A proof of Frankl–Kupavskii's conjecture on edge-union condition

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-01-07 DOI:10.1002/jgt.23073

Hongliang Lu, Xuechun Zhang

{"title":"A proof of Frankl–Kupavskii's conjecture on edge-union condition","authors":"Hongliang Lu, Xuechun Zhang","doi":"10.1002/jgt.23073","DOIUrl":null,"url":null,"abstract":"<p>A 3-graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> ${\\rm{ {\\mathcal F} }}$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>U</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>s</mi>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $U(s,2s+1)$</annotation>\n </semantics></math> if for any <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n </mrow>\n </mrow>\n <annotation> $s$</annotation>\n </semantics></math> edges <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>e</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <mi>…</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>e</mi>\n \n <mi>s</mi>\n </msub>\n \n <mo>∈</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${e}_{1},\\ldots ,{e}_{s}\\in E({\\rm{ {\\mathcal F} }})$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mo>∣</mo>\n \n <msub>\n <mi>e</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>∪</mo>\n \n <mtext>⋯</mtext>\n \n <mo>∪</mo>\n \n <msub>\n <mi>e</mi>\n \n <mi>s</mi>\n </msub>\n \n <mo>∣</mo>\n \n <mo>≤</mo>\n \n <mn>2</mn>\n \n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> $| {e}_{1}\\cup \\cdots \\cup {e}_{s}| \\le 2s+1$</annotation>\n </semantics></math>. Frankl and Kupavskii proposed the following conjecture: For any 3-graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> ${\\rm{ {\\mathcal F} }}$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> vertices, if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> ${\\rm{ {\\mathcal F} }}$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>U</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>s</mi>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $U(s,2s+1)$</annotation>\n </semantics></math>, then\n\n </p><p>In this paper, we confirm Frankl and Kupavskii's conjecture.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23073","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

A 3-graph $F$ is $U (s, 2 s + 1)$ if for any $s$ edges $e_{1}, \dots, e_{s} \in E (F)$ , $∣ e_{1} \cup ⋯ \cup e_{s} ∣ \leq 2 s + 1$ . Frankl and Kupavskii proposed the following conjecture: For any 3-graph $F$ with $n$ vertices, if $F$ is $U (s, 2 s + 1)$ , then

In this paper, we confirm Frankl and Kupavskii's conjecture.

查看原文本刊更多论文

弗兰克尔-库帕夫斯基关于边缘联合条件猜想的证明

一个 3 图 F${{rm{ {\mathcal F}U(s,2s+1)$U(s,2s+1)$ If for any s$s$ edges e1,...,es∈E(F)${e}_{1},\ldots ,{e}_{s}\in E({\rm{ {mathcal F} }})$, ∣e1∪⋯∪es∣≤2s+1$| {e}_{1}\cup \cdots \cup {e}_{s}| \le 2s+1$.弗兰克尔和库帕夫斯基提出了以下猜想：对于任意 3 图 F${\rm{ {\mathcal F}}$ 有 n$n$ 个顶点，如果 F${\rm{ {\mathcal F}}}$ 是 U(s,2s+1)$U(s,2s+1)$, 那么

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .