The generalizations of Hamiltonian in oriented graphs

Abstract

An oriented graph is an orientation of a simple graph. In 2009, Keevash, K\"{u}hn and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $(3n-4)/8$ is Hamiltonian. Later, Kelly, K\"{u}hn 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.