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Precise statement
Kotzig conjectured that for every even $n \ge 4$ the complete graph $K_n$ decomposes into $n-1$ perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: $K_n$ decomposes into $n-1$ perfect matchings of which $(1-o(1))n$ have the property that any pair forms a Hamilton cycle.
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fidelity, correctness, priority, or novelty.
What AI did
ChatGPT 5.4
The authors state that the construction in Theorem 1.3, a generalisation of the one in their introduction, was generated by ChatGPT 5.4, which also supplied a correct but long proof of it. The proof they present is a cleaner and substantially different one of their own, and the proof of the main theorem, Theorem 1.2, was obtained entirely by the authors. The model's construction is nonetheless load-bearing: the main result is built by deleting a random subset of the matchings it produces.