Clustered Colouring in Minor-Closed Classes

Author: 

Norin, S
Scott, A
Seymour, P
Wood, D

Publication Date: 

28 October 2019

Journal: 

COMBINATORICA

Last Updated: 

2019-12-01T14:01:04.307+00:00

DOI: 

10.1007/s00493-019-3848-z

abstract: 

© 2019, János Bolyai Mathematical Society and Springer-Verlag. 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

Download URL: 

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article