Packing colourings of claw-free subcubic graphs

@misc{romaniello2026packing,
  title = {{Packing colourings of claw-free subcubic graphs}},
  author = {Federico Romaniello},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33803},
  eprint = {2610.33803},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33803},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Packing colourings of claw-free subcubic graphs
AU  - Federico Romaniello
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33803
UR  - https://arxiv.org/abs/2610.33803
AB  - We prove that every claw-free subcubic graph admits a \packA-packing colouring, confirming a conjecture of Mortada, El Zein and Al Hajjar. The proof combines a reduction to line graphs of subdivided loopless cubic multigraphs with local colouring extensions. We also give a negative answer to their question on \packB-packing colourings by constructing an infinite family of connected bridgeless claw-free cubic counterexamples. The smallest graph in this family has 36 vertices. More generally, we characterise the simple cubic graphs whose triangle expansions admit a \packB-packing colouring in terms of gated independent odd-cycle transversals. This characterisation gives a classification for connected bridgeless claw-free cubic bases and shows that the double triangle expansion of a loopless cubic multigraph admits such a colouring if and only if the multigraph has a perfect matching.
ER  -