DeepAI AI Chat
Log In Sign Up

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

by   Kolja Knauer, et al.

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.


page 1

page 2

page 3

page 4


1-planar graphs with minimum degree at least 3 have bounded girth

We show that every 1-planar graph with minimum degree at least 4 has gir...

Partitioning edges of a planar graph into linear forests and a matching

We show that the edges of any planar graph of maximum degree at most 9 c...

Are highly connected 1-planar graphs Hamiltonian?

It is well-known that every planar 4-connected graph has a Hamiltonian c...

Planar Ramsey graphs

We say that a graph H is planar unavoidable if there is a planar graph G...

Tree-like distance colouring for planar graphs of sufficient girth

Given a multigraph G and a positive integer t, the distance-t chromatic ...

A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings

A plus-contact representation of a planar graph G is called c-balanced i...

A bijection for essentially 3-connected toroidal maps

We present a bijection for toroidal maps that are essentially 3-connecte...