Kotzig’s perfect one-factorisation conjecture
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For every sufficiently large even n, how many edge-disjoint perfect matchings of K_n can be chosen so every pair forms a Hamilton cycle? The asymptotic target is at least (1−ε)n for each ε>0.
Why it matters and what counts as progress
Why it matters: It seeks an almost-complete pairwise-perfect matching collection uniformly across all large even orders.
What counts as progress: Increase the pairwise-Hamiltonian perfect-matching collection toward a full factorization for every large even order.
How the problem developed
- 10 Jul 2026 · Asymptotic partial result [1]
Yangyang Cheng and Amedeo Sgueglia Constructed factorizations of Kₙ in which (1−o(1))n of the n−1 perfect matchings pair to form Hamilton cycles.
AI contribution
The perfect 1-factorisation conjecture holds asymptotically
For every ε>0 and all sufficiently large even n, the complete graph K_n contains at least (1−ε)n edge-disjoint perfect matchings such that every pair forms a Hamilton cycle. This is the model-origin construction of Theorem 1.3.
Sources
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Publication dates and editorial changes.
ChatGPT construction advances Kotzig’s factorisation conjecture