example, in quantum physics [@CJ87]. Many intractable counting problems have been added to the Valiant’s[@Va79] list of $\#$P-Complete, which is a natural correspondence of the concept NP-Complete for decision problems. Efficient approximating schemes called fully polynomial randomized approximation scheme(FPRAS) are naturally considered for the hard problems in counting. If $M$ is the true value, a randomized algorithm is called an FPRAS if it takes polynomial time of size of inputs, $\varepsilon^{-1}$ and $\log\delta^{-1}$ to obtain an output $\tilde{M}$. Here $\tilde{M}$ is the approximation of $M$, satisfying $$P((1-\varepsilon)M\leq \tilde{M}\leq (1+\varepsilon)M)\geq 1-\delta.$$

Due to the fact that the decision problem of whether a graph contains a Hamiltonian cycle is NP-Complete, there would be no FPRAS for counting the Hamiltonian cycles for general graphs unless NP=RP. Thus the FPRAS for counting Hamiltonian cycles are only possible for special or restricted graphs, for example, elementary recursive algorithms [@Ra94] for random digraphs; Markov Chain Monte Carlo(MCMC) methods for dense $undirected$ graphs[@DFJ94], for some random digraphs[@FS92] and random regular graphs[@FJMRW96].

Sequential acceptance/rejection method is introduced by Huber [@Hu06] for counting the number of the perfect matchings in a dense regular bipartite graph. Recently the regularity requirement is removed [@HL08]. The primary tool used in the algorithm 