Vertex partitions of K4,4-minor free graphs

We prove that a 4-connected K-4,K-4-minor free graph on n vertices has at most 4n-8 edges and we use this result to show that every K-4,K-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 K-[n+1/2],K-[n+1/2]-minor can be partitioned in n - m + 1 subgraphs without a K-m-minor and without a K-[m+1/2],K-[m+1/2]-minor.