A bijection for triangulations, quadrangulations, pentagulations, etc.
Résumé
A $d$-angulation is a planar map with faces of degree $d$. We present for each integer $d\geq 3$ a bijection between the class of $d$-angulations of girth~$d$ and a class of decorated plane trees. Each of the bijections is obtained by specializing a ``master bijection'' which extends an earlier construction of the first author. Bijections already existed for triangulations ($d=3$) and for quadrangulations ($d=4$). As a matter of fact, our construction unifies a bijection by Fusy, Poulalhon and Schaeffer for triangulations and a bijection by Schaeffer for quadrangulations. For $d\geq 5$, both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate $p$-gonal $d$-angulations, that is, $d$-angulations with a simple boundary of length $p$. We thereby recover bijectively the results of Brown for $p$-gonal triangulations and quadrangulations and establish new results for $d\geq 5$. A key ingredient in our proofs is a class of orientations characterizing $d$-angulations of girth $d$. Earlier results by Schnyder and by De Fraisseyx and Ossona de Mendez showed that triangulations of girth 3 and quadrangulations of girth 4 are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a $d$-angulation has girth $d$ if and only if the graph obtained by duplicating each edge $d-2$ times admits an orientation having indegree $d$ at each inner vertex.
Domaines
Combinatoire [math.CO]
Origine : Fichiers produits par l'(les) auteur(s)