Partitioning H-minor free graphs into three subgraphs with no large components

Abstract

We prove that for every graph H, if a graph G has no H minor, then V(G) can be partitioned into three sets such that the subgraph induced on each set has no component of size larger than a function of H and the maximum degree of G. This answers a question of Esperet and Joret and improves a result of Alon, Ding, Oporowski and Vertigan and a result of Esperet and Joret. As a corollary, for every positive integer t, if a graph G has no Kt+1 minor, then V(G) can be partitioned into 3t sets such that the subgraph induced on each set has no component of size larger than a function of t. This corollary improves a result of Wood.