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.

中文翻译：

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将没有 4 环和 5 环的平面图划分为有界度森林

1976 年，Steinberg 推测没有 $4$-cycles 和 $5$-cycles 的平面图是 $3$-colorable。这一猜想吸引了众多研究人员大约 40 年，直到最近被 Cohen-Addad 等人推翻。(2017)。然而，对循环长度有限制的平面图着色仍然是一个活跃的研究领域，并且对这个特定的图类的兴趣仍然存在。令 $G$ 是一个没有 $4$-cycles 和 $5$-cycles 的平面图。对于满足 $d_1+d_2\geq8$ 和 $d_2\geq d_1\geq 2$ 的整数 $d_1$ 和 $d_2$，已知 $V(G)$ 可以分为两个集合 $V_1$ 和 $V_2 $，其中每个 $V_i$ 诱导出一个最大度数最多为 $d_i$ 的图。由于 Steinberg 的猜想是错误的，因此不能保证将 $V(G)$ 划分为两组，其中一个导致空图，另一个导致森林。我们的主要定理位于上述两个研究方向的交集处。我们证明了 $V(G)$ 可以分为两个集合 $V_1$ 和 $V_2$，其中 $V_1$ 诱导最大度数为 $3$ 的森林，$V_2$ 诱导最大度数为 $4 的森林$; 这既是对 Steinberg 猜想的放松，也是对 Sittitrai 和 Nakprasit (2019) 以更强的形式进行的结果的强化。