  Rhombohedral multilayer graphene is a physical realization of the chiral
two-dimensional electron gas that can host zero-line modes (ZLMs), also known
as kink states, when the local ap opened by inversion symmetry breaking
potential changes sign in real space. Here we study how the variations in the
local stacking coordination of multilayer graphene affects the formation of the
ZLMs. Our analysis indicates that the valley Hall effect develops whenever an
interlayer potential difference is able to open up a band gap in stacking
faulted multilayer graphene, and that ZLMs can appear at the domain walls
separating two distinct regions with imperfect rhombohedral stacking
configurations. Based on a tight-binding formulation with distant hopping terms
between carbon atoms, we first show that topologically distinct domains
characterized by the valley Chern number are separated by a metallic region
connecting AA and AA$'$ stacking line in the layer translation vector space. We
find that gapless states appear at the interface between the two stacking
faulted domains with different layer translation or with opposite perpendicular
electric field if their valley Chern numbers are different.
