{
  "abstract": "Given a graph $G$ and a positive integer $k$, a packing $k$-domatic coloring of $G$ is a function $f: V(G) \\to \\{1,\\ldots,t\\}$ such that (1) for any color $j \\in \\{1,\\ldots, t\\}$ and any two distinct vertices $u,v\\in V(G)$ with $f(u)=f(v)=j$ we have $d_G(u,v)>j$, and (2) $V(G)$ admits a partition into $k$ sets $A_1,\\ldots,A_k$ such that for any $v\\in V(G)$ and any $i\\in\\{1,\\ldots,k\\}$ there exists a vertex $w\\in A_i$ such that $d_G(v,w)\\le f(w)$. The minimum integer $t$ such that $G$ admits a packing $k$-domatic coloring of $G$ using colors in $\\{1,\\ldots,t\\}$ is denoted by $χ_{ρ,k}(G)$. The condition (1) in the definition of a packing $k$-domatic coloring implies that $G$ is a packing coloring, hence $χ_{ρ,k}(G)\\ge χ_ρ(G)$ holds for any graph $G$, where $χ_ρ(G)$ is the packing chromatic number of $G$. On the other hand, the new concept also leads to a generalization of the domatic number of a graph due to which one can easily see that $χ_{ρ,2}(G)=χ_ρ(G)$ holds for any graph with no isolated vertices. One of the main result in this paper is that $χ_{ρ,3}(G)=χ_ρ(G)$ holds in any connected graph $G$ with minimum degree at least $2$, and the bound is best possible in two different senses. In addition, we provide several exact values and bounds on the new invariant in paths and cycles. We prove that $χ_{ρ,k}(P_n)=k$ for any $k\\in\\{3,4,5\\}$ as soon as $n\\ge 8$, and, in contrast, $χ_{ρ,k}(P_n)>k$ for any $n\\ge k\\ge 12$. We also prove the exact values of $χ_{ρ,k}(P_\\infty)$ when $k\\in \\{3,4,5\\}$ for the two-way infinite path $P_\\infty$, and exact values of $χ_{ρ,k}(C_n)$ when $k\\in\\{3,4\\}$ for all cycles $C_n$. We also consider a general framework of $S$-packing $k$-domatic colorings, where $S$ is an arbitrary sequence of non-negative integers, and present some basic results in this context.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03477",
  "author_total": 3,
  "authors": [
    {
      "name": "Boštjan Brešar"
    },
    {
      "name": "Jasmina Ferme"
    },
    {
      "name": "Wenjie Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1czvbj79pg133.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03477",
  "fulltext_url": null,
  "id": "1czvbj79pg133",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03477",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7220410699",
  "pdf_url": "https://arxiv.org/pdf/2610.03477",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1czvbj79pg133.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03477",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03477"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Partitioning an $S$-packing coloring into broadcast dominating sets",
  "title_text": "Partitioning an S-packing coloring into broadcast dominating sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/1czvbj79pg133",
  "venue": "arXiv",
  "versions": []
}