  A sufficiently strong Coulomb interaction may open an excitonic fermion gap
and thus drive a semimetal-insulator transition in graphene. In this paper, we
study the Eliashberg theory of excitonic transition by coupling the fermion gap
equation self-consistently to the equation of vacuum polarization function.
Including the fermion gap into polarization function increases the effective
strength of Coulomb interaction because it reduces the screening effects due to
the collective particle-hole excitations. Although this procedure does not
change the critical point, it leads to a significant enhancement of the
dynamical fermion gap in the excitonic insulating phase. The validity of the
Eliashberg theory is justified by showing that the vertex corrections are
suppressed at large $N$ limit.
