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 (odd) H minor, then its vertex set V(G) can be partitioned into three sets X1, X2, X3 such that for each i, the subgraph induced on Xi has no component of size larger than a function of H and the maximum degree of G. This improves a previous result of Alon, Ding, Oporowski and Vertigan (2003) [1] stating that V(G) can be partitioned into four such sets if G has no H minor. Our theorem generalizes a result of Esperet and Joret (2014) [9], who proved it for graphs embeddable on a fixed surface and asked whether it is true for graphs with no H minor.As a corollary, we prove that for every positive integer t, if a graph G has no Kt+1 minor, then its vertex set V(G) can be partitioned into 3t sets X1,…,X3t such that for each i, the subgraph induced on Xi has no component of size larger than a function of t. This corollary improves a result of Wood (2010) [21], which states that V(G) can be partitioned into ⌈3.5t+2⌉ such sets.