Fractional factors and component factors in graphs with isolated toughness smaller than 1

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-09-16 DOI:10.1002/jgt.23179

Isaak H. Wolf

{"title":"Fractional factors and component factors in graphs with isolated toughness smaller than 1","authors":"Isaak H. Wolf","doi":"10.1002/jgt.23179","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> be a simple graph and let <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>m</mi>\n </mrow>\n <annotation> $n,m$</annotation>\n </semantics></math> be two integers with <math>\n <semantics>\n <mrow>\n <mn>0</mn>\n \n <mo><</mo>\n \n <mi>m</mi>\n \n <mo><</mo>\n \n <mi>n</mi>\n </mrow>\n <annotation> $0\\lt m\\lt n$</annotation>\n </semantics></math>. We prove that <math>\n <semantics>\n <mrow>\n <mi>i</mi>\n \n <mi>s</mi>\n \n <mi>o</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>G</mi>\n \n <mo>−</mo>\n \n <mi>S</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mfrac>\n <mi>n</mi>\n \n <mi>m</mi>\n </mfrac>\n \n <mo>∣</mo>\n \n <mi>S</mi>\n \n <mo>∣</mo>\n </mrow>\n <annotation> $iso(G-S)\\le \\frac{n}{m}| S| $</annotation>\n </semantics></math> for every <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n \n <mo>⊂</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $S\\subset V(G)$</annotation>\n </semantics></math> if and only if <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> has a <math>\n <semantics>\n <mrow>\n <mfenced>\n <mrow>\n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>i</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <mi>T</mi>\n \n <mo>:</mo>\n \n <mn>1</mn>\n \n <mo>≤</mo>\n \n <mi>i</mi>\n \n <mo><</mo>\n \n <mfrac>\n <mi>m</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>−</mo>\n \n <mi>m</mi>\n </mrow>\n </mfrac>\n \n <mo>,</mo>\n \n <mi>T</mi>\n \n <mo>∈</mo>\n \n <msub>\n <mi>T</mi>\n \n <mfrac>\n <mi>n</mi>\n \n <mi>m</mi>\n </mfrac>\n </msub>\n </mrow>\n </mfenced>\n </mrow>\n <annotation> $\\left\\{{C}_{2i+1},T:1\\le i\\lt \\frac{m}{n-m},T\\in {{\\mathscr{T}}}_{\\frac{n}{m}}\\right\\}$</annotation>\n </semantics></math>-factor, where <math>\n <semantics>\n <mrow>\n <mi>i</mi>\n \n <mi>s</mi>\n \n <mi>o</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>G</mi>\n \n <mo>−</mo>\n \n <mi>S</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $iso(G-S)$</annotation>\n </semantics></math> denotes the number of isolated vertices of <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n \n <mo>−</mo>\n \n <mi>S</mi>\n </mrow>\n <annotation> $G-S$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <msub>\n <mi>T</mi>\n \n <mfrac>\n <mi>n</mi>\n \n <mi>m</mi>\n </mfrac>\n </msub>\n </mrow>\n <annotation> ${{\\mathscr{T}}}_{\\frac{n}{m}}$</annotation>\n </semantics></math> is a special family of trees. Furthermore, we characterize the trees in <math>\n <semantics>\n <mrow>\n <msub>\n <mi>T</mi>\n \n <mfrac>\n <mi>n</mi>\n \n <mi>m</mi>\n </mfrac>\n </msub>\n </mrow>\n <annotation> ${{\\mathscr{T}}}_{\\frac{n}{m}}$</annotation>\n </semantics></math> in terms of their bipartition.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"274-287"},"PeriodicalIF":0.9000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23179","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23179","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

Abstract

Let $G$ be a simple graph and let $n, m$ be two integers with $0 < m < n$ . We prove that $i s o (G - S) \leq \frac{n}{m} ∣ S ∣$ for every $S \subset V (G)$ if and only if $G$ has a $(C_{2 i + 1}, T : 1 \leq i < \frac{m}{n - m}, T \in T_{\frac{n}{m}})$ -factor, where $i s o (G - S)$ denotes the number of isolated vertices of $G - S$ and $T_{\frac{n}{m}}$ is a special family of trees. Furthermore, we characterize the trees in $T_{\frac{n}{m}}$ in terms of their bipartition.

Abstract Image

查看原文本刊更多论文

孤立韧性小于 1 的图形中的分数因子和分量因子

让是一个简单图，让是两个整数，且。我们证明，对于每一个当且仅当有-因子时，其中表示孤立顶点的数量，并且是一个特殊的树族。此外，我们还用树的二分法来描述树的特征。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

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 .