Marthe Bonamy , Marc Heinrich , Clément Legrand-Duchesne , Jonathan Narboni
{"title":"关于Hadwiger猜想的一个变色版本","authors":"Marthe Bonamy , Marc Heinrich , Clément Legrand-Duchesne , Jonathan Narboni","doi":"10.1016/j.jctb.2023.10.001","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that for any <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, for any large enough <em>t</em>, there is a graph that admits no <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-minor but admits a <span><math><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo><mi>t</mi></math></span>-colouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 364-370"},"PeriodicalIF":1.3000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On a recolouring version of Hadwiger's conjecture\",\"authors\":\"Marthe Bonamy , Marc Heinrich , Clément Legrand-Duchesne , Jonathan Narboni\",\"doi\":\"10.1016/j.jctb.2023.10.001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that for any <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, for any large enough <em>t</em>, there is a graph that admits no <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>-minor but admits a <span><math><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo><mi>t</mi></math></span>-colouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.</p></div>\",\"PeriodicalId\":54865,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series B\",\"volume\":\"164 \",\"pages\":\"Pages 364-370\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series B\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0095895623000837\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2023/10/30 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series B","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895623000837","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/10/30 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
We prove that for any , for any large enough t, there is a graph that admits no -minor but admits a -colouring that is “frozen” with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.
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