Claimed AI-led

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

Zenodocycles of length divisible by five in graphs of minimum degree five↗DOIZenodo: computational supplement, version 1.0.1↗

Independent commentary

VibeMathedDean's conjecture for k = 5↗

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.

Independent checks

vibemathed.com certificate replay: Downloaded the Zenodo computational supplement, matched all 86 files against its SHA-256 manifest and ran the driver end to end: all 47 certificate runs passed. This covers the nine finite proposition families only, not the prose reductions from arbitrary graphs to them. · link ↗

The open problem

MathDBDean's conjecture on cycles divisible by the minimum degree

374541

Nothing mechanically. It records what a problem says, what is known about it, and the standing of any claimed solution.

Names the same three models this entry credits and records k = 5 as the last open case, which is the framing the entry rests on.

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.

Disagree with these grades?

Bring a citation: a grade moves on evidence, not on argument.

Challenge this grade Submit a check

Or on GitHub: submit a check challenge the grade send a correction or send a pull request

History of this entry

2026-09-24CheckedRecorded vibemathed's full replay of the computational supplement, which passes. The graph reductions that carry the result beyond the finite cases remain unread by anyone independent, so no grade moved. source ↗
2026-09-10CorrectedRecorded the MathDB record for this result as a registration. It names the same three models and confirms k = 5 as the last open case. No grade moved. source ↗
2026-09-03AddedEntered 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

Plain text

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-k5

BibTeX

@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}
}

Related findings

ClaimedHamilton cycles in connected Cayley graphs of polylogarithmic degree, toward the Lovász conjectureIndependent · 2026-09-26 ClaimedBollobas-Nikiforov conjecture claimed in full, in a weighted formIndependent · 2026-09-06 Formally verifiedErdos-Sos conjecture proved in Lean, in a form marginally weaker than the classical statementOpenAI / Epoch AI · 2026-09-03

graph-theory combinatorics computer-assisted argument

← All mathematics findings in the registry