  We calculate the density of states (DOS) in graphene for electrons coupled to
a phonon in an external magnetic field. We find that coupling to an Einstein
mode of frequency $\omega_E$ not only shifts and broadens the Landau levels
(LLs), but radically alters the DOS by introducing a new set of peaks at
energies $E_n\pm\omega_E$, where $E_n$ is the energy of the $n$th LL. If one of
these new peaks lies sufficiently close to a LL, it causes the LL to split in
two; if the system contains an energy gap, a LL may be split in three. The new
peaks occur outside the interval $(-\omega_E,\omega_E)$, leaving the LLs in
that interval largely unaffected. If the chemical potential is greater than the
phonon frequency, the zeroth LL lies outside the interval and can be split,
eliminating its association with a single Dirac point. We find that coupling to
an extended phonon distribution such as a Lorentzian or Debye spectrum does not
qualitatively alter these results.
