{"title":"Removable Edges in Claw-Free Bricks","authors":"","doi":"10.1007/s00373-024-02769-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>An edge <em>e</em> in a matching covered graph <em>G</em> is <em>removable</em> if <span> <span>\\(G-e\\)</span> </span> is matching covered. Removable edges were introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A <em>brick</em> is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Lovász proved that every brick other than <span> <span>\\(K_4\\)</span> </span> and <span> <span>\\(\\overline{C_6}\\)</span> </span> has a removable edge. It is known that every 3-connected claw-free graph with even number of vertices is a brick. By characterizing the structure of adjacent non-removable edges, we show that every claw-free brick <em>G</em> with more than 6 vertices has at least 5|<em>V</em>(<em>G</em>)|/8 removable edges.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"138 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Graphs and Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02769-6","RegionNum":4,"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
An edge e in a matching covered graph G is removable if \(G-e\) is matching covered. Removable edges were introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A brick is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Lovász proved that every brick other than \(K_4\) and \(\overline{C_6}\) has a removable edge. It is known that every 3-connected claw-free graph with even number of vertices is a brick. By characterizing the structure of adjacent non-removable edges, we show that every claw-free brick G with more than 6 vertices has at least 5|V(G)|/8 removable edges.
期刊介绍：
Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers, the journal also features survey articles from authors invited by the editorial board.
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