Proof that complement of HAMPATH is not in NP
The complement of HAMPATH is the following problem:


  Given an undirected graph, determine whether it doesn't contain a Hamiltonian path.


A YES instance for the complement of HAMPATH is a graph which doesn't contain a Hamiltonian path. It is not clear what a witness for this property would be. Contrast that with HAMPATH itself, in which the witness for having a Hamiltonian path is simply a Hamiltonian path.



It is not known whether the complement of HAMPATH is in NP, or equivalently, whether HAMPATH is in coNP.

However, it is considered unlikely that HAMPATH is in coNP, since this would imply that NP=coNP, which is considered unlikely by most researchers in the area.