{
  "id": "2202.07719",
  "version": "v1",
  "published": "2022-02-15T20:39:35.000Z",
  "updated": "2022-02-15T20:39:35.000Z",
  "title": "Matchings in matroids over abelian groups",
  "authors": [
    "Mohsen Aliabadi",
    "Shira Zerbib"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-15T20:39:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian group",
      "finite subsets",
      "matching property",
      "additive number theory",
      "group theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}