  In this paper we consider the possibility of chiral (charge or spin density
wave) symmetry breaking in graphene due to long-range Coulomb interaction by
comparing the results of the Bethe-Salpeter and functional
renormalization-group approaches. The former approach performs summation of
ladder diagrams in the particle-hole channel, and reproduces the results of the
Schwinger-Dyson approach for the critical interaction strength of the quantum
phase transition. The renormalization-group approach combines the effect of
different channels and allows to study the role of vertex corrections. The
critical interaction strength, which is necessary to induce the symmetry
breaking in the latter approach is found in the static approximation to be
$\alpha _{c}=e^{2}/(\epsilon v_{F})\approx 1.05$ without considering the Fermi
velocity renormalization, and $\alpha _{c}=3.7$ with accounting the
renormailzation of the Fermi velocity. The latter value is expected to be
however reduced, when the dynamic screening effects are taken into account,
yielding the critical interaction, which is comparable to the one in freely
suspended graphene. We show that the vertex corrections are crucially important
to obtain the mentioned values of critical interactions.
