{"data":{"id":"lf-hilton-cycle-count-conjecture-below-order-440","statement_oneline":"Every Hamiltonian n-vertex graph with more than floor(n²/4)+1 edges has at least n-ℓ+2 cycles of each length 3≤ℓ≤n.","statement_full_latex":"Every Hamiltonian n-vertex graph with more than floor(n²/4)+1 edges has at least n-ℓ+2 cycles of each length 3≤ℓ≤n. Open residue: Only orders n<440 remain.","source":{"title":"The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture","arxiv":"2606.16114v1","url":"https://arxiv.org/abs/2606.16114","year":2026,"area":"Hamiltonian graph theory"},"renown":{"score":1,"rationale":"Recent research-paper conjecture; renown scored at catalog level per Rubric v1.","scored_by":"rubric-v1"},"attackability":{"score":4,"rationale":"The paper proves the statement for n≥440, leaving a bounded residue; any violation is a finite graph with a countable cycle certificate.","scored_by":"rubric-v1","subscores":{"finite_witness":5,"oracle_cost":2,"freshness":5,"seedability":5}},"verification_tier":null,"ai_systems":[],"posed_by":null,"years_open":0,"notability":null,"source_trackers":[],"status":"open","ranking_score":4,"last_verified_open":"2026-07-27","added":"2026-07-27","resolution_type":null,"resolver":null,"resolution_date":null,"evidence_url":null,"evidence_note":null,"method_note":null,"submitted_by":"Luke Francis","verification_note":"Generate dense Hamiltonian graphs, count ℓ-cycles exactly, and retain the graph plus cycle counts; proof may use SAT/flag-algebra bounds. — Current arXiv revision checked; source-stated open/partial. Independent literature and author confirmation still required. The remaining range is finite (n<440), but the raw graph search is expensive; Tier 1 reflects boundedness and exact verifiability, not low compute cost. Editorially screened candidate; not independently certified open. Author confirmation pending.","openconjecture_id":"3695","effective_status":"open","claims":[],"status_events":[{"id":"seed-lf-hilton-cycle-count-conjecture-below-order-440","conjecture_id":"lf-hilton-cycle-count-conjecture-below-order-440","status":"open","date":"2026-07-27T00:00:00.000Z","actor":"human-admin","rationale":"Imported Tier 1 contributor candidate from Luke Francis.","note":"Editorially screened candidate; author confirmation pending."}],"public_actions":[{"name":"claim","method":"POST","href":"/c/lf-hilton-cycle-count-conjecture-below-order-440/claim","verification":"email magic link"},{"name":"report_resolution","method":"POST","href":"/c/lf-hilton-cycle-count-conjecture-below-order-440/resolve","verification":"admin review"}],"resolutions":[]},"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}