Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cycle-factors in oriented graphs

Zhilan Wang, Jin Yan, Jie Zhang

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 $k\geq4$. Additionally, under the same hypotheses, we also show that for any sequence $n_1, \ldots, n_t$ with $\sum^t_{i=1}n_i=n$ and the number of the $n_i$ equal to $3$ is $αn$, where $α$ is any real number with $0<α<1/3$, the oriented graph contains $t$ disjoint cycles of lengths $n_1, \ldots, n_t$. This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov.

Cycle-factors in oriented graphs

Abstract

Let be a positive integer. A -cycle-factor of an oriented graph is a set of disjoint cycles of length that covers all vertices of the graph. In this paper, we prove that there exists a positive constant such that for sufficiently large, any oriented graph on vertices with both minimum out-degree and minimum in-degree at least contains a -cycle-factor for any . Additionally, under the same hypotheses, we also show that for any sequence with and the number of the equal to is , where is any real number with , the oriented graph contains disjoint cycles of lengths . This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov.
Paper Structure (9 sections, 21 theorems, 72 equations, 4 figures)

This paper contains 9 sections, 21 theorems, 72 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['songbaobao']}
  3. Preparation for Theorem \ref{['t1']}
  4. Terminology and notation
  5. Proof overview of Theorem \ref{['t1']}
  6. Tools when $S$ is not ($H(D), \beta, l$)-closed
  7. The proof of Theorem \ref{['t1']}
  8. Properties of the partition $\mathcal{P}_1$
  9. Completion of Theorem \ref{['t1']}

Key Result

Theorem 1.1

(Keevash et al. K1) There exists a number $n_0$ such that any oriented graph $D$ on $n\geq n_0$ vertices with $\delta^0(D)\geq\lceil(3n-4)/8\rceil$ contains a Hamiltonian cycle.

Figures (4)

  • Figure 1: Definition \ref{['aaaaa']} for $\sigma=+$.
  • Figure 2: In the figure, the blue arcs form a $k$-cycle.
  • Figure 3: In the figure, the blue arcs form a $k$-cycle.
  • Figure 4: Examples for Cases 1-3.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 28 more