Enumeration of extensions of the cycle matroid of a complete graph

TL;DR

This paper determines the asymptotic number of single-element extensions of the cycle matroid of a complete graph, using a characterization of extensions as linear subclasses, revealing exponential growth related to binomial coefficients.

Contribution

It provides an asymptotic enumeration of extensions of the cycle matroid of a complete graph, introducing a characterization via linear subclasses.

Findings

Number of extensions grows as 2^{binomial(n, n/2)(1+o(1))}

Extensions characterized as linear subclasses

Asymptotic enumeration of matroid extensions

Abstract

We prove that the number of single element extensions of $M(K_{n+1})$ is $2^{{n\choose n/2}(1+o(1))}$. This is done using a characterization of extensions as "linear subclasses".