Triangle-Free Graphs of Toughness Approaching Two Without a 2-Factor

[2608.14500] Triangle-Free Graphs of Toughness Approaching Two Without a 2-Factor Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.14500 (math) [Submitted on 14 Aug 2026] Title: Triangle-Free Graphs of Toughness Approaching Two Without a 2-Factor Authors: Songling Shan View a PDF of the paper titled Triangle-Free Graphs of Toughness Approaching Two Without a 2-Factor, by Songling Shan View PDF HTML (experimental) Abstract: By work of Enomoto, Jackson, Katerinis, and Saito from 1985, every $2$-tough graph has a $2$-factor, and this toughness bound is best possible: for every $\varepsilon>0$, there exist $(2-\varepsilon)$-tough graphs with no $2$-factor. It is natural to ask whether the latter statement remains true for triangle-free graphs. Bauer, van den Heuvel, and Schmeichel conjectured this in 1996. In the same paper, they proposed an infinite family of triangle-free graphs with no $2$-factor whose toughness they believed approaches $2$, but the required toughness bound was not established. In this paper, we confirm their conjecture. For every even integer $q\ge 6$, we construct a triangle-free graph $G_q$ with no $2$-factor and with toughness \[ \tau(G_q) =\frac{2q^2-q-2}{q^2+q} =2-\frac{3q+2}{q^2+q}. \] In particular, $\tau(G_q)\to 2$ as $q\to\infty$, showing that the threshold $2$ for the existence of a $2$-factor remains best possible even within the class of triangle-free graphs. Subjects: Combinatorics (math.CO) Cite as: arXiv:2608.14500 [math.CO] (or arXiv:2608.14500v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.14500 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Songling Shan [ view email ] [v1] Fri, 14 Aug 2026 17:15:47 UTC (11 KB) Full-text links: Access Paper: View a PDF of the paper titled Triangle-Free Graphs of Toughness Approaching Two Without a 2-Factor, by Songling Shan View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 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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