  We consider the 4-precoloring extension problem in \emph{planar
near-Eulerian-triangulations}, i.e., plane graphs where all faces except
possibly for the outer one have length three, all vertices not incident with
the outer face have even degree, and exactly the vertices incident with the
outer face are precolored. We give a necessary topological condition for the
precoloring to extend, and give a complete characterization when the outer face
has length at most five and when all vertices of the outer face have odd degree
and are colored using only three colors.
