  It is proved that the vertex set of any simple graph $G$ can be equitably
partitioned into $k$ subsets for any integer
$k\geq\max\{\big\lceil\frac{\Delta(G)+1}{2}\big\rceil,\big\lceil\frac{|G|}{4}\big\rceil\}$
so that each of them induces a linear forest.

Combinatorics;Discrete Mathematics