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.
Disclosure statement
No potential conflict of interest was reported by the authors.