{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:01:06Z","timestamp":1760058066313,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T00:00:00Z","timestamp":1741564800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007834","name":"Ningbo Natural Science Foundation","doi-asserted-by":"publisher","award":["2023J186","Y202351606","2090011540015"],"award-info":[{"award-number":["2023J186","Y202351606","2090011540015"]}],"id":[{"id":"10.13039\/100007834","id-type":"DOI","asserted-by":"publisher"}]},{"name":"General Scientific Research Foundation of Zhejiang Province Education Department","award":["2023J186","Y202351606","2090011540015"],"award-info":[{"award-number":["2023J186","Y202351606","2090011540015"]}]},{"name":"Ph.D Research Foundation of Ningbo University of Technology","award":["2023J186","Y202351606","2090011540015"],"award-info":[{"award-number":["2023J186","Y202351606","2090011540015"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The n-dimensional enhanced hypercube Qn,k(1\u2264k\u2264n\u22121) is a well-known variation of hypercube networks. Its structure can be obtained from the hypercube by adding 2n\u22121 complementary edges. We denote a network G to be a conditionally faulty model if every fault-free vertex of G connects at least two fault-free edges. Let Fv and Fe be the set of faulty vertices and faulty edges in Qn,k(1\u2264k\u2264n\u22121), respectively. In this paper, for the conditionally faulty Qn,k with |Fv|+|Fe|\u22642n\u22125, where n(\u22653) and k have different parity, I prove that Qn,k\u2212Fv\u2212Fe contains a fault-free cycle with every odd length l, where n\u2212k+4\u2264l\u22642n\u22122|Fv|\u22121.<\/jats:p>","DOI":"10.3390\/axioms14030201","type":"journal-article","created":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T05:46:52Z","timestamp":1741585612000},"page":"201","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Odd Cycles in Conditionally Faulty Enhanced Hypercube Networks"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1317-8935","authenticated-orcid":false,"given":"Min","family":"Liu","sequence":"first","affiliation":[{"name":"School of Statistics and Data Science, Ningbo University of Technology, Ningbo 315211, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Leighton, F.T. 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