Matching Stability for 3-Partite 3-Uniform Hypergraphs

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2026-10-12 Epub Date: 2026-06-19 DOI:10.1002/jgt.70089

Hongliang Lu, Xinxin Ma

{"title":"Matching Stability for 3-Partite 3-Uniform Hypergraphs","authors":"Hongliang Lu, Xinxin Ma","doi":"10.1002/jgt.70089","DOIUrl":null,"url":null,"abstract":"<div>\n \n Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>s</mi>\n </mrow>\n </mrow>\n </semantics></math> be three integers such that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n \n <mo>≥</mo>\n \n <mi>s</mi>\n \n <mo>≥</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n </semantics></math>. Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n </semantics></math> be a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n </semantics></math>-partite <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n </semantics></math>-uniform hypergraph with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n </semantics></math> vertices in each class. Aharoni (2017) showed that if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>e</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>></mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>s</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <msup>\n <mi>n</mi>\n \n <mrow>\n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n </mrow>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n </semantics></math> has a matching of size <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n </mrow>\n </mrow>\n </semantics></math>. In this paper, we give a stability result for 3-partite 3-uniform hypergraphs: if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> is a 3-partite 3-uniform hypergraph with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n \n <mo>≥</mo>\n \n <mn>162</mn>\n </mrow>\n </mrow>\n </semantics></math> vertices in each class, <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>e</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>s</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>+</mo>\n \n <mn>3</mn>\n \n <mi>n</mi>\n \n <mo>−</mo>\n \n <mi>s</mi>\n </mrow>\n </mrow>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> contains no matching of size <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> has a vertex cover of size <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n </mrow>\n </mrow>\n </semantics></math>. Our bound is also tight.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 4","pages":"650-662"},"PeriodicalIF":1.0000,"publicationDate":"2026-10-12","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.70089","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/6/19 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $n, k, s$ be three integers such that $k \geq 2$ and $n \geq s \geq 1$ . Let $H$ be a $k$ -partite $k$ -uniform hypergraph with $n$ vertices in each class. Aharoni (2017) showed that if $e (H) > (s - 1) n^{k - 1}$ , then $H$ has a matching of size $s$ . In this paper, we give a stability result for 3-partite 3-uniform hypergraphs: if $G$ is a 3-partite 3-uniform hypergraph with $n \geq 162$ vertices in each class, $e (G) \geq (s - 1) n^{2} + 3 n - s$ and $G$ contains no matching of size $s + 1$ , then $G$ has a vertex cover of size $s$ . Our bound is also tight.

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3部3-一致超图的匹配稳定性

设n k，S为k≥2且n≥的三个整数S≥1。设H是一个k部k一致超图每个类有N个顶点。Aharoni（2017）表明，如果e (H) > (s)−1)n k−1，则H的匹配大小为s。本文给出了3部3-一致超图的一个稳定性结果：如果G是一个3部3-一致超图，每一类中有n≥162个顶点，e (G)≥(s−1)n 2 + 3 n−s和G不包含大小为s + 1的匹配，那么G的顶点覆盖大小为s。我们的边界也很紧。

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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 .