For every varepsilon>0, there exists a constant C_0>0 such that, for every digraph H with h arcs and no isolated vertices, every n-vertex digraph D satisfying n≥ C_0h and delta^0(D)≥ (1/2+varepsilon)n contains a spanning subdivision of H in which the directed subdivision paths replacing the arcs of H have lengths differing by at most one.

Pavez-Signé conjecture and length-control question for spanning H-subdivisions

Pavez-Signé asked whether spanning subdivisions could be required to have nearly equal replacement-path lengths. Lee settled the underlying existence conjecture in 2023, but his method allowed one path to be much longer than the others.

Known results

Lee (2023): every sufficiently large digraph with minimum semidegree at least $(1/2+\varepsilon)n$ contains a spanning subdivision of every $m$ -arc digraph $H$ without isolated vertices.
Pavez-Signé et al. (2023): proved balancing for a restricted class of regular graphs, not arbitrary $H$ .
Wang, Wei, and Yan (2024): obtained stronger dense-digraph subdivision and prescribed-size results, including an almost-balancing claim.

August 2026 nearly balanced theorem

Wang, Wei, and Yan state that for every $\varepsilon>0$ , sufficiently large digraphs with minimum semidegree greater than $(1/2+\varepsilon)n$ contain spanning $H$ -subdivisions with nearly equal path lengths. The arXiv preprint provides the first direct resolution of the length-control question under these asymptotic hypotheses.

Current status (as of August 2026): The asymptotic dense-digraph length-control question is resolved by the 2026 preprint; versions without its density, size, or near-balance hypotheses remain outside the reported theorem.

Pavez-Signé conjecture and length-control question for spanning H-subdivisions

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Additional references

Progress summary

Solutions 0