The zigzag edges of single- or few-layer graphene are perfect one-dimensional conductors owing to a set of gapless states that are topologically protected against backscattering. Direct experimental evidence of these states has been limited so far to their local thermodynamic and magnetic properties, determined by the competing effects of edge topology and electron–electron interaction. However, experimental signatures of edge-bound electrical conduction have remained elusive, primarily due to the lack of graphitic nanostructures with low structural and/or chemical edge disorder. Here, we report the experimental detection of edge-mode electrical transport in suspended atomic-scale constrictions of single and multilayer graphene created during nanomechanical exfoliation of highly oriented pyrolytic graphite. The edge-mode transport leads to the observed quantization of conductance close to multiples of G$_0$ = 2e$^2$/h. At the same time, conductance plateaux at G$_0$/2 and a split zero-bias anomaly in non-equilibrium transport suggest conduction via spin-polarized states in the presence of an electron–electron interaction.