Partitioning sparse plane graphs into two induced subgraphs of small degree

A graph $G$ is said to be $(a,b)$-partitionable for positive integers $a$, $b$ if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ any path contains at most a vertices and in $G[V_2]$ any path contains at most $b$ vertices. We prove that every planar graph of girth $8$ is $(2,2)$-partitionable.