{"title":"The number of perfect matchings in a brick","authors":"Fuliang Lu, Huali Pan","doi":"10.1016/j.disc.2024.114365","DOIUrl":null,"url":null,"abstract":"<div><div>A 3-connected graph is a <em>brick</em> if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovász and Plummer.</div><div>Lucchesi and Murty conjectured that there exists a positive integer <em>N</em> such that for every <span><math><mi>n</mi><mo>≥</mo><mi>N</mi></math></span>, every brick on <em>n</em> vertices has at least <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> perfect matchings. We present an infinite family of bricks such that for each even integer <em>n</em> (<span><math><mi>n</mi><mo>></mo><mn>17</mn></math></span>), there exists a brick with <em>n</em> vertices in this family that contains at most <span><math><mo>⌈</mo><mn>0.625</mn><mi>n</mi><mo>⌉</mo></math></span> perfect matchings, showing that this conjecture fails.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114365"},"PeriodicalIF":0.7000,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004965","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/12/18 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovász and Plummer.
Lucchesi and Murty conjectured that there exists a positive integer N such that for every , every brick on n vertices has at least perfect matchings. We present an infinite family of bricks such that for each even integer n (), there exists a brick with n vertices in this family that contains at most perfect matchings, showing that this conjecture fails.
期刊介绍：
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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