The $k$-fold circuit property for matroids

TL;DR

This paper generalizes the concept of double circuits to k-fold circuits in matroids, showing that many matroid families satisfy this property for all k, which may measure how close their lattice of flats is to being modular.

Contribution

It introduces the k-fold circuit property for all natural numbers k and proves that several known matroid families satisfy this property for all k.

Findings

Many matroid families satisfy the k-fold circuit property for all k

The k-fold circuit property relates to the modularity of the lattice of flats

Extends the classical double circuit property to a broader context

Abstract

Double circuits were introduced by Lov\'{a}sz in 1980 as a fundamental tool in his derivation of a min-max formula for the size of a maximum matching in linear matroids. This formula was extended to all matroids satisfying the so-called `double circuit property' by Dress and Lov\'{a}sz in 1987. We extend these notions to $k$-fold circuits for all natural numbers $k$ and show, in particular that several families of matroids which are known to satisfy the double circuit property, satisfy the $k$-fold circuit property for all natural numbers $k$. These families include all pseudomodular matroids (such as full linear, algebraic and transversal matroids) and certain families of count matroids. These results suggest that the $k$-fold circuit property can be used as a measure of how close the lattice of flats of a matroid is to being a modular lattice.