A bijection for essentially 4-connected toroidal triangulations

TL;DR

This paper extends the concept of transversal structures from planar to toroidal triangulations, revealing a lattice structure and providing a bijective enumeration method for essentially 4-connected toroidal triangulations.

Contribution

It introduces balanced transversal structures for toroidal triangulations, demonstrating they form a single distributive lattice and enabling bijective enumeration.

Findings

Balanced transversal structures form a single distributive lattice.

The minimal element of the lattice allows bijective enumeration.

The approach generalizes planar results to toroidal cases.

Abstract

Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications (drawing algorithm, random generation, enumeration ...). In this paper we introduce and study a generalization of these objects for the toroidal case. Contrary to what happens in the plane, the set of toroidal transversal structures of a given toroidal triangulation is partitioned into several distributive lattices. We exhibit a subset of toroidal transversal structures, called balanced, and show that it forms a single distributive lattice. Then, using the minimal element of the lattice, we are able to enumerate bijectively essentially 4-connected toroidal triangulations.