{
  "abstract": "For a graph $G$ and an integer $d\\geq 0$, the defective chromatic polynomial $χ_d(G;k)$ counts the $k$-colorings of $G$ in which each vertex has at most $d$ neighbors of its own color. We investigate which structural properties of $G$ are determined by the full family $\\{χ_d(G;k)\\}_{d\\geq 0}$. We establish a contraction formula expressing $χ_d(G;k)$ as a sum of ordinary chromatic polynomials of the edge contractions of $G$. As a first application, we prove that for triangle-free graphs, the full family determines the degree sequence. For trees, we show further that the family $\\{χ_d(T;k)\\}_{d\\geq 0}$ determines the path-subgraph counts $N(P_j,T)$ for $j=1,2,3,4$, but not for $j=5$. For each $n\\geq 9$, we construct a pair of nonisomorphic trees of order $n$ that share the same defective chromatic polynomials for every $d\\geq 0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.05550",
  "author_total": 3,
  "authors": [
    {
      "name": "Shamil Asgarli"
    },
    {
      "name": "Tamsen Whitehead McGinley"
    },
    {
      "name": "Nicholas Xue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/73548cg1dwk1y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.05550",
  "fulltext_url": null,
  "id": "73548cg1dwk1y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.05550",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7160634916",
  "pdf_url": "https://arxiv.org/pdf/2605.05550",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/73548cg1dwk1y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.05550",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.05550"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Defective chromatic polynomials",
  "title_text": "Defective chromatic polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/73548cg1dwk1y",
  "venue": "arXiv",
  "versions": []
}