Defective coloring is perfect for minors
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916520944205824 |
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| author | Liu, Chun-Hung |
| author_facet | Liu, Chun-Hung |
| contents | The defective chromatic number of a graph class is the infimum $k$ such that there exists an integer $d$ such that every graph in this class can be partitioned into at most $k$ induced subgraphs with maximum degree at most $d$. Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger's conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear relation between the clustered chromatic number of any minor-closed family and the tree-depth of its forbidden minors, improving an earlier exponential bound proved by Norin, Scott, Seymour, and Wood and confirming the planar case of their conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_10729 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Defective coloring is perfect for minors Liu, Chun-Hung Combinatorics The defective chromatic number of a graph class is the infimum $k$ such that there exists an integer $d$ such that every graph in this class can be partitioned into at most $k$ induced subgraphs with maximum degree at most $d$. Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger's conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear relation between the clustered chromatic number of any minor-closed family and the tree-depth of its forbidden minors, improving an earlier exponential bound proved by Norin, Scott, Seymour, and Wood and confirming the planar case of their conjecture. |
| title | Defective coloring is perfect for minors |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2208.10729 |