A unified bijective method for maps: application to two classes with boundaries

Olivier Bernardi,Eric Fusy

doi:10.46298/dmtcs.2869

Abstract

Based on a construction of the first author, we present a general bijection between certain decorated plane trees and certain orientations of planar maps with no counterclockwise circuit. Many natural classes of maps (e.g. Eulerian maps, simple triangulations,...) are in bijection with a subset of these orientations, and our construction restricts in a simple way on the subset. This gives a general bijective strategy for classes of maps. As a non-trivial application of our method we give the first bijective proofs for counting (rooted) simple triangulations and quadrangulations with a boundary of arbitrary size, recovering enumeration results found by Brown using Tutte's recursive method. En nous appuyant sur une construction du premier auteur, nous donnons une bijection générale entre certains arbres décorés et certaines orientations de cartes planaires sans cycle direct. De nombreuses classes de cartes (par exemple les eulériennes, les triangulations) sont en bijection avec un sous-ensemble de ces orientations, et notre construction se spécialise de manière simple sur le sous-ensemble. Cela donne un cadre bijectif général pour traiter les familles de cartes. Comme application non-triviale de notre méthode nous donnons les premières preuves bijectives pour l'énumération des triangulations et quadrangulations simples (enracinées) ayant un bord de taille arbitraire, et retrouvons ainsi des formules de comptage trouvées par Brown en utilisant la méthode récursive de Tutte.

Highlights

The enumeration of planar maps has received a lot of attention since the seminal work of Tutte in the 60’s [Tut63]
Even if it has been successfully applied to many classes, e.g. in [PS06, PS03, FPS08, BFG04], the bijective method for maps is up to now not as systematic as Tutte’s recursive method, since for each class of maps one has to “guess” the tree family to match with, and one has to specify the construction from trees to maps
Based on a construction of the first author [Ber07, BC10], we provide in Section 3 a general bijection Φ between a set D of certain decorated plane trees which we call mobile(i) and a set O of certain orientations on planar maps with no counterclockwise circuit

Summary

Introduction

The enumeration of planar maps (connected graphs embedded on the sphere) has received a lot of attention since the seminal work of Tutte in the 60’s [Tut63]. Tutte’s recursive method consists in translating the decomposition of a class of maps (typically obtained by deleting an edge) into a functional equation satisfied by the corresponding generating function. Based on a construction of the first author [Ber, BC10], we provide in Section 3 a general bijection Φ between a set D of certain decorated plane trees which we call mobile(i) and a set O of certain orientations on planar maps with no counterclockwise circuit. As it turns out, a map class is often in bijection with a subfamily S of O on which our construction restricts nicely; typically the orientations in S are characterized by degree constraints which can be traced through our construction and yields a degree characterization of the associated mobiles. The case of triangulations with boundaries has received a partial bijective interpretation, different from ours, in [PS06] (only one direction is given, from trees to maps, which by injection shows that tn,k is at least the number above, but does not suffice to prove equality)

Maps and orientations

Bijections between mobiles and orientations

Bijection between 1-mobiles and minimal accessible orientations

Bijective counting of triangulations with boundaries

Bijective counting of quadrangulations with boundaries