Cycle-residue stability at minimum degree five
Classify the finite simple graphs of minimum degree at least five whose cycle lengths fail to represent every residue class modulo five. Is residue two the only possible missing residue, and can all such graphs be characterized through an explicit family of exceptional end-blocks together with a condition on the remaining blocks?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Graph Theory
- Posed by
- Luo, Ma and Zhao, whose stability theorem covers every k >= 6 and leaves k = 5
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-04
- Model
- GPT-5.6 Sol, GPT-6 Astra
- Vendor
- OpenAI
- Collaborators
- Elias Botsford
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The preprint claims an exact classification. Write for the residues modulo five represented by cycle lengths in . Let
,
where is obtained from by deleting edges incident with one vertex in the part of size .
For every finite simple graph with minimum degree at least five, exactly one alternative holds: ; or every end-block belongs to and every non-end-block contains no cycle of length congruent to two modulo five. Every member of has cycle-residue spectrum .
The proof combines structural arguments with finite computational checks. It uses the separately established Dean–5 theorem and its weak-graph strengthening as inputs. The contribution is the stronger stability classification; independent expert review remains pending.
What the AI did
Large language models contributed substantially to developing and auditing the proof, repairing intermediate arguments, and implementing computational checks. The work included checking rooted-path and cycle constructions, identifying gaps in structural reductions, developing replacement lemmas and corrected attachment arguments, and revising finite residue calculations and literal-cycle geometry checks.
Verification
Unreviewed. The 82-page preprint (Zenodo 10.5281/zenodo.22311785, version 1.0.0 of 4 September 2026) was text-extracted and its introduction, main theorem and disclosure read here; the computational supplement (10.5281/zenodo.22311412) reports 23 passing operations and was not replayed on this site, unlike the Dean-5 supplement, which was. Two dependencies a reader should hold in mind: the proof imports the author's own modulus-five Dean theorem, version 1.0.1, which is itself Candidate here pending independent review; and the earlier Zenodo version of this preprint was withdrawn by the author for errors before this one, which is stated in the submission and matches the deleted record. Tier changed from AI-discovered to AI co-developed: the paper's own disclosure says the results were obtained "with substantial assistance from large language models" and that the author reviewed the arguments and takes responsibility, which is the co-developed pattern on this site, not discovery.
Sources
Submitted by eli on