An extension of Nash-Williams and Tutte's Theorem

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-09-26 DOI:10.1002/jgt.23189

Xuqian Fang, Daqing Yang

{"title":"An extension of Nash-Williams and Tutte's Theorem","authors":"Xuqian Fang, Daqing Yang","doi":"10.1002/jgt.23189","DOIUrl":null,"url":null,"abstract":"The celebrated Nash-Williams and Tutte's Theorem states that a graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</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> edge-disjoint spanning trees if and only if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>ν</mi>\n \n <mi>f</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> ${\\nu }_{f}(G)\\ge k$</annotation>\n </semantics></math>, where\n\n In this paper, we prove that, for integers <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>0</mn>\n </mrow>\n </mrow>\n <annotation> $k\\ge 0$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n \n <mo>≥</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> $d\\ge 1$</annotation>\n </semantics></math>, if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>ν</mi>\n \n <mi>f</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>></mo>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mfrac>\n <mrow>\n <mi>d</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mi>d</mi>\n </mfrac>\n </mrow>\n </mrow>\n <annotation> ${\\nu }_{f}(G)\\gt k+\\frac{d-1}{d}$</annotation>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</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> edge-disjoint spanning trees and another forest <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n \n <mo>></mo>\n \n <mfrac>\n <mrow>\n <mi>d</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mi>d</mi>\n </mfrac>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\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 \n <mo>∣</mo>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $| E(F)| \\gt \\frac{d-1}{d}(| V(G)| -1)$</annotation>\n </semantics></math>, and if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> is not a spanning tree, then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> has a component with at least <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n </mrow>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math> edges. Moreover, the bound of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>ν</mi>\n \n <mi>f</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${\\nu }_{f}(G)$</annotation>\n </semantics></math> is sharp.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"361-367"},"PeriodicalIF":0.9000,"publicationDate":"2024-09-26","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.23189","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

The celebrated Nash-Williams and Tutte's Theorem states that a graph $G$ contains $k$ edge-disjoint spanning trees if and only if $ν_{f} (G) \geq k$ , where

In this paper, we prove that, for integers $k \geq 0$ , $d \geq 1$ , if $ν_{f} (G) > k + \frac{d - 1}{d}$ , then $G$ contains $k$ edge-disjoint spanning trees and another forest $F$ with $∣ E (F) ∣ > \frac{d - 1}{d} (∣ V (G) ∣ - 1)$ , and if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Moreover, the bound of $ν_{f} (G)$ is sharp.

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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 .