{"title":"根据图形的叶片数估算图形的周长","authors":"Jingru Yan","doi":"10.1007/s13226-024-00682-5","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(\\mathcal {T}\\)</span> be the set of spanning trees of a graph <i>G</i> and let <i>L</i>(<i>T</i>) be the number of leaves in a tree <i>T</i>. The leaf number <i>L</i>(<i>G</i>) of <i>G</i> is defined as <span>\\(L(G)=\\max \\{L(T)|T\\in \\mathcal {T}\\}\\)</span>. Let <i>G</i> be a connected graph of order <i>n</i> and minimum degree <span>\\(\\delta \\)</span> such that <span>\\(L(G)\\le 2\\delta -1\\)</span>. We show that the circumference of <i>G</i> is at least <span>\\(n-1\\)</span>, and that if <i>G</i> is regular then <i>G</i> is hamiltonian.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Estimating the circumference of a graph in terms of its leaf number\",\"authors\":\"Jingru Yan\",\"doi\":\"10.1007/s13226-024-00682-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\(\\\\mathcal {T}\\\\)</span> be the set of spanning trees of a graph <i>G</i> and let <i>L</i>(<i>T</i>) be the number of leaves in a tree <i>T</i>. The leaf number <i>L</i>(<i>G</i>) of <i>G</i> is defined as <span>\\\\(L(G)=\\\\max \\\\{L(T)|T\\\\in \\\\mathcal {T}\\\\}\\\\)</span>. Let <i>G</i> be a connected graph of order <i>n</i> and minimum degree <span>\\\\(\\\\delta \\\\)</span> such that <span>\\\\(L(G)\\\\le 2\\\\delta -1\\\\)</span>. We show that the circumference of <i>G</i> is at least <span>\\\\(n-1\\\\)</span>, and that if <i>G</i> is regular then <i>G</i> is hamiltonian.</p>\",\"PeriodicalId\":501427,\"journal\":{\"name\":\"Indian Journal of Pure and Applied Mathematics\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indian Journal of Pure and Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s13226-024-00682-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indian Journal of Pure and Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s13226-024-00682-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
让 \(\mathcal {T}\)是图 G 的生成树集合,让 L(T) 是树 T 中叶子的数量。让 G 是一个阶数为 n 且最小度数为 \(\delta \)的连通图,使得 \(L(G)\le 2\delta -1\).我们证明 G 的周长至少是 \(n-1\),如果 G 是正则图,那么 G 就是哈密顿图。
Estimating the circumference of a graph in terms of its leaf number
Let \(\mathcal {T}\) be the set of spanning trees of a graph G and let L(T) be the number of leaves in a tree T. The leaf number L(G) of G is defined as \(L(G)=\max \{L(T)|T\in \mathcal {T}\}\). Let G be a connected graph of order n and minimum degree \(\delta \) such that \(L(G)\le 2\delta -1\). We show that the circumference of G is at least \(n-1\), and that if G is regular then G is hamiltonian.