{
  "abstract": "Tutte's $3$-flow conjecture asserts that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In 2013, Lovász, Thomassen, Wu, and Zhang proved that every odd-$7$-edge-connected graph admits a nowhere-zero $3$-flow; consequently, the conjecture holds for $4$-edge-connected graphs with no edge-cut of size $5$. We consider the complementary situation in which $5$-edge-cuts are allowed. We show that, when edge-cuts of size $5$ are permitted, Tutte's $3$-flow conjecture holds for graphs with no edge-cut of any size from $6$ to $k$, where $k$ is an absolute constant. In fact, $k=40$ suffices and we prove a stronger version in which only nontrivial edge-cuts of those sizes are forbidden. A graph is called essentially $t$-edge-connected if deleting any set of at most $t-1$ edges leaves at most one nontrivial component. Motivated by Jaeger's weak $3$-flow conjecture, we prove an analogous result for essential edge connectivity: every $4$-edge-connected, essentially $41$-edge-connected graph admits a nowhere-zero $3$-flow.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08377",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiaao Li"
    },
    {
      "name": "Xinyuan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n8st8fgdrkqp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08377",
  "fulltext_url": null,
  "id": "2n8st8fgdrkqp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08377",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212002889",
  "pdf_url": "https://arxiv.org/pdf/2609.08377",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n8st8fgdrkqp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08377",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08377"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nowhere-zero $3$-flows in graphs with forbidden edge-cuts",
  "title_text": "Nowhere-zero $3$-flows in graphs with forbidden edge-cuts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2n8st8fgdrkqp",
  "venue": "arXiv",
  "versions": []
}