A bijection for triangulations, quadrangulations, pentagulations, etc.

Abstract

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 (i.e., with no cycle of length less than 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. Our construction unifies known bijections by Fusy, Poulalhon and Schaeffer for triangulations ( d = 3 ) and by Schaeffer for quadrangulations ( d = 4 ). 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 ( d-angulations with a simple boundary of length p) of girth d. We thereby recover bijectively the results of Brown for simple p-gonal triangulations and simple 2 p-gonal quadrangulations and establish new results for d ⩾ 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 Fraysseix and Ossona de Mendez showed that simple triangulations and simple quadrangulations 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.