Cycle tilings and $H$-factors in directed graphs

TL;DR

This paper establishes minimum degree conditions in directed graphs that guarantee the existence of cycle tilings and $H$-factors, advancing understanding of digraph structures and their spanning subgraphs.

Contribution

It provides asymptotically optimal degree thresholds for cycle tilings and $H$-factors in digraphs, including new results for odd cycles, trees, and transitive tournaments.

Findings

Minimum semi-degree condition for cycle tilings in digraphs.

Asymptotic degree thresholds for $H$-factors including trees and anti-directed cycles.

An asymptotically exact Ore-type condition for transitive tournament factors.

Abstract

We prove several results concerning cycle tilings and $H$-factors in digraphs. We provide a minimum semi-degree condition for forcing a digraph to contain a given spanning collection of vertex-disjoint orientations of cycles. Our result is asymptotically best possible for odd cycles and can be viewed as a digraph analogue of the El-Zahar conjecture. In addition, we asymptotically determine the minimum degree threshold for forcing an $H$-factor in a digraph for a range of digraphs $H$, including the cases when $H$ is a tree or anti-directed cycle. Furthermore, an asymptotically exact Ore-type result for forcing a transitive tournament factor in a digraph is proven. Several related open problems are also highlighted.