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Proper Conflict-Free Coloring on Structured Graph Classes: Bounds, Exact Algorithms, and Hardness — arXiv:2611.03906

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.03906
License (as archived): CC0-1.0
Archived: 2026-11-04T12:48:58.906Z
SHA-256 of the archived text: 6cc6f1101596158dd47de6b8f46b60739a31ba89680e8e50fa1e8d2a3b2ca42e
Stable URL of this page: https://metraipolis.com/documento/dc_ccb60f55168c

How to cite

Proper Conflict-Free Coloring on Structured Graph Classes: Bounds, Exact Algorithms, and Hardness — arXiv:2611.03906 — arXiv (en), https://arxiv.org/abs/2611.03906 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ccb60f55168c (Metraipolis archive dc_ccb60f55168c, sha256 6cc6f1101596158dd47de6b8f46b60739a31ba89680e8e50fa1e8d2a3b2ca42e)

BibTeX

@misc{dcccb60f55168c,
  title = {Proper Conflict-Free Coloring on Structured Graph Classes: Bounds, Exact Algorithms, and Hardness — arXiv:2611.03906},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.03906},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ccb60f55168c; sha256 6cc6f1101596158dd47de6b8f46b60739a31ba89680e8e50fa1e8d2a3b2ca42e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.03906. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\omega(G)\leq\pcf(G)\leq\omega(G)+1$ and give an $O(n^2)$ construction from a partition tree. Every assignment of lists of at least $\omega(G)+1$ colors also admits a PCF coloring that respects the lists. Deciding whether $\omega(G)$ 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 $\omega(G)+1$ always suffice, improving the previous ordinary-coloring bound of $\omega(G)+2$. We use matchings to characterize when $\omega(G)$ colors suffice and give an endpoint algorithm taking $2^{O(\omega(G)^2)}n^{O(1)}$ 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.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Discrete Mathematics (cs.DM) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2611.03906 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-04T12:32:32.050Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).