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Seymour's Second Neighborhood Conjecture

Combinatorics · posed by Paul Seymour, 1990 · partial

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Statement

Seymour conjectured that every oriented graph has a vertex xx with ∣N++(x)∣≥∣N+(x)∣|N^{++}(x)| \ge |N^{+}(x)|. It holds for oriented graphs of minimum out-degree exactly 77, the first improvement to the out-degree threshold since Kaneko and Locke settled degree 66 in 2001.

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

  1. computation · #1

    Arpan Sadhukhan, R. B. Sandeep and Sagnik Sen, using ChatGPT 5.5 Pro

    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 CP-SAT models behind the computational part were developed with assistance from ChatGPT 5.5 Pro; the resulting OR-Tools encodings were then run and independently checked by the authors.

    minimum out-degree 7; the conjecture is open in general

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open in Claude →open in ChatGPT →

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