On (1,2⁴) and (1,2⁵)-packing edge-coloring of sparse subcubic graphs

@misc{kim2026125packing,
  title = {{On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs}},
  author = {Seog-Jin Kim and Xujun Liu and Boyan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29163},
  eprint = {2608.29163},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29163},
}

TY  - UNPB
TI  - On (1,2^4) and (1,2^5)-packing edge-coloring of sparse subcubic graphs
AU  - Seog-Jin Kim
AU  - Xujun Liu
AU  - Boyan Xu
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29163
UR  - https://arxiv.org/abs/2608.29163
AB  - 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 ≤ k_1 ≤ 12 and 6 ≤ k_2 ≤ 16.
ER  -