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 , 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 , and show that any such condition has intrinsic complexity which grows superpolynomially with the order of G .