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On the size of k-irreducible triangulations

Vincent Delecroix, Oscar Fontaine, Arnaud de Mesmay

Abstract

A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of a surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].

On the size of k-irreducible triangulations

Abstract

A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of a surface of genus g has triangles, which is optimal. This is an improvement over the previous best bound of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].
Paper Structure (9 sections, 12 theorems, 2 equations, 6 figures)

This paper contains 9 sections, 12 theorems, 2 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Surfaces and curves
  4. Combinatorial surfaces
  5. Systems of curves
  6. From triangulations to system of curves
  7. Short filling systems for geodesically k-covered systems of curves
  8. Upper bounds for geodesically k-covered systems of curves
  9. Non-orientable surfaces

Key Result

Theorem 1.1

Any $k$-irreducible triangulation of an orientable surface $S$ of genus $g$ has at most $966 k^2 g = O(k^2 g)$ edges.

Figures (6)

  • Figure 1: The triangulation in blue, a walk in red, its corresponding noose in green and the medial graph in orange.
  • Figure 2: A smoothing at $v$
  • Figure 3: A graph $G$ in blue on a genus $2$ surface and its medial graph in orange
  • Figure 4: The contraction of an edge in $G$. The blue graph is $G$, the orange one is $\mathcal{C}$ and the red curve is $c$.
  • Figure 5: Wedding of two curves $s$ and $s'$.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Remark 2.1
  • Theorem 2.2: schrijver1991decomposition
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • proof
  • ...and 23 more