{"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":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-024-01144-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","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.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.