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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/2509.04044 |
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| _version_ | 1866912570176176128 |
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| author | Deniz, Zakir Guler, Hakan |
| author_facet | Deniz, Zakir Guler, Hakan |
| contents | A total coloring of a graph $G$ is a coloring of the vertices and edges such that two adjacent or incident elements receive different colors. The minimum number of colors required for a total coloring of a graph $G$ is called the total chromatic number, denoted by $χ''(G)$. Let $G$ be a planar graph of maximum degree eight. It is known that $9\leq χ''(G) \leq 10$. We here prove that $χ''(G)=9$ when the graph does not contain any subgraph isomorphic to a $4$-fan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A sufficient condition for planar graphs with maximum degree eight to be totally 9-colorable Deniz, Zakir Guler, Hakan Combinatorics A total coloring of a graph $G$ is a coloring of the vertices and edges such that two adjacent or incident elements receive different colors. The minimum number of colors required for a total coloring of a graph $G$ is called the total chromatic number, denoted by $χ''(G)$. Let $G$ be a planar graph of maximum degree eight. It is known that $9\leq χ''(G) \leq 10$. We here prove that $χ''(G)=9$ when the graph does not contain any subgraph isomorphic to a $4$-fan. |
| title | A sufficient condition for planar graphs with maximum degree eight to be totally 9-colorable |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.04044 |