树中最大广义 4 个独立集合的最大数量

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-05-30 DOI:10.1002/jgt.23122

Pingshan Li, Min Xu

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Zito proved that the maximum number of maximum generalized 2-independent sets in a tree of order <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n <msup>\n <mn>2</mn>\n <mfrac>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>3</mn>\n </mrow>\n <mn>2</mn>\n </mfrac>\n </msup>\n </mrow>\n <annotation> ${2}^{\\frac{n-3}{2}}$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> is odd, and <math>\n <semantics>\n <mrow>\n <msup>\n <mn>2</mn>\n <mfrac>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n <mn>2</mn>\n </mfrac>\n </msup>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n <annotation> ${2}^{\\frac{n-2}{2}}+1$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> is even. Tu et al. showed that the maximum number of maximum generalized 3-independent sets in a tree of order <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n <msup>\n <mn>3</mn>\n <mrow>\n <mfrac>\n <mi>n</mi>\n <mn>3</mn>\n </mfrac>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mfrac>\n <mi>n</mi>\n <mn>3</mn>\n </mfrac>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n <annotation> ${3}^{\\frac{n}{3}-1}+\\frac{n}{3}+1$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>≡</mo>\n <mn>0</mn>\n <mo> </mo>\n <mrow>\n <mo>(</mo>\n <mtext>mod 3</mtext>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $n\\equiv 0\\unicode{x02007}(\\text{mod 3})$</annotation>\n </semantics></math>, and <math>\n <semantics>\n <mrow>\n <msup>\n <mn>3</mn>\n <mrow>\n <mfrac>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <mn>3</mn>\n </mfrac>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n <annotation> ${3}^{\\frac{n-1}{3}-1}+1$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>≡</mo>\n <mn>1</mn>\n <mo> </mo>\n <mrow>\n <mo>(</mo>\n <mtext>mod 3</mtext>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $n\\equiv 1\\unicode{x02007}(\\text{mod 3})$</annotation>\n </semantics></math>, and <math>\n <semantics>\n <mrow>\n <msup>\n <mn>3</mn>\n <mrow>\n <mfrac>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n <mn>3</mn>\n </mfrac>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation> ${3}^{\\frac{n-2}{3}-1}$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>≡</mo>\n <mn>2</mn>\n <mo> </mo>\n <mrow>\n <mo>(</mo>\n <mtext>mod 3</mtext>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $n\\equiv 2\\unicode{x02007}(\\text{mod 3})$</annotation>\n </semantics></math> and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-independent sets in a tree for a general integer <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>. As applications, we show that the maximum number of generalized 4-independent sets in a tree of order <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo> </mo>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>≥</mo>\n <mn>4</mn>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $n\\unicode{x02007}(n\\ge 4)$</annotation>\n </semantics></math> is\n\n ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"359-380"},"PeriodicalIF":0.9000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The maximum number of maximum generalized 4-independent sets in trees\",\"authors\":\"Pingshan Li, Min Xu\",\"doi\":\"10.1002/jgt.23122\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A generalized <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-independent set is a set of vertices such that the induced subgraph contains no trees with <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-vertices, and the generalized <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-independence number is the cardinality of a maximum <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-independent set in <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>. Zito proved that the maximum number of maximum generalized 2-independent sets in a tree of order <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> is <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mn>2</mn>\\n <mfrac>\\n <mrow>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>3</mn>\\n </mrow>\\n <mn>2</mn>\\n </mfrac>\\n </msup>\\n </mrow>\\n <annotation> ${2}^{\\\\frac{n-3}{2}}$</annotation>\\n </semantics></math> if <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> is odd, and <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mn>2</mn>\\n <mfrac>\\n <mrow>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>2</mn>\\n </mrow>\\n <mn>2</mn>\\n </mfrac>\\n </msup>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation> ${2}^{\\\\frac{n-2}{2}}+1$</annotation>\\n </semantics></math> if <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> is even. Tu et al. showed that the maximum number of maximum generalized 3-independent sets in a tree of order <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> is <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mn>3</mn>\\n <mrow>\\n <mfrac>\\n <mi>n</mi>\\n <mn>3</mn>\\n </mfrac>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mo>+</mo>\\n <mfrac>\\n <mi>n</mi>\\n <mn>3</mn>\\n </mfrac>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation> ${3}^{\\\\frac{n}{3}-1}+\\\\frac{n}{3}+1$</annotation>\\n </semantics></math> if <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>≡</mo>\\n <mn>0</mn>\\n <mo> </mo>\\n <mrow>\\n <mo>(</mo>\\n <mtext>mod 3</mtext>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $n\\\\equiv 0\\\\unicode{x02007}(\\\\text{mod 3})$</annotation>\\n </semantics></math>, and <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mn>3</mn>\\n <mrow>\\n <mfrac>\\n <mrow>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <mn>3</mn>\\n </mfrac>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation> ${3}^{\\\\frac{n-1}{3}-1}+1$</annotation>\\n </semantics></math> if <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>≡</mo>\\n <mn>1</mn>\\n <mo> </mo>\\n <mrow>\\n <mo>(</mo>\\n <mtext>mod 3</mtext>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $n\\\\equiv 1\\\\unicode{x02007}(\\\\text{mod 3})$</annotation>\\n </semantics></math>, and <math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mn>3</mn>\\n <mrow>\\n <mfrac>\\n <mrow>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>2</mn>\\n </mrow>\\n <mn>3</mn>\\n </mfrac>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation> ${3}^{\\\\frac{n-2}{3}-1}$</annotation>\\n </semantics></math> if <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>≡</mo>\\n <mn>2</mn>\\n <mo> </mo>\\n <mrow>\\n <mo>(</mo>\\n <mtext>mod 3</mtext>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $n\\\\equiv 2\\\\unicode{x02007}(\\\\text{mod 3})$</annotation>\\n </semantics></math> and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-independent sets in a tree for a general integer <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>. As applications, we show that the maximum number of generalized 4-independent sets in a tree of order <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo> </mo>\\n <mrow>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>≥</mo>\\n <mn>4</mn>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $n\\\\unicode{x02007}(n\\\\ge 4)$</annotation>\\n </semantics></math> is\\n\\n \",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"107 2\",\"pages\":\"359-380\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-05-30\",\"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.23122\",\"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.23122","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

广义-独立集是这样一个顶点集，即诱导子图不包含有-顶点的树，广义-独立数是.中最大-独立集的心数。齐托证明，如果是奇数，如果是偶数，阶树中的最大广义 2-independent 集的最大个数是。Tu 等人证明了阶树状图中最大广义 3-independent 集的最大个数是，且，且，且，且他们描述了所有极值图的特征。受这两个漂亮结果的启发，我们建立了四个关于树中最大广义-独立集的结构定理，对于一般整数。作为应用，我们证明了有序树中最大广义 4-independent 集的个数是，并且我们还描述了具有最大广义 4-independent 集的最大个数的所有极值树的结构。

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The maximum number of maximum generalized 4-independent sets in trees

A generalized $k$ -independent set is a set of vertices such that the induced subgraph contains no trees with $k$ -vertices, and the generalized $k$ -independence number is the cardinality of a maximum $k$ -independent set in $G$ . Zito proved that the maximum number of maximum generalized 2-independent sets in a tree of order $n$ is $2^{\frac{n - 3}{2}}$ if $n$ is odd, and $2^{\frac{n - 2}{2}} + 1$ if $n$ is even. Tu et al. showed that the maximum number of maximum generalized 3-independent sets in a tree of order $n$ is $3^{\frac{n}{3} - 1} + \frac{n}{3} + 1$ if $n \equiv 0 (mod 3)$ , and $3^{\frac{n - 1}{3} - 1} + 1$ if $n \equiv 1 (mod 3)$ , and $3^{\frac{n - 2}{3} - 1}$ if $n \equiv 2 (mod 3)$ and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized $k$ -independent sets in a tree for a general integer $k$ . As applications, we show that the maximum number of generalized 4-independent sets in a tree of order $n (n \geq 4)$ is

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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 .