  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.
