{
  "abstract": "We study simultaneous color densities in packing colorings of integer distance graphs. For $D(1,6)$, we determine several exact density regions and prove that colors $1$ through $7$ have maximum combined density $211/252$. When this maximum is approached, the seven individual color frequencies are forced to converge to a specified vector. On an optimal low-color layer, some density vectors have nonperiodic realizations but no periodic realization; we determine how much accumulated density loss is necessary for switching between the relevant configurations. For sufficiently large additional color indices, a fixed finite graph describes the joint density region. In particular, we determine a seven-vertex region for every $i\\equiv8\\pmod{14}$ with $i\\ge36$ and prove that $36$ is the first stable index in this residue class. The proofs combine finite-state integer certificates with explicit constructions and limit arguments. Applications give $17\\leχ_ρ(D(1,6))\\le20$, $18\\leχ_ρ(D(1,8))\\le22$, and $χ_ρ(D(1,9))\\le17$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09018",
  "author_total": 1,
  "authors": [
    {
      "name": "Enkai Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dqzg8ahb5x50m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09018",
  "fulltext_url": null,
  "id": "dqzg8ahb5x50m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09018",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7212068076",
  "pdf_url": "https://arxiv.org/pdf/2609.09018",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dqzg8ahb5x50m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.09018",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09018"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Density regions, integer certificates and packing colorings of distance graphs",
  "title_text": "Density regions, integer certificates and packing colorings of distance graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/dqzg8ahb5x50m",
  "venue": "arXiv",
  "versions": []
}