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arXiv · 1707.08191

A bijection for essentially 4-connected toroidal triangulations

Nicolas Bonichon·Benjamin Lévêque
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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.

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Nicolas Bonichon, Benjamin Lévêque. 2017-07-24. A bijection for essentially 4-connected toroidal triangulations. https://arxiv.org/abs/1707.08191

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