Journal title
Combinatorica
DOI
10.1007/s00493-019-3848-z
Issue
6
Volume
39
Last updated
2023-12-19T10:56:20.33+00:00
Page
1387-1412
Abstract
The clustered chromatic number of a class of graphs is the minimum integer k such that for some integer c every graph in the class is k-colourable with monochromatic components of size at most c. We prove that for every graph H, the clustered chromatic number of the class of H-minor-free graphs is tied to the tree-depth of H. In particular, if H is connected with tree-depth t, then every H-minor-free graph is (2t+1–4)-colourable with monochromatic components of size at most c(H). This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of H-minor-free graphs. If t = 3, then we prove that 4 colours suffie, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class.
Symplectic ID
974344
Submitted to ORA
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Publication type
Journal Article
Publication date
28 Oct 2019