{
  "abstract": "A sub-quorum coloring is a partial vertex coloring in which every colored vertex sees at least half of its colored closed neighborhood in its own color. We study the maximum number of colors $\\psq(G)$ from structural, algorithmic and extremal viewpoints. Every $2$-independent set yields a sub-quorum coloring, so $\\psq(G)\\geβ_2(G)$. We prove the degree bound $\\psq(G)\\le Δ(G)|V(G)|/(Δ(G)+δ(G)-1)$ for $δ(G)\\ge2$, and obtain an exact matching characterization for graphs of maximum degree at most three. Consequently every cubic graph of order $n$ satisfies $n/2\\le\\psq(G)\\le3n/5$, with a stability identity at the upper endpoint. For trees we give an edge representation and a linear-time dynamic program that computes both $\\psq(T)$ and an optimal coloring; we also show on the double star $S_{2,2}$ that $\\psq$ can exceed the dissociation number even on trees. We prove NP-completeness even for bipartite graphs and determine exact values for several classical families. For rectangular grids, we formulate the conjecture $\\psq(G_{m,n})=β_2(G_{m,n})=F(m,n)$, recover the dissociation-number formula from known $3$-path vertex-cover results, prove the ladder case directly, and verify the conjecture for every fixed width $2\\le m\\le11$; widths $3$ through $11$ use exact transfer certificates. For hypercubes we prove $\\psq(Q_n)=β_2(Q_n)=2^{n-1}$ for every $n\\ge2$. Finally, a Nordhaus--Gaddum argument gives $\\psq(G)+\\psq(\\overline G)>\\log_2 |V(G)|$ and leads to a complementary upper-bound problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25128",
  "author_total": 3,
  "authors": [
    {
      "name": "Haobo Ma"
    },
    {
      "name": "Rafik Sahbi"
    },
    {
      "name": "Zhang Wenlin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6aknp0j9gcn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25128",
  "fulltext_url": null,
  "id": "d6aknp0j9gcn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25128",
  "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": "W7214726930",
  "pdf_url": "https://arxiv.org/pdf/2609.25128",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6aknp0j9gcn3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.25128",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25128"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sub-quorum colorings of graphs",
  "title_text": "Sub-quorum colorings of graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/d6aknp0j9gcn3",
  "venue": "arXiv",
  "versions": []
}