Nisan's answer to this question shows the Impossiblity of efficient sampling from random non-Hamiltonian graphs (unless $NP=coNP$). I am interested in the implications of this conjecture.
Does the Impossibility of efficient sampling from random non-Hamiltonian graphs imply that $NP \ne coNP$?
I am also interested in other complexity implications.
EDIT July 4: My intuition is that efficient sampling from random non-Hamiltonian graphs is possible if and only if $NP=coNP$. It hints a probabilistic characterization of class $NP$ (possibly connected to PCP theorem).
I would be very interested in published works about (preferably efficient) sampling from random non-Hamiltonian graphs (and any possible connection to PCP theorem).
