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{"title":"纳什-威廉姆斯和图特定理的推广","authors":"Xuqian Fang, Daqing Yang","doi":"10.1002/jgt.23189","DOIUrl":null,"url":null,"abstract":"<p>The celebrated Nash-Williams and Tutte's Theorem states that a graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> contains <span></span><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 <span></span><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 </p><p>In this paper, we prove that, for integers <span></span><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>, <span></span><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 <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> contains <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> with <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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.</p>","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":"{\"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\":\"<p>The celebrated Nash-Williams and Tutte's Theorem states that a graph <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> contains <span></span><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 <span></span><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 </p><p>In this paper, we prove that, for integers <span></span><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>, <span></span><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 <span></span><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 <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> contains <span></span><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 <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>F</mi>\\n </mrow>\\n </mrow>\\n <annotation> $F$</annotation>\\n </semantics></math> with <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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.</p>\",\"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}","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}
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