  We calculate from first principles the electron-phonon coupling strength in
ferromagnetic iron-cobalt Fe$_{1-x}$Co$_x$ alloys for compositions ranging from
$x=0$ to $x=0.75$. We find strong, spin-dependent variation of the
electron-phonon coupling strength with alloy composition. The minimum of the
electron-phonon interaction is found near the composition $x=0.25$. We analyze
the variation of the electron-phonon interaction with composition, as a
function of electron spin, density of states, electron-phonon matrix elements,
and phonon frequencies. Our results are in good qualitative agreement with
magnetization dynamics experiments.
