Infinite matroids in graphs

TL;DR

This paper explores the theory of infinite matroids, demonstrating how they can be axiomatized similarly to finite matroids, and investigates their applications to infinite graphs, including duality and representability.

Contribution

It introduces a new axiomatization for infinite matroids, applies it to graph-related matroids, and discusses their duals and conditions for representability.

Findings

Infinite matroids can be axiomatized similarly to finite matroids.

Cycle and bond matroids of infinite graphs are analyzed.

Conditions for matroid representability in the infinite case are discussed.

Abstract

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals. In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to infinite matroids. Which graphic matroids are representable in this sense remains an open question.