  A zigzag edge of a graphene nanoribbon supports localized zero modes,
ignoring interactions. Based mainly on mean field arguments and numerical
approaches, it has been suggested that interactions can produce a large
magnetic moment on the edges. By considering the Hubbard model in the weak
coupling limit, U<<t, for bearded as well as zigzag edges, we argue for such a
magnetic state, based on Lieb's theorem. Projecting the Hubbard interactions
onto the flat edge band, we then prove that resulting 1 dimensional model has a
fully polarized ferromagnetic ground state. We also study excitons and the
effects of second neighbor hopping as well as a potential energy term acting on
the edge only, proposing a simple and possibly exact phase diagram with the
magnetic moment varying smoothly to zero. Finally, we consider corrections of
second order in U arising from integrating out the gapless bulk Dirac
excitations.
