A Graph Invariant and 2-factorizations of a graph

A spanning subgraph of a graph G is called a [0,2]-factor of G, if for . is a union of some disjoint cycles, paths and isolate vertices, that span the graph G. It is easy to get a [0,2]-factor of G and there would be many of [0,2]-factors for a G.A characteristic number for a [0,2]-factor, which reflect the number of the paths and isolate vertices in it,is defineted. The [0,2]-factor of G is called maximum if its characteristic number is minimum, and is called characteristic number of G.It to be proved that characteristic number of graph is a graph invariant and a polynomial time algorithm for computing a maximum [0,2]-factor of a graph G has been given in this paper. A [0,2]-factor is Called a 2-factor, if its characteristic number is zero. That is, a 2-factor is a set of some disjoint cycles, that span G.We propose a A polynomial time algorism for computing 2-factor from a [0,2]-factor,which can be got easily. A HAMILTON Cycle is a 2-factor, therefore a necessary condition of a HAMILTON Graph is that, the graph have a 2-factor or the characteristic number of the graph is zero. The algorism, given in this paper, make it possible to examine the condition in polynomial time.