  In 1976, Steinberg conjectured that planar graphs without $4$-cycles and
$5$-cycles are $3$-colorable. This conjecture attracted numerous researchers
for about 40 years, until it was recently disproved by Cohen-Addad et al.
(2017). However, coloring planar graphs with restrictions on cycle lengths is
still an active area of research, and the interest in this particular graph
class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For
integers $d_1$ and $d_2$ satisfying $d_1+d_2\geq8$ and $d_2\geq d_1\geq 2$, it
is known that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where
each $V_i$ induces a graph with maximum degree at most $d_i$. Since Steinberg's
Conjecture is false, a partition of $V(G)$ into two sets, where one induces an
empty graph and the other induces a forest is not guaranteed. Our main theorem
is at the intersection of the two aforementioned research directions. We prove
that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where $V_1$
induces a forest with maximum degree at most $3$ and $V_2$ induces a forest
with maximum degree at most $4$; this is both a relaxation of Steinberg's
conjecture and a strengthening of results by Sittitrai and Nakprasit (2019) in
a much stronger form.
