{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T06:37:22Z","timestamp":1716878242242},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2020,9,8]],"date-time":"2020-09-08T00:00:00Z","timestamp":1599523200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2021,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this note we study the emergence of Hamiltonian Berge cycles in random <jats:italic>r<\/jats:italic>-uniform hypergraphs. For <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000437_inline1.png\" \/><jats:tex-math>\n$r\\geq 3$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> we prove an optimal stopping time result that if edges are sequentially added to an initially empty <jats:italic>r<\/jats:italic>-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erd\u0151s\u2013R\u00e9nyi random <jats:italic>r<\/jats:italic>-graph, and we also show that the <jats:italic>2<\/jats:italic>-out random <jats:italic>r<\/jats:italic>-graph with high probability has such a cycle. We obtain similar results for <jats:italic>weak Berge<\/jats:italic> cycles as well, thus resolving a conjecture of Poole.<\/jats:p>","DOI":"10.1017\/s0963548320000437","type":"journal-article","created":{"date-parts":[[2020,9,8]],"date-time":"2020-09-08T07:36:48Z","timestamp":1599550608000},"page":"228-238","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Hamiltonian Berge cycles in random hypergraphs"],"prefix":"10.1017","volume":"30","author":[{"given":"Deepak","family":"Bal","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ross","family":"Berkowitz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pat","family":"Devlin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mathias","family":"Schacht","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2020,9,8]]},"reference":[{"key":"S0963548320000437_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(83)90021-3"},{"key":"S0963548320000437_ref4","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20272"},{"key":"S0963548320000437_ref20","unstructured":"[20] Poole, D. (2014) On weak Hamiltonicity of a random hypergraph. arXiv:1410.7446"},{"key":"S0963548320000437_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/1098-2418(200007)16:4<369::AID-RSA6>3.0.CO;2-J"},{"key":"S0963548320000437_ref18","unstructured":"[18] K\u00fchn, D. and Osthus, D. (2014) Hamilton cycles in graphs and hypergraphs: an extremal perspective. Proceedings of the International Congress of Mathematicians\u2014Seoul 2014. Vol. IV, 381\u2013406, Kyung Moon Sa, Seoul."},{"key":"S0963548320000437_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511814068"},{"key":"S0963548320000437_ref11","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20404"},{"key":"S0963548320000437_ref13","doi-asserted-by":"publisher","DOI":"10.37236\/2523"},{"key":"S0963548320000437_ref2","doi-asserted-by":"publisher","DOI":"10.37236\/8339"},{"key":"S0963548320000437_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.endm.2016.09.032"},{"key":"S0963548320000437_ref3","unstructured":"[3] Berge, C. (1970) Graphes et Hypergraphes, Vol. 37 of Monographies Universitaires de Math\u00e9matiques. Dunod."},{"key":"S0963548320000437_ref15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316339831"},{"key":"S0963548320000437_ref9","doi-asserted-by":"publisher","DOI":"10.37236\/6890"},{"key":"S0963548320000437_ref19","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20690"},{"key":"S0963548320000437_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579431"},{"key":"S0963548320000437_ref1","first-page":"173","volume-title":"Studies","author":"Ajtai","year":"1985"},{"key":"S0963548320000437_ref17","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316479988.002"},{"key":"S0963548320000437_ref5","unstructured":"[5] Bollob\u00e1s, B. (1984) The evolution of sparse graphs. In Graph Theory and Combinatorics (Cambridge, 1983), pp. 35\u201357. Academic Press."},{"key":"S0963548320000437_ref10","doi-asserted-by":"publisher","DOI":"10.37236\/535"},{"key":"S0963548320000437_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2017.11.009"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548320000437","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,4]],"date-time":"2021-03-04T12:52:04Z","timestamp":1614862324000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548320000437\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,8]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["S0963548320000437"],"URL":"https:\/\/doi.org\/10.1017\/s0963548320000437","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,8]]},"assertion":[{"value":"\u00a9 The Author(s), 2020. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}