  The Landau level spectrum of graphene superlattices is studied using a
tight-binding approach. We consider non-interacting particles moving on a
hexagonal lattice with an additional one-dimensional superlattice made up of
periodic square potential barriers, which are oriented along the zig-zag or
along the arm-chair directions of graphene. In the presence of a perpendicular
magnetic field, such systems can be described by a set of one-dimensional
tight-binding equations, the Harper equations. The qualitative behavior of the
energy spectrum with respect to the strength of the superlattice potential
depends on the relation between the superlattice period and the magnetic
length. When the potential barriers are oriented along the arm-chair direction
of graphene, we find for strong magnetic fields that the zeroth Landau level of
graphene splits into two well separated sublevels, if the width of the barriers
is smaller than the magnetic length. In this situation, which persists even in
the presence of disorder, a plateau with zero Hall conductivity can be observed
around the Dirac point. This Landau level splitting is a true lattice effect
that cannot be obtained from the generally used continuum Dirac-fermion model.
