A bijection for triangulations, quadrangulations, pentagulations, etc

A d-angulation is a planar map with faces of degree d. We present for each integer d ≥ 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 ≥ 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 ≥ 5. A key ingredient in our proofs is a class of orientations characterizing dangulations 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.