Seymour's Second Neighborhood Conjecture
Statement
Seymour conjectured that every oriented graph has a vertex with . It holds for oriented graphs of minimum out-degree exactly , the first improvement to the out-degree threshold since Kaneko and Locke settled degree in 2001.
Context
minimum out-degree 7; the conjecture is open in general
One of the best known open problems on tournaments and oriented graphs, attacked steadily for three decades with the out-degree threshold as the standard measure of progress.
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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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