  Monolayer graphene at neutrality in the quantum Hall regime has many
competing ground states with various types of ordering. The outcome of this
competition is modified by the presence of the sample boundaries. In this paper
we use a Hartree-Fock treatment of the electronic correlations allowing for
space-dependent ordering. The edge influence is modeled by a simple
perturbative effective magnetic field in valley space. We find that all phases
found in the bulk of the sample, ferromagnetic, canted antiferromagnetic,
charge-density wave and Kekul$\'e$ distortion are smoothly connected to a
Kekul$\'e$-distorted edge. The single-particle excitations are computed taking
into account the spatial variation of the order parameters. An eventual
metal-insulator transition as a function of the Zeeman energy is not simply
related to the type of bulk order.
