English

Every planar graph with $\Delta\geqslant 8$ is totally $(\Delta+2)$-choosable

Discrete Mathematics 2022-12-12 v4 Combinatorics

Abstract

Total coloring is a variant of edge coloring where both vertices and edges are to be colored. A graph is totally kk-choosable if for any list assignment of kk colors to each vertex and each edge, we can extract a proper total coloring. In this setting, a graph of maximum degree Δ\Delta needs at least Δ+1\Delta+1 colors. In the planar case, Borodin proved in 1989 that Δ+2\Delta+2 colors suffice when Δ\Delta is at least 9. We show that this bound also holds when Δ\Delta is 88.

Keywords

Cite

@article{arxiv.1904.12060,
  title  = {Every planar graph with $\Delta\geqslant 8$ is totally $(\Delta+2)$-choosable},
  author = {Marthe Bonamy and Théo Pierron and Éric Sopena},
  journal= {arXiv preprint arXiv:1904.12060},
  year   = {2022}
}

Comments

64 pages, 77 figures