{"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/longest-cycles-in-vertex-transitive-and","title":"Longest cycles in vertex-transitive and highly connected graphs","arxiv_id":"2408.04618","date":"2024-08-08","proceeding":null,"authors":["Carla Groenland","Sean Longbrake","Raphael Steiner","Jérémie Turcotte","Liana Yepremyan"],"abstract":"We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for $r\\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\\geq 3$ vertices contains a cycle (and hence path) of length at least $\\Omega(n^{13/21})$, improving on $\\Omega(n^{3/5})$ from [DeVos, \\emph{arXiv:2302:04255}, 2023]. Second, we show that in every $r$-connected graph with $r\\geq 2$, any two longest cycles meet in at least $\\Omega(r^{5/8})$ vertices, improving on $\\Omega(r^{3/5})$ from [Chen, Faudree and Gould, \\emph{J. Combin. Theory, Ser.~ B}, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming.","url_abs":"https://arxiv.org/abs/2408.04618v1","url_pdf":"https://arxiv.org/pdf/2408.04618v1.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":"longest-cycles-in-vertex-transitive-and","repo_url":"https://github.com/tjeremie/long-cycles","is_official":1,"mentioned_in_paper":1,"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}