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 v a k ≡ ( G ) , is the strong equitable vertex k -arboricity. In this paper, we give sharp upper bounds for v a 1 ≡ ( K n , n ) and v a k ≡ ( K n , n ) , and prove that v a ∞ ≡ ( G ) ≤ 3 for every planar graph G with girth at least 5 and v a ∞ ≡ ( G ) ≤ 2 for every planar graph G with girth at least 6 and for every outerplanar graph.