Clustered colouring in minor-closed classes

Author: 

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

Publication Date: 

28 October 2019

Journal: 

Combinatorica

Last Updated: 

2020-06-02T22:28:00.333+01:00

Issue: 

6

Volume: 

39

DOI: 

10.1007/s00493-019-3848-z

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: 

Submitted

Publication Type: 

Journal Article