Ore-type conjecture for all orientations of Hamilton cycles

An oriented graph is a directed graph with at most one directed edge between any pair of vertices. For an oriented graph GG, let V(G)V(G) be its vertex set, let n=V(G)n=|V(G)|, and write σ+(G)\sigma_{+-}(G) for the minimum, over all directed edges uvu\to v, of d+(u)+d(v)d^+(u)+d^-(v). An orientation of an undirected Hamilton cycle is obtained by assigning a direction to every edge of a Hamilton cycle on V(G)V(G); saying that GG contains all possible orientations means that it contains a copy of every such orientation. All-orientations conjecture. There exists an integer n0n_0 such that every oriented graph GG on nn0n\geqslant n_0 vertices with σ+(G)(3n+2)/4\sigma_{+-}(G)\geqslant(3n+2)/4 contains all possible orientations of a (undirected) Hamilton cycle. The problem asks whether this Ore-type condition forces every orientation pattern of a spanning cycle, extending Hamiltonicity results beyond a single prescribed cycle orientation.

Sources & referencesView supporting material

Primary source

Junqing Cai, Guanghui Wang, Yun Wang and Zhiwei Zhang, “Ore-type condition for antidirected Hamilton cycles in oriented graphs”, arXiv:2511.11302 (2025).

Additional references

5 papers in this index state this conjecture (2011–2025). The statement above is taken from the most recent of them; the others are arXiv:2202.06388, arXiv:1607.03348, arXiv:1403.0776, arXiv:1111.4292.

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