Proper conflict-free degree-choosability of graphs with maximum average degree less than 8/3

@misc{xu2026proper,
  title = {{Proper conflict-free degree-choosability of graphs with maximum average degree less than $8/3$}},
  author = {Huijuan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41668},
  eprint = {2610.41668},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41668},
}

TY  - UNPB
TI  - Proper conflict-free degree-choosability of graphs with maximum average degree less than 8/3
AU  - Huijuan Xu
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41668
UR  - https://arxiv.org/abs/2610.41668
AB  - A proper coloring φ of a graph G is called proper conflict-free if, for every non-isolated vertex v, some color appears exactly once in its open neighborhood N_G(v). For a nonnegative integer k, a graph G is proper conflict-free (degree+k)-choosable if, for every list assignment L satisfying |L(v)|≥ d_G(v)+k for each v∈ V(G), there exists a proper conflict-free coloring φ of G with φ(v)∈ L(v) for each v∈ V(G). We prove that every connected graph other than the $5$-cycle C_5 with maximum average degree less than 8/3 is proper conflict-free (degree+2)-choosable. This improves the maximum average degree bound 18/7 of Kashima, Škrekovski and Xu, and removes the assumption g(G)≥7 from the 8/3 theorem of Qi, Song and Cao. As an immediate consequence, our theorem also yields that every planar graph of girth at least $8$ is proper conflict-free (degree+2)-choosable.
ER  -