On triangle meshes with valence $6$ dominant vertices

TL;DR

This paper investigates special triangulations of a disk with one irregular vertex of valence not divisible by 6, proving uniqueness in certain cases and constructing multiple triangulations with irregular vertices of valence multiple of 6.

Contribution

It establishes a uniqueness theorem for such triangulations when the irregular vertex's valence isn't a multiple of 6 and constructs examples with irregular vertices of valence 6k.

Findings

Proves uniqueness of triangulations with irregular vertex valence not divisible by 6.

Constructs non-isomorphic triangulations with irregular vertex valence 6k.

Uses flat singular Riemannian metrics to analyze triangulations.

Abstract

We study triangulations $\cal T$ defined on a closed disc $X$ satisfying the following condition: In the interior of $X$, the valence of all vertices of $\cal T$ except one of them (the irregular vertex) is $6$. By using a flat singular Riemannian metric adapted to $\cal T$, we prove a uniqueness theorem when the valence of the irregular vertex is not a multiple of $6$. Moreover, for a given integer $k >1$, we exhibit non isomorphic triangulations on $X$ with the same boundary, and with a unique irregular vertex whose valence is $6k$.