Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graphs without a 3-connected subgraph are 4-colorable

Édouard Bonnet, Carl Feghali, Tung Nguyen, Alex Scott, Paul Seymour, Stéphan Thomassé, Nicolas Trotignon

Abstract

In 1972, Mader showed that every graph without a 3-connected subgraph is 4-degenerate and thus 5-colorable}. We show that the number 5 of colors can be replaced by 4, which is best possible.

Graphs without a 3-connected subgraph are 4-colorable

Abstract

In 1972, Mader showed that every graph without a 3-connected subgraph is 4-degenerate and thus 5-colorable}. We show that the number 5 of colors can be replaced by 4, which is best possible.
Paper Structure (6 sections, 4 theorems, 3 equations, 5 figures)

This paper contains 6 sections, 4 theorems, 3 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}
  3. Conclusion and open questions
  4. Fragile graphs have average degree less than 5
  5. Girth conditions
  6. Algorithms

Key Result

Theorem 1.1

For every integer $k\geq 1$, every graph with average degree at least $4k$ contains a $(k + 1)$-connected subgraph.

Figures (5)

  • Figure 1: Graphs with no 3-connected subgraph.
  • Figure 2: Colourings obtained in the proof of \ref{['l:c1']}.
  • Figure 3: The graph $G_1$.
  • Figure 4: The graph $G_2$.
  • Figure 5: A fragile graph with no cycle of length 4 and chromatic number 4.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Claim 3
  • proof
  • ...and 4 more