  Graphene on two dimensional (2D) noncommutative (NC) plane in the presence of
a constant background magnetic field has been studied. To handel the
gauge-invariance issue we start our analysis by a effective massles NC Dirac
field theory where we incorporate the Seiberg-Witten (SW) map along with the
Moyal star ($\star$) product. The gauge-invariant Hamiltonian of a massless
Dirac particle is then computed which is used to study the relativistic Landau
problem of graphene on NC plane. Specifically we study the quantum dynamics of
a massless relativistic electron moves on monolayer graphene, in the presence
of a constant background magnetic field, on NC plane. We also compute the
energy spectrum of the NC Landau system in graphene. The results obtained are
corrected by the spatial NC parameter $\theta$. Finally we visit the Weyl
equation for electron in graphene on NC plane. Interestingly, in this case
helicity is found to be $\theta$ modified.
