{
  "abstract": "Thomassen (J. Combin. Theory Ser. B 108 (2014), 81-91) showed that every graph admitting a nowhere-zero $3$-flow also admits an $S^1$-flow. He also proved that the converse holds for cubic graphs, but constructed counterexamples showing it fails in general. Wang et al. (SIAM J. Discrete Math. 29 (2015), 2166-2178) presented a couple of sufficient conditions under which the existence of an $S^1$-flow guarantees the existence of a nowhere-zero 3-flow. In this paper, we first prove that a graph with maximum degree at most four admits a nowhere-zero $3$-flow if and only if it admits an $S^1$-flow. We then develop some reduction techniques for $S^1$-flows based on graph operations including bull-growth, $2$-sums, and contractions. Finally, we apply those techniques to characterize triangularly connected graphs and graphs containing a spanning triangle-tree that admit $S^1$-flows, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18725",
  "author_total": 4,
  "authors": [
    {
      "name": "Chenxing Li"
    },
    {
      "name": "Jiaao Li"
    },
    {
      "name": "Rong Luo"
    },
    {
      "name": "Bo Su"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/21jary1q1r1kr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18725",
  "fulltext_url": null,
  "id": "21jary1q1r1kr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18725",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7203899622",
  "pdf_url": "https://arxiv.org/pdf/2608.18725",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/21jary1q1r1kr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2608.18725",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18725"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Reduction Operations and Characterizations of $S^1$-Flows in Graphs",
  "title_text": "Reduction Operations and Characterizations of S^1-Flows in Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/21jary1q1r1kr",
  "venue": "arXiv",
  "versions": []
}