Extending total colorings in planar graphs
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914052268097536 |
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| author | Henderschedt, Owen McDonald, Jessica |
| author_facet | Henderschedt, Owen McDonald, Jessica |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extending total colorings in planar graphs Henderschedt, Owen McDonald, Jessica Combinatorics 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. |
| title | Extending total colorings in planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.18940 |