Vertex Partitions of K4,4-Minor Free Graphs

Abstract.

 We prove that a 4-connected K _4,4-minor free graph on n vertices has at most 4n−8 edges and we use this result to show that every K _4,4-minor free graph has vertex-arboricity at most 4. This improves the case (n,m)=(7,3) of the following conjecture of Woodall: the vertex set of a graph without a K _n -minor and without a -minor can be partitioned in n−m+1 subgraphs without a K _m -minor and without a -minor.