English

Extending total colorings in planar graphs

Combinatorics 2025-09-24 v1

Abstract

We initiate the study of total-coloring extensions, and focus our attention on planar graphs, asking: ``When can a total-kk-coloring of some subgraph HH of a planar graph GG be extended to a total-kk-coloring of GG?'' We prove that if HH is a matching, then any total-(Δ+3)(\Delta+3)-coloring of HH in GG extends to GG provided Δ28\Delta\geq 28; this number of colors is best-possible without introducing a distance condition on HH. We also prove that if HH is a set of distance-3 cliques then any total-(Δ+1)(\Delta+1)-coloring of HH extends to GG provided Δ27\Delta\geq 27; this distance condition cannot be lowered.

Keywords

Cite

@article{arxiv.2509.18940,
  title  = {Extending total colorings in planar graphs},
  author = {Owen Henderschedt and Jessica McDonald},
  journal= {arXiv preprint arXiv:2509.18940},
  year   = {2025}
}
R2 v1 2026-07-01T05:51:58.080Z