The generalizations of Hamiltonian in oriented graphs

Abstract

An oriented graph is an orientation of a simple graph. In 2009, Keevash, Kühn and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $(3n-4)/8$ is Hamiltonian. Later, Kelly, Kühn and Osthus showed that it is also pancyclic. Inspired by this, we show that for any given constant $t$ and positive integer partition $n = n_1 + \cdots + n_t$, if $D$ is an oriented graph on $n$ vertices with minimum semidegree at least $(3n-4)/8$, then it contains $t$ disjoint cycles of lengths $n_1,\ldots , n_t$. Also, we determine the bounds on the semidegree of sufficiently large oriented graphs that are strongly Hamiltonian-connected, $k$-ordered Hamiltonian and spanning $k$-linked.

Figures (2)

• Figure 1: A member $D$ of extremal family $\mathcal{F}$ of order $n$, where the number of arcs from $D_i$ to $D_{i+1}$ are close to $n^2/16$, for $i\in [4]$ (shown in bold); the number of arcs from $D_2$ to $D_4$ and the number of arcs from $D_4$ to $D_2$ are close to $n^2/32$ (shown in bold).
• Figure 2: An oriented graph $D$ with $\delta^0(D)\geq n/4+3k/2-5/2$ that is not $k$-linked.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.1