{
  "abstract": "A proper conflict-free coloring is a proper coloring in which each nonisolated vertex has a neighbor whose color occurs exactly once in its open neighborhood. We prove that deciding whether the clique number of colors suffices is NP-complete on undirected path graphs, even with a supplied representation by paths in a star. One extra color suffices on every instance in the reduction. We also prove NP-completeness on chordal bipartite graphs for every fixed palette size $k\\ge4$. For strongly chordal graphs, reverse simple elimination gives a constructive upper bound of one more than the clique number. We obtain exact polynomial-time algorithms under several conditions on the maximum cliques, including when they induce a connected subtree of a clique tree. For strictly chordal graphs we characterize the optimum and construct an optimal coloring in $O(n+m)$ time, where $n$ and $m$ count the vertices and edges. For strongly chordal graphs with clique-tree separators of size at most $s$, we give an exact algorithm in $n^{2^{O(s+1)}}$ time, with no bound on the clique number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08534",
  "author_total": 2,
  "authors": [
    {
      "name": "Gautam K. Das"
    },
    {
      "name": "Kamal Santra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtjw4kwe4eqn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08534",
  "fulltext_url": null,
  "id": "dtjw4kwe4eqn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08534",
  "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/2611.08534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtjw4kwe4eqn3.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proper Conflict-Free Coloring of Undirected Path, Chordal Bipartite, and Strongly Chordal Graphs",
  "title_text": "Proper Conflict-Free Coloring of Undirected Path, Chordal Bipartite, and Strongly Chordal Graphs",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/dtjw4kwe4eqn3",
  "venue": "arXiv",
  "versions": []
}