We consider graphs which contain both directed and undirected edges (partially directed graphs). We show that the problem of covering the edges of such graphs with a minimum number of edge-disjoint directed paths respecting the orientations of the directed edges is polynormally solvable. We exhibit a good characterization for this problem in the form of a min-max theorem. We introduce a more general problem including weights on possible orientations of the undirected edges. We show that this more general weighted formulation is equivalent to the weighted bipartite b-factor problem. This implies the existence of a strongly polynomial algorithm for this weighted generalization of Euler's problem to partially directed graphs (compare this with the negative results for the mixed Chinese postman problem). We also provide a compact linear programming formulation for the weighted generalization that we propose.