Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically
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Statement
Kotzig conjectured that for every even the complete graph decomposes into perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: decomposes into perfect matchings of which have the property that any pair forms a Hamilton cycle.
Context
asymptotic form only; Kotzig's conjecture itself remains far from solved
A famous 1960s conjecture in design theory and graph decomposition, attacked for six decades and known throughout combinatorics.
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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.
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