Long directed cycles in diregular digraphs

Status open medium confidence

Jackson's 1981 conjecture that every strong oriented graph with minimum indegree and outdegree at least $d$ contains a directed cycle of length at least $2d+1$ remains open. The 1981 paper established directed paths of length at least $2d$, but the cycle version for strongly connected graphs is unresolved. Related progress has been made on a distinct Jackson conjecture: Lo, Patel, and Yıldız (2023) proved for sufficiently large $n$ that every $d$-regular oriented graph on $n \leq 4d+1$ vertices contains a Hamilton cycle, but this does not settle the general minimum-semidegree conjecture.

Cited literature (1)

• partial Cycle Partitions in Dense Regular Digraphs and Oriented Graphs (2025)
  Allan Lo, Viresh Patel, Mehmet Akif Yıldız · Forum of Mathematics, Sigma · arXiv:2309.11677 · doi:10.1017/fms.2025.28

  Proves for sufficiently large $n$ that every $d$-regular oriented graph on $n \leq 4d+1$ vertices has a Hamilton cycle (a related but distinct Jackson conjecture from 1981), and that $d$-regular oriented graphs on $n$ vertices can be covered by at most $n/(2d+1)$ vertex-disjoint cycles; does not address the OPG conjecture for general strong oriented graphs with minimum semidegree $d$.

Reviewer notes. The OPG conjecture (cycles of length ≥ 2d+1 in strong oriented graphs with minimum semidegree d) is distinct from the Jackson Hamilton-cycle conjecture (Hamilton cycles in d-regular oriented graphs on n ≤ 4d+1 vertices). Most search results and papers found address the latter or the Caccetta–Häggkvist / Kelly–Kühn–Osthus conjectures. No post-2013 paper directly proving or disproving the OPG conjecture was found after exhausting 6 search queries. The Lo–Patel–Yıldız result is the closest related progress but addresses a different (more restricted) setting.

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