{
  "abstract": "An $r$-dynamic coloring of a graph $G$ is a proper vertex coloring in which every vertex $v$ sees at least $\\min\\{r,d(v)\\}$ distinct colors in its neighborhood. The minimum number of colors in such a coloring is the $r$-dynamic chromatic number $χ_r^d(G)$. We study dynamic colorings of regular graphs. A straighforward observation shows that $χ_r^d(G)=χ(G^2)$, for any $r$-regular graph $G$. We prove that $χ_3^d(G)=χ(G^2)\\le6$ for every claw-free cubic graph $G$, and the bound is sharp. For Hamiltonian claw-free cubic graphs, we improve the bound to $5$ apart from four explicit exceptions. We also determine exactly the $r$-dynamic chromatic number of the $4$-regular circulant graph $C_p(1,3)$ for each $r\\in\\{2,3,4\\}$. In particular, the case $r=4$ determines $χ(C^2_p(1,3))$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09565",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Gutierrez"
    },
    {
      "name": "Grover Ugarte"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8kb8mhfk9w8b8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09565",
  "fulltext_url": null,
  "id": "8kb8mhfk9w8b8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09565",
  "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|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7220976856",
  "pdf_url": "https://arxiv.org/pdf/2610.09565",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8kb8mhfk9w8b8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.09565",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09565"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Dynamic Coloring and Graph Squares of Regular Graphs",
  "title_text": "Dynamic Coloring and Graph Squares of Regular Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8kb8mhfk9w8b8",
  "venue": "arXiv",
  "versions": []
}