Equitable vertex arboricity of graphs

An equitable (t, k)-tree-coloring of a graph G is a coloring of vertices of G such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most k. The minimum t such that G has an equitable (t', k)-tree-coloring for every t' >= t, denoted by va(k)()(G), is the strong equitable vertex k-arboricity. In this paper, we give sharp upper bounds for va(1)()(K-n,K-n) and va(k)()(K-n,K-n), and prove that va(infinity)()(G) <= 3 for every planar graph G with girth at least 5 and va(infinity)()(G) <= 2 for every planar graph G with girth at least 6 and for every outerplanar graph. (C) 2013 Elsevier B.V. All rights reserved.