无 C 2 k + 1 图形的最小度稳定性

Pub Date : 2024-02-11 DOI:10.1002/jgt.23086

Xiaoli Yuan, Yuejian Peng

{"title":"无 C 2 k + 1 图形的最小度稳定性","authors":"Xiaoli Yuan, Yuejian Peng","doi":"10.1002/jgt.23086","DOIUrl":null,"url":null,"abstract":"We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${C}_{2k+1}$</annotation>\n </semantics></math>-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a <math>\n <semantics>\n <mrow>\n <mo>{</mo>\n \n <msub>\n <mi>C</mi>\n \n <mn>3</mn>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>C</mi>\n \n <mn>5</mn>\n </msub>\n \n <mo>,</mo>\n \n <mtext>…</mtext>\n \n <mo>,</mo>\n \n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>}</mo>\n </mrow>\n <annotation> $\\{{C}_{3},{C}_{5},\\ldots ,{C}_{2k+1}\\}$</annotation>\n </semantics></math>-free graph on <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> vertices has minimum degree greater than <math>\n <semantics>\n <mrow>\n <mfrac>\n <mn>2</mn>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>3</mn>\n </mrow>\n </mfrac>\n \n <mi>n</mi>\n </mrow>\n <annotation> $\\frac{2}{2k+3}n$</annotation>\n </semantics></math>, then it is bipartite. Häggkvist showed that for <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n \n <mo>∈</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mn>3</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n <annotation> $k\\in \\{1,2,3,4\\}$</annotation>\n </semantics></math>, if a <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${C}_{2k+1}$</annotation>\n </semantics></math>-free graph on <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> vertices has minimum degree greater than <math>\n <semantics>\n <mrow>\n <mfrac>\n <mn>2</mn>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>3</mn>\n </mrow>\n </mfrac>\n \n <mi>n</mi>\n </mrow>\n <annotation> $\\frac{2}{2k+3}n$</annotation>\n </semantics></math>, then it is bipartite. Häggkvist also pointed out that this result cannot be extended to <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>5</mn>\n </mrow>\n <annotation> $k\\ge 5$</annotation>\n </semantics></math>. In this paper, we give a complete answer for any <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>5</mn>\n </mrow>\n <annotation> $k\\ge 5$</annotation>\n </semantics></math>. We show that if <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>5</mn>\n </mrow>\n <annotation> $k\\ge 5$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is an <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-vertex <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${C}_{2k+1}$</annotation>\n </semantics></math>-free graph with <math>\n <semantics>\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 <mfrac>\n <mi>n</mi>\n \n <mn>6</mn>\n </mfrac>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $\\delta (G)\\ge \\frac{n}{6}+1$</annotation>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is bipartite, and the bound <math>\n <semantics>\n <mrow>\n <mfrac>\n <mi>n</mi>\n \n <mn>6</mn>\n </mfrac>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $\\frac{n}{6}+1$</annotation>\n </semantics></math> is tight.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Minimum degree stability of \\n \\n \\n \\n C\\n \\n 2\\n k\\n +\\n 1\\n \\n \\n \\n ${C}_{2k+1}$\\n -free graphs\",\"authors\":\"Xiaoli Yuan, Yuejian Peng\",\"doi\":\"10.1002/jgt.23086\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> ${C}_{2k+1}$</annotation>\\n </semantics></math>-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a <math>\\n <semantics>\\n <mrow>\\n <mo>{</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mn>3</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mn>5</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <mtext>…</mtext>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>C</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n \\n <mo>}</mo>\\n </mrow>\\n <annotation> $\\\\{{C}_{3},{C}_{5},\\\\ldots ,{C}_{2k+1}\\\\}$</annotation>\\n </semantics></math>-free graph on <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> vertices has minimum degree greater than <math>\\n <semantics>\\n <mrow>\\n <mfrac>\\n <mn>2</mn>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n </mfrac>\\n \\n <mi>n</mi>\\n </mrow>\\n <annotation> $\\\\frac{2}{2k+3}n$</annotation>\\n </semantics></math>, then it is bipartite. Häggkvist showed that for <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n \\n <mo>∈</mo>\\n \\n <mrow>\\n <mo>{</mo>\\n \\n <mrow>\\n <mn>1</mn>\\n \\n <mo>,</mo>\\n \\n <mn>2</mn>\\n \\n <mo>,</mo>\\n \\n <mn>3</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n <annotation> $k\\\\in \\\\{1,2,3,4\\\\}$</annotation>\\n </semantics></math>, if a <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> ${C}_{2k+1}$</annotation>\\n </semantics></math>-free graph on <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> vertices has minimum degree greater than <math>\\n <semantics>\\n <mrow>\\n <mfrac>\\n <mn>2</mn>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n </mfrac>\\n \\n <mi>n</mi>\\n </mrow>\\n <annotation> $\\\\frac{2}{2k+3}n$</annotation>\\n </semantics></math>, then it is bipartite. Häggkvist also pointed out that this result cannot be extended to <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>5</mn>\\n </mrow>\\n <annotation> $k\\\\ge 5$</annotation>\\n </semantics></math>. In this paper, we give a complete answer for any <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>5</mn>\\n </mrow>\\n <annotation> $k\\\\ge 5$</annotation>\\n </semantics></math>. We show that if <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>5</mn>\\n </mrow>\\n <annotation> $k\\\\ge 5$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> is an <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math>-vertex <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> ${C}_{2k+1}$</annotation>\\n </semantics></math>-free graph with <math>\\n <semantics>\\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 <mfrac>\\n <mi>n</mi>\\n \\n <mn>6</mn>\\n </mfrac>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $\\\\delta (G)\\\\ge \\\\frac{n}{6}+1$</annotation>\\n </semantics></math>, then <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> is bipartite, and the bound <math>\\n <semantics>\\n <mrow>\\n <mfrac>\\n <mi>n</mi>\\n \\n <mn>6</mn>\\n </mfrac>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $\\\\frac{n}{6}+1$</annotation>\\n </semantics></math> is tight.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-02-11\",\"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.23086\",\"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.23086","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

我们考虑的是禁止奇数循环的图的最小度稳定性：要保证不含 C2k+1${C}_{2k+1}$ 的图的结构继承于极值图（平衡的完整二叉图），最小度数的严格约束是什么？Andrásfai、Erdős 和 Sós 发现，如果 n$n$ 个顶点上的{C3,C5,...,C2k+1}${{C}_{3},{C}_{5},\ldots ,{C}_{2k+1}\}$ 无顶图的最小度大于 22k+3n$/frac{2}{2k+3}n$，那么它是双向图。Häggkvist 证明了对于 k∈{1,2,3,4}$k\in \{1,2,3,4/}$，如果 n$n$ 个顶点上的无 C2k+1${C}_{2k+1}$ 图的最小度大于 22k+3n$/frac{2}{2k+3}n$，那么它是双方形的。海格奎斯特还指出，这一结果无法扩展到 k≥5$k\ge 5$。在本文中，我们给出了任意 k≥5$k\ge 5$ 的完整答案。我们证明，如果 k≥5$k\ge 5$ 并且 G$G$ 是一个 n$n$-vertex C2k+1${C}_{2k+1}$-free graph，δ(G)≥n6+1$\delta (G)\ge \frac{n}{6}+1$，那么 G$G$ 是双分部图，并且约束 n6+1$\frac{n}{6}+1$ 是紧密的。

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Minimum degree stability of C 2 k + 1 ${C}_{2k+1}$ -free graphs

We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a $C_{2 k + 1}$ -free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a ${C_{3}, C_{5}, …, C_{2 k + 1}}$ -free graph on $n$ vertices has minimum degree greater than $\frac{2}{2 k + 3} n$ , then it is bipartite. Häggkvist showed that for $k \in {1, 2, 3, 4}$ , if a $C_{2 k + 1}$ -free graph on $n$ vertices has minimum degree greater than $\frac{2}{2 k + 3} n$ , then it is bipartite. Häggkvist also pointed out that this result cannot be extended to $k \geq 5$ . In this paper, we give a complete answer for any $k \geq 5$ . We show that if $k \geq 5$ and $G$ is an $n$ -vertex $C_{2 k + 1}$ -free graph with $δ (G) \geq \frac{n}{6} + 1$ , then $G$ is bipartite, and the bound $\frac{n}{6} + 1$ is tight.

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