Abstract
We study vertex partitions of graphs according to some minor-monotone graph parameters. Ding et al. [J Combin Theory Ser B 79(2) (2000), 221-246] proved that some minor-monotone parameters ρ are such that, any graph G with ρ(G)⩾2 admits a vertex partition into two graphs with parameter ρ at most ρ(G) − 1. Here we prove that some of these parameters ρ are such that, any graph G with ρ(G)⩾3 admits a vertex partition into three graphs with parameter ρ at most ρ(G) − 2.