{"title":"The Frank number and nowhere-zero flows on graphs","authors":"Jan Goedgebeur , Edita Máčajová , Jarne Renders","doi":"10.1016/j.ejc.2025.104127","DOIUrl":null,"url":null,"abstract":"<div><div>An edge <span><math><mi>e</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is called <em>deletable</em> for some orientation <span><math><mi>o</mi></math></span> if the restriction of <span><math><mi>o</mi></math></span> to <span><math><mrow><mi>G</mi><mo>−</mo><mi>e</mi></mrow></math></span> is a strong orientation. Inspired by a problem of Frank, in 2021 Hörsch and Szigeti proposed a new parameter for 3-edge-connected graphs, called the Frank number, which refines <span><math><mi>k</mi></math></span>-edge-connectivity. The <em>Frank number</em> is defined as the minimum number of orientations of <span><math><mi>G</mi></math></span> for which every edge of <span><math><mi>G</mi></math></span> is deletable in at least one of them. They showed that every 3-edge-connected graph has Frank number at most 7 and that in case these graphs are also 5-edge-colourable the parameter is at most 3. Here we strengthen both results by showing that every 3-edge-connected graph has Frank number at most 4 and that every graph which is 3-edge-connected and 3-edge-colourable has Frank number 2. The latter also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number 2 and use them in an algorithm to computationally show that the Petersen graph is the only cyclically 4-edge-connected cubic graph up to 36 vertices having Frank number greater than 2.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104127"},"PeriodicalIF":1.0000,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669825000095","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
Abstract
An edge of a graph is called deletable for some orientation if the restriction of to is a strong orientation. Inspired by a problem of Frank, in 2021 Hörsch and Szigeti proposed a new parameter for 3-edge-connected graphs, called the Frank number, which refines -edge-connectivity. The Frank number is defined as the minimum number of orientations of for which every edge of is deletable in at least one of them. They showed that every 3-edge-connected graph has Frank number at most 7 and that in case these graphs are also 5-edge-colourable the parameter is at most 3. Here we strengthen both results by showing that every 3-edge-connected graph has Frank number at most 4 and that every graph which is 3-edge-connected and 3-edge-colourable has Frank number 2. The latter also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number 2 and use them in an algorithm to computationally show that the Petersen graph is the only cyclically 4-edge-connected cubic graph up to 36 vertices having Frank number greater than 2.
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.