{
  "abstract": "A $(1^j,2^k)$-packing edge-coloring of a graph $G$ is a partition of the edge set $E(G)$ into $j$ matchings and $k$ induced matchings. Hocquard, Lajou, and Lu\\v zar found a subcubic planar graph of girth $3$ that has no $(1,2^5)$-packing edge-coloring and also conjectured that every subcubic planar graph has a $(1,2^6)$-packing edge-coloring. We also notice that for every fixed positive integer $k$ there exists a subcubic planar graph with girth $k$ that is not $(1,2^3)$-packing edge-colorable. It is natural to consider what is the minimum positive integer $k_1$ such that every subcubic planar graph with girth at least $k_1$ is $(1,2^5)$-packing edge-colorable. Furthermore, we also consider what is the minimum positive integer $k_2$ such that every subcubic planar graph with girth at least $k_2$ is $(1,2^4)$-packing edge-colorable. In this paper, we show both $k_1$ and $k_2$ are finite, and in fact $5 \\le k_1 \\le 12$ and $6 \\le k_2 \\le 16$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29163",
  "author_total": 3,
  "authors": [
    {
      "name": "Seog-Jin Kim"
    },
    {
      "name": "Xujun Liu"
    },
    {
      "name": "Boyan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bagt2snq1sdb4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29163",
  "fulltext_url": null,
  "id": "bagt2snq1sdb4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29163",
  "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|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7204914475",
  "pdf_url": "https://arxiv.org/pdf/2608.29163",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bagt2snq1sdb4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29163",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29163"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs",
  "title_text": "On (1,2^4) and (1,2^5)-packing edge-coloring of sparse subcubic graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/bagt2snq1sdb4",
  "venue": "arXiv",
  "versions": []
}