Partial duals of orientable hypermaps are Eulerian iff they correspond to crossing-total directions on the medial map with E' = D(Ω) ∪ T' (T' ⊆ T(Ω)), and bipartite iff they correspond to all-crossing directions with E' = C(Φ), plus the obstruction that all original hyperedges must have even length
Theory of maps on orie ntable surfaces
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Characterizations of bipartite and Eulerian partial duals of orientable hypermaps
Partial duals of orientable hypermaps are Eulerian iff they correspond to crossing-total directions on the medial map with E' = D(Ω) ∪ T' (T' ⊆ T(Ω)), and bipartite iff they correspond to all-crossing directions with E' = C(Φ), plus the obstruction that all original hyperedges must have even length