Eulerian and Bipartite Partial Duals of Hypermaps

Abstract

We study hyperedge partial duals of finite hypermaps in a purely combinatorial framework, without assuming orientability. A hypermap is represented by three fixed-point-free involutions $(\tau_0,\tau_1,\tau_2)$ on its flag set. We first give an explicit construction of the medial map from this model: $02$-orbits become the medial vertex discs, while $\tau_1$-transpositions become the medial bands; a local orientation system and its twist data then provide a signed rotation description of the medial map. We next prove that the state circles associated with a chosen set of hyperedges are in natural bijection with the vertex orbits of the corresponding partial dual, yielding a crossing-total characterization of all Eulerian hyperedge partial duals. For bipartiteness, the twist data lead to a modified medial map in which inserted bars record the obstruction to a global orientation. We prove that a partial dual is bipartite if and only if its dualized hyperedge set is exactly the set of $c$-type hyperedges identified by an all-crossing orientation of this modified medial map. When the hypermap is orientable, these constructions specialize to the known orientable-hypermap results; when every hyperedge has valence two, they specialize to the ribbon-graph results.