{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,13]],"date-time":"2025-12-13T06:58:04Z","timestamp":1765609084205,"version":"3.40.5"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2024,5,13]],"date-time":"2024-05-13T00:00:00Z","timestamp":1715558400000},"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":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline1.png\"\/><jats:tex-math>\n$\\delta \\in (0,1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> there exists <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline2.png\"\/><jats:tex-math>\n$C = C(\\delta ) \\gt 0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that the following holds. Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline3.png\"\/><jats:tex-math>\n$D_0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be an <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline4.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-vertex digraph with minimum semidegree at least <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline5.png\"\/><jats:tex-math>\n$\\delta n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and suppose that each edge of the union of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline6.png\"\/><jats:tex-math>\n$D_0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with a copy of the random digraph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline7.png\"\/><jats:tex-math>\n$\\mathbf{D}(n,C\/n)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> on the same vertex set gets a colour in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline8.png\"\/><jats:tex-math>\n$[n]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> independently and uniformly at random. Then, with high probability, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline9.png\"\/><jats:tex-math>\n$D_0 \\cup \\mathbf{D}(n,C\/n)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has a rainbow directed Hamilton cycle.<\/jats:p><jats:p>This improves a result of Aigner-Horev and Hefetz ((2021) <jats:italic>SIAM J. Discrete Math.<\/jats:italic><jats:bold>35<\/jats:bold>(3) 1569\u20131577), who proved the same in the undirected setting when the edges are coloured uniformly in a set of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000130_inline10.png\"\/><jats:tex-math>\n$(1 + \\varepsilon )n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> colours.<\/jats:p>","DOI":"10.1017\/s0963548324000130","type":"journal-article","created":{"date-parts":[[2024,5,13]],"date-time":"2024-05-13T08:57:16Z","timestamp":1715590636000},"page":"624-642","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Rainbow Hamiltonicity in uniformly coloured perturbed digraphs"],"prefix":"10.1017","volume":"33","author":[{"given":"Kyriakos","family":"Katsamaktsis","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1193-2017","authenticated-orcid":false,"given":"Shoham","family":"Letzter","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0629-5431","authenticated-orcid":false,"given":"Amedeo","family":"Sgueglia","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2024,5,13]]},"reference":[{"key":"S0963548324000130_ref15","first-page":"495","article-title":"Une condition suffisante d\u2019existence d\u2019un circuit Hamiltonien","volume":"251","author":"Ghouila-Houri","year":"1960","journal-title":"Comptes Rendus Hebdomadaires des S\u00e9ances de l\u2019Acad\u00e9mie des Sciences"},{"key":"S0963548324000130_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20594"},{"key":"S0963548324000130_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548323000391"},{"key":"S0963548324000130_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548301005004"},{"key":"S0963548324000130_ref1","doi-asserted-by":"publisher","DOI":"10.1137\/20M1332992"},{"key":"S0963548324000130_ref11","article-title":"Closing gaps in problems related to Hamilton cycles in random graphs and hypergraphs","volume":"22","author":"Ferber","year":"2015","journal-title":"Electron. 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Trudov Novosibirsk"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548324000130","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T22:46:11Z","timestamp":1728427571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548324000130\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,13]]},"references-count":25,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["S0963548324000130"],"URL":"https:\/\/doi.org\/10.1017\/s0963548324000130","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2024,5,13]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. 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