Matchings in matroids over abelian groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909105244864512 |
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| author | Aliabadi, Mohsen Zerbib, Shira |
| author_facet | Aliabadi, Mohsen Zerbib, Shira |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_07719 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Matchings in matroids over abelian groups Aliabadi, Mohsen Zerbib, Shira Combinatorics 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. |
| title | Matchings in matroids over abelian groups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2202.07719 |