{
  "abstract": "A proper conflict-free coloring is a proper vertex coloring in which every non-isolated vertex has a neighbor whose color appears exactly once in its open neighborhood. We study this coloring on chordal and ordered bipartite graph classes. For well-partitioned chordal graphs, we prove the tight bound $ω(G)\\leq\\pcf(G)\\leqω(G)+1$ and give an $O(n^2)$ construction from a partition tree. Every assignment of lists of at least $ω(G)+1$ colors also admits a PCF coloring that respects the lists. Deciding whether $ω(G)$ colors suffice is NP-complete, even when the partition tree is a star. This also proves NP-completeness on chordal graphs when the number of colors is part of the input. For interval graphs, we prove that lists of size $ω(G)+1$ always suffice, improving the previous ordinary-coloring bound of $ω(G)+2$. We use matchings to characterize when $ω(G)$ colors suffice and give an endpoint algorithm taking $2^{O(ω(G)^2)}n^{O(1)}$ time. We also characterize the exact value when the clique number is at most three and give a linear-time algorithm from sorted interval endpoints. For biconvex graphs, we prove a tight five-color bound and give a direct exact algorithm taking $O(n^{33})$ time. For bipartite permutation graphs, a smaller dynamic program gives an exact $O(n^{17})$ algorithm. Finally, if a bipartite graph is convex over an ordered shore $X$, we prove the sharp bound $\\pcf(G)\\leq\\lceil\\log_2(|X|+1)\\rceil+3$ and construct a coloring in linear time from the order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03906",
  "author_total": 1,
  "authors": [
    {
      "name": "Kamal Santra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c8kqygd28dp6e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03906",
  "fulltext_url": null,
  "id": "c8kqygd28dp6e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03906",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03906",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c8kqygd28dp6e.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03906",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03906"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proper Conflict-Free Coloring on Structured Graph Classes: Bounds, Exact Algorithms, and Hardness",
  "title_text": "Proper Conflict-Free Coloring on Structured Graph Classes: Bounds, Exact Algorithms, and Hardness",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/c8kqygd28dp6e",
  "venue": "arXiv",
  "versions": []
}