{
  "abstract": "Extending the notions of Kneser graphs and Schrijver graphs, we define the $G$-stable Kneser graph $KG(G,k)$ (where $k\\geq 2$ and $G$ is a graph on $n\\geq 2k$ vertices) as the graph whose vertices are the $k$-element independent sets in $G$ and whose edges join pairs of disjoint independent sets. The well-known result of Lovász on the chromatic number of Kneser graphs implies that $χ(KG(G,k)) \\leq n-2k+2$ for any $G$. We call $G$ $k$-tight if equality holds, and we study the following question: Which graphs are $k$-tight? A result of Daneshpajouh et al.\\ implies a characterization of $2$-tight graphs. We prove a necessary condition for a graph to be $k$-tight for $k\\geq 3$ which extends this characterization, and we conjecture it to be sufficient. Another result gives a lower bound to the chromatic number of the $G$-stable Kneser graphs when $G$ is a disjoint union of cycles. We use this result to give a short proof of the fact that $4$-regular graphs on $3k$ vertices consisting of two vertex-disjoint cycles and $k$ cycles of length $3$ admit an independent set of size $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17884",
  "author_total": 4,
  "authors": [
    {
      "name": "Carmen Arana"
    },
    {
      "name": "Penny Haxell"
    },
    {
      "name": "Tomáš Kaiser"
    },
    {
      "name": "Matěj Stehlík"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19z8tr9cpfev8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17884",
  "fulltext_url": null,
  "id": "19z8tr9cpfev8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17884",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221772855",
  "pdf_url": "https://arxiv.org/pdf/2610.17884",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19z8tr9cpfev8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17884",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17884"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The chromatic number of $G$-stable Kneser graphs",
  "title_text": "The chromatic number of G-stable Kneser graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/19z8tr9cpfev8",
  "venue": "arXiv",
  "versions": []
}