Defective and Clustered Colouring of Graphs with Given Girth
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918201084870656 |
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| author | Briański, Marcin Hickingbotham, Robert Wood, David R. |
| author_facet | Briański, Marcin Hickingbotham, Robert Wood, David R. |
| contents | The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$. Similarly, the clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $c$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has at most $c$ vertices. This paper determines or establishes bounds on the defective and clustered chromatic numbers of graphs with given girth in minor-closed classes defined by the following parameters: Hadwiger number, treewidth, pathwidth, treedepth, circumference, and feedback vertex number. One striking result is that for any integer $k$, for the class of triangle-free graphs with treewidth $k$, the defective chromatic number, clustered chromatic number and chromatic number are all equal. The same result holds for graphs with treedepth $k$, and generalises for graphs with no $K_p$ subgraph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_14940 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Defective and Clustered Colouring of Graphs with Given Girth Briański, Marcin Hickingbotham, Robert Wood, David R. Combinatorics The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$. Similarly, the clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $c$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has at most $c$ vertices. This paper determines or establishes bounds on the defective and clustered chromatic numbers of graphs with given girth in minor-closed classes defined by the following parameters: Hadwiger number, treewidth, pathwidth, treedepth, circumference, and feedback vertex number. One striking result is that for any integer $k$, for the class of triangle-free graphs with treewidth $k$, the defective chromatic number, clustered chromatic number and chromatic number are all equal. The same result holds for graphs with treedepth $k$, and generalises for graphs with no $K_p$ subgraph. |
| title | Defective and Clustered Colouring of Graphs with Given Girth |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.14940 |