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Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically

Combinatorics · posed by Anton Kotzig, 1964 · partial

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Statement

Kotzig conjectured that for every even n≥4n \ge 4 the complete graph KnK_n decomposes into n−1n-1 perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: KnK_n decomposes into n−1n-1 perfect matchings of which (1−o(1))n(1-o(1))n have the property that any pair forms a Hamilton cycle.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Yangyang Cheng and Amedeo Sgueglia, using ChatGPT 5.4

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    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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