  We demonstrate that an undoped two-dimensional carbon plane (graphene) whose
bulk is in the integer quantum Hall regime supports a non-chiral Luttinger
liquid at an armchair edge. This behavior arises due to the unusual dispersion
of the non-interacting edges states, causing a crossing of bands with different
valley and spin indices at the edge. We demonstrate that this stabilizes a
domain wall structure with a spontaneously ordered phase degree of freedom.
This coherent domain wall supports gapless charged excitations, and has a power
law tunneling $I-V$ with a non-integral exponent. In proximity to a bulk lead,
the edge may undergo a quantum phase transition between the Luttinger liquid
phase and a metallic state when the edge confinement is sufficiently strong
relative to the interaction energy scale.
