A Bijection for Essentially 4-Connected Toroidal Triangulations

Nicolas Bonichon, Benjamin Lévêque
2019 Electronic Journal of Combinatorics  
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
more » ... roidal 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.
doi:10.37236/7897 fatcat:ottfqj3zzjganiv5e3gucdjo4u