Matchings in matroids over abelian groups

TL;DR

This paper extends the concept of matchings from abelian groups to matroids over abelian groups, providing criteria for their existence using classical theorems in matroid theory, group theory, and additive number theory.

Contribution

It introduces a new notion of matchings in matroids over abelian groups and establishes criteria for their existence, generalizing previous group-based results.

Findings

Characterization of matroid matchings over abelian groups

Criteria for existence of matchings in this setting

Application of classical theorems to matroid matchings

Abstract

We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group $(G,+)$ is a bijection $f:A\to B$ between two finite subsets $A,B$ of $G$ satisfying $a+f(a)\notin A$ for all $a\in A$. A group $G$ has the matching property if for every two finite subsets $A,B \subset G$ of the same size with $0 \notin B$, there exists a matching from $A$ to $B$. In [19] it was proved that an abelian group has the matching property if and only if it is torsion-free or cyclic of prime order. Here we consider a similar question in a matroid setting. We introduce an analogous notion of matching between matroids whose ground sets are subsets of an abelian group $G$, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.