{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T09:17:55Z","timestamp":1760347075293},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2022,5,30]],"date-time":"2022-05-30T00:00:00Z","timestamp":1653868800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Hadwiger\u2019s conjecture asserts that every graph without a <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline1.png\" \/><jats:tex-math>\n$K_t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-minor is <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline2.png\" \/><jats:tex-math>\n$(t-1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-colourable. It is known that the exact version of Hadwiger\u2019s conjecture does not extend to list colouring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline3.png\" \/><jats:tex-math>\n$c$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that every graph with no <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline4.png\" \/><jats:tex-math>\n$K_t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-minor has list chromatic number at most <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline5.png\" \/><jats:tex-math>\n$ct$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. More specifically, they also conjectured that this holds for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline6.png\" \/><jats:tex-math>\n$c=\\frac{3}{2}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p><jats:p>Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline7.png\" \/><jats:tex-math>\n$K_t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-minor is at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline8.png\" \/><jats:tex-math>\n$(2-o(1))t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and hence <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline9.png\" \/><jats:tex-math>\n$c \\ge 2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in the above conjecture is necessary. This improves the previous best lower bound by Bar\u00e1t, Joret and Wood (2011), who proved that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548322000116_inline10.png\" \/><jats:tex-math>\n$c \\ge \\frac{4}{3}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Our lower-bound examples are obtained via the probabilistic method.<\/jats:p>","DOI":"10.1017\/s0963548322000116","type":"journal-article","created":{"date-parts":[[2022,5,30]],"date-time":"2022-05-30T10:07:57Z","timestamp":1653905277000},"page":"1070-1075","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":6,"title":["Improved lower bound for the list chromatic number of graphs with no <i>K<sub>t<\/sub><\/i> minor"],"prefix":"10.1017","volume":"31","author":[{"given":"Raphael","family":"Steiner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,5,30]]},"reference":[{"key":"S0963548322000116_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01202354"},{"key":"S0963548322000116_ref22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01594196"},{"key":"S0963548322000116_ref24","unstructured":"[24] Open Problem Garden. 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(2020) An even better density increment theorem and its application to Hadwiger\u2019s conjecture. arXiv preprint, arXiv: 2006.14945."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548322000116","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T04:52:01Z","timestamp":1665636721000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548322000116\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,30]]},"references-count":24,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["S0963548322000116"],"URL":"https:\/\/doi.org\/10.1017\/s0963548322000116","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,30]]},"assertion":[{"value":"\u00a9 The Author(s), 2022. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}