{
  "abstract": "Nash--Williams's orientation theorem converts undirected edge-connectivity into directed connectivity. We extend it to prescribed outdegrees modulo $q$, whose residues must sum to the number of edges modulo $q$. For integers $q,k\\ge1$, every such prescription has a realization with at least $k$ edges in each direction across each nontrivial cut if the graph is $(2q+2k-2)$-edge-connected or contains $q+2k-1$ edge-disjoint spanning trees. The method combines a cycle-space approximation bound with rounding that preserves the imbalance across every cut. Fractional spanning-tree packing gives an exact criterion: every compatible prescription has a strongly connected realization if and only if the fractional packing number exceeds $q$. This yields a polynomial-time recognition test. With no directed connectivity requirement, $q$ disjoint spanning trees suffice, proving Hasanvand's prescribed-residue conjecture. For every odd prime $p$, this proves the conjectured statement that the union of $p$ graphic bases over the field with $p$ elements is an additive basis. For every integer $p\\ge5$, the theorem and the counterexamples of Han, Li, Wu and Zhang give the exact edge-connectivity threshold $4p+2$ for a modulo-$(2p+1)$ orientation. We also show that every six-edge-connected graph has circular flow number strictly below three, as conjectured by Li, Thomassen, Wu and Zhang. For three-flow-critical graphs on $n$ vertices and $m$ edges, we improve the previous general bound $m\\le4n-10$ to $m\\le3n-6$, and to $m\\le3n-7$ when $n\\ge5$. Further applications include Seymour's six-flow theorem, star decompositions, antisymmetric flows, and signed modular orientations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07034",
  "author_total": 1,
  "authors": [
    {
      "name": "Amir Jafari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9p6y5sez2gd9w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07034",
  "fulltext_url": null,
  "id": "9p6y5sez2gd9w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07034",
  "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": "W7220919875",
  "pdf_url": "https://arxiv.org/pdf/2610.07034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9p6y5sez2gd9w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.07034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graphic additive bases and graph flows",
  "title_text": "Graphic additive bases and graph flows",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:27Z",
  "url": "https://research.draconitic.com/paper/9p6y5sez2gd9w",
  "venue": "arXiv",
  "versions": []
}