{"title":"The average size of maximal matchings in graphs","authors":"Alain Hertz, Sébastien Bonte, Gauvain Devillez, Hadrien Mélot","doi":"10.1007/s10878-024-01144-8","DOIUrl":null,"url":null,"abstract":"<p>We investigate the ratio <span>\\(\\mathcal {I}(G)\\)</span> of the average size of a maximal matching to the size of a maximum matching in a graph <i>G</i>. If many maximal matchings have a size close to <span>\\(\\nu (G)\\)</span>, this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, <span>\\(\\mathcal {I}(G)\\)</span> approaches <span>\\(\\frac{1}{2}\\)</span>. We propose a general technique to determine the asymptotic behavior of <span>\\(\\mathcal {I}(G)\\)</span> for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of <span>\\(\\mathcal {I}(G)\\)</span> which were typically obtained using generating functions, and we then determine the asymptotic value of <span>\\(\\mathcal {I}(G)\\)</span> for other families of graphs, highlighting the spectrum of possible values of this graph invariant between <span>\\(\\frac{1}{2}\\)</span> and 1.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-024-01144-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the ratio \(\mathcal {I}(G)\) of the average size of a maximal matching to the size of a maximum matching in a graph G. If many maximal matchings have a size close to \(\nu (G)\), this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, \(\mathcal {I}(G)\) approaches \(\frac{1}{2}\). We propose a general technique to determine the asymptotic behavior of \(\mathcal {I}(G)\) for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of \(\mathcal {I}(G)\) which were typically obtained using generating functions, and we then determine the asymptotic value of \(\mathcal {I}(G)\) for other families of graphs, highlighting the spectrum of possible values of this graph invariant between \(\frac{1}{2}\) and 1.