co-NP but not NP problems
co-NP is the set of complements of problems that are in NP. So co-NP contains problems such as non-3-colourability, Boolean unsatisfiability and so on.
Most complexity theorists believe that NP$\,\neq\,$co-NP and one consequence of this is that the complement of any NP-complete problem would be in co-NP but not in NP.

Wikipedia has more information on co-NP.