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

TL;DR

This paper proves that 4-connected planar triangulations can be decomposed into a Hamiltonian path and two trees, leading to improved decompositions of planar graphs into forests with bounded maximum degree.

Contribution

It introduces new decomposition methods for 4-connected planar triangulations into a Hamiltonian path and two trees, improving previous bounds on forest decompositions of planar graphs.

Findings

Decomposition of 4-connected planar triangulations into a Hamiltonian path and two trees.

Every 4-connected planar graph decomposes into three forests, one with maximum degree at most 2.

Improved decomposition of Hamiltonian planar triangulations into two trees and one bounded-degree spanning tree.

Abstract

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 Gon\c{c}alves [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.