Define a decision problem H as follows. The input of H is a pair (G1,G2) of graphs, and the problem is to verify whether the number of Hamiltonian cycles in G1 is greater than the number of Hamiltonian cycles in G2. Is there a known complexity class C such that H is in C and H is C-hard under many-one reductions? Clearly H is in P^{#P}, and it is possible to prove that H is PP-hard under many-one reductions. But is H in PP? Or is H hard for the class P^{#P} under many-one reductions?
A decision problem related to the problem of counting Hamiltonian cycles
counting complexity;hamiltonian paths