  Recently, Borodin, Kostochka, and Yancey (On $1$-improper $2$-coloring of
sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of
each planar graph of girth at least $7$ can be $2$-colored so that each color
class induces a subgraph of a matching. We prove that any planar graph of girth
at least $6$ admits a vertex coloring in $2$ colors such that each
monochromatic component is a path of length at most $14$. Moreover, we show a
list version of this result. On the other hand, for each positive integer
$t\geq 3$, we construct a planar graph of girth $4$ such that in any coloring
of vertices in $2$ colors there is a monochromatic path of length at least $t$.
It remains open whether each planar graph of girth $5$ admits a $2$-coloring
with no long monochromatic paths.
