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Loose elements in binary and ternary matroids

Jagdeep Singh, Thomas Zaslavsky

TL;DR

This paper investigates loose elements in binary and ternary matroids, introducing paving matroids and building on prior work by Acketa and Oxley. It completely characterizes simple binary matroids with a loose element: such a matroid must be isomorphic to $L_r$ or $J_r$, or be a restriction of $M_r$ or $N_r$ containing the loose element, with proofs leveraging the standard binary representation. For ternary matroids, the authors prove a linear-size bound in rank: if $r \ge 5$ and $M$ is a simple ternary matroid with a loose element and no coloops, then $|E(M)| \le \left\lfloor \frac{41r-101}{2} \right\rfloor$ for $r>10$ (and $|E(M)| \le \left\lfloor \frac{35r-35}{2} \right\rfloor$ for $r\le 10$), via a detailed analysis of the representation matrix $P=[I_r|Q]$ and the column types of $Q-e^P$. The paper further provides a partial characterization of GF($q$)-matroids with two loose elements, showing $r(M) \le 2q$ unless $\\{e,f\} $ is a cocircuit, and connects these findings to Rajpal’s paving matroid results. Collectively, these results yield structural and quantitative constraints on loose elements, with implications for paving matroids and small-field representability.

Abstract

We call a matroid element "loose" if it is contained in no circuits of size less than the rank of the matroid. A matroid in which all elements are loose is a paving matroid. Acketa determined all binary paving matroids, while Oxley specified all ternary paving matroids. We characterize the binary matroids that contain a loose element. For ternary matroids with a loose element, we show that their size is linear in terms of their rank. Moreover, for a prime power $q$, we give a partial characterization of $GF(q)$-representable matroids that have two or more loose elements; we note Rajpal's partial characterization of $GF(q)$-representable paving matroids as a consequence.

Loose elements in binary and ternary matroids

TL;DR

This paper investigates loose elements in binary and ternary matroids, introducing paving matroids and building on prior work by Acketa and Oxley. It completely characterizes simple binary matroids with a loose element: such a matroid must be isomorphic to or , or be a restriction of or containing the loose element, with proofs leveraging the standard binary representation. For ternary matroids, the authors prove a linear-size bound in rank: if and is a simple ternary matroid with a loose element and no coloops, then for (and for ), via a detailed analysis of the representation matrix and the column types of . The paper further provides a partial characterization of GF()-matroids with two loose elements, showing unless is a cocircuit, and connects these findings to Rajpal’s paving matroid results. Collectively, these results yield structural and quantitative constraints on loose elements, with implications for paving matroids and small-field representability.

Abstract

We call a matroid element "loose" if it is contained in no circuits of size less than the rank of the matroid. A matroid in which all elements are loose is a paving matroid. Acketa determined all binary paving matroids, while Oxley specified all ternary paving matroids. We characterize the binary matroids that contain a loose element. For ternary matroids with a loose element, we show that their size is linear in terms of their rank. Moreover, for a prime power , we give a partial characterization of -representable matroids that have two or more loose elements; we note Rajpal's partial characterization of -representable paving matroids as a consequence.
Paper Structure (4 sections, 10 theorems, 4 equations, 4 figures)

This paper contains 4 sections, 10 theorems, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. loose elements in binary matroids
  3. loose elements in ternary matroids
  4. Two (or more) loose elements in $GF(q)$-matroids

Key Result

Theorem 1.1

Let $M$ be a simple binary matroid of rank $r$ that has no coloops. Then $M$ has a loose element $e$ if and only if one of the following holds.

Figures (4)

  • Figure 1: Representation of $L_r$ over $GF(2)$.
  • Figure 2: Representation of $J_r$ over $GF(2)$.
  • Figure 3: Representation of $M_r$ over $GF(2)$.
  • Figure 4: Representation of $N_r$ over $GF(2)$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Rajpal rajpal
  • Theorem 1.4
  • Proposition 2.1: oxjag
  • proof : Proof of Theorem \ref{['almost_loose_binary_intro']}
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 8 more