  We study the electronic band structure of monolayer graphene when Rashba
spin-orbit coupling is present. We show that if the Rashba spin-orbit coupling
is stronger than the intrinsic spin-orbit coupling, the low energy bands
undergo trigonal-warping deformation and that for energies smaller than the
Lifshitz energy, the Fermi circle breaks up into separate parts. The effect is
very similar to what happens in bilayer graphene at low energies. We discuss
the possible experimental implications, such as threefold increase of the
minimal conductivity for low electron densities, the wavenumber dependence of
the band splitting and the spin polarization structure. Our theoretical
predictions are in agreement with recent experimental results.
