Vertex Partitions of K4,4-Minor Free Graphs

 We prove that a 4-connected K4,4-minor free graph on n vertices has at most 4n−8 edges and we use this result to show that every K4,4-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 Kn-minor and without a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-minor can be partitioned in n−m+1 subgraphs without a Km-minor and without a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-minor.