A Hamilton cycle is a cycle containing every vertex of a graph. A graph is called Hamiltonian if it contains a Hamilton cycle. The Hamilton cycle problem is to find the sufficient and necessary condition that a graph is Hamiltonian. In this paper, we give out some new kind of definitions of the subgraphs and determine the Hamiltoncity of edges according to the existence of the subgraphs in a graph, and then obtain a new property of Hamilton graphs as being a necessary and sufficient condition

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... aracterized in the connectivity of the subgraph that induced from the cycle structure of a given graph.