Some sufficient conditions on Hamiltonian digraph
  Z-mapping graph is a balanced bipartite graph $G$ of a digraph $D$ by split each vertex of $D$ into a pair of vertices of $G$. Based on the property of the $G$, it is proved that if $D$ is strong connected and $G$ is Hamiltonian, then $D$ is Hamiltonian. It is also proved if $D$ is Hamiltonian, then $G$ contains at least a perfect matching. Thus some existence sufficient conditions for Hamiltonian digraph and Hamiltonian graph are proved to be equivalent, and two sufficient conditions of disjoint Hamiltonian digraph are given in this paper. 
Guohun Zhu