Dean's Conjecture

Dean's conjecture (Dean 1991) asserts that, for every integer , every finite simple graph with minimum vertex degree at least contains a cycle whose length is divisible by . For odd integers , the complete bipartite graphs with show that the minimum vertex degree cannot be decreased to (Luo et al. 2026).

The cases and were proved by Chen and Saito (1994) and Dean et al. (1993), respectively. Luo et al. (2026) proved the conjecture for all , leaving as the only unresolved case.

Botsford (2026b) claimed a proof for . Version 1.0.1 supplies a complete two-case argument for the exceptional-leaf boundary-deletion step, covering whether the deletion witness lies on the shortest odd cycle or outside it. The revision changes no claimed result, computational predicate, or certificate output.

Nine finite configuration propositions in the proof were established by exhaustive search in a separate computational supplement (Botsford 2026a). VibeMathed (2026) replayed all 47 certificate runs successfully, but the reductions connecting arbitrary graphs to those finite configurations have not received independent verification. Botsford (2026b) credits GPT-5.6 Sol with the primary contribution to the proof, with additional assistance from Claude Opus 5 and GLM 5.3 Flash.

Botsford (2026c) also proposed a structural classification of graphs of minimum vertex degree 5 having no cycle of length congruent to 2 modulo 5. This is a different residue question from the original divisibility conjecture. The proposed classification depends on the author's argument, whose reductions remain independently unverified as of Sep. 7, 2026.