  We study the disorder effect on the excitonic gap generation caused by strong
Coulomb interaction in graphene. By solving the self-consistently coupled
equations of dynamical fermion gap $m$ and disorder scattering rate $\Gamma$,
we found a critical line on the plane of interaction strength $\lambda$ and
disorder strength $g$. The phase diagram is divided into two regions: in the
region with large $\lambda$ and small $g$, $m \neq 0$ and $\Gamma = 0$; in the
other region, $m = 0$ and $\Gamma \neq 0$ for nonzero $g$. In particular, there
is no coexistence of finite fermion gap and finite scattering rate. These
results imply a strong competition between excitonic gap generation and
disorder scattering. This conclusion does not change when an additional contact
four-fermion interaction is included. For sufficiently large $\lambda$, the
growing disorder may drive a quantum phase transition from an excitonic
insulator to a metal.
