  In this paper, we show that every planar graph without $4$-cycles and
$6$-cycles has a partition of its vertex set into two sets, where one set
induces a forest, and the other induces a forest with maximum degree at most
$2$ (equivalently, a disjoint union of paths).
  Note that we can partition the vertex set of a forest into two independent
sets. However, a pair of independent sets combined may not induce a forest.
Thus our result extends the result of Wang and Xu (2013) stating that the
vertex set of every planar graph without $4$-cycles and $6$-cycles can be
partitioned into three sets, where one induces a graph with maximum degree two,
and the remaining two are independent sets.
