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A construction of infinite sets of intertwines for pairs of matroids

Joseph E. Bonin

Abstract

An intertwine of a pair of matroids is a matroid such that it, but none of its proper minors, has minors that are isomorphic to each matroid in the pair. For pairs for which neither matroid can be obtained, up to isomorphism, from the other by taking free extensions, free coextensions, and minors, we construct a family of rank-k intertwines for each sufficiently large integer k. We also treat some properties of these intertwines.

A construction of infinite sets of intertwines for pairs of matroids

Abstract

An intertwine of a pair of matroids is a matroid such that it, but none of its proper minors, has minors that are isomorphic to each matroid in the pair. For pairs for which neither matroid can be obtained, up to isomorphism, from the other by taking free extensions, free coextensions, and minors, we construct a family of rank-k intertwines for each sufficiently large integer k. We also treat some properties of these intertwines.

Paper Structure

This paper contains 10 sections, 18 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Intertwines
  4. Further Results
  5. Sizes of intertwines
  6. Representable matroids
  7. Transversal matroids and gammoids
  8. Uniform minors
  9. Connectivity
  10. The Relation to Vertigan's Construction

Key Result

Proposition 2.1

Let $\mathcal{Z}$ be a collection of subsets of a set $S$ and let $r$ be an integer-valued function on $\mathcal{Z}$. There is a matroid for which $\mathcal{Z}$ is the collection of cyclic flats and $r$ is the rank function restricted to the sets in $\mathcal{Z}$ if and only if

Theorems & Definitions (24)

  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 14 more