Twelve-critical graphs with (2/5 + o(1))n^2 edges

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proposition_3: From a K5-saturated graph on v >= 5 vertices with maximum degree d < v - 1 and an odd h >= 11 with v > 40hd, builds a 12-critical graph on 5(a + h + 2v) vertices, a = 2hv, with at least 10a^2(1 - d/v) edges; unreviewed preprint.

remark_p6: Version 7's unnumbered remark sketches the same construction for every even k >= 8, with edge density tending to (k - 4)/(2(k - 2)), and after joining a vertex for every odd k >= 9, with density (k - 5)/(2(k - 3)); a sketch only.

theorem_1: Claims a sequence of 12-critical graphs whose order tends to infinity and whose edge count divided by the square of the order tends to 2/5, so that f_12(n) is not asymptotic to 3n^2/8; unreviewed preprint.

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Qiyuan Gu, Twelve-critical graphs with $(2/5+o(1))n^2$ edges. The selected citation artifact is Zenodo v7, record 22569201, whose metadata has publication date 6 September 2026 and whose record was created 7 September 2026. Zenodo groups it with the historical v3 deposit under concept DOI 10.5281/zenodo.22348353. The official latest-version endpoint was checked and returned this exact v7 metadata and file identities. That is a time-bounded selection record, not a promise that v7 remains the newest version later. No notice is printed in gu_2026_twelve_critical_graphs.pdf; the Zenodo record it was selected from (https://zenodo.org/api/records/22569201, version v7, read 2026-10-02) names the Creative Commons Attribution 4.0 license, though the record's metadata and files have changed since the selection and it no longer serves the held bytes. No notice is printed in the v3 PDF read for this card; its Zenodo record 22352283 (version v3) had been removed when read on 2026-10-02 (https://zenodo.org/records/22352283 returns HTTP 410 Gone with the tombstone "Record deleted", removal reason "personal-data"), the concept's versions listing no longer includes it, and no page now states its license; the term is unstated.

Reported mathematical scope

The selected manuscript's Theorem 1 (p. 1) claims that some sequence of twelve-critical graphs $G_s$ satisfies

$$|V(G_s)|\longrightarrow\infty, \qquad \frac{e(G_s)}{|V(G_s)|^2}\longrightarrow\frac25.$$

The manuscript presents this as a counterexample at $k=12$ to the general asymptotic proposed in #917, whose predicted coefficient there is $3/8$. If the claim survives independent mathematical review, its scope is the general formula at $k=12$ only. It does not settle the $k=6$ asymptotic, and it is not used here to change any status. Outside Theorem 1, the remark "Other chromatic numbers" on p. 6 of v7, absent from v3, sketches the same construction for every even $k\ge8$, with $e(G)/|V(G)|^2$ tending to $(k-4)/(2(k-2))$, and, after joining one vertex, for every odd $k\ge9$, with limit $(k-5)/(2(k-3))$. By a comparison made here, not in the remark, these limits exceed the proposed coefficient $\frac12(1-1/\lfloor k/3\rfloor)$ at every multiple of $3$ from $12$ on and equal it at $k=9$; the remark too leaves $k=6$ untouched. Both versions disclose generative-AI assistance. No peer-review, named community acceptance, complete-proof review, or independent verification evidence is claimed by this filing.

Result pages

Labels and pages are those of the selected v7 PDF.

• Theorem 1 (p. 1): the twelve-critical sequence with density tending to $2/5$.
• Proposition 3 (p. 3): the reduction from $K_5$-saturated graphs of small maximum degree, with the exact edge count (3.4) of the Remark on p. 4.
• Remark "Other chromatic numbers" (p. 6): the sketch for every $k\ge8$.

Read status. Claims checked: the statements of Theorem 1, Proposition 3, Lemmas 2 and 4, display (3.4) and the p. 6 remark were read clause by clause on the v7 PDF. No proof was checked.

Local artifacts and version boundary

• The selected folder-basename PDF is Zenodo record 22569201, version v7: seven physical pages, 94,324 bytes.
• The distinct historical PDF, read for this card but not held, is Zenodo record 22352283, version v3, publication date 5 September 2026: six physical pages, 328,666 bytes.

The PDFs are not byte-identical. The checked site-linked record for problem 917 is the historical v3 deposit; v7 is the later selected record. V3 uses edge-deletion criticality directly. V7 defines every-proper-subgraph criticality and states its equivalence to edge-deletion criticality for graphs without isolated vertices; it also adds formalization claims and further material. These observations identify versions and do not certify either proof or full-text equivalence. The source record pins both artifacts and the manual same-work decision.

The PDF byline is Qiyuan Gu, while both Zenodo metadata records give the creator string Gu, Qiyuan. Those are ordinary orderings of the same bibliographic name and are both preserved. The checked problem submission is associated instead with the site account Alex Gu. An account label is not a mathematical-author assertion: this filing records the source-qualified names and does not claim that the account and the manuscript author are either the same or different people.

Lean and archive limits

V7 and its Zenodo metadata report a Lean development, a ZIP archive, a GitHub commit, a toolchain version, and a FORMALIZATION.md file. The pinned ZIP's static 36-entry listing contains README.md but no FORMALIZATION.md; this is a metadata/archive inconsistency, not a mathematical finding. The ZIP is deliberately not adopted into this source home and was not executed or built. This filing did not inspect theorem fidelity, verify the claimed placeholder or axiom checks, or certify the archive. All such descriptions remain manuscript/deposit claims and confer no formalization or proof credit.

Bears on. #917, third question: Theorem 1 claims that the proposed asymptotic fails at $k=12$, where it predicts $3n^2/8$, and the p. 6 remark sketches limits that, by the comparison made here, exceed the proposed coefficient at every multiple of $3$ from $12$ on. Both are an unreviewed source lead only; neither touches $k=6$.

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.