  The fine structure of the Dirac energy spectrum in graphene induced by
electron-optical phonon coupling is investigated in the portion of the spectrum
near the phonon emission threshold. The derived new dispersion equation in the
immediate neighborhood below the phonon threshold corresponds to an
electron-phonon bound state. We find that the singular vertex corrections
beyond perturbation theory increase strongly the electron-phonon binding
energy. The predicted enhancement of the effective electron-phonon coupling can
be measured using angle-resolved spectroscopy.
