Dean’s cycle-length divisibility conjecture

Does every finite simple graph of minimum degree at least five have a simple cycle whose length is divisible by five? This is the remaining five-divisibility case in Dean’s cycle-length conjecture.

Why it matters and what counts as progress

Why it matters: The five-divisibility case is the remaining minimum-degree cycle-length target in Dean’s conjecture.

What counts as progress: Prove the cycle-existence assertion in the full class or exhibit a minimum-degree-five counterexample.

How the problem developed

1993 · Partial theorem [1][3]
Nathaniel Dean, Linda Lesniak and Akira Saito Published results on cycles of length divisible by four, cited by the later Dean-conjecture literature as establishing the k=4 case.
1994 · Partial theorem [2][3]
Guantao Chen and Akira Saito Proved that every graph with minimum degree at least three contains a cycle whose length is divisible by three.
2026 · Reported proof [4]
Dean conjecture k=5 repository authors Posted a manuscript claiming the remaining k=5 case, which together with the earlier cases would settle the conjecture.
20 Jan 2026 · Partial theorem [3]
Yufan Luo, Jie Ma and Ziyuan Zhao Resolved the conjecture for every k≥6 and proved stronger cycle-residue conclusions under the stated end-block conditions.

AI contribution

Cycles of length divisible by five in graphs of minimum degree five - The k=5 case of Dean's conjecture

Elias Botsford

Botsford’s revised theorem says every finite simple graph with minimum degree at least five has a simple cycle whose length is divisible by five. The proof reduces a minimal counterexample to 5-weak graphs and treats bipartite, triangular and triangle-free cases.

AI involvementAI-led

ImportanceMajor

Technical verificationPartial

Human understandingFew