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), the edge set E can be partitioned into four subsets (E-i) (1 <= i <= 4) in such a way that G[E-i], for 1 <= 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. (c) 2006 Published by Elsevier B.V.