Random planar maps and graphs with minimum degree two and three

TL;DR

This paper provides detailed asymptotic estimates for the number and structure of random planar maps and graphs with minimum degree constraints, revealing the size of the largest attached tree scales logarithmically with the graph size.

Contribution

It introduces precise asymptotic formulas for counting such graphs and analyzes structural properties of random instances, especially the size of the largest attached tree.

Findings

Largest attached tree size is of order c log(n)

Provides explicit asymptotic counts for planar maps and graphs

Enhances understanding of the structure of random planar graphs

Abstract

We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order c log(n) for an explicit constant c. These results provide new information on the structure of random planar graphs.