Partitioning planar graphs without 4-cycles and 5-cycles into two forests with a specific condition

Let A be the family of planar graphs without 4-cycles and 5-cycles. In 2013, Hill et al. proved that every graph G E A has a partition dividing V(G) into three sets, where two of them are independent, and the other induces a graph with a maximum degree at most 3. In 2021 Cho, Choi, and Park conjectured that every graph G E A has a partition dividing V(G) into two sets, where one set induces a forest, and the other induces a forest with a maximum degree at most 2. In this paper, we show that every graph G E A has a partition dividing V(G) into two sets, where one set induces a forest, and the other induces a disjoint union of paths and subdivisions of K1,3. The result improves the aforementioned result by Hill et al. and yields progress toward the conjecture of Cho, Choi, and Park. (c) 2023 Elsevier B.V. All rights reserved.