无限图的Nash-Williams方向猜想

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-10-08 DOI:10.1002/jgt.23192

Amena Assem

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He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mn>8</mn>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0003\" wiley:location=\"equation/jgt23192-math-0003.png\"><mrow><mrow><mn>8</mn><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-edge-connected infinite graphs admit a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0004\" wiley:location=\"equation/jgt23192-math-0004.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-arc-connected orientation. Here we improve this result for the class of one-ended locally finite graphs and show that an edge-connectivity of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mn>4</mn>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0005\" wiley:location=\"equation/jgt23192-math-0005.png\"><mrow><mrow><mn>4</mn><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math> is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"608-619"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23192","citationCount":"0","resultStr":"{\"title\":\"Towards Nash-Williams orientation conjecture for infinite graphs\",\"authors\":\"Amena Assem\",\"doi\":\"10.1002/jgt.23192\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1960 Nash-Williams proved that an edge-connectivity of <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" altimg=\\\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0001\\\" wiley:location=\\\"equation/jgt23192-math-0001.png\\\"><mrow><mrow><mn>2</mn><mi>k</mi></mrow></mrow></math></annotation>\\n </semantics></math> is sufficient for a finite graph to have a <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" altimg=\\\"urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0002\\\" wiley:location=\\\"equation/jgt23192-math-0002.png\\\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\\n </semantics></math>-arc-connected orientation. 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引用次数: 0

摘要

1960年，Nash-Williams证明了2 k的边连通性<；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0001”威利:位置= "方程/ jgt23192 -数学- 0001. png”祝辞& lt; mrow> & lt; mrow> & lt; mn> 2 & lt; / mn> & lt; mi> k< / mi> & lt; / mrow> & lt; / mrow> & lt; / math>对于有限图来说，k< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0002" wiley:location="equation/jgt23192-math-0002.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math>；-arc-connected取向。然后他推测无限图也是如此。2016年，托马森利用他自己在辅助提升图上的研究结果，证明了8 k<； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0003" wiley:location="equation/jgt23192-math-0003.png"><mrow><mrow>< mrow><mrow> k</ mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></math>；-边连接无限图承认k<； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0004" wiley:location="equation/jgt23192-math-0004.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></math>；-arc-connected取向。这里，我们对一类单端局部有限图改进了这一结果，并证明了4 k的边连通性<；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23192:jgt23192-math-0005”威利:位置= "方程/ jgt23192 -数学- 0005. png”祝辞& lt; mrow> & lt; mrow> & lt; mn> 4 & lt; / mn> & lt; mi> k< / mi> & lt; / mrow> & lt; / mrow> & lt; / math>在这种情况下就足够了。这一改进的关键是本文同一作者在另一篇论文中提出的关于提升图关键概念的结果，扩展了Ok， Richter和Thomassen的结果。

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Towards Nash-Williams orientation conjecture for infinite graphs

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Towards Nash-Williams orientation conjecture for infinite graphs

In 1960 Nash-Williams proved that an edge-connectivity of $2 k$ is sufficient for a finite graph to have a $k$ -arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that $8 k$ -edge-connected infinite graphs admit a $k$ -arc-connected orientation. Here we improve this result for the class of one-ended locally finite graphs and show that an edge-connectivity of $4 k$ is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen.

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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 .