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arXiv · 1711.01247

Degree-regular triangulations of surfaces

Basudeb Datta·Subhojoy Gupta
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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.

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Basudeb Datta, Subhojoy Gupta. 2017-11-03. Degree-regular triangulations of surfaces. https://arxiv.org/abs/1711.01247

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