Extremal problems about the order and size of nonhamiltonian locally linear graphs

[2402.16066] Extremal problems about the order and size of nonhamiltonian locally linear graphs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2402.16066 (math) [Submitted on 25 Feb 2024 ( v1 ), last revised 15 Sep 2026 (this version, v4)] Title: Extremal problems about the order and size of nonhamiltonian locally linear graphs Authors: Feng Liu , Leilei Zhang View a PDF of the paper titled Extremal problems about the order and size of nonhamiltonian locally linear graphs, by Feng Liu and 1 other authors View PDF HTML (experimental) Abstract: The relation between local structure and global cycle properties is a classical topic in graph theory. A graph $G$ is locally linear if $G[N(v)]$ is a path for every $v\in V(G)$. It is locally Hamiltonian or locally traceable if every vertex neighborhood induces a Hamiltonian or traceable graph, respectively. Earlier work by Pareek and Skupień, Skupień, Davies and Thomassen, Asratian and Oksimets, and de Wet and van Aardt studied extremal questions for these related graph classes. We prove that the minimum order of a nonhamiltonian locally linear graph is $12$ and that, for every integer $n\geq 12$, the minimum size of such a graph of order $n$ is $2n$. We also prove that every nontraceable locally linear graph of order $n$ has at least $2n+3$ edges. Comments: 16 pages, 4 figures Subjects: Combinatorics (math.CO) MSC classes: 05C35, 05C45, 05C69 Cite as: arXiv:2402.16066 [math.CO] (or arXiv:2402.16066v4 [math.CO] for this version) https://doi.org/10.48550/arXiv.2402.16066 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Feng Liu [ view email ] [v1] Sun, 25 Feb 2024 11:27:58 UTC (78 KB) [v2] Wed, 15 Apr 2026 03:24:55 UTC (18 KB) [v3] Wed, 9 Sep 2026 05:35:41 UTC (17 KB) [v4] Tue, 15 Sep 2026 09:17:22 UTC (24 KB) Full-text links: Access Paper: View a PDF of the paper titled Extremal problems about the order and size of nonhamiltonian locally linear graphs, by Feng Liu and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2024-02 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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