{"title":"三流临界图的稀疏性","authors":"Zdeněk Dvořák , Sergey Norin","doi":"10.1016/j.ejc.2025.104174","DOIUrl":null,"url":null,"abstract":"<div><div>A connected graph <span><math><mi>G</mi></math></span> is 3-flow-critical if <span><math><mi>G</mi></math></span> does not have a nowhere-zero 3-flow, but every proper contraction of <span><math><mi>G</mi></math></span> does. We prove that every <span><math><mi>n</mi></math></span>-vertex 3-flow-critical graph other than <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> has at least <span><math><mrow><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi></mrow></math></span> edges. This bound is tight up to lower-order terms, answering a question of Li et al. (2022). It also generalizes the result of Koester (1991) on the maximum average degree of 4-critical planar graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104174"},"PeriodicalIF":0.9000,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sparsity of 3-flow-critical graphs\",\"authors\":\"Zdeněk Dvořák , Sergey Norin\",\"doi\":\"10.1016/j.ejc.2025.104174\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>A connected graph <span><math><mi>G</mi></math></span> is 3-flow-critical if <span><math><mi>G</mi></math></span> does not have a nowhere-zero 3-flow, but every proper contraction of <span><math><mi>G</mi></math></span> does. We prove that every <span><math><mi>n</mi></math></span>-vertex 3-flow-critical graph other than <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> has at least <span><math><mrow><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi></mrow></math></span> edges. This bound is tight up to lower-order terms, answering a question of Li et al. (2022). It also generalizes the result of Koester (1991) on the maximum average degree of 4-critical planar graphs.</div></div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":\"128 \",\"pages\":\"Article 104174\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2025-05-14\",\"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/S0195669825000587\",\"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/S0195669825000587","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
A connected graph is 3-flow-critical if does not have a nowhere-zero 3-flow, but every proper contraction of does. We prove that every -vertex 3-flow-critical graph other than and has at least edges. This bound is tight up to lower-order terms, answering a question of Li et al. (2022). It also generalizes the result of Koester (1991) on the maximum average degree of 4-critical planar graphs.
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.