{"title":"关于哈密顿相交图族的说明","authors":"","doi":"10.1016/j.disc.2024.114160","DOIUrl":null,"url":null,"abstract":"<div><p>How many graphs on an <em>n</em>-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly <span><math><mn>1</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> of all graphs. Our aim in this short note is to give a ‘directed’ version of this result; we show that a family of oriented graphs such that any two have strongly-connected intersection has size at most <span><math><mn>1</mn><mo>/</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> of all oriented graphs. We also show that a family of graphs such that any two have Hamiltonian intersection has size at most <span><math><mn>1</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> of all graphs, verifying a conjecture of the above authors.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Note on Hamiltonian-intersecting families of graphs\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114160\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>How many graphs on an <em>n</em>-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly <span><math><mn>1</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> of all graphs. Our aim in this short note is to give a ‘directed’ version of this result; we show that a family of oriented graphs such that any two have strongly-connected intersection has size at most <span><math><mn>1</mn><mo>/</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> of all oriented graphs. We also show that a family of graphs such that any two have Hamiltonian intersection has size at most <span><math><mn>1</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> of all graphs, verifying a conjecture of the above authors.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-15\",\"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/S0012365X24002917\",\"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/S0012365X24002917","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A Note on Hamiltonian-intersecting families of graphs
How many graphs on an n-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly of all graphs. Our aim in this short note is to give a ‘directed’ version of this result; we show that a family of oriented graphs such that any two have strongly-connected intersection has size at most of all oriented graphs. We also show that a family of graphs such that any two have Hamiltonian intersection has size at most of all graphs, verifying a conjecture of the above authors.
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.