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 (τ0,τ1,τ2) 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 τ1-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.
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BibTeX
@misc{han2026eulerian,
title = {{Eulerian and Bipartite Partial Duals of Hypermaps}},
author = {Yufan Han and Metrose Metsidik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31327},
eprint = {2609.31327},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31327},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eulerian and Bipartite Partial Duals of Hypermaps
AU - Yufan Han
AU - Metrose Metsidik
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31327
UR - https://arxiv.org/abs/2609.31327
AB - 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 (τ_0,τ_1,τ_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 τ_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.
ER -