On the number of graphs not containing $K_{3,3}$ as a minor

TL;DR

This paper provides precise asymptotic counts and probabilistic properties for graphs excluding the $K_{3,3}$ minor, advancing understanding of their structure and enumeration.

Contribution

It introduces exact asymptotic formulas and limit laws for $K_{3,3}$-minor-free graphs, including edge-maximal cases and graphs excluding $K_{3,3}$ plus an edge.

Findings

Asymptotic estimates for the number of $K_{3,3}$-minor-free graphs

Limit laws for parameters like the expected number of edges

Precise enumeration of graphs excluding $K_{3,3}$ plus an edge

Abstract

We derive precise asymptotic estimates for the number of labelled graphs not containing $K_{3,3}$ as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random $K_{3,3}$-minor-free graphs, like the expected number of edges. To establish these results, we translate a decomposition for the corresponding graph class into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph $K_{3,3}$ plus an edge as a minor.