Abstract

Consider the following relaxation of the Hadwiger Conjecture: For each t there exists Nt such that every graph with no Kt-minor admits a vertex partition into ⌈αt+β ⌉ parts, such that each component of the subgraph induced by each part has at most Nt vertices. The Hadwiger Conjecture corresponds to the case α = 1, β = −1 and Nt = 1. Kawarabayashi and Mohar [J. Combin. Theory Ser. B, 2007] proved this relaxation with α = 31 2 α = 7 2 and β = 0 (and Nt a huge function of t). This paper proves this relaxation with and β = −3. The main ingredients in the proof are: (1) a list colouring argument 2 due to Kawarabayashi and Mohar, (2) a recent result of Norine and Thomas that says that every sufficiently large (t + 1)-connected graph contains a Kt-minor, and (3) a new sufficient condition for a graph to have a set of edges whose contraction increases the connectivity.