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Cyclic ordering of split matroids

Kristóf Bérczi, Áron Jánosik, Bence Mátravölgyi

Abstract

There is a long list of open questions rooted in the same underlying problem: understanding the structure of bases or common bases of matroids. These conjectures suggest that matroids may possess much stronger structural properties than are currently known. One example is related to cyclic orderings of matroids. A rank-$r$ matroid is called cyclically orderable if its ground set admits a cyclic ordering such that any interval of $r$ consecutive elements forms a basis. In this paper, we show that if the ground set of a split matroid decomposes into pairwise disjoint bases, then it is cyclically orderable. This result answers a conjecture of Kajitani, Ueno, and Miyano in a special case, and also strengthens Gabow's conjecture for this class of matroids. Our proof is algorithmic, hence it provides a procedure for determining a cyclic ordering in question using a polynomial number of independence oracle calls.

Cyclic ordering of split matroids

Abstract

There is a long list of open questions rooted in the same underlying problem: understanding the structure of bases or common bases of matroids. These conjectures suggest that matroids may possess much stronger structural properties than are currently known. One example is related to cyclic orderings of matroids. A rank- matroid is called cyclically orderable if its ground set admits a cyclic ordering such that any interval of consecutive elements forms a basis. In this paper, we show that if the ground set of a split matroid decomposes into pairwise disjoint bases, then it is cyclically orderable. This result answers a conjecture of Kajitani, Ueno, and Miyano in a special case, and also strengthens Gabow's conjecture for this class of matroids. Our proof is algorithmic, hence it provides a procedure for determining a cyclic ordering in question using a polynomial number of independence oracle calls.

Paper Structure

This paper contains 9 sections, 4 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation.
  4. Split matroids.
  5. Proof of Theorem \ref{['thm:main']}
  6. Further remarks and open problems
  7. Comments on Conjecture \ref{['conj:cyclic']}
  8. Exchange distance of basis sequences
  9. Acknowledgement

Key Result

Theorem 1

Conjecture conj:main is true for split matroids.

Theorems & Definitions (27)

  • Conjecture 1: Gabow
  • Conjecture 2: Kajitani, Ueno, and Miyano
  • Conjecture 3
  • Theorem 1
  • Remark 2
  • Lemma 3: Bérczi, Király, Schwarcz, Yamaguchi and Yokoi
  • Lemma 4: Bérczi, Király, Schwarcz, Yamaguchi and Yokoi
  • proof : Proof of Theorem \ref{['thm:main']}
  • Claim 5
  • proof
  • ...and 17 more