Circuit decompositions of binary matroids

TL;DR

This paper investigates the minimum number of disjoint circuits needed to decompose simple Eulerian binary matroids, providing bounds for complete and general cases, and analyzing asymptotic behavior.

Contribution

It establishes new bounds on the circuit decomposition of binary matroids, including specific results for complete matroids and general cases, and studies asymptotic properties.

Findings

For complete binary matroids, at most |M| / (rank(M) + 1) circuits suffice.

For general binary matroids, O(2^{rank(M)} / (rank(M) + 1)) circuits are enough.

The asymptotic behavior of the minimum number of circuits in odd-covers is characterized.

Abstract

Given a simple Eulerian binary matroid $M$, what is the minimum number of disjoint circuits necessary to decompose $M$? We prove that $|M| / (\operatorname{rank}(M) + 1)$ many circuits suffice if $M = \mathbb F_2^n \setminus \{0\}$ is the complete binary matroid, for certain values of $n$, and that $\mathcal{O}(2^{\operatorname{rank}(M)} / (\operatorname{rank}(M) + 1))$ many circuits suffice for general $M$. We also determine the asymptotic behaviour of the minimum number of circuits in an odd-cover of $M$.