{
  "abstract": "If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a mapping $f: E(G)\\rightarrow E(H)$, such that for every $v\\in V(G)$ there is a vertex $x\\in V(H)$, such that $f(\\partial_G(v))=\\partial_H(x)$. If $G$ admits an $H$-coloring then it is common to write $H\\prec G$. In this paper, for a fixed connected cubic graph $H$, we consider a decision problem, which we call $H$-coloring. In this problem, on the input one is given a connected cubic graph $G$. The goal is to decide whether $H\\prec G$. A conjecture by Hakobyan and the author states that $H$-coloring is polynomial time solvable, if $H$ does not admit a $P_{12}$-coloring. Here $P_{12}$ is the unique cubic graph obtained from the Petersen graph by expanding one of its vertices into a triangle. In this paper, we disprove this conjecture under $P\\neq NP$, by showing that $S_6$-coloring is NP-complete. Here $S_6$ is the unique cubic graph on six vertices having exactly one bridge. Our second result is an algorithm that solves $H$-coloring problem in time $O^*(c^n)$, where $H$ is a simple connected cubic graph, $n$ is the number of vertices of the input cubic graph $G$ and $c=\\sqrt{8}\\approx 2.8284$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35794",
  "author_total": 1,
  "authors": [
    {
      "name": "Vahan Mkrtchyan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa93am78bt29r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35794",
  "fulltext_url": null,
  "id": "fa93am78bt29r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35794",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7223851543",
  "pdf_url": "https://arxiv.org/pdf/2610.35794",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa93am78bt29r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.35794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35794"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Computational complexity of the $H$-coloring problem in cubic graphs",
  "title_text": "Computational complexity of the H-coloring problem in cubic graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/fa93am78bt29r",
  "venue": "arXiv",
  "versions": []
}