{
  "abstract": "A strong odd $k$-coloring of a graph $G$ is a proper $k$-coloring such that every color appearing in the neighborhood of a non-isolated vertex appears an odd number of times. The minimum $k$ for which $G$ admits a strong odd $k$-coloring is the \\emph{strong odd chromatic number}, denoted by $χ_{\\text{so}}(G)$, of $G$. Given a graph $G$ and an integer $k$, \\textsc{strong odd $k$-colorability} problem asks whether $G$ admits a strong odd $k$-coloring. It is known that STRONG ODD $k$-COLORABILITY is NP-complete in general graphs. In this paper, we prove that the problem is NP-complete on perfect elimination bipartite graphs for $k\\geq3$, which is a subclass of bipartite graphs. Furthermore, we show that $χ_{\\text{so}}(G)$ is inapproximable within a factor of $O(n^{\\frac{1}{2}-\\varepsilon})$ for every $\\varepsilon>0$. On the positive side, we obtain a linear time algorithm to compute an optimal strong odd coloring for block graphs. From a parameterized perspective, we present an FPT algorithm for STRONG ODD $k$-COLORABILITY when parameterized by treewidth. Moreover, we show that the problem cannot be solved in time $(k-\\varepsilon)^{\\texttt{tw}}n^{O(1)}$ for every $k\\geq3$ and $\\varepsilon>0$ when parameterized by treewidth under SETH. Furthermore, we show that STRONG ODD $k$-COLORABILITY does not admit a polynomial kernel when parameterized by feedback vertex set. Lastly, we prove that STRONG ODD $k$-COLORABILITY is W[1]-hard when parameterized by clique-width.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01441",
  "author_total": 3,
  "authors": [
    {
      "name": "Dinabandhu Pradhan"
    },
    {
      "name": "Vaishali Sharma"
    },
    {
      "name": "Shaily Verma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jtzhhw5cbsdm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01441",
  "fulltext_url": null,
  "id": "6jtzhhw5cbsdm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01441",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7216305406",
  "pdf_url": "https://arxiv.org/pdf/2610.01441",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jtzhhw5cbsdm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01441",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01441"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Classical and Parameterized Complexity of Strong Odd Coloring",
  "title_text": "On the Classical and Parameterized Complexity of Strong Odd Coloring",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/6jtzhhw5cbsdm",
  "venue": "arXiv",
  "versions": []
}