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 2p-gonal 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 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. *

Bijections for planar maps with boundaries

“…For d = 2, the internal girth condition is void for bipartite maps, and restricting the non-boundary faces to have degree d + 2 = 4 gives our result for bipartite quadrangulations with boundaries. For the values of d ≥ 3, the case of a single boundary with all the internal faces of degree d corresponds to the results obtained in [2] (bijections for d-angulations of girth d ≥ 3 with at most one boundary). For d = 2, the case of a single boundary with all the internal faces of degree 3 gives a bijection for loopless triangulations (i.e.…”

“…where b := r i=1 a i is the half-total boundary length, k := r + m − 2, and e = 3b + 2k is the number of edges. Equations (1) and (2) are generalizations of classical formulas. Indeed, the doubly degenerate case m = 0 and r = 1 of (1) gives the well-known Catalan formula for the number of triangulations of a polygon without interior points |T [0; a]| = Cat(a − 2) = (2a−4)!…”

Blossoming bijection for higher-genus maps

“…We define similarly e f (B) to be the first visited edge (or stem) around B. Again, since it has to be outgoing, it is either e 1 or e 7 .…”

“…As emphasized by Bernardi [5] in the planar case and generalized by Bernardi and Chapuy [6], a map endowed with a spanning unicellular embedded graph (whose genus can be smaller than the genus of the initial surface) can also be viewed as a map endowed with an orientation of its edges with specific properties. The general theory of α-orientations developed by Felsner in the planar case [18] has been successfully combined with the result of [5] to give general bijective schemes in the planar case [7,8,1], which enables to recover the previously known bijections. It would be highly desirable to obtain systematic bijective schemes in higher genus by combining Bernardi and Chapuy's result together with the theory of c-orientations introduced by Propp [24] or its extension by Felsner and Knauer [19].…”

A Bijection for Essentially 4-Connected Toroidal Triangulations

“…Bijections for planar maps with no girth nor connectivity 1 conditions rely on geodesic labellings [31,10] and can be extended to any fixed genus [15,13,25] (where the map is encoded by a decorated unicellular map). On the other hand, when a girth condition or connectivity condition is imposed the bijections typically rely (see [2,6] for general methodologies) on the existence of a certain 'canonical' orientation with prescribed outdegree conditions, such as Schnyder orientations (introduced by Schnyder [33] for simple planar triangulations, and later generalized to 3-connected planar maps [18] and to d-angulations of girth d [6,7]), separating decompositions [16] or transversal structures [23,19]; and it is not known how to specify such a canonical orientation in any fixed genus (we also mention the powerful approach by Bouttier and Guitter [11] based on decomposing so-called slices; this method, which yields a unified combinatorial decomposition for irreducible maps with control on the girth and face-degrees, is yet to be extended to higher genus).…”