# Decomposing 4-connected planar triangulations into two trees and one path

Refining a classical proof of Whitney, we show that any 4-connected planar triangulation can be decomposed into a Hamiltonian path and two trees. Therefore, every 4-connected planar graph decomposes into three forests, one having maximum degree at most 2. We use this result to show that any Hamiltonian planar triangulation can be decomposed into two trees and one spanning tree of maximum degree at most 3. These decompositions improve the result of Gonçalves [Covering planar graphs with forests, one having bounded maximum degree. J. Comb. Theory, Ser. B, 100(6):729--739, 2010] that every planar graph can be decomposed into three forests, one of maximum degree at most 4. We also show that our results are best-possible.

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