It is well-known that a plane graph $G$ is Eulerian if and only if its (geometric) dual $G^*$ is bipartite.
I am interested in generalisations of this result to cellular embeddings of Eulerian graphs in orientable surfaces.
One generalisation is given by Metsidik [1]: A cellularly embedded graph $G$ is Eulerian if and only if its dual graph $G^*$ is an even-face graph.
(even-face graph means that each face has a boundary of even length)
What conditions ensure that the dual of a cellularly embedded Eulerian graph is bipartite?
Is this studied? (References to papers dealing with related notions would be appreciated).
I understand that there is significant amount on work on topics such as partial duals and ribbon graphs. But, since I am not familar with those works, I don't know if results on them answer this question (at least indirectly).
PS: One necessary and sufficient condition is the existence of an Eulerian orientation of $G$ such that in-edges and out-edges alternate in the rotation at each vertex. This type of embedding of an oriented Eulerian graph is called an 'embedding' of Eulerian digraph in [2].
References
[1] Metsidik, M. Eulerian and Even-Face Graph Partial Duals. Symmetry 2021, 13, 1475. https://doi.org/10.3390/sym13081475
[2] Archdeacon, Dan; Bonnington, C. Paul; Mohar, Bojan, Embedding quartic Eulerian digraphs on the plane, Australas. J. Comb. 67, Part 2, 364-377 (2017). ZBL1375.05112.
