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

TL;DR

This paper proves optimal degree conditions for spanning and disjoint $H$-subdivisions in dense digraphs, extending and strengthening previous results and conjectures in the field.

Contribution

It establishes sharp minimum semi-degree thresholds for spanning and multiple disjoint $H$-subdivisions in dense digraphs, confirming conjectures and generalizing recent findings.

Findings

Minimum semi-degree at least (n+h)/2 - 1 guarantees a spanning H-subdivision.

Existence of constants C, α, β for disjoint H-subdivisions with specified sizes under degree conditions.

Bounds on semi-degree are proven to be optimal.

Abstract

Given a (di)graph $H$, we say that a (di)graph $H^\prime$ is an $H$-subdivision if $H^\prime$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Sign\'{e} conjectured that for every $\varepsilon>0$, 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 $n\geq C_0h$ vertices and minimum degree $\delta(G)\geq(1+\varepsilon)\frac{n}{2}$, then $G$ contains a spanning $H$-subdivision. This conjecture was later resolved by Lee [European J. Combin. \textbf{124} (2025), 104059]. In this paper, we strengthen Lee's result. Specifically, we prove that for any digraph $D$ on $n\geq C_0h$ 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 $n=n_1+\cdots+n_m\geq Cm$ with $n_i\geq|V(H)|+3h$ for each $i$, and $\sum_{n_i<\alpha n}n_i\leq\beta n$, if a digraph of order $n\geq Cm$ has minimum semi-degree at least $\frac{n+m+h}{2}-1$, then it contains $m$ vertex-disjoint $H$-subdivisions whose orders are $n_1, \ldots, n_m$, respectively. The bound $\frac{n+m+h}{2}-1$ is also optimal. This work partly answers a conjecture of Lee [Combin. Probab. Comput. \textbf{34} (2025), 421--444] and generalizes a recent result from the same paper.