Vertex Partitions of K4, 4-Minor Free Graphs

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 vertexset of a graph without a K n-minor and without a K b n+1 2 c;d n+1 2 e-minor can be partitioned in n ? m + 1 subgraphs without a K m-minor and without a K b m+1 2 c;d m+1 2 e-minor.