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 in the minimal conductivity for low electron densities, anisotropic, wave-number-dependent spin splitting of the bands, and the spin-polarization structure.