs computed

Extends classical results from 1991 to new graph classes

Abstract

In 1991, Bollob\'{a}s and Frieze showed that the threshold for $G_{n,p}$ to contain a spanning maximal planar subgraph is very close to $p = n^{-1/3}$. In this paper, we compute similar threshold ranges for $G_{n,p}$ to contain a maximal bipartite planar subgraph and for $G_{n,p}$ to contain a maximal planar subgraph of fixed girth $g$.