Vertex Arboricity and Vertex Degrees

The vertex arboricity of a graph G, denoted a(G), is the minimum number of subsets into which V(G) can be partitioned so that each subset induces an acyclic graph. We first give a vertex degree condition to guarantee a(G)≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(G) \le k$$\end{document}, which is best possible in the same sense as Chvátal’s well-known hamiltonian degree condition. We then explore comparably strong degree conditions for a(G)≥k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(G) \ge k$$\end{document}, and show that any such condition has intrinsic complexity which grows superpolynomially with the order of G.