Nearly balanced spanning subdivisions in dense digraphs

Abstract

Pavez-Signé [Combin. Probab. Comput. 33 (2024), 121--128] conjectured a Dirac-type condition for spanning $H$-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths. Lee [European J. Combin. 124 (2025), 104059] resolved the existence conjecture in the stronger setting of digraphs. We answer the length-control question in this stronger directed setting: 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 $D$ with $n\ge C_0h$ and $δ^0(D)\ge(1/2+\varepsilon)n$ contains a spanning $H$-subdivision whose subdivision paths have lengths differing by at most one.

Disclosure

“d no isolated vertices and every n-vertex digraph D with n ≥ C0 h and δ 0 (D) ≥ (n + h)/2 − 1, the digraph D contains a spanning H-subdivision whose subdivision paths have lengths differing by at most one? Acknowledgment. The authors used ChatGPT 5.6 to assist in the development of the probabilistic partition argument in Lemma 3.1, as well as for some language polishing. All mathematical arguments were independently verified by the authors, who take full responsibility for the cont”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:15:48.918990Z

Structural counts