Dean's conjecture for k = 5, cycles of length divisible by five
A preprint claims every finite simple graph of minimum degree at least five contains a cycle whose length is divisible by five, the last open case of Dean's 1988 conjecture.
- Lab
- Independent
- Model
- GPT-5.6 Sol; Claude Opus 5; GLM 5.3 Flash
- Field
- Mathematics
- Date
- 2026-08-29
- Human collaborators
- Elias Botsford
- Problem posed
- 1988 · open 38 yrs
Sources
Original work
Independent commentary
What was found
Dean's conjecture asserts that every finite simple graph of minimum degree at least k contains a cycle of length divisible by k. The cases k = 3 and k = 4 were known, and Luo, Ma and Zhao proved every k at least 6, which left k = 5 as the single open case; if this proof stands, the conjecture holds for all k at least 3. Nine finite configuration propositions in the bipartite and triangle-free branches are computer-assisted, with verifier sources and certificate data archived in a separate computational supplement.
Novelty check
The landscape is fixed by an independent recent paper rather than by the claimant: Luo, Ma and Zhao (arXiv:2601.13552) state in their own abstract that the conjecture was known for k in {3, 4} and prove it for every k at least 6, which identifies k = 5 as the remaining case at the time of writing. No competing resolution of k = 5 appears in that literature or in the vibemathed record. The claim is new work on an identified gap.
Caveats and known objections
A Zenodo preprint days old at entry, unrefereed, with no independent check on record. The computer-assisted part is the narrow part: nine finite configuration families have verifiers and certificates, and certificate runs are reported as passing, but the reductions from arbitrary graphs to those finite state spaces are ordinary prose arguments that are not machine-checked and that only the submitter has audited. That is why the grade is claimed despite the artifact. Autonomy is graded ai-led rather than autonomous because a human submitted, framed and archived the work; the Zenodo record names Elias Botsford as the sole creator, a detail the registry rows carrying this result omit.
Also recorded at
vibematheddean-s-conjecture-for-k-5
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.
Or on GitHub: submit a check challenge the grade send a correction or send a pull request
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Entry history (1 event)
- AddedEntered the registry graded Claimed and AI-led.
Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.
Graded claimed for verification and ai-led for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Dean's conjecture for k = 5, cycles of length divisible by five. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-29-dean-conjecture-k5BibTeX
@misc{whataifound-independent-2026-k5,
title = {Dean's conjecture for k = 5, cycles of length divisible by five},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Claimed. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-29-dean-conjecture-k5}
}