{
  "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 $(τ_0,τ_1,τ_2)$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31327",
  "author_total": 2,
  "authors": [
    {
      "name": "Yufan Han"
    },
    {
      "name": "Metrose Metsidik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e7xhtfjk72gjk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31327",
  "fulltext_url": null,
  "id": "e7xhtfjk72gjk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31327",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214655445",
  "pdf_url": "https://arxiv.org/pdf/2609.31327",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e7xhtfjk72gjk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31327",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31327"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Eulerian and Bipartite Partial Duals of Hypermaps",
  "title_text": "Eulerian and Bipartite Partial Duals of Hypermaps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/e7xhtfjk72gjk",
  "venue": "arXiv",
  "versions": []
}