In a random graph, the number of paths that cannot be further extended in an hamiltonian path or an hamiltonian cycle, can be roughly be separated in two equal parts?

I have an idea for solve for some class of graphs, for example random graphs, the Hamiltonian cycle problem in a very efficient way, but to proof this I need to separate about equally in two parts the paths of a graph that do not contain the same vertex more than one time, and that cannot be extended in length to cover all vertices. This separation could be for example based on paths length, for example even and odd legth not Hamiltonian paths. And under which assumptions on the graph this works. I would like of some proof of this.

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