APPROXIMATELY COUNTING HAMILTON PATHS AND CYCLES IN DENSE GRAPHS

We describe fully polynomial randomized approximation schemes for the problems of determining the number of Hamilton paths and cycles in an n-vertex graph with minimum degree (1/2 + alpha)n, for any fixed alpha > 0. We show that the exact counting problems are #P-complete. We als o describe fully polynomial randomized approximation schemes for count ing paths and cycles of all sizes in such graphs.