  We study effects of a repulsive Coulomb interaction on the spectral gap in
monolayer and bilayer graphene in the vicinity of the charge neutrality point
by employing the functional renormalization-group technique. In both cases
Coulomb interaction supports the gap once it is open. For monolayer graphene we
correctly reproduce results obtained previously by several authors, e.g., an
apparent logarithmic divergence of the Fermi velocity and the gap as well as a
fixed point corresponding to a quantum phase transition at infinitely large
Coulomb interaction. On the other hand, we show that the gap introduces an
additional length scale at which renormalization flow of diverging quantities
saturates. An analogous analysis is also performed for bilayer graphene with
similar results. We find an additional fixed point in the gapless regime with
linear spectrum corresponding to the vanishing electronic band mass. This fixed
point is unstable with respect to gap fluctuations and can not be reached as
soon as the gap is opened. This preserves the quadratic scaling of the spectrum
and finite electronic band mass.
