Ordering Circuits of Matroids

TL;DR

This paper investigates which matroids allow a consistent cyclic ordering of their circuits, providing a complete characterization for non-binary matroids and linking orderability to graph representability in binary matroids.

Contribution

The paper characterizes orderable matroids for non-binary cases and establishes that binary matroids are orderable if and only if they are graphic, highlighting the complexity of the problem.

Findings

Complete characterization for non-binary matroids

Binary matroids are orderable iff they are graphic

Examples illustrating the difficulty of the general problem

Abstract

The cycles of a graph give a natural cyclic ordering to their edge-sets, and these orderings are consistent in that two edges are adjacent in one cycle if and only if they are adjacent in every cycle in which they appear together. An orderable matroid is one whose set of circuits admits such a consistent ordering. In this paper, we consider the question of determining which matroids are orderable. Although we are able to answer this question for non-binary matroids, it remains open for binary matroids. We give examples to provide insight into the potential difficulty of this question in general. We also show that, by requiring that the ordering preserves the three arcs in every theta-graph restriction of a binary matroid $M$, we guarantee that $M$ is orderable if and only if $M$ is graphic.