{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T18:39:17Z","timestamp":1649097557784},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2017,8,1]],"date-time":"2017-08-01T00:00:00Z","timestamp":1501545600000},"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":[[2017,11]]},"abstract":"<jats:p>A subset<jats:italic>C<\/jats:italic>of edges in a<jats:italic>k<\/jats:italic>-uniform hypergraph<jats:italic>H<\/jats:italic>is a<jats:italic>loose Hamilton cycle<\/jats:italic>if<jats:italic>C<\/jats:italic>covers all the vertices of<jats:italic>H<\/jats:italic>and there exists a cyclic ordering of these vertices such that the edges in<jats:italic>C<\/jats:italic>are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random<jats:italic>k<\/jats:italic>-uniform hypergraph<jats:italic>H<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic><\/jats:sup><jats:sub><jats:italic>n,p<\/jats:italic><\/jats:sub>has vertex set [<jats:italic>n<\/jats:italic>] and an edge set<jats:italic>E<\/jats:italic>obtained by adding each<jats:italic>k<\/jats:italic>-tuple<jats:italic>e<\/jats:italic>\u2208 (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548317000402_inline1\" \/><jats:tex-math>$\\binom{[n]}{k}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>) to<jats:italic>E<\/jats:italic>with probability<jats:italic>p<\/jats:italic>, independently at random.<\/jats:p><jats:p>Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but<jats:italic>o<\/jats:italic>(|<jats:italic>E<\/jats:italic>|) edges, referred to as the<jats:italic>packing problem<\/jats:italic>. While it is known that the threshold probability of the appearance of a loose Hamilton cycle in<jats:italic>H<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic><\/jats:sup><jats:sub><jats:italic>n,p<\/jats:italic><\/jats:sub>is<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548317000402_eqnU1\" \/><jats:tex-math>$p=\\Theta\\biggl(\\frac{\\log n}{n^{k-1}}\\biggr),$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group>the best known bounds for the packing problem are around<jats:italic>p<\/jats:italic>= polylog(<jats:italic>n<\/jats:italic>)\/<jats:italic>n<\/jats:italic>. Here we make substantial progress and prove the following asymptotically (up to a polylog(<jats:italic>n<\/jats:italic>) factor) best possible result: for<jats:italic>p<\/jats:italic>\u2265 log<jats:sup><jats:italic>C<\/jats:italic><\/jats:sup><jats:italic>n<\/jats:italic>\/<jats:italic>n<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic>\u22121<\/jats:sup>, a random<jats:italic>k<\/jats:italic>-uniform hypergraph<jats:italic>H<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic><\/jats:sup><jats:sub><jats:italic>n,p<\/jats:italic><\/jats:sub>with high probability contains<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548317000402_eqnU2\" \/><jats:tex-math>$N:=(1-o(1))\\frac{\\binom{n}{k}p}{n\/(k-1)}$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group>edge-disjoint loose Hamilton cycles.<\/jats:p><jats:p>Our proof utilizes and modifies the idea of \u2018online sprinkling\u2019 recently introduced by Vu and the first author.<\/jats:p>","DOI":"10.1017\/s0963548317000402","type":"journal-article","created":{"date-parts":[[2017,8,1]],"date-time":"2017-08-01T04:48:18Z","timestamp":1501562898000},"page":"839-849","source":"Crossref","is-referenced-by-count":0,"title":["Packing Loose Hamilton Cycles"],"prefix":"10.1017","volume":"26","author":[{"given":"ASAF","family":"FERBER","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"KYLE","family":"LUH","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"DANIEL","family":"MONTEALEGRE","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"OANH","family":"NGUYEN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,8,1]]},"reference":[{"key":"S0963548317000402_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12788-9_6"},{"key":"S0963548317000402_ref5","unstructured":"Ferber A. and Vu V. 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