{
  "abstract": "Chen, Mohar and Wu [J. Combin. Theory Ser. B 161 (2023)] conjectured that every graph of orientable genus $g$ has proper orientation number $O(\\sqrt g)$. We prove this conjecture. Our main result extends the potential-outdegree method from independent color classes to three parts of bounded internal degree. Let $k,D\\in\\mathbb Z_{\\ge0}$. If $G$ has maximum average degree at most $2k$ and $V(G)=V_1\\cup V_2\\cup V_3$, where each $V_i$ induces a subgraph of maximum degree at most $D$, then $$\\pchi(G)\\le k+9D+6+\\floor{\\frac32\\ceil{\\frac{2D+2}{5}}}\\le k+\\floor{\\frac{48D}{5}}+8.$$ When the partition is given, we construct such an orientation in polynomial time. For every $D\\ge1$, we also construct a graph with a partition into three parts of internal maximum degree at most $D$ for which $\\pchi(G)-\\ceil{\\mad(G)/2}\\ge6D+5$; hence the linear dependence on $D$ is unavoidable. Finally, Woodall's defective-coloring theorem yields $\\pchi(G)=O(\\sqrt{γ+1})$ for every graph of Euler genus $γ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30315",
  "author_total": 2,
  "authors": [
    {
      "name": "Qi Wu"
    },
    {
      "name": "Yong Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1nsq19javcy5v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30315",
  "fulltext_url": null,
  "id": "1nsq19javcy5v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30315",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214653393",
  "pdf_url": "https://arxiv.org/pdf/2609.30315",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1nsq19javcy5v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30315",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30315"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounded-Defect Proper Orientations and Graphs on Surfaces",
  "title_text": "Bounded-Defect Proper Orientations and Graphs on Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/1nsq19javcy5v",
  "venue": "arXiv",
  "versions": []
}