{
  "abstract": "The square $G^2$ of a graph joins two vertices at distance at most two; a proper coloring of $G^2$ is a 2-distance coloring of $G$. For a cactus $G$ (every edge on at most one cycle) we determine both the 2-distance chromatic number $χ(G^2)$ and the choice number $\\mathrm{ch}(G^2)$ exactly: they are always equal, and the common value is $Δ+1$ if $Δ\\ge4$, is 4 if $Δ=3$ and $G$ has no block equal to $C_5$, is 5 if $Δ=3$ and $G$ has a $C_5$ block, and is the classical value if $Δ\\le2$. For $Δ\\ge6$ the value $Δ+1$ is already known, since cacti are outerplanar and hence $K_{2,3}$-minor-free (Hetherington-Woodall; Agnarsson-Halldorsson). Our contribution is the small-degree regime. For subcubic cacti we obtain a complete classification in both the ordinary and list settings, with the 5-cycle as the unique obstruction; the list statement has no prior analogue and is a genuine positive instance of the List Square Coloring Conjecture, which is false in general. A single elimination order then handles all $Δ\\ge4$ uniformly and, in particular, settles the two cases $Δ\\in\\{4,5\\}$ that the superclass bounds leave at $Δ+2$. The number 5 turns out to be one obstruction wearing three disguises: $C_5^2=K_5$, the Frobenius number of $\\{3,4\\}$, and a degenerate $K_4$ list-coloring instance.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20204",
  "author_total": 1,
  "authors": [
    {
      "name": "Vaibhav Suvagiya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/etwzdvvv6k9c4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20204",
  "fulltext_url": null,
  "id": "etwzdvvv6k9c4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20204",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213581495",
  "pdf_url": "https://arxiv.org/pdf/2609.20204",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/etwzdvvv6k9c4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20204",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20204"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Two-distance and list-two-distance coloring of cacti: the subcubic case and the C5 obstruction",
  "title_text": "Two-distance and list-two-distance coloring of cacti: the subcubic case and the C5 obstruction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/etwzdvvv6k9c4",
  "venue": "arXiv",
  "versions": []
}