Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

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 ≥ 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.