On vertex partitions and some minor‐monotone graph parameters

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. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 49‐56, 2010