Loose elements in binary and ternary matroids

TL;DR

This paper investigates loose elements in binary and ternary matroids, characterizing their structure, size constraints, and partial classifications of representable matroids with multiple loose elements.

Contribution

It provides a characterization of binary matroids with loose elements, size bounds for ternary matroids with loose elements, and partial classifications of GF(q)-representable matroids with multiple loose elements.

Findings

Binary matroids with loose elements are characterized.

Ternary matroids with loose elements have size linear in rank.

Partial classification of GF(q)-representable matroids with multiple loose elements.

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.