Partially broken orientations of Eulerian graphs on closed surfaces

In this paper, we discuss orientations given to edges of Eulerian graphs embedded on closed surfaces, and give a characterization of such a graph having a good orientation, i.e., incoming edges and outgoing edges appear alternately around each vertex of the graph. Furthermore, we consider the partially broken orientations of Eulerian graphs embedded on closed surfaces, that is, one in which only for each of specified vertices, incoming edges and outgoing edges appear alternately around it. We discuss a little about good orientations defined for Eulerian graphs drawn on closed surfaces with crossings, and for general connected graphs, i.e., not necessarily Eulerian graphs, embedded on closed surfaces.