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Flip graphs of coloured triangulations of convex polygons

Karin Baur, Diana Bergerova, Jenni Voon, Lejie Xu

Abstract

A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other diagonal in the quadrilateral it defines. In this paper, we study coloured triangulations and coloured flips. In this more general situation, it is no longer true that any two triangulations can be linked by a sequence of (coloured) flips. In this paper, we study the connected components of the coloured flip graphs of triangulations. The motivation for this is a result of Gravier and Payan proving that the Four-Colour Theorem is equivalent to the connectedness of the flip graph of 2-coloured triangulations.

Flip graphs of coloured triangulations of convex polygons

Abstract

A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other diagonal in the quadrilateral it defines. In this paper, we study coloured triangulations and coloured flips. In this more general situation, it is no longer true that any two triangulations can be linked by a sequence of (coloured) flips. In this paper, we study the connected components of the coloured flip graphs of triangulations. The motivation for this is a result of Gravier and Payan proving that the Four-Colour Theorem is equivalent to the connectedness of the flip graph of 2-coloured triangulations.
Paper Structure (10 sections, 17 theorems, 20 figures)

This paper contains 10 sections, 17 theorems, 20 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Coloured triangulations, coloured flips
  4. Triangulations of polygons and the Four-Colour Theorem
  5. Connected components of coloured flip graphs
  6. Structure of the flip graph
  7. Observations and a conjecture
  8. Proof of Theorem \ref{['thm1']}
  9. Connected components of flip graphs
  10. Component sizes

Key Result

Lemma 2.3

The number of triangulations of a convex $(n+2)$-gon is given by the $n$-th Catalan number $C_{n} = \frac{1}{n+1}\binom{2n}{n}$.

Figures (20)

  • Figure 1: A fan triangulation of a hexagon.
  • Figure 2: Each possible triangulation of the octagon falls into one of these 6 types.
  • Figure 3: A flip sequence with two colours, translated into a sequence with four colours (with $\sigma =$ (red, yellow, cyan, green))
  • Figure 4: Connected components of the coloured flip graph of $P_6$ up to isometry, excluding the isolated points.
  • Figure 5: A coloured fan triangulation of a decagon.
  • ...and 15 more figures

Theorems & Definitions (54)

  • Definition 2.1: Triangulation
  • Example 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Colouring
  • Definition 2.7: Coloured flip
  • Definition 2.8
  • Lemma 2.9
  • ...and 44 more