Ore-type conjecture for all orientations of Hamilton cycles
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 , let be its vertex set, let , and write for the minimum, over all directed edges , of . An orientation of an undirected Hamilton cycle is obtained by assigning a direction to every edge of a Hamilton cycle on ; saying that contains all possible orientations means that it contains a copy of every such orientation. All-orientations conjecture. There exists an integer such that every oriented graph on vertices with 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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