{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/matchings-in-hypercubes-extend-to-long-cycles-1","title":"Matchings in Hypercubes Extend to Long Cycles","arxiv_id":"2501.19029","date":"2025-01-31","proceeding":null,"authors":["Jiří Fink","Vojtěch Hotmar"],"abstract":"The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\\{1, \\ldots, n\\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional hypercube $Q_n$ can be extended into a Hamilton cycle. We prove that matchings of $Q_n$ containing edges spanning at most $d = 5$ directions can be extended into a Hamilton cycle. We also characterize when these matchings of most $d = 5$ directions can be extended into a Hamilton path between two prescribed vertices. Our proofs work for arbitrary $d$ and $n$ where $d \\le n$ assuming some extension properties hold in $Q_d$ which we verified by a computer for $d=5$.","url_abs":"https://arxiv.org/abs/2501.19029v1","url_pdf":"https://arxiv.org/pdf/2501.19029v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"matchings-in-hypercubes-extend-to-long-cycles-1","repo_url":"https://gitlab.com/jirka.fink/matching_of_five_directions","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}