{"title":"Matchings in matroids over abelian groups","authors":"Mohsen Aliabadi, Shira Zerbib","doi":"10.1007/s10801-024-01308-z","DOIUrl":null,"url":null,"abstract":"<p>We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group <span>\\((G,+)\\)</span> is a bijection <span>\\(f:A\\rightarrow B\\)</span> between two finite subsets <i>A</i>, <i>B</i> of <i>G</i> satisfying <span>\\(a+f(a)\\notin A\\)</span> for all <span>\\(a\\in A\\)</span>. A group <i>G</i> has the matching property if for every two finite subsets <span>\\(A,B \\subset G\\)</span> of the same size with <span>\\(0 \\notin B\\)</span>, there exists a matching from <i>A</i> to <i>B</i>. In Losonczy (Adv Appl Math 20(3):385–391, 1998) 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 <i>G</i>, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10801-024-01308-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group \((G,+)\) is a bijection \(f:A\rightarrow B\) 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 Losonczy (Adv Appl Math 20(3):385–391, 1998) 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.
期刊介绍:
The Journal of Algebraic Combinatorics provides a single forum for papers on algebraic combinatorics which, at present, are distributed throughout a number of journals. Within the last decade or so, algebraic combinatorics has evolved into a mature, established and identifiable area of mathematics. Research contributions in the field are increasingly seen to have substantial links with other areas of mathematics.
The journal publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems.