{
  "abstract": "We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its original one. We define the stable chromatic number $χ_\\mathrm{stable}(G)$ of a graph $G$ as the minimum integer $k$ such that every preference profile admits a stable $k$-coloring of $G$. We establish several upper and lower bounds. In particular, for any acyclic orientation of the edges of $G$, the largest number of vertices reachable from a vertex by directed paths, including the vertex itself, is an upper bound on $χ_\\mathrm{stable}(G)$. This shows that $χ_\\mathrm{stable}(G)$ is well-defined. We also show that $O(t \\log (1+n/t))$ colors suffice for an $n$-vertex graph $G$ of treewidth $t$, and complement this with a lower bound in terms of the Grundy number. Turning to the problem of finding a minimum stable coloring for a given profile, we show that stable $2$-colorability is polynomial-time solvable, whereas stable $k$-colorability is NP-complete for every fixed $k\\ge 3$. Using the treewidth bound, we give a fixed-parameter tractable algorithm parameterized by treewidth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00569",
  "author_total": 3,
  "authors": [
    {
      "name": "Tomohiro Koana"
    },
    {
      "name": "Yeeseok Oh"
    },
    {
      "name": "Hirotaka Yoneda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51xadgx1genn5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00569",
  "fulltext_url": null,
  "id": "51xadgx1genn5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00569",
  "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",
      "cs.GT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7206185944",
  "pdf_url": "https://arxiv.org/pdf/2609.00569",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51xadgx1genn5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00569",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00569"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graph Coloring with Color Preferences",
  "title_text": "Graph Coloring with Color Preferences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/51xadgx1genn5",
  "venue": "arXiv",
  "versions": []
}