{
  "abstract": "The local clique cover number $lcc(G)$ is the minimum valency of an edge-clique cover of $G$. We prove the conjectured inequality $lcc(G)+χ(G)\\le |V(G)|+1$ for every finite simple graph. The proof gives an independent set improvement of an endpoint-cover estimate and applies the resulting construction to an induced four-vertex path. We further prove that the cover can be chosen so that every nonuniversal vertex has valency at most $|V(G)|-χ(G)$. Consequently, every graph attaining equality has a universal vertex. The stronger statement follows by reducing a counterexample of minimum order to a prime double-critical graph and refining the induced path extension. We also show that an induced matching of size $m\\ge2$ yields $lcc(G)+χ(G)\\le |V(G)|+3-m$, which improves the general bound when $m\\ge3$, and determine the restrictions imposed by equality on deletion of an induced $2K_2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07988",
  "author_total": 1,
  "authors": [
    {
      "name": "Akbar Davoodi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dvtkdbbp2f45x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07988",
  "fulltext_url": null,
  "id": "dvtkdbbp2f45x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07988",
  "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": "W7212017009",
  "pdf_url": "https://arxiv.org/pdf/2609.07988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dvtkdbbp2f45x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Local Clique Covers and Chromatic Number",
  "title_text": "Local Clique Covers and Chromatic Number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/dvtkdbbp2f45x",
  "venue": "arXiv",
  "versions": []
}