Seymour's conjecture on powers of Hamilton cycles
Seymour's conjecture on powers of Hamilton cycles
Let be a graph on vertices, and let the th power of a Hamilton cycle be the graph obtained by joining every pair of vertices whose distance along the cycle is at most . Let denote the minimum degree of .
Seymour's conjecture. For positive integers and with and , if
then contains the th power of a Hamilton cycle.
This generalizes Pósa's conjecture and Dirac's theorem by prescribing a minimum-degree threshold for every power of a Hamilton cycle. The supplied text does not state whether it has been resolved.
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Sources & referencesView supporting material
Primary source
Zhilan Wang, Shuo Wei and Jin Yan, “The exact total degree threshold for the square of a Hamilton cycle in digraphs”, arXiv:2607.13831 (2026).
Additional references
14 papers in this index state this conjecture (2010–2026). The statement above is taken from the most recent of them; the others are arXiv:2512.14285, arXiv:2511.14304, arXiv:2411.01753, arXiv:2406.10814, arXiv:1901.10316, arXiv:1811.03807, arXiv:1610.03999, arXiv:1509.01823, arXiv:1403.0776, arXiv:1110.3490, arXiv:1104.4367, arXiv:1011.4476, and 1 more.
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