Abstract
An equitable -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 , is the smallest t such that G has an equitable-tree-colouring for every . In this paper, we give upper bounds for when G is a balanced complete bipartite graph and . For some special cases, we determine the exact values. We also prove that: (1) for every planar graph without 4-cycles, 5-cycles and 6-cycles; (2) for every planar graph with neither 3-cycles nor adjacent 4-cycles.
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