Triangulated spheres with holes in triangulated surfaces

TL;DR

This paper addresses when a triangulation of a surface contains a spanning subgraph that triangulates a sphere with $h$ holes, $\mathbb{S}_h$. It first gives a short torus cylinder proof and then establishes a high facewidth result for orientable surfaces: if a triangulation on a surface of Euler genus $g$ has facewidth at least $\gamma(h)$, then it contains a spanning subgraph triangulating $\mathbb{S}_{2h}$, with $\gamma(h)$ explicitly bounded by $\gamma(h) = (4h+1) \cdot 2^{h-1} \cdot (2h-3)!! + \sum_{i=0}^{h-2} ((20i+11) \cdot 2^{i} \cdot (2i-1)!!)$. This bound is asymptotically $O(h 2^{h} (2h-1)!!)$ and is shown to be tight in the sense that hole-count cannot be reduced in general. Additionally, the authors prove a sharp obstruction: for every $0 \le g' < g$ and every $w$, there exists a triangulation of facewidth at least $w$ on a surface of genus $g$ that does not admit a spanning subgraph triangulating $\mathbb{S}_{g'}$. A complementary, short torus proof and applications to rigidity theory underpin the results, highlighting concrete 2-dimensional rigidity implications and providing computable bounds in high-genus settings.

Abstract

Let $\mathbb{S}_h$ denote a sphere with $h$ holes. Given a triangulation $G$ of a surface $\mathbb{M}$, we consider the question of when $G$ contains a spanning subgraph $H$ such that $H$ is a triangulated $\mathbb{S}_h$. We give a new short proof of a theorem of Nevo and Tarabykin that every triangulation $G$ of the torus contains a spanning subgraph which is a triangulated cylinder. For arbitrary surfaces, we prove that every high facewidth triangulation of a surface with $h$ handles contains a spanning subgraph which is a triangulated $\mathbb{S}_{2h}$. We also prove that for every $0 \leq g' < g$ and $w \in \mathbb{N}$, there exists a triangulation of facewidth at least $w$ of a surface of Euler genus $g$ that does not have a spanning subgraph which is a triangulated $\mathbb{S}_{g'}$. Our results are motivated by, and have applications for, rigidity questions in the plane.

Theorems & Definitions (15)

• Theorem 1.1: NN15 and NT
• Theorem 1.2: NT
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 2.1: schrijver93
• proof
• proof : Proof of \ref{['thm:toruscylinder']}
• Theorem 3.1: BMR96