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arXiv · 2012.09201

Trees and treelike structures in dense digraphs

Richard Mycroft·Tássio Naia
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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.

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Richard Mycroft, Tássio Naia. 2020-12-16. Trees and treelike structures in dense digraphs. https://arxiv.org/abs/2012.09201

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