Encoding toroidal triangulations

TL;DR

This paper generalizes a linear encoding method for planar triangulations to toroidal triangulations using a bijection with unicellular maps, extending the combinatorial framework to higher genus surfaces.

Contribution

It introduces a novel encoding technique for toroidal triangulations based on a generalization of Schnyder woods, linking them to unicellular maps.

Findings

Established a bijection between toroidal triangulations and unicellular maps.

Extended the planar encoding method to the torus case.

Provided a combinatorial framework for higher genus surface triangulations.

Abstract

Poulalhon and Schaeffer introduced an elegant method to linearly encode a planar triangulation optimally. The method is based on performing a special depth-first search algorithm on a particular orientation of the triangulation: the minimal Schnyder wood. Recent progress toward generalizing Schnyder woods to higher genus enables us to generalize this method to the toroidal case. In the plane, the method leads to a bijection between planar triangulations and some particular trees. For the torus we obtain a similar bijection but with particular unicellular maps (maps with only one face).