  We address the enumeration of planar 4-valent maps equipped with an Eulerian
orientation by two different methods, and compare the solutions we thus obtain.
With the first method we enumerate these orientations as well as a restricted
class which we show to be in bijection with general Eulerian orientations. The
second method, based on the work of Kostov, allows us to enumerate these
4-valent orientations with a weight on some vertices, corresponding to the six
vertex model. We prove that this result generalises both results obtained using
the first method, although the equivalence is not immediately clear.
