{"title":"在具有大最小半度的图中计算定向树","authors":"Felix Joos, Jonathan Schrodt","doi":"10.1016/j.jctb.2024.05.004","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>T</em> be an oriented tree on <em>n</em> vertices with maximum degree at most <span><math><msup><mrow><mi>e</mi></mrow><mrow><mi>o</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></msup></math></span>. If <em>G</em> is a digraph on <em>n</em> vertices with minimum semidegree <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>n</mi></math></span>, then <em>G</em> contains <em>T</em> as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of <em>T</em> the digraph <em>G</em> contains. Our main result states that every such <em>G</em> contains at least <span><math><mo>|</mo><mrow><mi>Aut</mi><mi>(</mi><mi>T</mi><mi>)</mi></mrow><mspace></mspace><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>n</mi><mo>!</mo></math></span> copies of <em>T</em>, which is optimal. This implies the analogous result in the undirected case.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 236-270"},"PeriodicalIF":1.2000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000431/pdfft?md5=7f84c56186e46b0ae787c373b4164785&pid=1-s2.0-S0095895624000431-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Counting oriented trees in digraphs with large minimum semidegree\",\"authors\":\"Felix Joos, Jonathan Schrodt\",\"doi\":\"10.1016/j.jctb.2024.05.004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <em>T</em> be an oriented tree on <em>n</em> vertices with maximum degree at most <span><math><msup><mrow><mi>e</mi></mrow><mrow><mi>o</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></msup></math></span>. If <em>G</em> is a digraph on <em>n</em> vertices with minimum semidegree <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>n</mi></math></span>, then <em>G</em> contains <em>T</em> as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree <span><math><mi>o</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of <em>T</em> the digraph <em>G</em> contains. Our main result states that every such <em>G</em> contains at least <span><math><mo>|</mo><mrow><mi>Aut</mi><mi>(</mi><mi>T</mi><mi>)</mi></mrow><mspace></mspace><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>n</mi><mo>!</mo></math></span> copies of <em>T</em>, which is optimal. This implies the analogous result in the undirected case.</p></div>\",\"PeriodicalId\":54865,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series B\",\"volume\":\"168 \",\"pages\":\"Pages 236-270\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-05-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0095895624000431/pdfft?md5=7f84c56186e46b0ae787c373b4164785&pid=1-s2.0-S0095895624000431-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/S0095895624000431\",\"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/S0095895624000431","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
设 T 是 n 个顶点上的定向树,其最大度最多为 eo(logn)。如果 G 是 n 个顶点上的数图,最小半度 δ0(G)≥(12+o(1))n,那么 G 包含作为生成树的 T,正如 Kathapurkar 和 Montgomery 最近证明的那样(事实上,他们只要求最大度为 o(n/logn))。这推广了 Komlós、Sárközy 和 Szemerédi 对图的相应结果。我们研究了数图 G 包含多少份 T 的自然问题。我们的主要结果表明,每个这样的 G 至少包含 T 的 |Aut(T)|-1(12-o(1))nn!这意味着无向情况下的类似结果。
Counting oriented trees in digraphs with large minimum semidegree
Let T be an oriented tree on n vertices with maximum degree at most . If G is a digraph on n vertices with minimum semidegree , then G contains T as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree ). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of T the digraph G contains. Our main result states that every such G contains at least copies of T, which is optimal. This implies the analogous result in the undirected case.
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.