{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T11:53:11Z","timestamp":1768218791029,"version":"3.49.0"},"reference-count":37,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:00:00Z","timestamp":1761609600000},"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":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The P\u00f3sa\u2013Seymour conjecture determines the minimum degree threshold for forcing the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100230_inline1.png\"\/>\n                        <jats:tex-math>$k$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    th power of a Hamilton cycle in a graph. After numerous partial results, Koml\u00f3s, S\u00e1rk\u00f6zy, and Szemer\u00e9di proved the conjecture for sufficiently large graphs. In this paper, we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100230_inline2.png\"\/>\n                        <jats:tex-math>$k$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100230_inline3.png\"\/>\n                        <jats:tex-math>$k$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Tur\u00e1n-type problems for oriented graphs are also discussed.\n                  <\/jats:p>","DOI":"10.1017\/s0963548325100230","type":"journal-article","created":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T09:33:52Z","timestamp":1761644032000},"page":"101-133","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Powers of Hamilton cycles in oriented and directed graphs"],"prefix":"10.1017","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7569-7952","authenticated-orcid":false,"given":"Louis","family":"DeBiasio","sequence":"first","affiliation":[{"name":"Miami University"}]},{"given":"Jie","family":"Han","sequence":"additional","affiliation":[{"name":"Beijing Institute of Technology"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9767-2863","authenticated-orcid":false,"given":"Allan","family":"Lo","sequence":"additional","affiliation":[{"name":"University of Birmingham"}]},{"given":"Theodore","family":"Molla","sequence":"additional","affiliation":[{"name":"University of South Florida"}]},{"given":"Sim\u00f3n","family":"Piga","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, Czech Academy of Sciences"}]},{"given":"Andrew","family":"Treglown","sequence":"additional","affiliation":[{"name":"University of Birmingham"}]}],"member":"56","published-online":{"date-parts":[[2025,10,28]]},"reference":[{"key":"S0963548325100230_ref14","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480192232254"},{"key":"S0963548325100230_ref20","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20344"},{"key":"S0963548325100230_ref32","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548305007042"},{"key":"S0963548325100230_ref17","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199611)23:3<241::AID-JGT4>3.0.CO;2-S"},{"key":"S0963548325100230_ref16","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1996.0039"},{"key":"S0963548325100230_ref21","first-page":"495","article-title":"Une condition suffisante d\u2019existence d\u2019un circuit hamiltonien","volume":"251","author":"Ghouila-Houri","year":"1960","journal-title":"C.R. 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(2025) On the Tur\u00e1nability and tileability of oriented graphs, arXiv preprint arXiv: 2507.13267."},{"key":"S0963548325100230_ref18","first-page":"197215","volume-title":"Recent Advances in Graph Theory","author":"Faudree","year":"1991"},{"key":"S0963548325100230_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548321000067"},{"key":"S0963548325100230_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01626028"},{"key":"S0963548325100230_ref29","doi-asserted-by":"publisher","DOI":"10.37236\/7759"},{"key":"S0963548325100230_ref12","first-page":"159","volume-title":"Theory of Graphs and its Applications","author":"Erd\u0151s","year":"1964"},{"key":"S0963548325100230_ref1","first-page":"141pp","article-title":"Blow-up lemmas for sparse graphs","volume":"8","author":"Allen","year":"2025","journal-title":"Disc. 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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, provided the original article 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"}]}}