Trees and treelike structures in dense digraphs

TL;DR

The paper proves that bounded-degree oriented trees can be embedded as spanning subdigraphs in dense directed graphs, extending to more complex tree-like structures and cycles.

Contribution

It generalizes a classical theorem to include a wider class of spanning structures in dense digraphs, including cycles and subdivisions.

Findings

Every bounded-degree oriented tree on n vertices appears in dense digraphs with minimum semidegree > n/2.

The result extends to collections of disjoint cycles and subdivided graphs with specific size constraints.

The approach generalizes known theorems to broader classes of spanning structures in directed graphs.

Abstract

We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+o(n)$. This can be seen as a directed graph analogue of a well-known theorem of Koml\'os, S\'ark\"ozy and Szemer\'edi. Our result for trees follows from a more general result, allowing the embedding of arbitrary orientations of a much wider class of spanning ``tree-like'' structures, such as collections of at most $O(n^{0.99})$ pairwise vertex-disjoint cycles and subdivisions of graphs $H$ with $|H| < \exp (\sqrt{O(\log n)})$ in which each edge is subdivided at least once.