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Partitioning a Planar Graph into two Triangle-Forests

Kolja Knauer, Clément Rambaud, Torsten Ueckerdt

Abstract

We show that the vertices of every planar graph can be partitioned into two sets, each inducing a so-called triangle-forest, i.e., a graph with no cycles of length more than three. We further discuss extensions to locally planar graphs. After finishing the paper we noticed that our main result was already proved much earlier by Carsten Thomassen [Decomposing a Planar Graph into Degenerate Graphs, JCTB 1995].

Partitioning a Planar Graph into two Triangle-Forests

Abstract

We show that the vertices of every planar graph can be partitioned into two sets, each inducing a so-called triangle-forest, i.e., a graph with no cycles of length more than three. We further discuss extensions to locally planar graphs. After finishing the paper we noticed that our main result was already proved much earlier by Carsten Thomassen [Decomposing a Planar Graph into Degenerate Graphs, JCTB 1995].
Paper Structure (4 sections, 9 theorems, 3 figures)

This paper contains 4 sections, 9 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Vertex-Partitioning a Planar Graph into two Triangle-Forests
  3. Tightness and possible strengthenings
  4. Graphs on surfaces

Key Result

Lemma 1

Let $C$ be the outer face of a $2$-connected planar graph $G$ and $u,v,e\in C$ be two vertices and an edge of $C$. Then there exists a Tutte path from $u$ to $v$ through $e$ in $G$.

Figures (3)

  • Figure 1: A connected triangle forest.
  • Figure 2: The monochromatic and heterochromatic case of construction of Tutte cycles in the proof of \ref{['4connected']}.
  • Figure 3: A projective planar graph (with g6-code J|tyIlxJGb?) that cannot be vertex-partitioned into two triangle-forests.

Theorems & Definitions (14)

  • Lemma 1: Tutte 1956 Tut56
  • Lemma 2: Three-Edge-Lemma San96TY94
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • Lemma 5
  • proof
  • ...and 4 more