Circuit-partition of infinite matroids

TL;DR

This paper simplifies the proof of a theorem on partitioning infinite finitary matroids into circuits and extends the concept to directed circuits in binary oriented matroids, connecting to infinite graph cycle partitions.

Contribution

Provides a shorter proof of a key theorem on circuit partitions in finitary matroids and characterizes directed circuit partitions in binary oriented matroids, extending previous conjectures.

Findings

A simplified proof of the compactness theorem for finitary matroids.

Characterization of when a binary oriented matroid can be partitioned into directed circuits.

A counterexample showing the necessity of 'binary' in the directed circuit partition result.

Abstract

Komj\'ath, Milner, and Polat investigated when a finitary matroid admits a partition into circuits. They defined the class of ``finite matching extendable'' matroids and showed in their compactness theorem that those matroids always admit such a partition. Their proof is based on Shelah's singular compactness technique and a careful analysis of certain $\triangle$-systems. We provide a short, simple proof of their theorem. Then we show that a finitary binary oriented matroid can be partitioned into directed circuits if and only if, in every cocircuit, the cardinality of the negative and positive edges is the same. This generalizes an earlier conjecture of Thomassen, settled affirmatively by the second author, about partitioning the edges of an infinite directed graph into directed cycles. As side results, a Laviolette theorem for finitary matroids and a Farkas lemma for finitary binary oriented matroids are proven. An example is given to show that, in contrast to finite oriented matroids, `binary' is essential in the latter result.