Triangulated spheres with holes in triangulated surfaces

TL;DR

This paper investigates conditions under which triangulated surfaces contain spanning subgraphs that are triangulated spheres with holes, providing new proofs and constructions relevant to surface topology and rigidity.

Contribution

It offers a new proof of a theorem on triangulated cylinders in torus triangulations and extends results to higher genus surfaces, exploring the existence of specific spanning subgraphs.

Findings

Every triangulation of a torus contains a spanning triangulated cylinder.

High facewidth triangulations of surfaces with h handles contain spanning triangulated spheres with 2h holes.

There exist triangulations with high facewidth that do not contain certain triangulated spheres with fewer holes.

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.