{
  "abstract": "For a given graph $F$, a graph $G$ is $F$-irregular if all its vertices have pairwise distinct $F$-degrees, where the $F$-degree of a vertex $v$ is the number of subgraphs of $G$ containing $v$ and isomorphic to $F$. While the existence of $F$-irregular graphs has been actively studied for connected graphs $F$, the case where $F$ is disconnected remains entirely unexplored. In this paper, we provide a complete solution to this problem. We prove that for every disconnected graph $F$ with $|E(F)| \\ge 2$, there exist infinitely many $F$-irregular graphs. Conversely, if $|E(F)| \\le 1$, we show that no non-trivial $F$-irregular graph exists.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18349",
  "author_total": 1,
  "authors": [
    {
      "name": "Tatiana Dovzhenok"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80en7801gp1hb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18349",
  "fulltext_url": null,
  "id": "80en7801gp1hb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18349",
  "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": "W7221791778",
  "pdf_url": "https://arxiv.org/pdf/2610.18349",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80en7801gp1hb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18349",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18349"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the existence of $F$-irregular graphs for disconnected graphs $F$",
  "title_text": "On the existence of F-irregular graphs for disconnected graphs F",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/80en7801gp1hb",
  "venue": "arXiv",
  "versions": []
}