Non-locality for graph states

The possibility of preparing two-photon hyper-entangled states encoding three or more qubits in each photon, and six-photon six-qubit states, lead to the following problem: If we distribute N qubits between two parties, what quantum pure states and distributions of qubits allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell's theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs, and provide all possible proofs up to N=7 qubits. On the other hand, we provide all optimal Mermin-like Bell inequalities for all graph states up to N=5 qubits, and some optimal Bell inequalities for certain relevant classes of graph states. These Bell inequalities are interesting for testing how nonlocality grows with the size for different graph states.