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The $k$-fold circuit property for matroids

Bill Jackson, Anthony Nixon, Ben Smith

TL;DR

The paper generalizes Lovász's double circuit concept to $k$-fold circuits in matroids, connecting circuit structure to the lattice of flats via a principal partition. It defines the $k$-fold circuit property and a balanced notion, establishing a key rank-closure inequality that extends the Dress–Lovász framework. The authors prove that several natural matroid classes satisfy the $k$-fold circuit property for all $k$, including full linear matroids, pseudomodular matroids, and full count matroids under suitable parameters, and they relate this property to modular sublattices. They also provide constructions showing limitations for certain $k$ and discuss implications for modularity and ear decompositions, suggesting the property as a refined measure of how close a matroid’s lattice of flats is to modular.

Abstract

Double circuits were introduced by Lovász 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 Lovász 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.

The $k$-fold circuit property for matroids

TL;DR

The paper generalizes Lovász's double circuit concept to -fold circuits in matroids, connecting circuit structure to the lattice of flats via a principal partition. It defines the -fold circuit property and a balanced notion, establishing a key rank-closure inequality that extends the Dress–Lovász framework. The authors prove that several natural matroid classes satisfy the -fold circuit property for all , including full linear matroids, pseudomodular matroids, and full count matroids under suitable parameters, and they relate this property to modular sublattices. They also provide constructions showing limitations for certain and discuss implications for modularity and ear decompositions, suggesting the property as a refined measure of how close a matroid’s lattice of flats is to modular.

Abstract

Double circuits were introduced by Lovász 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 Lovász in 1987. We extend these notions to -fold circuits for all natural numbers and show, in particular that several families of matroids which are known to satisfy the double circuit property, satisfy the -fold circuit property for all natural numbers . 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 -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.

Paper Structure

This paper contains 13 sections, 31 theorems, 56 equations, 4 figures.

Table of Contents

  1. Keywords:
  2. MSC classes:
  3. Introduction
  4. Main results
  5. Preliminaries
  6. Matroids and lattices
  7. Count matroids
  8. Double circuits
  9. $k$-fold circuits
  10. The $k$-fold circuit property
  11. Pseudomodular matroids
  12. The $k$-fold circuit property in count matroids
  13. Concluding remarks

Key Result

Lemma 2.1

Let ${\mathcal{M}} = (E, r)$ be a matroid, $X \in {\mathcal{L}}({\mathcal{M}})$ and $\{Y_1, \dots, Y_\ell\}$ be the set of flats that cover $X$. Then $\{Y_1 \setminus X, \dots, Y_\ell \setminus X\}$ is a partition of $E \setminus X$.

Figures (4)

  • Figure 1: Four examples of double circuits in ${\mathcal{M}}(2,3)$.
  • Figure 2: The lattice ${\mathcal{D}}({\mathcal{M}})$ of cyclic sets of ${\mathcal{M}}$ and the lattice ${\mathcal{L}}({\mathcal{M}}^*)$ of flats of ${\mathcal{M}}^*$. The two lattices are anti-isomorphic via the map $D \mapsto E\setminus D$. Moreover, the covering axiom for flats in ${\mathcal{L}}({\mathcal{M}}^*)$ implies the notion of the principal partition for $k$-fold circuits.
  • Figure 3: The left figure gives the 'diamond' condition for modularity. The right figure gives the 'diamond prism' condition for pseudomodularity.
  • Figure 4: The posets analysed in the proof of Theorem \ref{['thm:modular+substructure']}.

Theorems & Definitions (74)

  • Lemma 2.1
  • Lemma 2.2: CH
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • ...and 64 more