Degree-regular triangulations of surfaces

TL;DR

This paper proves that degree-regular triangulations of surfaces are geometric and classifies their possibilities, showing that for the plane, such triangulations are unique for degrees greater than 6.

Contribution

It establishes the geometric nature of degree-regular triangulations on surfaces and proves the uniqueness of such triangulations of the plane for degrees above 6.

Findings

Degree-regular triangulations are geometric with constant curvature.

Uniqueness of $d$-regular triangulations of the plane for $d>6$.

Classification of possible degree-regular triangulations on surfaces.

Abstract

A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on $S$, and we list the possibilities. A key ingredient of the proof is to show that any two $d$-regular triangulations of the plane for $d> 6 $ are combinatorially equivalent. The proof of this uniqueness result, which is of independent interest, is based on an inductive argument involving some combinatorial topology.