具有可数端点的图的纳什-威廉姆斯定向定理

IF 1 3区数学 Q1 MATHEMATICS European Journal of Combinatorics Pub Date : 2024-08-21 DOI:10.1016/j.ejc.2024.104043

Amena Assem , Marcel Koloschin , Max Pitz

{"title":"具有可数端点的图的纳什-威廉姆斯定向定理","authors":"Amena Assem , Marcel Koloschin , Max Pitz","doi":"10.1016/j.ejc.2024.104043","DOIUrl":null,"url":null,"abstract":"<div>Nash-Williams proved in 1960 that a finite graph admits a <math><mi>k</mi></math>-arc-connected orientation if and only if it is <math><mrow><mn>2</mn><mi>k</mi></mrow></math>-edge-connected, and conjectured that the same result should hold for all infinite graphs, too.Progress on Nash-Williams’s problem was made by C. Thomassen, who proved in 2016 that all <math><mrow><mn>8</mn><mi>k</mi></mrow></math>-edge-connected infinite graphs admit a <math><mi>k</mi></math>-arc connected orientation, and by the first author, who recently showed that edge-connectivity of <math><mrow><mn>4</mn><mi>k</mi></mrow></math> suffices for locally-finite, 1-ended graphs.In the present article, we establish the optimal bound <math><mrow><mn>2</mn><mi>k</mi></mrow></math> in Nash-Williams’s conjecture for all locally finite graphs with countably many ends.</div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001288/pdfft?md5=aee14e19cbba10a0a057111710d01339&pid=1-s2.0-S0195669824001288-main.pdf","citationCount":"0","resultStr":"{\"title\":\"The Nash-Williams orientation theorem for graphs with countably many ends\",\"authors\":\"Amena Assem , Marcel Koloschin , Max Pitz\",\"doi\":\"10.1016/j.ejc.2024.104043\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>Nash-Williams proved in 1960 that a finite graph admits a <math><mi>k</mi></math>-arc-connected orientation if and only if it is <math><mrow><mn>2</mn><mi>k</mi></mrow></math>-edge-connected, and conjectured that the same result should hold for all infinite graphs, too.Progress on Nash-Williams’s problem was made by C. Thomassen, who proved in 2016 that all <math><mrow><mn>8</mn><mi>k</mi></mrow></math>-edge-connected infinite graphs admit a <math><mi>k</mi></math>-arc connected orientation, and by the first author, who recently showed that edge-connectivity of <math><mrow><mn>4</mn><mi>k</mi></mrow></math> suffices for locally-finite, 1-ended graphs.In the present article, we establish the optimal bound <math><mrow><mn>2</mn><mi>k</mi></mrow></math> in Nash-Williams’s conjecture for all locally finite graphs with countably many ends.</div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0195669824001288/pdfft?md5=aee14e19cbba10a0a057111710d01339&pid=1-s2.0-S0195669824001288-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"European Journal of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0195669824001288\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669824001288","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

纳什-威廉姆斯（Nash-Williams）在 1960 年证明，当且仅当一个有限图是 2k 边连接时，它才会有一个 k 弧连接的方向，并猜想同样的结果也应该适用于所有无限图。托马森（C. Thomassen）在 2016 年证明了所有 8k 边连接的无限图都承认 k 弧连接的方向，而第一作者最近也证明了对于局部有限的 1 端图，4k 的边连接性就足够了。在本文中，我们为所有具有可数端点的局部有限图建立了 Nash-Williams 猜想中的最优约束 2k。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

The Nash-Williams orientation theorem for graphs with countably many ends

Nash-Williams proved in 1960 that a finite graph admits a $k$ -arc-connected orientation if and only if it is $2 k$ -edge-connected, and conjectured that the same result should hold for all infinite graphs, too.

Progress on Nash-Williams’s problem was made by C. Thomassen, who proved in 2016 that all $8 k$ -edge-connected infinite graphs admit a $k$ -arc connected orientation, and by the first author, who recently showed that edge-connectivity of $4 k$ suffices for locally-finite, 1-ended graphs.

In the present article, we establish the optimal bound $2 k$ in Nash-Williams’s conjecture for all locally finite graphs with countably many ends.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

European Journal of Combinatorics 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.

期刊最新文献

A combinatorial PROP for bialgebras Signed Mahonian polynomials on derangements in classical Weyl groups Degree conditions for Ramsey goodness of paths Bounded unique representation bases for the integers On the faces of unigraphic 3-polytopes