Directed path of length twice the minimum outdegree

Status partial high confidence

Thomassé's conjecture — every oriented graph with minimum outdegree $k$ contains a directed path of length $2k$ — remains open. Cheng and Keevash (2024) proved the first non-trivial lower bound: every oriented graph with minimum outdegree $\delta$ contains a directed path of length $\frac{3}{2}\delta$, leaving a gap to the conjectured $2\delta$. The stronger generalisation to digraphs with girth $g \geq 4$ was disproved by counterexamples, but the $g=3$ (oriented-graph) case is still open.

Cited literature (2)

• partial On the length of directed paths in digraphs (2024)
  Yangyang Cheng, Peter Keevash · arXiv preprint · arXiv:2402.16776

  Proves that every oriented graph with minimum outdegree $\delta$ contains a directed path of length $\frac{3}{2}\delta$ (first non-trivial bound), while also disproving Thomassé's general conjecture for all girth $g \geq 4$.

• survey Oriented trees and paths in digraphs (2024)
  Maya Stein · arXiv preprint · arXiv:2310.18719

  Survey noting that Thomassé's conjecture (directed path of length $2k$ in oriented graphs with minimum outdegree $k$) 'seems wide open' and reviewing related partial results.

Reviewer notes. arXiv:2510.11311 ('Extending Thomassen's conjecture to directed graphs', 2025) refers to Thomassen's conjecture (undirected graphs), not Thomassé's conjecture for oriented graphs — not directly relevant. The Springer Combinatorica paper (doi:10.1007/s00493-021-4750-z, 'Oriented Cycles in Digraphs of Large Outdegree') was inaccessible due to a paywall redirect. Bai and Manoussakis counterexamples (mentioned in Cheng–Keevash) concern girth g≥4, not the g=3 case reviewed here.

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