(F1,F)-partition of plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles

An $(F_k,F)$-partition of a graph $G$ is a partition of $V\left(G\right)$ into two sets $V_1$ and $V_2$ such that $G\left[V_1\right]$ is a forest of maximum degree at most $k$ and $G\left[V_2\right]$ is just a forest. A plane graph is a planar graph $G$ together with an embedding of $G$ into the Euclidean plane. A cycle $C$ of a plane graph is ext-triangular if it is adjacent to a triangle $T$ such that the interiors of $C$ and $T$ have no intersection. Recently, Liu and Yu (2020) proposed a problem whether planar graphs without cycles of length from 4 to 7 are $(F_0,F)$-partitionable. In this paper, we prove that plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are $(F_1,F)$-partitionable. As a consequence, planar graphs without cycles of length 4, 5 or $l$ for any $l\in \{7,8\}$ are $(F_1,F)$-partitionable.