Lovász conjecture (1969): long paths in vertex-transitive graphs pushed to n^(1−o(1)), with GPT-5.6 Sol supplying key steps
On Sept 29, 2026 Bowen Li and Abhishek Methuku posted a proof that every connected vertex-transitive graph on n vertices contains a cycle of length n^(1−o(1)). The previous bound was n^(2/3−o(1)), set earlier in 2026. This is the strongest general progress so far towards Lovász's 1969 conjecture that every such graph has a Hamiltonian path. The paper's AI statement says GPT-5.6 Sol pointed the authors to two key tools (the Tessera–Tointon structure theorem and Babai's contraction lemma) and "supplied the required group-theoretic arguments" for the last missing lemma.
Key facts
- arXiv 2609.38135 (math.CO), 'A nearly linear bound for the Lovász conjecture', Bowen Li and Abhishek Methuku, Sept 29, 2026
- Result: every connected vertex-transitive graph on n vertices has a cycle (and path) of length n^(1−o(1)); the previous record n^(2/3−o(1)) was by Bucić, Christoph, Pokrovskiy and Steiner (arXiv 2606.09742, June 2026); Babai's 1979 bound was about √n
- AI statement: 'In July 2026, when the authors asked OpenAI's GPT-5.6 Sol whether a decomposition into vertex-transitive parts is possible, it directed them to the structure theorem of Tessera and Tointon'
- AI statement: for the small-orbit case 'GPT pointed them to Babai's contraction lemma', and 'GPT supplied the required group-theoretic arguments for controlling both the number and the word lengths of the required generators (Lemma 6.1) … This provided the key remaining step'
- Human part (per the paper): the 2025 spanning-tree traversal plus Lovász local lemma strategy and the path-lifting bound (Lemma 5.7) were the authors' own
- Context: Christoph, Hunter and Sudakov (arXiv 2609.30165, Sept 24, no AI disclosure) separately showed (1−o(1))n cycles when the degree is polylogarithmically large
Science result
- Field
- mathematics / combinatorics / graph theory
- Problem
- Lovász conjecture (1969): every connected vertex-transitive graph has a Hamiltonian path; quantitative version, the longest guaranteed cycle (open since 1969)
- Result
- Every connected vertex-transitive graph on n vertices contains a cycle of length n^(1−o(1)), up from n^(2/3−o(1)).
- AI system
- GPT-5.6 Sol
- Human role
- Human-led with substantive AI input: the authors' framework; GPT-5.6 Sol found two key literature tools and supplied the group-theoretic Lemma 6.1
- Verification
- Unrefereed preprint
- Status
- pending
What happened
Lovász asked in 1969 whether every connected vertex-transitive graph has a Hamiltonian path. For decades the best general guarantee was a cycle of length about √n (Babai, 1979). In 2026 the bound moved quickly: first n^(2/3−o(1)) by Bucić, Christoph, Pokrovskiy and Steiner (June), and now n^(1−o(1)) by Li and Methuku.
The paper has a detailed "Statement of AI use". The authors had a 2025 strategy: traverse a spanning tree of the quotient graph repeatedly and use the Lovász local lemma to join random short paths. It worked only if the vertex set could be split into large vertex-transitive parts. In July 2026 GPT-5.6 Sol directed them to the Tessera–Tointon structure theorem, which supplies such a partition, and the large-orbit cases followed "quickly". For the small-orbit case GPT pointed them to Babai's contraction lemma, which reduces the problem to Cayley graphs of nilpotent groups. When the authors could not bound the word lengths of the generators they needed, "GPT supplied the required group-theoretic arguments" (Lemma 6.1), "the key remaining step".
Why it matters
This is a large quantitative step on a famous 57-year-old problem in graph theory. The disclosure follows a pattern seen across 2026: the model did not produce the whole proof, but it supplied the right literature tools and one hard lemma. The conjecture itself (a full Hamiltonian path) remains open. The result is an unrefereed preprint, and no expert reactions were found as of Oct 5.
Changelog
- 2026-10-05: created (sweep 2026-10-05 listed it with no AI disclosure; a PDF scan found the AI statement at the end of the paper)
Related events
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
- Chvátal's 1972 conjecture proved (Chang–Liu–Liu, ChatGPT-assisted), then a GPT-6 Astra 'proof from The Book' and a Codex-built Lean formalization ★★★★
Sources (3)
- paperarXiv 2609.38135: A nearly linear bound for the Lovász conjecture
- paperarXiv 2606.09742: Towards the Lovász conjecture via sublinear expanders (previous n^(2/3) bound)
- paperarXiv 2609.30165: Thinning and sprinkling, from robust sampling to almost Hamiltonicity (Christoph, Hunter, Sudakov)
id: 2026-09-29-lovasz-conjecture-nearly-linear-bound-gpt-5-6-sol · updated 2026-10-05 · open in the interactive timeline