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Sivaraman's Perfect-Divisibility Characterization Question

Combinatorics · posed by Vaidy Sivaraman, 2026 · disproved

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Statement

Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph GG perfectly divisible if and only if χ(H)≤(ω(H)+12)\chi(H) \le \binom{\omega(H)+1}{2} for every induced subgraph HH of GG? False: the Paley graph P(17)P(17) satisfies the chromatic bound hereditarily but is not perfectly divisible.

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

  1. construction · #1

    Zhiyu Wang and Weihao Xia, using ChatGPT

    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 co developed
    — a person and a model developed the result together.

    The acknowledgement states that the Paley graph P(17) - the counterexample witness - was identified during exploratory use of ChatGPT and subsequently verified by the authors, with ChatGPT also used for language polishing and generating the verification code. All arguments and computations were checked by the authors.

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