{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:20:39Z","timestamp":1758824439437},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,9,24]],"date-time":"2014-09-24T00:00:00Z","timestamp":1411516800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>We establish a relation between two uniform models of random<jats:italic>k<\/jats:italic>-graphs (for constant<jats:italic>k<\/jats:italic>\u2a7e 3) on<jats:italic>n<\/jats:italic>labelled vertices: \u210d<jats:sup><jats:italic>(k)<\/jats:italic><\/jats:sup><jats:italic>(n,m)<\/jats:italic>, the random<jats:italic>k<\/jats:italic>-graph with exactly<jats:italic>m<\/jats:italic>edges, and \u210d<jats:sup><jats:italic>(k)<\/jats:italic><\/jats:sup><jats:italic>(n,d)<\/jats:italic>, the random<jats:italic>d<\/jats:italic>-regular<jats:italic>k<\/jats:italic>-graph. By extending the switching technique of McKay and Wormald to<jats:italic>k<\/jats:italic>-graphs, we show that, for some range of<jats:italic>d = d(n)<\/jats:italic>and a constant<jats:italic>c<\/jats:italic>&gt; 0, if<jats:italic>m<\/jats:italic>~<jats:italic>cnd<\/jats:italic>, then one can couple \u210d<jats:sup><jats:italic>(k)<\/jats:italic><\/jats:sup><jats:italic>(n,m)<\/jats:italic>and \u210d<jats:sup><jats:italic>(k)<\/jats:italic><\/jats:sup><jats:italic>(n,d)<\/jats:italic>so that the latter contains the former with probability tending to one as<jats:italic>n<\/jats:italic>\u2192 \u221e. In view of known results on the existence of a loose Hamilton cycle in \u210d<jats:sup><jats:italic>(k)<\/jats:italic><\/jats:sup><jats:italic>(n,m)<\/jats:italic>, we conclude that \u210d<jats:sup><jats:italic>(k)<\/jats:italic><\/jats:sup><jats:italic>(n,d)<\/jats:italic>contains a loose Hamilton cycle when<jats:italic>d<\/jats:italic>\u226b log<jats:italic>n<\/jats:italic>(or just<jats:italic>d<\/jats:italic>\u2a7e<jats:italic>C<\/jats:italic>log<jats:italic>n<\/jats:italic>, if<jats:italic>k<\/jats:italic>= 3) and<jats:italic>d<\/jats:italic>=<jats:italic>o<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>1\/2<\/jats:sup>).<\/jats:p>","DOI":"10.1017\/s0963548314000406","type":"journal-article","created":{"date-parts":[[2014,9,24]],"date-time":"2014-09-24T10:35:38Z","timestamp":1411554938000},"page":"179-194","source":"Crossref","is-referenced-by-count":3,"title":["Loose Hamilton Cycles in Regular Hypergraphs"],"prefix":"10.1017","volume":"24","author":[{"given":"ANDRZEJ","family":"DUDEK","sequence":"first","affiliation":[]},{"given":"ALAN","family":"FRIEZE","sequence":"additional","affiliation":[]},{"given":"ANDRZEJ","family":"RUCI\u0143SKI","sequence":"additional","affiliation":[]},{"given":"MATAS","family":"\u0160ILEIKIS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,9,24]]},"reference":[{"key":"S0963548314000406_ref13","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"S0963548314000406_ref5","first-page":"23","volume-title":"Random Graphs '83: Pozna\u0144 1983","author":"Bollob\u00e1s","year":"1985"},{"key":"S0963548314000406_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(78)90059-6"},{"key":"S0963548314000406_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(80)80030-8"},{"key":"S0963548314000406_ref9","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20404"},{"key":"S0963548314000406_ref12","doi-asserted-by":"crossref","DOI":"10.37236\/477","article-title":"Loose Hamilton cycles in random 3-uniform hypergraphs","volume":"17","author":"Frieze","year":"2010","journal-title":"Electron. J. Combin."},{"key":"S0963548314000406_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(90)90029-E"},{"key":"S0963548314000406_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2013.07.018"},{"key":"S0963548314000406_ref17","first-page":"195","volume-title":"Algorithms and Combinatorics","author":"McDiarmid","year":"1998"},{"key":"S0963548314000406_ref1","volume-title":"Oxford Studies in Probability","author":"Barbour","year":"1992"},{"key":"S0963548314000406_ref7","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1017\/S0963548301005090","article-title":"Random regular graphs of non-constant degree: Connectivity and Hamiltonicity","volume":"11","author":"Cooper","year":"2002","journal-title":"Combin. Probab. Comput."},{"key":"S0963548314000406_ref4","volume-title":"Cambridge Studies in Advanced Mathematics","author":"Bollob\u00e1s","year":"2001"},{"key":"S0963548314000406_ref16","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.1013"},{"key":"S0963548314000406_ref21","first-page":"239","volume-title":"London Mathematical Society Lecture Note Series","author":"Wormald","year":"1999"},{"key":"S0963548314000406_ref15","doi-asserted-by":"publisher","DOI":"10.1145\/1008328.1008329"},{"key":"S0963548314000406_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2003.10.007"},{"key":"S0963548314000406_ref10","doi-asserted-by":"crossref","first-page":"#44","DOI":"10.37236\/2523","article-title":"Optimal divisibility conditions for loose Hamilton cycles in random hypergraphs","volume":"19","author":"Dudek","year":"2012","journal-title":"Electron. J. Combin."},{"key":"S0963548314000406_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240020103"},{"key":"S0963548314000406_ref8","doi-asserted-by":"crossref","first-page":"#48","DOI":"10.37236\/535","article-title":"Loose Hamilton cycles in random uniform hypergraphs","volume":"18","author":"Dudek","year":"2011","journal-title":"Electron. J. Combin."},{"key":"S0963548314000406_ref19","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240030202"},{"key":"S0963548314000406_ref20","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240050209"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548314000406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,24]],"date-time":"2020-08-24T21:43:57Z","timestamp":1598305437000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548314000406\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,24]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["S0963548314000406"],"URL":"https:\/\/doi.org\/10.1017\/s0963548314000406","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,9,24]]}}}