{"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/few-hamiltonian-cycles-in-graphs-with-one-or","title":"Few hamiltonian cycles in graphs with one or two vertex degrees","arxiv_id":"2211.08105","date":"2022-11-15","proceeding":null,"authors":["Jan Goedgebeur","Jorik Jooken","On-Hei Solomon Lo","Ben Seamone","Carol T. Zamfirescu"],"abstract":"We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe's computational enumerative results from [Experim. Math. 27 (2018) 426-430]. Thereafter, we use the Lov\\'asz Local Lemma to extend Thomassen's independent dominating set method. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen. Motivated by an observation of Aldred and Thomassen, we prove that for every $\\kappa \\in \\{ 2, 3 \\}$ and any positive integer $k$, there are infinitely many non-regular graphs of connectivity $\\kappa$ containing exactly one hamiltonian cycle and in which every vertex has degree $3$ or $2k$.","url_abs":"https://arxiv.org/abs/2211.08105v1","url_pdf":"https://arxiv.org/pdf/2211.08105v1.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":"few-hamiltonian-cycles-in-graphs-with-one-or","repo_url":"https://github.com/jorikjooken/hamiltonian_cycles","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}