Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles

@misc{zhu2026totalcoloring,
  title = {{Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles}},
  author = {Enqiang Zhu and Yangyang Zhou and Jin Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19503},
  eprint = {2609.19503},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19503},
}

TY  - UNPB
TI  - Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles
AU  - Enqiang Zhu
AU  - Yangyang Zhou
AU  - Jin Xu
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.19503
UR  - https://arxiv.org/abs/2609.19503
AB  - The Total Coloring Conjecture (TCC) is a challenging unsolved problem posed by Behzad and Vizing independently, which states that every simple graph G admits a (Δ(G) +2)-total-coloring, where Δ(G) denotes the maximum degree of G. This conjecture has been confirmed for graphs with Δ(G)≤ 5. However, for planar graphs, the only open case is Δ(G)=6. It was known that planar graphs with maximum degree 6 and without 4-cycles are 7-totally-colorable. In this paper, we improve this result by showing that any planar graph G of maximum degree 6, which does not contain some special 4-cycles, is 7-totally-colorable.
ER  -