{"title":"关于最长路径与循环相交的两个猜想","authors":"Juan Gutiérrez , Christian Valqui","doi":"10.1016/j.disc.2024.114148","DOIUrl":null,"url":null,"abstract":"<div><p>A conjecture attributed to Smith states that every two longest cycles in a <em>k</em>-connected graph intersect in at least <em>k</em> vertices. In this paper, we show that every two longest cycles in a <em>k</em>-connected graph on <em>n</em> vertices intersect in at least <span><math><mi>min</mi><mo></mo><mo>{</mo><mi>n</mi><mo>,</mo><mn>8</mn><mi>k</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>16</mn><mo>}</mo></math></span> vertices, which confirms Smith's conjecture when <span><math><mi>k</mi><mo>≥</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>16</mn><mo>)</mo><mo>/</mo><mn>7</mn></math></span>. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either <span><math><mi>k</mi><mo>≤</mo><mn>7</mn></math></span> or <span><math><mi>k</mi><mo>≥</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>9</mn><mo>)</mo><mo>/</mo><mn>7</mn></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On two conjectures about the intersection of longest paths and cycles\",\"authors\":\"Juan Gutiérrez , Christian Valqui\",\"doi\":\"10.1016/j.disc.2024.114148\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A conjecture attributed to Smith states that every two longest cycles in a <em>k</em>-connected graph intersect in at least <em>k</em> vertices. In this paper, we show that every two longest cycles in a <em>k</em>-connected graph on <em>n</em> vertices intersect in at least <span><math><mi>min</mi><mo></mo><mo>{</mo><mi>n</mi><mo>,</mo><mn>8</mn><mi>k</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>16</mn><mo>}</mo></math></span> vertices, which confirms Smith's conjecture when <span><math><mi>k</mi><mo>≥</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>16</mn><mo>)</mo><mo>/</mo><mn>7</mn></math></span>. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either <span><math><mi>k</mi><mo>≤</mo><mn>7</mn></math></span> or <span><math><mi>k</mi><mo>≥</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>9</mn><mo>)</mo><mo>/</mo><mn>7</mn></math></span>.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-03\",\"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/S0012365X24002796\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24002796","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
Smith 提出的一个猜想是:在 k 个连通图中,每两个最长循环至少在 k 个顶点上相交。在本文中,我们证明了当 k≥(n+16)/7 时,n 个顶点上 k 个连接图中的每两个最长循环至少相交于 min{n,8k-n-16} 个顶点,这证实了 Smith 的猜想。希普钦(Hippchen)针对路径而非循环提出了类似猜想。通过简单的还原,我们将这两个猜想联系起来,证明当 k≤7 或 k≥(n+9)/7 时,希普钦的猜想是有效的。
On two conjectures about the intersection of longest paths and cycles
A conjecture attributed to Smith states that every two longest cycles in a k-connected graph intersect in at least k vertices. In this paper, we show that every two longest cycles in a k-connected graph on n vertices intersect in at least vertices, which confirms Smith's conjecture when . An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either or .
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.