Nearly balanced spanning subdivisions at the optimal additive minimum semi-degree threshold

Determine whether there exists a constant C0 > 0 such that every digraph H with h arcs and no isolated vertices and every n-vertex digraph D with n ≥ C0h and minimum semi-degree δ0(D) ≥ (n + h)/2 − 1 contains a spanning H-subdivision whose subdivision paths have lengths differing by at most one.

Background

The paper proves that, for every ε > 0, a sufficiently large n-vertex digraph with minimum semi-degree at least (1/2 + ε)n contains a spanning subdivision of any h-arc digraph H without isolated vertices in which the subdivision-path lengths differ by at most one. This establishes nearly balanced spanning subdivisions under a linear εn surplus above the one-half threshold.

A result of Wang, Cheng, and Yan gives an exact existence threshold: δ0(D) ≥ (n + h)/2 − 1 is sufficient for a spanning H-subdivision when n is sufficiently large compared with h. The authors therefore ask whether their nearly balanced conclusion remains valid at this optimal additive threshold, replacing the εn surplus by the term h/2 − 1.

References

It is natural to ask whether the εn surplus in the minimum semi-degree condition can be replaced by the optimal additive term. In view of the exact spanning subdivision theorem of Wang, Cheng and Yan [13],the following question seems particularly natural. Question 4.1. Does there exist a constant C0 > 0 such that, for every digraph H with h arcs and no isolated vertices and every n-vertex digraph D with n ≥ C0h and δ0(D) ≥ (n + h)/2 − 1, the digraph D contains a spanning H-subdivision whose subdivision paths have lengths differing by at most one?

— Nearly balanced spanning subdivisions in dense digraphs  (2608.14432 - Wang et al., 14 Aug 2026) in Question 4.1, Section 4 (Concluding remarks)