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

Abstract

Let A be the family of planar graphs without 4-cycles and 5-cycles. In 2013, Hill et al. proved that every graph G∈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∈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∈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.