高连接三元组和马德猜想

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-06-17 DOI:10.1002/jgt.23144

Qinghai Liu, Kai Ying, Yanmei Hong

{"title":"高连接三元组和马德猜想","authors":"Qinghai Liu, Kai Ying, Yanmei Hong","doi":"10.1002/jgt.23144","DOIUrl":null,"url":null,"abstract":"Mader proved that, for any tree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>T</mi>\n </mrow>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> of order <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>m</mi>\n </mrow>\n </mrow>\n <annotation> $m$</annotation>\n </semantics></math>, every <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-connected graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>δ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n \n <msup>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>+</mo>\n \n <mi>m</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mn>2</mn>\n </msup>\n \n <mo>+</mo>\n \n <mi>m</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> $\\delta (G)\\ge 2{(k+m-1)}^{2}+m-1$</annotation>\n </semantics></math> contains a subtree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>T</mi>\n \n <mo>′</mo>\n </msup>\n \n <mo>≅</mo>\n \n <mi>T</mi>\n </mrow>\n </mrow>\n <annotation> ${T}^{^{\\prime} }\\cong T$</annotation>\n </semantics></math> such that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>−</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>T</mi>\n \n <mo>′</mo>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $G-V({T}^{^{\\prime} })$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-connected. We proved that any graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with minimum degree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>δ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $\\delta (G)\\ge 2k$</annotation>\n </semantics></math> contains <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-connected triples. As a corollary, we prove that, for any tree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>T</mi>\n </mrow>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> of order <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>m</mi>\n </mrow>\n </mrow>\n <annotation> $m$</annotation>\n </semantics></math>, every <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-connected graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>δ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>4</mn>\n \n <mi>m</mi>\n \n <mo>−</mo>\n \n <mn>6</mn>\n </mrow>\n </mrow>\n <annotation> $\\delta (G)\\ge 3k+4m-6$</annotation>\n </semantics></math> contains a subtree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>T</mi>\n \n <mo>′</mo>\n </msup>\n \n <mo>≅</mo>\n \n <mi>T</mi>\n </mrow>\n </mrow>\n <annotation> ${T}^{^{\\prime} }\\cong T$</annotation>\n </semantics></math> such that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>−</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>T</mi>\n \n <mo>′</mo>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $G-V({T}^{^{\\prime} })$</annotation>\n </semantics></math> is still <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-connected, improving Mader's condition to a linear bound.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"478-484"},"PeriodicalIF":0.9000,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Highly connected triples and Mader's conjecture\",\"authors\":\"Qinghai Liu, Kai Ying, Yanmei Hong\",\"doi\":\"10.1002/jgt.23144\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Mader proved that, for any tree <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> of order <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>m</mi>\\n </mrow>\\n </mrow>\\n <annotation> $m$</annotation>\\n </semantics></math>, every <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-connected graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> with <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>δ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≥</mo>\\n \\n <mn>2</mn>\\n \\n <msup>\\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mi>m</mi>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mn>2</mn>\\n </msup>\\n \\n <mo>+</mo>\\n \\n <mi>m</mi>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\delta (G)\\\\ge 2{(k+m-1)}^{2}+m-1$</annotation>\\n </semantics></math> contains a subtree <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msup>\\n <mi>T</mi>\\n \\n <mo>′</mo>\\n </msup>\\n \\n <mo>≅</mo>\\n \\n <mi>T</mi>\\n </mrow>\\n </mrow>\\n <annotation> ${T}^{^{\\\\prime} }\\\\cong T$</annotation>\\n </semantics></math> such that <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n \\n <mo>−</mo>\\n \\n <mi>V</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <msup>\\n <mi>T</mi>\\n \\n <mo>′</mo>\\n </msup>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n <annotation> $G-V({T}^{^{\\\\prime} })$</annotation>\\n </semantics></math> is <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-connected. We proved that any graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> with minimum degree <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>δ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≥</mo>\\n \\n <mn>2</mn>\\n \\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\delta (G)\\\\ge 2k$</annotation>\\n </semantics></math> contains <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-connected triples. As a corollary, we prove that, for any tree <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> of order <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>m</mi>\\n </mrow>\\n </mrow>\\n <annotation> $m$</annotation>\\n </semantics></math>, every <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-connected graph <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> with <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>δ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>G</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≥</mo>\\n \\n <mn>3</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>4</mn>\\n \\n <mi>m</mi>\\n \\n <mo>−</mo>\\n \\n <mn>6</mn>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\delta (G)\\\\ge 3k+4m-6$</annotation>\\n </semantics></math> contains a subtree <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msup>\\n <mi>T</mi>\\n \\n <mo>′</mo>\\n </msup>\\n \\n <mo>≅</mo>\\n \\n <mi>T</mi>\\n </mrow>\\n </mrow>\\n <annotation> ${T}^{^{\\\\prime} }\\\\cong T$</annotation>\\n </semantics></math> such that <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n \\n <mo>−</mo>\\n \\n <mi>V</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <msup>\\n <mi>T</mi>\\n \\n <mo>′</mo>\\n </msup>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n <annotation> $G-V({T}^{^{\\\\prime} })$</annotation>\\n </semantics></math> is still <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-connected, improving Mader's condition to a linear bound.\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"107 3\",\"pages\":\"478-484\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-06-17\",\"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.23144\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23144","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

摘要

作为推论，我们证明对于任何阶数为 m $m$ 的树 T $T$ ，每一个 k $k$ -connected graph G $G$ δ ( G ) ≥ 3 k + 4 m - 6 $\delta (G)\ge 3k+4m-6$ 都包含一个子树 T ′ ≅ T ${T}^{^{\prime} }\cong T$，使得 G - V ( T ′ ) $G-V({T}^{^{\prime} })$ 仍然是 k $k$ -connected 的，从而将马德的条件改进为线性约束。

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Highly connected triples and Mader's conjecture

Mader proved that, for any tree $T$ of order $m$ , every $k$ -connected graph $G$ with $δ (G) \geq 2 {(k + m - 1)}^{2} + m - 1$ contains a subtree $T^{'} ≅ T$ such that $G - V (T^{'})$ is $k$ -connected. We proved that any graph $G$ with minimum degree $δ (G) \geq 2 k$ contains $k$ -connected triples. As a corollary, we prove that, for any tree $T$ of order $m$ , every $k$ -connected graph $G$ with $δ (G) \geq 3 k + 4 m - 6$ contains a subtree $T^{'} ≅ T$ such that $G - V (T^{'})$ is still $k$ -connected, improving Mader's condition to a linear bound.

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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 .