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?

(Also posted in mathoverflow: https://mathoverflow.net/q/445347/114739)

Interestingly, dual being bipartite is independent of the specific surface and the specific embedding. Existence of a restricted type of Eulerian orientation of the graph is necessary and sufficient to get bipartite dual (see https://mathoverflow.net/q/445347/114739 for details).