Spanning subdivisions in dense digraphs

TL;DR

This work addresses whether a dense digraph with large minimum semi-degree necessarily contains a spanning subdivision of any given sparse digraph without isolated vertices. The authors prove that for every $\varepsilon>0$ there exists a constant $C$ such that if a digraph $D$ on $n$ vertices satisfies $\delta^0(D)\ge(\tfrac12+\varepsilon)n$ and $n\ge Cm$, then $D$ contains a spanning subdivision of every $m$-arc digraph $H$ with no isolated vertices; this result is asymptotically tight and settles a stronger form of a conjecture by Pavez-Signé. The proof builds on the absorption method, constructing an absorbing path and a small connector set, embedding the branch vertices of $H$ into the digraph, and using a directed Hamiltonian path on the remainder to realize subdivisions before absorbing leftovers back into the absorber. The contribution extends Dirac-type spanning-structure results to subdivisions in digraphs, providing a tight minimum semi-degree threshold and a robust framework via $(n,t,d)$-tuple systems for absorption and connection steps.

Abstract

We prove that an $n$-vertex digraph $D$ with minimum semi-degree at least $\left(\frac{1}{2} + \varepsilon \right)n$ and $n \geq C m$ contains a subdivision of all $m$-arc digraphs without isolated vertices. Here, $C$ is a constant only depending on $\varepsilon.$ This is the best possible and settles a conjecture raised by Pavez-Signé in a stronger form.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2: Ghouila-Houri ghouilahouri1960condition
• Definition 3.1
• Lemma 3.2
• proof : Proof of \ref{['lem:ntd-absorbing']}
• Lemma 4.1
• proof : Proof of \ref{['lem:absorber']}
• Claim 1
• Lemma 4.2
• proof : Proof of \ref{['lem:connect']}