Splitting a planar graph of girth 5 into two forests with trees of small diameter

In 1985, Mihok and recently Axenovich, Ueckerdt, and Weiner asked about the minimum integer g∗>3 such that every planar graph with girth at least g∗ admits a 2-colouring of its vertices where the length of every monochromatic path is bounded from above by a constant. By results of Glebov and Zambalaeva and of Axenovich et al., it follows that 5≤g∗≤6. In this paper we establish that g∗=5. Moreover, we prove that every planar graph of girth at least 5 admits a 2-colouring of its vertices such that every monochromatic component is a tree of diameter at most 6. We also present the list version of our result.