On the size of $(K_{t_1}, \ldots, K_{t_k})$-co-critical graphs

[2608.07422] On the size of $(K_{t_1}, \ldots, K_{t_k})$-co-critical graphs Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2608.07422 (math) [Submitted on 7 Aug 2026 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: On the size of $(K_{t_1}, \ldots, K_{t_k})$-co-critical graphs Authors: Zi-Xia Song View PDF HTML (experimental) Abstract: Given integers $k\ge2$ and $t_1, \ldots, t_k\ge2$, we write \emph{$G \rightarrow (K_{t_1}, \ldots, K_{t_k})$} if every $k$-coloring of the edges of a graph $G$ contains a monochromatic copy of $K_{t_i}$ in color $i$ for some $i\in\{1, \ldots, k\}$. A non-complete graph $G$ is \emph{$(K_{t_1}, \ldots, K_{t_k})$-co-critical} if $G \nrightarrow (K_{t_1}, \ldots, K_{t_k})$, but $G+e\rightarrow (K_{t_1}, \ldots, K_{t_k})$ for every edge $e\notin E(G)$. Let $r=R(K_{t_1}, \ldots, K_{t_k})$ denote the Ramsey number. In 1987, Hanson and Toft conjectured that every $(K_{t_1}, \ldots, K_{t_k})$-co-critical graph $G$ on $n\ge r$ vertices satisfies \[|E(G)|\ge (r-2)n- \binom{r- 1}{2}.\] This bound is best possible for every $n\ge r$. More recently, the present author conjectured that every such graph has minimum degree at least $r-2$. Using the $q$-neighbor bootstrap percolation closure method, here we prove that the Hanson-Toft Conjecture holds asymptotically, provided that the minimum-degree conjecture is true; more precisely, If every $(K_{t_1},\ldots,K_{t_k})$-co-critical graph has minimum degree at least $r-2$, then there is a constant $C=C(r,k)$ such that every $(K_{t_1},\ldots,K_{t_k})$-co-critical graph $G$ on $n\ge r$ vertices satisfies $|E(G)|\ge (r-2)n-C$. Subjects: Combinatorics (math.CO) Cite as: arXiv:2608.07422 [math.CO] (or arXiv:2608.07422v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.07422 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Zi-Xia Song [ view email ] [v1] Fri, 7 Aug 2026 17:10:18 UTC (9 KB) [v2] Mon, 24 Aug 2026 18:45:54 UTC (9 KB) Full-text links: Access Paper: 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 SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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