Total eight-colorings of maximal planar graphs of maximum degree six

@misc{zhu2026total,
  title = {{Total eight-colorings of maximal planar graphs of maximum degree six}},
  author = {Enqiang Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32717},
  eprint = {2610.32717},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32717},
}

TY  - UNPB
TI  - Total eight-colorings of maximal planar graphs of maximum degree six
AU  - Enqiang Zhu
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.32717
UR  - https://arxiv.org/abs/2610.32717
AB  - We prove that every finite simple maximal planar graph of maximum degree at most six admits a total eight-coloring. We first show that every proper vertex four-coloring of a subcubic graph extends to a total five-coloring. A curvature inequality for boundary walks then gives a matching decomposition of four-connected triangulations with degrees five and six, extending their prescribed vertex four-colorings to total eight-colorings. In a minimum counterexample, two local boundary colorings force the degree-four vertices to have disjoint neighborhoods. A characterization of the local extension obstructions permits these vertices to be restored simultaneously. The proof uses Hamiltonian-connectedness of four-connected planar graphs and Kardoš's Hamiltonicity theorem for three-connected cubic planar graphs.
ER  -