  Motivated by the surge in research activities on graphene, we investigate
instabilities of electrons on the honeycomb lattice, interacting by onsite and
nearest-neighbor terms, using a renormalization group scheme. Near half
band-filling, critical minimal interaction strengths are required for
instabilities toward antiferromagnetic or charge-density wave order. Away from
half filling, f-wave triplet pairing and d+id singlet pairing instabilities are
found to emerge out of density-wave regimes.
