On triangle meshes with valence $6$ dominant vertices

Abstract

We study triangulations $\mathcal{T}$ defined on a closed disc $X$ satisfying the following condition: In the interior of $X$, the valence of all vertices of $\mathcal{T}$ except one of them (the irregular vertex) is $6$. By using a flat singular Riemannian metric adapted to $\mathcal{T}$, 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 $6k$.

13 pages, 7 figures