Equitable vertex arboricity of graphs

An equitable (t,k)(t,k)-tree-coloring of a graph GG is a coloring of vertices of GG 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 kk. The minimum tt such that GG has an equitable (t′,k)(t′,k)-tree-coloring for every t′≥tt′≥t, denoted by vak≡(G), is the strong equitable vertex kk-arboricity. In this paper, we give sharp upper bounds for va1≡(Kn,n) and vak≡(Kn,n), and prove that va∞≡(G)≤3 for every planar graph GG with girth at least 5 and va∞≡(G)≤2 for every planar graph GG with girth at least 6 and for every outerplanar graph.