{
  "abstract": "Duke's interpolation theorem states that the orientable genera of a connected graph form an integer interval, and Stahl established the corresponding result for nonorientable embeddings. In 1991, Širáň showed that this interpolation property fails for signed graph embeddings: the Euler-genus spectrum of a signed graph may contain gaps. He subsequently asked whether all such gaps must occur at the lower end of the spectrum. In this paper, we establish a characterization of the Euler-genus spectrum of a connected signed graph. We prove that, for each parity class, the Euler genera form a step-two interval. Moreover, whenever both parity classes are nonempty, their maximum elements differ by one. As an consequence, if two consecutive integers $k$ and $k+1$ belong to the Euler-genus spectrum, then every integer from $k$ to the maximum Euler-genus also belongs to the spectrum, thereby answering Širáň's question affirmatively. Our proof uses the pre-signed graph representation of signed embeddings together with ordered adjacent-exchange operations and a matching interpretation of face numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27628",
  "author_total": 2,
  "authors": [
    {
      "name": "Yichao Chen"
    },
    {
      "name": "Yan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f24yjtag9wma8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27628",
  "fulltext_url": null,
  "id": "f24yjtag9wma8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27628",
  "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": "W7214191384",
  "pdf_url": "https://arxiv.org/pdf/2609.27628",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f24yjtag9wma8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27628",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27628"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generalized Duke's theorem for signed Graphs",
  "title_text": "Generalized Duke's theorem for signed Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/f24yjtag9wma8",
  "venue": "arXiv",
  "versions": []
}