Cycle-factors in oriented graphs

Abstract

Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$ sufficiently large, any oriented graph on $n$ vertices with both minimum out-degree and minimum in-degree at least $(1/2-c)n$ contains a $k$-cycle-factor for any $k\ge 4$. Additionally, under the same hypotheses, we also show that for any sequence $n_1,\dots ,n_t$ with ${\sum }_{i=1}^t{n}_i=n$ and the number of the $n_i$ equal to $3$ is $\alpha n$, where $\alpha$ is any real number with $0<\alpha <1/3$, the oriented graph contains $t$ disjoint cycles of lengths $n_1,\dots ,n_t$. This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov.

28 pages, 4 figures