Defective correspondence coloring of planar graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915031403200512 |
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| author | Anderson, James |
| author_facet | Anderson, James |
| contents | Defective coloring (also known as relaxed or improper coloring) is a generalization of proper coloring defined as follows: for $d \in \mathbb{N}$, a coloring of a graph is $d$-defective if every vertex is colored the same as at most $d$ of its neighbors. We investigate defective coloring of planar graphs in the context of correspondence coloring, a generalization of list coloring introduced by Dvořák and Postle. First we show there exists a planar graph that is not $3$-defective $3$-correspondable, strengthening a recent result of Cho, Choi, Kim, Park, Shan, and Zhu. Then we construct a planar graph that is $1$-defective $3$-correspondable but not $4$-correspondable, thereby extending a recent result of Ma, Xu, and Zhu from list coloring to correspondence coloring. Finally we show all outerplanar graphs are $3$-defective $2$-correspondence colorable, with 3 defects being best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15336 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Defective correspondence coloring of planar graphs Anderson, James Combinatorics Defective coloring (also known as relaxed or improper coloring) is a generalization of proper coloring defined as follows: for $d \in \mathbb{N}$, a coloring of a graph is $d$-defective if every vertex is colored the same as at most $d$ of its neighbors. We investigate defective coloring of planar graphs in the context of correspondence coloring, a generalization of list coloring introduced by Dvořák and Postle. First we show there exists a planar graph that is not $3$-defective $3$-correspondable, strengthening a recent result of Cho, Choi, Kim, Park, Shan, and Zhu. Then we construct a planar graph that is $1$-defective $3$-correspondable but not $4$-correspondable, thereby extending a recent result of Ma, Xu, and Zhu from list coloring to correspondence coloring. Finally we show all outerplanar graphs are $3$-defective $2$-correspondence colorable, with 3 defects being best possible. |
| title | Defective correspondence coloring of planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.15336 |