{"title":"在 3-Uniform 超图中嵌入松散生成树","authors":"Yanitsa Pehova , Kalina Petrova","doi":"10.1016/j.jctb.2024.04.003","DOIUrl":null,"url":null,"abstract":"<div><p>In 1995, Komlós, Sárközy and Szemerédi showed that every large <em>n</em>-vertex graph with minimum degree at least <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>γ</mi><mo>)</mo><mi>n</mi></math></span> contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all <em>γ</em> and Δ, and <em>n</em> large, every <em>n</em>-vertex 3-uniform hypergraph of minimum vertex degree <span><math><mo>(</mo><mn>5</mn><mo>/</mo><mn>9</mn><mo>+</mo><mi>γ</mi><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span> contains every loose spanning tree <em>T</em> with maximum vertex degree Δ. This bound is asymptotically tight, since some loose trees contain perfect matchings.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 47-67"},"PeriodicalIF":1.2000,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000303/pdfft?md5=4e333586884c0a88ecc3b2284d18ce92&pid=1-s2.0-S0095895624000303-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Embedding loose spanning trees in 3-uniform hypergraphs\",\"authors\":\"Yanitsa Pehova , Kalina Petrova\",\"doi\":\"10.1016/j.jctb.2024.04.003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In 1995, Komlós, Sárközy and Szemerédi showed that every large <em>n</em>-vertex graph with minimum degree at least <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>γ</mi><mo>)</mo><mi>n</mi></math></span> contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all <em>γ</em> and Δ, and <em>n</em> large, every <em>n</em>-vertex 3-uniform hypergraph of minimum vertex degree <span><math><mo>(</mo><mn>5</mn><mo>/</mo><mn>9</mn><mo>+</mo><mi>γ</mi><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span> contains every loose spanning tree <em>T</em> with maximum vertex degree Δ. This bound is asymptotically tight, since some loose trees contain perfect matchings.</p></div>\",\"PeriodicalId\":54865,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series B\",\"volume\":\"168 \",\"pages\":\"Pages 47-67\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-05-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0095895624000303/pdfft?md5=4e333586884c0a88ecc3b2284d18ce92&pid=1-s2.0-S0095895624000303-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series B\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0095895624000303\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series B","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895624000303","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
1995 年,Komlós、Sárközy 和 Szemerédi 发现,每个最小度至少为 (1/2+γ)n 的大 n 顶点图都包含所有有界度的生成树。我们考虑将这一结果推广到 3 图中的松散生成树,即通过连续追加与前一条边共享一个顶点的边而得到的线性超图。我们证明,对于所有 γ 和 Δ 且 n 大的情况,最小顶点度 (5/9+γ)(n2) 的每个 n 顶点 3-Uniform 超图都包含最大顶点度 Δ 的每棵松散生成树 T。
Embedding loose spanning trees in 3-uniform hypergraphs
In 1995, Komlós, Sárközy and Szemerédi showed that every large n-vertex graph with minimum degree at least contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all γ and Δ, and n large, every n-vertex 3-uniform hypergraph of minimum vertex degree contains every loose spanning tree T with maximum vertex degree Δ. This bound is asymptotically tight, since some loose trees contain perfect matchings.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.