{"title":"Weak External Bisections of Regular Graphs","authors":"Juan Yan, Ya-Hong Chen","doi":"10.1007/s00373-024-02796-3","DOIUrl":null,"url":null,"abstract":"<p>Let <i>G</i> be a graph. A <i>bisection</i> of <i>G</i> is a bipartition of <i>V</i>(<i>G</i>) with <span>\\(V(G)=V_1\\cup V_2\\)</span>, <span>\\(V_1\\cap V_2=\\emptyset \\)</span> and <span>\\(||V_1|-|V_2||\\le 1\\)</span>. Bollobás and Scott conjectured that every graph admits a bisection such that for every vertex, its external degree is greater than or equal to its internal degree minus one. In this paper, we confirm this conjecture for some regular graphs. Our results extend a result given by Ban and Linial (J Graph Theory 83:5–18, 2016). We also give an upper bound of the maximum bisection of graphs.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","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/s00373-024-02796-3","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
Let G be a graph. A bisection of G is a bipartition of V(G) with \(V(G)=V_1\cup V_2\), \(V_1\cap V_2=\emptyset \) and \(||V_1|-|V_2||\le 1\). Bollobás and Scott conjectured that every graph admits a bisection such that for every vertex, its external degree is greater than or equal to its internal degree minus one. In this paper, we confirm this conjecture for some regular graphs. Our results extend a result given by Ban and Linial (J Graph Theory 83:5–18, 2016). We also give an upper bound of the maximum bisection of graphs.
让 G 是一个图。G 的一分为二是 V(G) 的二分，其中有\(V(G)=V_1\cup V_2\)、\(V_1\cap V_2=\emptyset \)和\(||V_1|-|V_2|||le 1\).Bollobás 和 Scott 猜想，每个图都有一个分段，使得每个顶点的外部度都大于或等于其内部度减一。在本文中，我们对一些规则图证实了这一猜想。我们的结果扩展了 Ban 和 Linial（J Graph Theory 83:5-18, 2016）给出的结果。我们还给出了图的最大平分上限。
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