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Sivaraman's Perfect-Divisibility Characterization Question

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sivaraman-perfect-divisibility-question-p17Combinatoricsposed by Vaidy Sivaraman, 2026recorded: 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.

Context

A natural characterization question, but posed by personal communication in 2026 and answered within the year. Below the Graffiti anchor; the value is the clean witness.

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  • #1

    Attempt 1

    constructionChatGPT with Zhiyu Wang, Weihao Xia ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT
    people
    Zhiyu Wang, Weihao Xia

    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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