{
  "abstract": "There are four covering numbers $\\mathrm{c}_g^{\\mathcal{G}}(H),\\mathrm{c}_u^{\\mathcal{G}}(H),\\mathrm{c}_l^{\\mathcal{G}}(H),\\mathrm{c}_f^{\\mathcal{G}}(H)$, each of which measures in a slightly different way how well the edges of a graph $H$ (called a host) can be covered with graphs of a class $\\mathcal{G}$ (called a guest class). If we require the graphs of $\\mathcal{G}$ to correspond to induced subgraphs of $H$, we obtain an induced variant $\\mathrm{ic}_x^{\\mathcal{G}}$ for each covering number $\\mathrm{c}_x^{\\mathcal{G}}$ which satisfies $\\mathrm{c}_x^{\\mathcal{G}}(H) \\leq \\mathrm{ic}_x^{\\mathcal{G}}(H)$ for every graph $H$. Yet, in general $\\mathrm{ic}_x^{\\mathcal{G}}$ cannot be bounded in terms of $\\mathrm{c}_x^{\\mathcal{G}}$. If there exists for a guest class $\\mathcal{G}$ and a host class $\\mathcal{H}$ a function $f$ such that $\\mathrm{ic}_x^{\\mathcal{G}}(H) \\leq f(\\mathrm{c}_x^{\\mathcal{G}}(H))$ for every graph $H \\in \\mathcal{H}$, we call $f$ a binding function. Within this work, we study for which structural properties of a guest class $\\mathcal{G}$ and a host class $\\mathcal{H}$ such binding functions exist. We consider guest classes $\\mathcal{G}$ that are monotone, hereditary, component-closed or neither, and have bounded maximum average degree, bounded chromatic number or neither. The host classes $\\mathcal{H}$ we consider have bounded treewidth, exclude some minor, have bounded maximum average degree, bounded chromatic number, or none of these properties. For $219$ out of the $240$ possible $3$-tuples of properties for $\\mathcal{G}$ and $\\mathcal{H}$ and covering numbers we either provide a binding function or an example where no such function exists. In particular, we show that such binding functions always exist for hereditary guest classes $\\mathcal{G}$ of bounded maximum average degree for three of the four covering numbers, but may not for the fourth kind.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09873",
  "author_total": 2,
  "authors": [
    {
      "name": "Miriam Goetze"
    },
    {
      "name": "Yidi Zang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7h2y56c6r1aha.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09873",
  "fulltext_url": null,
  "id": "7h2y56c6r1aha",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09873",
  "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": "W7212111617",
  "pdf_url": "https://arxiv.org/pdf/2609.09873",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7h2y56c6r1aha.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09873",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09873"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Boundedness and Separation Between Induced and Non-Induced Covering Numbers",
  "title_text": "Boundedness and Separation Between Induced and Non-Induced Covering Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/7h2y56c6r1aha",
  "venue": "arXiv",
  "versions": []
}