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.
@article{10_37236_7897,
author = {Nicolas Bonichon and Benjamin L\'ev\^eque},
title = {A bijection for essentially 4-connected toroidal triangulations},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {1},
doi = {10.37236/7897},
zbl = {1409.05107},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7897/}
}
TY - JOUR
AU - Nicolas Bonichon
AU - Benjamin Lévêque
TI - A bijection for essentially 4-connected toroidal triangulations
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/7897/
DO - 10.37236/7897
ID - 10_37236_7897
ER -
%0 Journal Article
%A Nicolas Bonichon
%A Benjamin Lévêque
%T A bijection for essentially 4-connected toroidal triangulations
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/7897/
%R 10.37236/7897
%F 10_37236_7897
Nicolas Bonichon; Benjamin Lévêque. A bijection for essentially 4-connected toroidal triangulations. The electronic journal of combinatorics, Tome 26 (2019) no. 1. doi: 10.37236/7897
