Equitable vertex arboricity of graphs

An equitable (t, k, d)-tree-coloring of a graph G is a coloring to 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 and diameter at most d. The minimum t such that G has an equitable (t′, k, d)-tree-coloring for every t′ ≥ t is called the strong equitable (k, d)-vertex-arboricity and denoted by vak,d(G). In this paper, we give sharp upper bounds for va1,1(Kn,n) and va ≡ k,∞(Kn,n) by showing that va ≡ 1,1(Kn,n) = O(n) and va ≡ k,∞(Kn,n) = O(n 1 2 ) for every k ≥ 2. It is also proved that va∞,∞(G) ≤ 3 for every planar graph G with girth at least 5 and va∞,∞(G) ≤ 2 for every planar graph G with girth at least 6 and for every outerplanar graph. We conjecture that va∞,∞(G) = O(1) for every planar graph and va ≡ ∞,∞(G) ≤ ⌈ ∆(G)+1 2 ⌉ for every graph G.