TY - JOUR

T1 - Clustered Colouring in Minor-Closed Classes

AU - Norin, Sergey

AU - Scott, Alex

AU - Seymour, Paul

AU - Wood, David R.

N1 - Publisher Copyright:
© 2019, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

PY - 2019/12/1

Y1 - 2019/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85074670491&partnerID=8YFLogxK

U2 - 10.1007/s00493-019-3848-z

DO - 10.1007/s00493-019-3848-z

M3 - Article

AN - SCOPUS:85074670491

SN - 0209-9683

VL - 39

SP - 1387

EP - 1412

JO - Combinatorica

JF - Combinatorica

IS - 6

ER -