On the Cogirth of Binary Matroids

TL;DR

This paper investigates the properties of the cogirth in binary matroids, establishing bounds and characterizations for cases of equality, and extending results to matroids over finite fields.

Contribution

It provides new bounds on the cogirth of binary matroids and characterizes those that attain equality, extending the results to finite fields.

Findings

g* (M) ≤ ½ |E(M)| for binary matroids unless M simplifies to a projective geometry

When equality holds, M simplifies to a Bose-Burton geometry

Results extend to matroids over arbitrary finite fields

Abstract

The cogirth, $g^\ast(M)$, of a matroid $M$ is the size of a smallest cocircuit of $M$. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that $g^\ast(M)\leq \frac{1}{2}\vert E(M)\vert$ when $M$ is binary, unless $M$ simplifies to a projective geometry. We also show that, when equality holds, $M$ simplifies to a Bose-Burton geometry, that is, a matroid of the form $PG(r-1,2)-PG(k-1,2)$. These results extend to matroids representable over arbitrary finite fields.