Further results on random cubic planar graphs

TL;DR

This paper refines the enumeration and probabilistic analysis of labelled cubic planar graphs, providing new asymptotic estimates, connectivity probabilities, and distribution results for subgraph counts, advancing understanding of their structural properties.

Contribution

It offers new asymptotic enumeration formulas and distribution results for cubic planar graphs, including the first asymptotic distribution for cycle-containing subgraphs.

Findings

Exact probability of connectivity in random cubic planar graphs

Asymptotic normality of the number of triangles

Linear expectation and variance for cycle subgraph counts

Abstract

We provide precise asymptotic estimates for the number of several classes of labelled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky et al. (Random Structures Algorithms 2007). We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.