Relaxation of Wegner's Planar Graph Conjecture for maximum degree 4
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arXiv
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| Natura: | Preprint |
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2022
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| author | Cho, Eun-Kyung Choi, Ilkyoo Lidický, Bernard |
| author_facet | Cho, Eun-Kyung Choi, Ilkyoo Lidický, Bernard |
| contents | The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on the chromatic number of the square $G^2$ of a planar graph $G$, depending on the maximum degree $Δ(G)$ of $G$. The only case that the conjecture is resolved is when $Δ(G)=3$, which was proven to be true by Thomassen, and independently by Hartke, Jahanbekam, and Thomas. For $Δ(G)=4$, Wegner's Planar Graph Conjecture states that the chromatic number of $G^2$ is at most 9; even this case is still widely open, and very recently Bousquet, de Meyer, Deschamps, and Pierron claimed an upper bound of 12.
We take a completely different approach, and show that a relaxation of properly coloring the square of a planar graph $G$ with $Δ(G)=4$ can be achieved with 9 colors. Instead of requiring every color in the neighborhood of a vertex to be unique, which is equivalent to a proper coloring of $G^2$, we seek a proper coloring of $G$ such that at most one color is allowed to be repeated in the neighborhood of a vertex of degree 4, but nowhere else. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_10643 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Relaxation of Wegner's Planar Graph Conjecture for maximum degree 4 Cho, Eun-Kyung Choi, Ilkyoo Lidický, Bernard Combinatorics The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on the chromatic number of the square $G^2$ of a planar graph $G$, depending on the maximum degree $Δ(G)$ of $G$. The only case that the conjecture is resolved is when $Δ(G)=3$, which was proven to be true by Thomassen, and independently by Hartke, Jahanbekam, and Thomas. For $Δ(G)=4$, Wegner's Planar Graph Conjecture states that the chromatic number of $G^2$ is at most 9; even this case is still widely open, and very recently Bousquet, de Meyer, Deschamps, and Pierron claimed an upper bound of 12. We take a completely different approach, and show that a relaxation of properly coloring the square of a planar graph $G$ with $Δ(G)=4$ can be achieved with 9 colors. Instead of requiring every color in the neighborhood of a vertex to be unique, which is equivalent to a proper coloring of $G^2$, we seek a proper coloring of $G$ such that at most one color is allowed to be repeated in the neighborhood of a vertex of degree 4, but nowhere else. |
| title | Relaxation of Wegner's Planar Graph Conjecture for maximum degree 4 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2212.10643 |