有界度超图中的许多团

R. Kirsch, Jamie Radcliffe
{"title":"有界度超图中的许多团","authors":"R. Kirsch, Jamie Radcliffe","doi":"10.1137/22m1507565","DOIUrl":null,"url":null,"abstract":"Recently Chase determined the maximum possible number of cliques of size $t$ in a graph on $n$ vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have $m$ edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For $s$-graphs with $s\\ge 3$ a number of issues arise that do not appear in the graph case. For instance, for general $s$-graphs we can assign degrees to any $i$-subset of the vertex set with $1\\le i\\le s-1$. We establish bounds on the number of $t$-cliques in an $s$-graph $\\mathcal{H}$ with $i$-degree bounded by $\\Delta$ in three contexts: $\\mathcal{H}$ has $n$ vertices; $\\mathcal{H}$ has $m$ (hyper)edges; and (generalizing the previous case) $\\mathcal{H}$ has a fixed number $p$ of $u$-cliques for some $u$ with $s\\le u \\le t$. When $\\Delta$ is of a special form we characterize the extremal $s$-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of F\\\"uredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Many Cliques in Bounded-Degree Hypergraphs\",\"authors\":\"R. Kirsch, Jamie Radcliffe\",\"doi\":\"10.1137/22m1507565\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Recently Chase determined the maximum possible number of cliques of size $t$ in a graph on $n$ vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have $m$ edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For $s$-graphs with $s\\\\ge 3$ a number of issues arise that do not appear in the graph case. For instance, for general $s$-graphs we can assign degrees to any $i$-subset of the vertex set with $1\\\\le i\\\\le s-1$. We establish bounds on the number of $t$-cliques in an $s$-graph $\\\\mathcal{H}$ with $i$-degree bounded by $\\\\Delta$ in three contexts: $\\\\mathcal{H}$ has $n$ vertices; $\\\\mathcal{H}$ has $m$ (hyper)edges; and (generalizing the previous case) $\\\\mathcal{H}$ has a fixed number $p$ of $u$-cliques for some $u$ with $s\\\\le u \\\\le t$. When $\\\\Delta$ is of a special form we characterize the extremal $s$-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of F\\\\\\\"uredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.\",\"PeriodicalId\":21749,\"journal\":{\"name\":\"SIAM J. Discret. Math.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-07-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM J. Discret. Math.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/22m1507565\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Discret. Math.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/22m1507565","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2

摘要

最近Chase在一个图形中确定了$n$顶点上具有给定最大度的大小为$t$的团的最大可能数量。不久之后,Chakraborti和Chen回答了这个问题的版本,我们要求图有$m$条边和固定的最大度(没有对顶点数量施加任何限制)。本文在超图上讨论了这些问题。对于使用$s\ge 3$的$s$ -graphs,会出现一些在图的情况下不会出现的问题。例如,对于一般的$s$ -图,我们可以用$1\le i\le s-1$为顶点集的任何$i$ -子集分配度数。我们在以下三种情况下建立了$s$ -图$\mathcal{H}$中$t$ -团的数量界限,其中$i$ -度由$\Delta$限定:$\mathcal{H}$有$n$个顶点;$\mathcal{H}$有$m$(超)边;并且(推广前面的情况)$\mathcal{H}$有固定数量的$p$$u$ -对于一些$u$和$s\le u \le t$的派系。当$\Delta$是一种特殊形式时,我们描述了$s$ -图的极值,并证明了边界是紧的。这些极端的例子是斯坦纳系统或填料的阴影。在证明我们的唯一性结果的过程中,我们将f redi和Griggs关于Kruskal-Katona的唯一性的结果从影子情形推广到团情形。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Many Cliques in Bounded-Degree Hypergraphs
Recently Chase determined the maximum possible number of cliques of size $t$ in a graph on $n$ vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have $m$ edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For $s$-graphs with $s\ge 3$ a number of issues arise that do not appear in the graph case. For instance, for general $s$-graphs we can assign degrees to any $i$-subset of the vertex set with $1\le i\le s-1$. We establish bounds on the number of $t$-cliques in an $s$-graph $\mathcal{H}$ with $i$-degree bounded by $\Delta$ in three contexts: $\mathcal{H}$ has $n$ vertices; $\mathcal{H}$ has $m$ (hyper)edges; and (generalizing the previous case) $\mathcal{H}$ has a fixed number $p$ of $u$-cliques for some $u$ with $s\le u \le t$. When $\Delta$ is of a special form we characterize the extremal $s$-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of F\"uredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.
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