高连接三元组和马德猜想

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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23144\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23144","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

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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