Matchings in matroids over abelian groups

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.

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Remark 1.4
• Proposition 1.7
• Theorem 2.1
• Remark 2.3
• Theorem 2.4
• Corollary 2.5
• Corollary 2.6