计算密集超图中的跨越子图

Richard Montgomery, Matías Pavez-Signé
{"title":"计算密集超图中的跨越子图","authors":"Richard Montgomery, Matías Pavez-Signé","doi":"10.1017/s0963548324000178","DOIUrl":null,"url":null,"abstract":"We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline1.png\"/> <jats:tex-math> $k\\geq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline2.png\"/> <jats:tex-math> $1\\leq \\ell \\leq k-1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline3.png\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-graph on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline4.png\"/> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vertices with minimum codegree at least<jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548324000178_eqnU1.png\"/> <jats:tex-math> \\begin{equation*} \\left \\{\\begin {array}{l@{\\quad}l} \\left (\\dfrac {1}{2}+o(1)\\right )n &amp; \\text { if }(k-\\ell )\\mid k,\\\\[5pt] \\left (\\dfrac {1}{\\lceil \\frac {k}{k-\\ell }\\rceil (k-\\ell )}+o(1)\\right )n &amp; \\text { if }(k-\\ell )\\nmid k, \\end {array} \\right . \\end{equation*} </jats:tex-math> </jats:alternatives> </jats:disp-formula>contains <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline5.png\"/> <jats:tex-math> $\\exp\\!(n\\log n-\\Theta (n))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> Hamilton <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline6.png\"/> <jats:tex-math> $\\ell$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-cycles as long as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline7.png\"/> <jats:tex-math> $(k-\\ell )\\mid n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. When <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline8.png\"/> <jats:tex-math> $(k-\\ell )\\mid k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline9.png\"/> <jats:tex-math> $(k-\\ell )\\nmid k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000178_inline10.png\"/> <jats:tex-math> $\\ell \\lt k/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Counting spanning subgraphs in dense hypergraphs\",\"authors\":\"Richard Montgomery, Matías Pavez-Signé\",\"doi\":\"10.1017/s0963548324000178\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline1.png\\\"/> <jats:tex-math> $k\\\\geq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline2.png\\\"/> <jats:tex-math> $1\\\\leq \\\\ell \\\\leq k-1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline3.png\\\"/> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-graph on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline4.png\\\"/> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vertices with minimum codegree at least<jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" position=\\\"float\\\" xlink:href=\\\"S0963548324000178_eqnU1.png\\\"/> <jats:tex-math> \\\\begin{equation*} \\\\left \\\\{\\\\begin {array}{l@{\\\\quad}l} \\\\left (\\\\dfrac {1}{2}+o(1)\\\\right )n &amp; \\\\text { if }(k-\\\\ell )\\\\mid k,\\\\\\\\[5pt] \\\\left (\\\\dfrac {1}{\\\\lceil \\\\frac {k}{k-\\\\ell }\\\\rceil (k-\\\\ell )}+o(1)\\\\right )n &amp; \\\\text { if }(k-\\\\ell )\\\\nmid k, \\\\end {array} \\\\right . \\\\end{equation*} </jats:tex-math> </jats:alternatives> </jats:disp-formula>contains <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline5.png\\\"/> <jats:tex-math> $\\\\exp\\\\!(n\\\\log n-\\\\Theta (n))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> Hamilton <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline6.png\\\"/> <jats:tex-math> $\\\\ell$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-cycles as long as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline7.png\\\"/> <jats:tex-math> $(k-\\\\ell )\\\\mid n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. When <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline8.png\\\"/> <jats:tex-math> $(k-\\\\ell )\\\\mid k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline9.png\\\"/> <jats:tex-math> $(k-\\\\ell )\\\\nmid k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000178_inline10.png\\\"/> <jats:tex-math> $\\\\ell \\\\lt k/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.\",\"PeriodicalId\":10503,\"journal\":{\"name\":\"Combinatorics, Probability and Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorics, Probability and Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/s0963548324000178\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorics, Probability and Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/s0963548324000178","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

我们给出了一种简单的方法来估计具有高最小度的超图中某些类别的跨越子图的不同副本的数量。特别是,对于每个 $k\geq 2$ 和 $1\leq \ell \leq k-1$ ,我们证明了在 $n$ 顶点上的每个 $k$ 图的最小度至少是 \begin{equation*}.\{left {array}{l@\{quad}l}\left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n &;\text { if }(k-\ell )\nmid k,\end {array}\right .\end{equation*} 包含 $\exp\!(n\log n-\Theta (n))$ Hamilton $\ell$ -cycles as long as $(k-\ell )\mid n$ .当 $(k-\ell )\mid k$ 时,这给出了格洛克(Glock)、古尔德(Gould)、乔斯(Joos)、库恩(Kühn)和奥斯特胡斯(Osthus)的一个结果的简单证明,而当 $(k-\ell )\nmid k$ 时,这给出了一个比费伯(Ferber)、哈迪曼(Hardiman)和蒙德(Mond)给出的,或当 $\ell \lt k/2$ 时费伯(Ferber)、克里夫列维奇(Krivelevich)和苏达科夫(Sudakov)给出的,或当 $\ell \lt k/2$ 时,费伯(Ferber)、克里夫列维奇(Krivelevich)和苏达科夫(Sudakov)给出的更弱的计数,但对于一个渐近最优的最小鳕鱼度边界来说,这个计数是成立的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Counting spanning subgraphs in dense hypergraphs
We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$ , we show that every $k$ -graph on $n$ vertices with minimum codegree at least \begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*} contains $\exp\!(n\log n-\Theta (n))$ Hamilton $\ell$ -cycles as long as $(k-\ell )\mid n$ . When $(k-\ell )\mid k$ , this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when $(k-\ell )\nmid k$ , this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when $\ell \lt k/2$ , by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信