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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.18940 |
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Table of Contents:
- We initiate the study of total-coloring extensions, and focus our attention on planar graphs, asking: ``When can a total-$k$-coloring of some subgraph $H$ of a planar graph $G$ be extended to a total-$k$-coloring of $G$?'' We prove that if $H$ is a matching, then any total-$(Δ+3)$-coloring of $H$ in $G$ extends to $G$ provided $Δ\geq 28$; this number of colors is best-possible without introducing a distance condition on $H$. We also prove that if $H$ is a set of distance-3 cliques then any total-$(Δ+1)$-coloring of $H$ extends to $G$ provided $Δ\geq 27$; this distance condition cannot be lowered.