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.
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Statement
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.
Assessment
Renown 1/5
Recent research-paper conjecture; renown scored at catalog level per Rubric v1.
Attackability 4/5
The paper proves the statement for n≥440, leaving a bounded residue; any violation is a finite graph with a countable cycle certificate.
- finite witness
- 5/5
- oracle cost
- 2/5
- freshness
- 5/5
- seedability
- 5/5
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.
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