Spanning $H$-subdivisions and perfect $H$-subdivision tilings in dense digraphs

Given a (di)graph $H$, we say that a (di)graph $H^{\mathrm{\prime }}$ is an $H$-subdivision if $H^{\mathrm{\prime }}$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Signé conjectured that for every $\epsilon >0$, there exists a constant $C_0>0$ such that for every graph $H$ with $h$ edges and no isolated vertices, if $G$ is a graph on $n\ge C_{0}h$ vertices and minimum degree $\delta (G)\ge (1+\epsilon )\frac{n}{2}$, then $G$ contains a spanning $H$-subdivision. This conjecture was later resolved by Lee [European J. Combin. $\mathbf{\text{124}}$ (2025), 104059]. In this paper, we strengthen Lee’s result. Specifically, we prove that for any digraph $D$ on $n\ge C_{0}h$ vertices, if the minimum semi-degree of $D$ is at least $\frac{n+h}{2}-1$, then $D$ contains a spanning $H$-subdivision. The lower bound on the minimum semi-degree is best possible. Furthermore, we show that there exist constants $C>0$ and $\alpha ,\beta \in (0,1)$ such that for any integer partition $n=n_1+\cdots +n_m\ge Cm$ with $n_i\ge \mathrm{\mid }V(H)\mathrm{\mid }+3h$ for each $i$, and ${\sum }_{n_i<\alpha n}{n}_i\le \beta n$, if a digraph of order $n\ge Cm$ has minimum semi-degree at least $\frac{n+m+h}{2}-1$, then it contains $m$ vertex-disjoint $H$-subdivisions whose orders are $n_1,\dots ,n_m$, respectively. The bound $\frac{n+m+h}{2}-1$ is also optimal. This work partly answers a conjecture of Lee [Combin. Probab. Comput. $\mathbf{\text{34}}$ (2025), 421–444] and generalizes a recent result from the same paper.