{
  "abstract": "Jaeger conjectured that, for every positive integer $p$, every $4p$-edge-connected graph admits an orientation with congruent indegree and outdegree modulo $2p+1$. We prove a stronger probabilistic counterpart: a uniformly random simple $(4p+1)$-regular graph has a strongly connected orientation with all outdegrees in $\\{p,3p+1\\}$ with probability tending to one as its even order grows. More generally, for every fixed odd $d\\ge9$, there is $\\varepsilon_d>0$ such that its circular flow number is less than $2+4/(d-1)-\\varepsilon_d$ with probability tending to one. We give explicit margins at the small degrees and prove the bound $91/33$ at degree seven, answering a fixed-upper-bound question of Li and Li. Gaps below $3$ at degree five and below $8/3$ at degree seven remain open. A uniform estimate for dense bisections also gives short intervals above the Li--Li maximum-bisection lower bound. Finally, we prove the one-unit bisection-gap conjecture of Greenhill, Isaev and Lewis: at every fixed degree $d\\ge3$, the typical minimum parameter for orientations with outdegrees in $\\{p,d-p\\}$ exceeds the threshold forced by the maximum bisection by at most one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03763",
  "author_total": 1,
  "authors": [
    {
      "name": "Amir Jafari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2kxb5y2m2sk0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03763",
  "fulltext_url": null,
  "id": "c2kxb5y2m2sk0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03763",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03763",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2kxb5y2m2sk0.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03763",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03763"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Circular flows and bisections of random regular graphs",
  "title_text": "Circular flows and bisections of random regular graphs",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/c2kxb5y2m2sk0",
  "venue": "arXiv",
  "versions": []
}