Some sufficient conditions on Hamiltonian digraph

TL;DR

This paper establishes new sufficient conditions linking Hamiltonian properties of digraphs and their associated bipartite graphs, providing insights into Hamiltonicity criteria and disjoint Hamiltonian digraphs.

Contribution

It proves that strong connectivity and Hamiltonicity of the Z-mapping graph imply Hamiltonicity of the digraph, and vice versa, establishing equivalence of certain Hamiltonian conditions.

Findings

If the Z-mapping graph is Hamiltonian, then the digraph is Hamiltonian.

A Hamiltonian digraph's Z-mapping graph contains at least one perfect matching.

The paper provides new sufficient conditions for the existence of disjoint Hamiltonian digraphs.

Abstract

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.