{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T19:32:28Z","timestamp":1773689548611,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T00:00:00Z","timestamp":1694044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Xinjiang, China","award":["2021D01C116"],"award-info":[{"award-number":["2021D01C116"]}]},{"name":"Natural Science Foundation of Xinjiang, China","award":["12261085"],"award-info":[{"award-number":["12261085"]}]},{"name":"National Natural Science Foundation of China","award":["2021D01C116"],"award-info":[{"award-number":["2021D01C116"]}]},{"name":"National Natural Science Foundation of China","award":["12261085"],"award-info":[{"award-number":["12261085"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e1,e2,\u2026,ek of G, there exist k vertex-disjoint cycles C1,C2,\u2026,Ck in G such that V(C1)\u222aV(C2)\u222a\u22ef\u222aV(Ck)=V(G) and ei\u2208E(Ci) for all 1\u2264i\u2264k. According to the definition, the problem of finding hamiltonian cycle focuses on k=1. The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube Qn is spanning k-edge-cyclable for 1\u2264k\u2264n\u22121 and n\u22652. This is the best possible result, in the sense that the n-dimensional hypercube Qn is not spanning n-edge-cyclable.<\/jats:p>","DOI":"10.3390\/axioms12090861","type":"journal-article","created":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T10:11:07Z","timestamp":1694081467000},"page":"861","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Embedding Spanning Disjoint Cycles in Hypercube Networks with Prescribed Edges in Each Cycle"],"prefix":"10.3390","volume":"12","author":[{"given":"Weiyan","family":"Wu","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eminjan","family":"Sabir","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2478\/ijmce-2023-0001","article-title":"Remarks on bipolar cubic fuzzy graphs and its chemical applications","volume":"1","author":"Lu","year":"2023","journal-title":"Int. J. Math. Comput. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"399","DOI":"10.2478\/amns.2022.2.00022","article-title":"The Control Relationship between the Enterprise\u2019s Electrical Equipment and Mechanical Equipment Based on Graph Theory","volume":"8","author":"Wang","year":"2022","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_3","first-page":"703","article-title":"Spatial-temporal graph neural network based on node attention","volume":"7","author":"Li","year":"2022","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2717","DOI":"10.2478\/amns.2021.2.00285","article-title":"Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model","volume":"8","author":"Wang","year":"2022","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2729","DOI":"10.2478\/amns.2021.2.00283","article-title":"Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction","volume":"8","author":"Sun","year":"2022","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.jpdc.2018.06.007","article-title":"Hamiltonian cycle and path embeddings in 3-ary n-cubes based on K1,3-structure faults","volume":"120","author":"Lv","year":"2018","journal-title":"J. Parallel Distr. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2944","DOI":"10.1093\/comjnl\/bxv019","article-title":"Hamiltonian Properties of DCell Networks","volume":"58","author":"Wang","year":"2015","journal-title":"Comput. J."},{"key":"ref_8","unstructured":"Leighton, F.T. (1991). Introduction to Parallel Algorithms and Architecture: Arrays, Trees, Hypercubes, Morgan Kaufmann Publishers Inc."},{"key":"ref_9","first-page":"127481","article-title":"Embedding spanning disjoint cycles in enhanced hypercube networks with prescribed vertices in each cycle","volume":"435","author":"Qiao","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2356","DOI":"10.1016\/j.disc.2012.04.003","article-title":"Solution to an open problem on 4-ordered Hamiltonian graphs","volume":"312","author":"Hsu","year":"2012","journal-title":"Discrete Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/S0095-8956(03)00033-9","article-title":"Hamiltonian decompositions of cayley graphs on abelian groups of even order","volume":"88","author":"Liu","year":"2003","journal-title":"J. Comb. Theory B"},{"key":"ref_12","first-page":"314","article-title":"Cycle embedding of augmented cubes","volume":"191","author":"Hsieh","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hsu, L.-H., and Lin, C.-K. (2008). Graph Theory and Interconnection Networks, CRC Press.","DOI":"10.1201\/9781420044829"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2992","DOI":"10.1016\/j.dam.2013.07.004","article-title":"Disjoint cycles in hypercubes with prescribed vertices in each cycle","volume":"161","author":"Lin","year":"2013","journal-title":"Discrete Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1002\/(SICI)1097-0118(199710)26:2<105::AID-JGT5>3.0.CO;2-W","article-title":"Covering a graph with cycles passing through given edges","volume":"26","author":"Wang","year":"1997","journal-title":"J. Graph Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s003730050005","article-title":"Vertex-disjoint cycles containing specified edges","volume":"16","author":"Egawa","year":"2000","journal-title":"Graphs Comb."},{"key":"ref_17","first-page":"115","article-title":"Covering a bipartite graph with cycles passing through given edges","volume":"19","author":"Wang","year":"1999","journal-title":"J. Graph Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1002\/jgt.10113","article-title":"Two-factors each component of which contains a specified vertex","volume":"43","author":"Egawa","year":"2003","journal-title":"J. Graph Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1002\/jgt.10090","article-title":"Vertex-disjoint cycles containing prescribed vertices","volume":"42","author":"Ishigami","year":"2003","journal-title":"J. Graph Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00373-017-1873-5","article-title":"Degree conditions for the existence of vertex-disjoint cycles and paths: A Survey","volume":"4","author":"Chiba","year":"2018","journal-title":"Graphs Comb."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6299","DOI":"10.1016\/j.disc.2008.04.017","article-title":"A look at cycles containing specified elements of a graph","volume":"309","author":"Gould","year":"2009","journal-title":"Discrete Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kung, T.-L., Hung, C.-N., Lin, C.-K., Chen, H.-C., Lin, C.-H., and Hsu, L.-H. (2016, January 6\u20138). A framework of cycle-based clustering on the crossed cube architecture. Proceedings of the International Conference on Innovation Mobile and Internet Services in Ubiquitous Computing, Fukuoka, Japan.","DOI":"10.1109\/IMIS.2016.72"},{"key":"ref_23","first-page":"283","article-title":"Disjoint cycles through prescribed vertices in multidimensional tori","volume":"4","author":"Shinde","year":"2021","journal-title":"J. Ramanujan Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"713","DOI":"10.7151\/dmgt.2150","article-title":"2-spanning cyclability problems of some generalized Petersen graphs","volume":"40","author":"Yang","year":"2020","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.dam.2022.12.005","article-title":"The spanning cyclability of Cayley graphs generated by transposition trees","volume":"328","author":"Qiao","year":"2023","journal-title":"Discrete Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.ipl.2006.08.011","article-title":"Path bipancyclicity of hypercubes","volume":"101","author":"Tsai","year":"2007","journal-title":"Inf. Process. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.ipl.2015.09.018","article-title":"Hamiltonian of hypercubes with faulty vertices","volume":"116","author":"Chen","year":"2016","journal-title":"Inf. Process. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1016\/S0167-8191(03)00058-9","article-title":"Fault-tolerant cycle embedding in the hypercube","volume":"29","author":"Fu","year":"2003","journal-title":"Parallel Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0020-0190(03)00258-8","article-title":"Bipannectivity and edge-fault-tolerant bipancyclicity of hypercubes","volume":"87","author":"Li","year":"2003","journal-title":"Inf. Process. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1109\/12.2234","article-title":"Topological properties of hypercubes","volume":"37","author":"Saad","year":"1988","journal-title":"IEEE Trans. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s11464-009-0017-5","article-title":"Survey on path and cycle embedding in some networks","volume":"4","author":"Xu","year":"2009","journal-title":"Front. Math. China"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1142\/S0219265907001941","article-title":"Hamiltonian laceability of faulty hypercubes","volume":"8","author":"Sun","year":"2007","journal-title":"J. Interconnect. Netw."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/9\/861\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:46:36Z","timestamp":1760129196000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/9\/861"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,7]]},"references-count":32,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["axioms12090861"],"URL":"https:\/\/doi.org\/10.3390\/axioms12090861","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,7]]}}}