English

Matchings in matroids over abelian groups

Combinatorics 2024-02-15 v3

Abstract

We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group (G,+)(G,+) is a bijection f:ABf:A\to B between two finite subsets A,BA,B of GG satisfying a+f(a)Aa+f(a)\notin A for all aAa\in A. A group GG has the matching property if for every two finite subsets A,BGA,B \subset G of the same size with 0B0 \notin B, there exists a matching from AA to BB. 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 GG, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.

Keywords

Cite

@article{arxiv.2202.07719,
  title  = {Matchings in matroids over abelian groups},
  author = {Mohsen Aliabadi and Shira Zerbib},
  journal= {arXiv preprint arXiv:2202.07719},
  year   = {2024}
}

Comments

To appear in Journal of Algebraic Combinatorics

R2 v1 2026-06-24T09:39:45.276Z