{
  "abstract": "We study several $k$-fold generalizations of the Lovász theta function associated with the maximum $k$-colorable induced subgraph problem. The first is the Narasimhan--Manber parameter $\\vartheta_k$. We prove that, for graphs whose adjacency matrix belongs to a homogeneous partially coherent algebra, this parameter is recovered by the theta number of the Cartesian product with the complete graph on $k$ vertices. This class includes distance-regular and $1$-walk-regular graphs, and thus our result generalizes a theorem by Sinjorgo and Sotirov (2022) for graphs that are vertex- and edge-transitive. We introduce a new parameter $φ_k$ obtained from orthonormal representations of graphs and show the inequality $φ_k \\leq \\vartheta_k$. For both parameters, we study the smallest $k$ for which the parameter is equal to the number of vertices; these saturation parameters yield lower bounds on the chromatic number. We determine which vertex-weighted versions of these parameters are gauges, and discuss a natural definition for the $k$-fold theta body of a graph. We conclude with open questions comparing $\\vartheta_k$, $φ_k$, $\\vartheta(G\\square K_k)$, and related convexifications.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06312",
  "author_total": 4,
  "authors": [
    {
      "name": "Marcel K Carli Silva"
    },
    {
      "name": "Gabriel Coutinho"
    },
    {
      "name": "Thiago Oliveira"
    },
    {
      "name": "Levent Tunçel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80npsptwpr0rp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06312",
  "fulltext_url": null,
  "id": "80npsptwpr0rp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06312",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212002746",
  "pdf_url": "https://arxiv.org/pdf/2609.06312",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80npsptwpr0rp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06312",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06312"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On some k-fold generalizations of Lovász theta and their sandwich theorems",
  "title_text": "On some k-fold generalizations of Lovász theta and their sandwich theorems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/80npsptwpr0rp",
  "venue": "arXiv",
  "versions": []
}