Path partitions of planar graphs

A graph $G$ is said to be $(a,b)$-partitionable for positive intergers $a,b$ if its vertices can be partitioned into subsets $V_1,V_2$ 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. Graph $G$ is $\tau$-partitionable if it is $(a,b)$-partitionable for any $a,b$ such that $a+b$ is the number of vertices in the longest path of $G$. We prove that every planar graph of girth $5$ is $\tau$-partitionable and that planar graphs with girth $8$, $9$ and $16$ are $(2,3)$-, $(2,2)$- and $(1,2)$-partitionable respectively.