Graphene, a two-dimensional network of carbon atoms, exhibits unique electronic properties because it supports low energy, massless, Dirac-like quasiparticles. The quantized Hall effect in this system has an unusual set of plateaus, whose locations may be interpreted in terms of a geometric “Berry’s phase” related to the chirality of the Dirac particles. The chiral nature of these states also leads to an unusual edge state structure, particularly near filling factor nu=0. We examine the transport behavior of the nu=0 graphene system in light of its unusual edge state structure. When electron-electrons interactions are included, we find a magnetic domain wall structure at the edge that is electrically conducting and behaves like a Luttinger liquid. Recent experiments at this filling reveal an unusual resistive state, with strong indications of a quantum phase transition driven by the magnetic field. We demonstrate that the behavior may be understood if localized magnetic moments are exchange-coupled to the domain wall, inducing a “chiral” Kondo effect. The various behaviors observed in experiment then emerge naturally as different limits of a competition between Kondo scattering and the critical behavior of a transition from an easy-axis to an easy-plane state of the domain wall.