Discoveries / combinatorics

Pavez-Signe's Length-Control Question for Spanning Subdivisions

Pavez-Signe (2024) conjectured a Dirac-type condition for spanning $H$-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths; Lee (2025) resolved the existence conjecture in the stronger digraph setting. Answered affirmatively with epsilon-room: for every $\varepsilon > 0$ there is $C_0$ such that every $n$-vertex digraph $D$ with $n \ge C_0 h$ and minimum semi-degree $\delta^0(D) \ge (1/2+\varepsilon)n$ contains a spanning $H$-subdivision whose path lengths differ by at most one, for every digraph $H$ with $h$ arcs and no isolated vertices.