Seymour's conjecture on powers of Hamilton cycles

From papers

Let GG be a graph on nn vertices, and let the kkth power of a Hamilton cycle be the graph obtained by joining every pair of vertices whose distance along the cycle is at most kk. Let δ(G)\delta(G) denote the minimum degree of GG.

Seymour's conjecture. For positive integers nn and kk with nk+1n\geq k+1 and k2k\geq2, if

δ(G)kk+1n,\delta(G)\geq\frac{k}{k+1}n,

then GG contains the kkth 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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