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Partitioning planar graphs without 4-cycles and 5-cycles into bounded degree forests. (English) Zbl 1455.05059
Summary: 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 V. Cohen-Addad et al. [J. Comb. Theory, Ser. B 122, 452–456 (2017; Zbl 1350.05018)]. 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 $$d_1 + d_2 \geq 8$$ 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 P. Sittitrai and K. Nakprasit [Discrete Math. 341, No. 8, 2142–2150 (2018; Zbl 1388.05072)] in a much stronger form.
##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C10 Planar graphs; geometric and topological aspects of graph theory
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