  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.

Combinatorics