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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 setinduces 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 independentsets. 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 thevertex set of every planar graph without $4$-cycles and $6$-cycles can bepartitioned into three sets, where one induces a graph with maximum degree two,and the remaining two are independent sets.

Partitioning planar graphs without 4-cycles and 6-cycles into a forest and a disjoint union of paths