A bijection for essentially 4-connected toroidal triangulations
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.
Cite
@article{arxiv.1707.08191,
title = {A bijection for essentially 4-connected toroidal triangulations},
author = {Nicolas Bonichon and Benjamin Lévêque},
journal= {arXiv preprint arXiv:1707.08191},
year = {2017}
}
Comments
67 pages. arXiv admin note: text overlap with arXiv:1702.07589