Caterpillar arboricity of planar graphs

We solve a conjecture of Roditty, Shoham and Yuster [P.J. Cameron (Ed.), Problems from the 17th British Combinatorial Conference, Discrete Math., 231 (2001) 469–478; Y. Roditty, B. Shoham, R. Yuster, Monotone paths in edge-ordered sparse graphs, Discrete Math. 226 (2001) 411–417] on the caterpillar arboricity of planar graphs. We prove that for every planar graph G=(V,E)G=(V,E), the edge set E   can be partitioned into four subsets (Ei)1⩽i⩽4(Ei)1⩽i⩽4 in such a way that G[Ei]G[Ei], for 1⩽i⩽41⩽i⩽4, is a forest of caterpillars. We also provide a linear-time algorithm which constructs for a given planar graph G, four forests of caterpillars covering the edges of G.