On the equitable vertex arboricity of graphs

An equitable (t, k)-tree-colouring of a graph G is a t-colouring of vertices of G such that the sizes of any two colour classes differ by at most one and the subgraph induced by each colour class is a forest of maximum degree at most k. The strong equitable vertex k-arboricity, denoted by va(k)(equivalent to) (G), is the smallest t such that G has an equitable (t', k)-tree-colouring for every t' >= t. In this paper, we give upper bounds for va(1)(equivalent to) (G) when G is a balanced complete bipartite graph K-n,K-n and n = 0, 1(mod3). For some special cases, we determine the exact values. We also prove that: (1) va(infinity)(equivalent to)(G) <= 12 for every planar graph without 4-cycles, 5-cycles and 6-cycles; (2) va(8)(equivalent to)(G) <= 6 for every planar graph with neither 3-cycles nor adjacent 4-cycles.