UNIFIED BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH

Transcription

1 UNIFIED BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH OLIVIER BERNARDI AND ÉRIC FUSY Abstract. This article presents unifie bijective constructions for planar maps, with control on the face egrees an on the girth. Recall that the girth is the length of the smallest cycle, so that maps of girth at least =,, are respectively the general, loopless, an simple maps. For each positive integer, we obtain a bijection for the class of plane maps (maps with one istinguishe root-face) of girth having a root-face of egree. We then obtain more general bijective constructions for annular maps (maps with two istinguishe root-faces) of girth at least. Our bijections associate to each map a ecorate plane tree, an non-root faces of egree k of the map correspon to vertices of egree k of the tree. As special cases we recover several known bijections for bipartite maps, loopless triangulations, simple triangulations, simple quarangulations, etc. Our work unifies an greatly extens these bijective constructions. In terms of counting, we obtain for each integer an expression for the generating function F (x,x +,x +,...) of plane maps of girth with root-face of egree, where the variable x k counts the non-root faces of egree k. The expression for F was alreay obtaine bijectively by Bouttier, Di Francesco an Guitter, but for the expression of F is new. We also obtain an expression for the generating function G p,q (,e) (x,x +,...) of annular maps with root-faces of egrees p an q, such that cycles separating the two root-faces have length at least e while other cycles have length at least. Our strategy is to obtain all the bijections as specializations of a single master bijection introuce by the authors in a previous article. In orer to use this approach, we exhibit certain canonical orientations characterizing maps with prescribe girth constraints.. Introuction A planar map is a connecte graph embee without ege-crossing in the sphere. There is a very rich literature on the enumeration of maps, going back to the seminal work of Tutte [9, ] using generating functions. The approach of Tutte applies to many families of maps (triangulations, bipartite maps, -connecte maps, etc.) but involves some technical calculations (the quaratic metho or its generalizations []; see also [8] for a more analytic approach). For many families of maps, the generating function turns out to be algebraic, an to have a simple expression in terms of the generating function of a family of trees. Enumerative results for maps can alternatively be obtaine by a matrix integral approach [5], an algebraic approach [], or a bijective approach [7]. Department of Mathematics, MIT, Cambrige, USA, bernari@math.mit.eu. Supporte by NSF grant DMS-6866, ANR project A an European project ExploreMaps. LIX, École Polytechnique, Palaiseau, France, fusy@lix.polytechnique.fr. Supporte by the European project ExploreMaps (ERC StG 847).

2 O. BERNARDI AND É. FUSY In the bijective approach one typically establishes a bijection between a class of maps an a class of ecorate plane trees (which are easy to count). This usually gives transparent proofs of the enumerative formulas together with algorithmic byproucts [4]. Moreover this approach has prove very powerful for stuying the metric properties of maps an solving statistical mechanics moels on maps [, 4]. There now exist bijections for many ifferent classes of maps [,,,, 7, 6]. In an attempt to unify several bijections the authors have recently efine a master bijection Φ for planar maps [8]. It was shown that for each integer the master bijection Φ can be specialize into a bijection for the class of - angulations of girth (the girth of a graph is the minimal length of its cycles). This approach has the avantage of unifying two known bijections corresponing to the cases = [7, Sec...4] an = 4 [, Thm. 4.]. More importantly, for 5 it gives new enumerative results which seem ifficult to obtain by a non-bijective approach. In the present article, we again use the master bijection strategy an obtain a consierable extension of the results in [8]. We first eal with plane maps, that is, planar maps with a face istinguishe as the root-face. For each positive integer we consier the class of plane maps of girth having a root-face of egree. We present a bijection between this class of maps an a class of plane trees which is easy to enumerate. Moreover it is possible to keep track of the istribution of the egrees of the faces of the map through the bijection. Consequently we obtain a system of algebraic equations specifying the (multivariate) generating function of plane maps of girth having a root-face of egree, counte accoring to the number of faces of each egree. The case = ha previously been obtaine by Bouttier, Di Francesco an Guitter []. Next we consier annular maps, that is, plane maps with a marke inner face. Annular maps have two girth parameters: the separating girth an the non-separating girth efine respectively as the minimum length of cycles separating an not separating the root face from the marke inner face. For each positive integer, we consier the class of annular maps of non-separating girth at least having separating girth equal to the egree of the root-face. We obtain a bijection between this class of maps an a class of plane trees which is easy to enumerate. Again it is possible to keep track of the istribution of the egrees of the faces of the map through the bijection. With some aitional work we obtain, for arbitrary positive integers, e, p, q, a system of algebraic equations specifying the multivariate generating function of roote annular maps of non-separating girth at least, separating girth at least e, root-face of egree p, an marke inner face of egree q, counte accoring to the number of faces of each egree. Using the above result, we prove a universal asymptotic behavior for the number of roote maps of girth at least with face-egrees belonging to a finite set. Precisely, the number c, (n) of such maps with n faces satisfies c, (n) κn 5/ γ n for certain computable constants κ, γ epening on an. This asymptotic behavior was alreay establishe by Bener an Canfiel [] in the case of bipartite maps without girth constraint (their statement actually hols for any set, not necessarily finite). We also obtain a (new) close formula for the number of roote simple bipartite maps with given number of faces of each egree. In orer to explain our strategy, we must point out that the master bijection Φ is a mapping between a certain class of oriente maps Õ an a class of ecorate

3 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH plane trees. Therefore, in orer to obtain a bijection for a particular class of maps C, one can try to efine a canonical orientation for each map in C so as to ientify the class C with a subset ÕC Õ on which the master bijection Φ specializes nicely. This is the approach we aopt in this paper, an our main ingreient is a proof that certain (generalize) orientations, calle /( )-orientations, characterize the maps of girth. A special case of /( )-orientations was alreay use in [8] to characterize -angulations of girth. These orientations are also relate to combinatorial structures known as Schnyer woos [9, 5]. Relation with known bijections. The bijective approach to maps was greatly evelope by Schaeffer [7] after initial constructions by Cori an Vauquelin [7], an Arquès []. Most of the bijections for maps are between a class of maps an a class of ecorate plane trees. These bijections can be ivie into two categories: (A) bijections in which the ecorate tree is a spanning tree of the map (an the ecorations are part of the eges not in the spanning trees), an (B) bijections in which the ecorate plane tree associate to a map M has vertices of two colors black an white corresponing respectively to the faces an vertices of the map (these bicolore trees are calle mobiles in several articles). The first bijection of type A is Schaeffer s construction for Eulerian maps [6]. The first bijection of type B is Schaeffer s construction for quarangulations [7] (which can be seen as a reformulation of [7]) later extene by Bouttier, Di Francesco an Guitter []. Bijections of both types requires one to first enow the maps with a canonical structure (typically an orientation) characterizing the class of maps: Schnyer woos for simple triangulations, -orientations for simple quarangulations, Eulerian orientations for Eulerian maps, etc. For several classes of maps, there exists both a bijection of type A an of type B. For instance, the bijections [6] an [] both allow one to count bipartite maps. The master bijection Φ obtaine in [8] can be seen as a meta construction for all the known bijections of type B (for maps without matter). The master bijection is actually a slight extension of a bijection introuce by the first author in [] an subsequently reformulate in [5] (an extene to maps on orientable surfaces). In [5] it was alreay shown that the master bijection can be specialize in orer to recover the bijection for bipartite maps presente in [, Sec. ]. In the present article, our bijection for plane maps of girth an outer face egree equal to generalizes several known bijections. In the case = our bijection (an the erive generating function expressions) coincies with the one escribe by Bouttier, Di Francesco an Guitter in []. In the case = (loopless maps), our bijection generalizes an unifies two bijections obtaine by Schaeffer in the ual setting. Inee the bijection for Eulerian maps escribe in [6] coincies via uality with our bijection for = applie to bipartite maps, an the bijection for brigeless cubic maps escribe in [7, Sec...4] (which is also escribe an extene in []) coincies via uality with our bijection for = applie to triangulations. For all, our bijection generalizes the bijection for -angulations This classification comes with two subtleties. First, there are two ual versions for bijections of type B: in one version the ecorations of the mobiles are some angling half-eges, while in the ual version the ecorations are some labellings of the vertices; see [5, Sec. 7]. Secon, it sometimes happens that a bijection of type A can be ientifie with a egenerate form of a bijection of type B in which all the white vertices of the mobiles are leaves; see Section 7.

4 4 O. BERNARDI AND É. FUSY of girth given in [8]. This inclues the cases = an = 4 (simple triangulations an simple quarangulations) previously obtaine in [, Thm. 4.] an [7, Sec...]. Lastly, a slight reformulation of our construction allows us to inclue the case =, recovering a bijection escribe in [] for vertex-pointe maps. In two articles in preparation [6, 7], we further generalize the results presente here. In [6] we exten the master bijection to hypermaps an count hypermaps with control on a certain girth parameter (which extens the efinition of girth of a map) an on the egrees of the hypereges an of the faces. In [7], relying on more involve orientations, we count so-calle irreucible maps (an hypermaps), an recover in particular the bijections for irreucible triangulations [] an for irreucible quarangulations []. Outline. In Section we gather useful efinitions on maps an orientations. In Section, we recall the master bijection introuce in [8] between a set Õ of (weighte) oriente maps an a set of (weighte) mobiles. From there, our strategy is to obtain bijections for (plane an annular) maps with control on the girth an face-egrees by specializing the master bijection. As explaine above, this requires the efinition of some (generalize) orientations characterizing the ifferent classes of maps. Section4ealswiththeclassofplanemapsofgirthwithroot-faceegree. We efine a class of (weighte) orientations, calle /( )-orientations, an show that a plane map M of root-face egree has girth if an only if it amits a /( )- orientation. Moreover in this case there is a unique /( )-orientation such that M enowe with this orientation is in Õ. The class of plane maps of girth with rootface egree is thus ientifie with a subset of Õ. Moreover, the master bijection Φ specializesnicelyonthissubset, sothat foreach weobtainabijection between plane maps of girth with root-face egree, an a family of ecorate plane trees calle -branching mobiles specifiable by certain egree constraints. Through this bijection, each inner face of egree k in the map correspons to a black vertex of egree k in the associate mobile. Some simplifications occur for the subclass of bipartite maps when = b (in particular one can use simpler orientations calle b/(b )-orientations) an our presentation actually starts with this simpler case. In Section 5, we exten our bijections to annular maps. More precisely, for any integers p,q, we obtain a bijection for annular maps with root-faces of egrees p an q, with separating girth p an non-separating girth. The strategy parallels the one of the previous section. In Section 6, we enumerate the families of mobiles associate to the above mentione families of plane maps an annular maps. Concerning plane maps, we obtain, for each, an explicit system of algebraic equations characterizing the series F (x,x +,x +,...) counting roote plane maps of girth with root-face of egree, where each variable x k counts the non-root faces of egree k (as alreay mentione, only the case = was known so far []). Concerning annular maps, we obtain for each quaruple p,q,,e of positive integers, an expression for the series G (p,q),e (x,x +,...) counting roote annular maps of non-separating girth at least an separating girth at least e with root-faces of egrees p an q, where the variable x k marks the number of non-root faces of egree k. From these expressions we obtain asymptotic enumerative results. Aitionally we obtain a close formula for the number of roote simple bipartite maps with given number of faces of each

5 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH 5 egree, an give an alternative erivation of the enumerative formula obtaine in [] for loopless maps. In Section 7, we takeacloserlook at the casesb = an =,of ourbijections an explain the relations with bijections escribe in [, 7, 6]. We also escribe a slight reformulation which allows us to inclue the further case = an explain the relation with []. In Section 8, we prove the missing results about /( )-orientations an b/(b )- orientations for plane maps an annular maps.. Maps, biorientations an mobiles This section gathers efinitions about maps, orientations, an mobiles. Maps. A planar map is a connecte planar graph embee (without egecrossing) in the oriente sphere an consiere up to continuous eformation. The faces are the connecte components of the complement of the graph. A plane tree is a map without cycles (it has a unique face). The numbers v, e, f of vertices, eges an faces of a map are relate by the Euler relation: v e+f =. Cutting an ege e at its mile point gives two half-eges, each incient to an enpoint of e (they are both incient to the same vertex if e is a loop). A corner is the angular section between two consecutive half-eges aroun a vertex. The egree of a vertex or face x, enote eg(x), is the number of incient corners. A -angulation is a map such that every face has egree. Triangulations an quarangulations correspon to the cases = an = 4 respectively. The girth of a graph is the minimum length of its cycles. Obviously, a map of girth oes not have faces of egree less than. Note that a map is loopless if an only if it has girth at least an is simple (has no loops or multiple eges) if an only if it has girth at least. A graph is bipartite if its vertices can be bicolore in such a way that every ege connects two vertices of ifferent colors. Clearly, the girth of a bipartite graph is even. Lastly, it is easy to see that a planar map is bipartite if an only if every face has even egree. A plane map (also calle face-roote map) is a planar map with a marke face, calle the root-face. See Figure (a). We think of a plane map as embee in the plane with the root-face taken as the (infinite) outer face. A roote map (also calle corner-roote map) is a map with a marke corner, calle the root; in this case the root-face an root-vertex are the face an vertex incient to the root. The outer egree of a plane (or roote) map is the egree of the root-face. The faces istinct from the root-face are calle inner faces. The vertices, eges, an corners are calle outer if they are incient to the root-face an inner otherwise. A half-ege is outer if it lies on an outer ege, an is inner otherwise. An annular map is a plane map with a marke inner face. See Figure (b). Equivalently, it is a planar map with two istinguishe root-faces calle outer rootface an inner root-face respectively. There are two types of cycles in an annular map: those enclosing the inner root-face are calle separating an those not enclosing the inner root-face are calle non-separating. Accoringly, there are two girth parameters: the separating (resp. non-separating) girth is the minimal length of a separating (resp. non-separating) cycle. We say that an annular map is roote if a corner is marke in each of the root-faces.

6 6 O. BERNARDI AND É. FUSY root-face outer root-face inner root-face (a) (b) Figure. (a) A plane map of outer egree 7 an girth (ue to the cycle in bol eges). (b) An annular map of outer egree 6, separating girth 4 (ue to the cycle in black bol eges) an nonseparating girth (ue to the cycle in gray bol eges). Biorientations. A biorientation of a map G, is a choice of an orientation for each half-ege of G: each half-ege can be either ingoing (oriente towar the vertex), or outgoing (oriente towar the mile of the ege). For i {,,}, we call an ege i-way if it has exactly i ingoing half-eges. Our convention for representing -way, -way, an -way eges is given in Figure (a). The orinary notion of orientation correspons to biorientations having only -way eges. The inegree of a vertex v of G is the number of ingoing half-eges incient to v. Given a biorientation O of a map G, a irecte path of O is a path P = (v,...,v k ) of G such that for all i {,...,k } the ege {v i,v i+ } is either -way or -way from v i to v i+. The orientation O is sai to be accessible from a vertex v if any vertex is reachable from v by a irecte path. If O is a biorientation of a plane map, a clockwise circuit of O is a simple cycle C of G such that each ege of C is either -way or -way with the interior of C on its right. A counterclockwise circuit is efine similarly. A biorientation of a plane map is sai to be minimal if it has no counterclockwise circuit. -way ege outgoing outgoing -way ege outgoing ingoing u -way ege ingoing ingoing (a) (b) v Figure. (a) Convention for representing -way, -way an - way eges. (b) A biorientation, which is not minimal (a counterclockwise circuit is inicate in bol eges). This orientation is accessible from the vertex u but not from the vertex v. A biorientation is weighte if a weight is associate to each half-ege h (in this article the weights will be integers). The weight of an ege is the sum of the

7 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH 7 weights of its half-eges. The weight of a vertex v is the sum of the weights of the ingoing half-eges incient to v. The weight of a face f, enote w(f), is the sum of the weights of the outgoing half-eges incient to f an having f on their right; see Figure. A Z-biorientation is a weighte biorientation where the weight of each half-ege h is an integer which is positive if h is ingoing an non-positive if h is outgoing. An N-biorientation is a Z-biorientation where the weights are non-negative (positive for ingoing half-eges, an zero for outgoing half-eges). A weighte biorientation of a plane map is sai to be amissible if the contour of the outer face is a simple cycle of -way eges with weights an on the outgoing an ingoing half-eges, an the inner half-eges incient to the outer vertices are outgoing. Definition. A Z-biorientation of a plane map is sai to be suitable if it is minimal, amissible, an accessible from every outer vertex (see for instance Figure (a)). We enote by Õ the set of suitably Z-bioriente plane maps v f -4 - (a) - u - (b) - v - - Figure. (a) A suitably Z-bioriente plane map. The vertex v has weight 4+ = 7, the face f has weight 4 = 6. (b) A Z-mobile. The white vertex u has weight ++ = 4, the black vertex v has weight = an has egree 6. Mobiles. A mobile is a plane tree with vertices colore either black or white, an where the black vertices can be incient to some angling half-eges calle bus. Bus are represente by outgoing arrows as in Figure (b). The egree of a black vertex is its number of incient half-eges (incluing the bus). The excess of a mobile is the total number of half-eges incient to the white vertices minus the total number of bus. A Z-mobile is a mobile where each non-bu half-ege h carries a weight which is a positive integer if h is incient to a white vertex, an a non-positive integer if h is incient to a black vertex, see Figure (b). The weight of an ege is the sum of the weight of its half-eges. The weight of a vertex is the sum of weights of all its incient (non-bu) half-eges.. Master bijection between bioriente maps an mobiles In this section we recall the master bijection Φ efine in [8] (where it is enote Φ ) between the set Õ of suitably Z-bioriente plane maps an a set of Z-mobiles. The bijection Φ is illustrate in Figure 5. It will be specialize in Sections 4 an 5 to count classes of plane an annular maps.

8 8 O. BERNARDI AND É. FUSY Definition. Let M be a suitably Z-bioriente plane map (Definition ) with rootface f. We view the vertices of M as white an place a black vertex b f in each face f of M. The embee graph Φ(M) with black an white vertices is obtaine as follows: Reverse the orientation of all the eges of the root-face (which is a clockwise irecte cycle of -way eges). For each ege e, perform the following operation represente in Figure 4. Let h an h be the half-eges of e with respective weights w an w. Let v an v be respectively the vertices incient to h an h, let c, c be the corners preceing h, h in clockwise orer aroun v, v, an let f, f be the faces containing these corners. If e is -way, then create an ege between the black vertices b f an b f across e, an give weight w an w to the half-eges incient to b f an b f respectively. Then, elete the ege e. If e is -way with h being the ingoing half-ege, then create an ege joining the black vertex b f to the white vertex v in the corner c, an give weight w an w to the half-eges incient to v an b f respectively. Then, glue a bu on b f in the irection of c, an elete the ege e. If e is -way, then glue bus on b f an b f in the irections of the corners c an c respectively (an leave intact the weighte ege e). Delete the black vertex b f, the outer vertices of M, an the eges between them (no other ege or bu is incient to these vertices). v -way ege w h w h v v -way ege w h w h v v -way ege w h w h v v b f w v v b f w w v v w b f w v w b f b f b f Figure 4. Local transformation of -way, -way an -way eges one by the bijection Φ. The following theorem is prove in [8]: Theorem. The mapping Φ is abijection between the set Õ of suitably Z-bioriente plane maps (Definition ) an the set of Z-mobiles of negative excess, with the parameter-corresponence given in Figure 6. For M a suitably Z-bioriente plane map of outer egree, an T = Φ(M) the corresponing mobile, we call expose the bus of the mobile T = Φ(M) create by applying the local transformation to the outer eges of M (which have preliminarily been returne). The following aitional claim, prove in [8], will be useful for counting purposes. Claim 4. Let M be a suitably Z-bioriente plane map of outer egree, an let T = Φ(M) be the corresponing mobile. There is a bijection between the set M of

9 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH Figure 5. The master bijection Φ applie to a suitably Z- bioriente plane map. Bioriente map M Õ Mobile Φ(M) inner vertex same weight white vertex inner face inner ege { -way outer egree same egree black vertex same weight same weight ege -way black-black } black-white -way white-white excess Figure 6. Parameter-corresponence of the master bijection Φ. all istinct corner-roote maps obtaine from M by marking an outer corner (note that the carinality of M can be less than ue to symmetries), an the set T of all istinct mobiles obtaine from T by marking one of the expose bus. Moreover, there is a bijection between the set T of mobiles obtaine from T by marking a non-expose bu, an the set T of mobiles obtaine from T by marking a half-ege incient to a white vertex. Before we close this section we recall from [8] how to recover the map starting from a mobile (this escription will be useful in Section 7 to compare our bijection with other known bijections). Let T be a mobile (weighte or not) with negative excess δ. The corresponing fully blossoming mobile T is obtaine from T by first inserting a fake black vertex in the mile of each white-white ege, an then by inserting a angling half-ege calle stem in each corner preceing a black-white ege e in clockwise orer aroun the black extremity of e. A fully blossoming mobile is represente in soli lines in Figure 7 (bus an stems are respectively inicate by outgoing an ingoing arrows). Turning in counterclockwise irection aroun the mobile T, one sees a sequence of bus an stems. The partial closure of T is obtaine by rawing an ege from each bu to the next available stem in counterclockwise orer aroun T (these eges can be rawn without crossings). This leaves δ bus unmatche (since the excess δ is equal to the number of stems minus the number of bus). The complete closure Ψ(T) of T is the vertex-roote bioriente map obtaine from the partial closure by first creating a root-vertex v in the face containing the unmatche bus an joining it to all the unmatche

10 O. BERNARDI AND É. FUSY bus, an then eleting all the white-white an black-white eges of the mobile T an erasing the fake black vertices (these were at the mile of some eges); see Figure 7. (a) (b) (c) () Figure 7. The mapping Ψ. (a) A mobile T. (b) The fully blossoming mobile T (rawn in soli lines with bus represente as outgoing arrows, an stems represente as ingoing arrows) an its partial closure (rawn in ashe lines). (c) The complete closure Ψ(T). () The ual of Ψ(T). Proposition 5 ([8]). Let M be suitably Z-bioriente plane map, let T = Φ(M) be the associate mobile, an let N = Ψ(T) be its complete closure. Then the plane map unerlying M is ual to the vertex-roote map unerlying N. 4. Bijections for maps with one root-face In this section, we present our bijections for plane maps. For each positive integer, we consier the class C of plane maps of outer egree an girth. We efine some Z-biorientations that characterize the maps in C. This allows us to ientify the class C with a subset of Õ. We then specialize the master bijection to this subset of Õ an obtain a bijection for maps in C with controlon the number of inner faces in each egree i. For the sake of clarity we start with the bipartite case, where the orientations an bijections are simpler. 4.. Bipartite case. In this section, b is a fixe positive integer. We start with the efinition of the Z-biorientations that characterize the bipartite maps in C b. Definition 6. Let M be a bipartite plane map of outer egree b having no face of egree less than b. A b/(b )-orientation of M is an amissible Z-biorientation such that every outgoing half-ege has weight or - an (i) each inner ege has weight b, (ii) each inner vertex has weight b, (iii) each inner face f has egree an weight satisfying eg(f)/+w(f) = b. Figure 9 shows some b/(b )-orientations for b = an b =. Observe that for b, a b/(b )-orientation has no -way eges, while for b it has no -way eges. Definition 6 of b/(b )-orientations actually generalizes the one given in [8] for b-angulations. Note that b/(b )-orientations of b-angulations are in fact N-biorientations since Conition (iii) implies that the weight of every outgoing half-ege is.

11 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH Theorem 7. Let M be a bipartite plane map of outer egree b having no face of egree less than b. Then M amits a b/(b )-orientation if an only if M has girth b. In this case, there exists a unique suitable b/(b )-orientation of M. The proof of Theorem 7 (which extens a result given in [8] for b-angulations) is elaye to Section 8. We now efine the class of Z-mobiles that we will show to be in bijection with bipartite maps in C b. Definition 8. A b-ibranching mobile is a Z-mobile such that half-eges incient to black vertices have weight or an (i) each ege has weight b, (ii) each white vertex has weight b, (iii) each black vertex v has egree an weight satisfying eg(v)/+w(v) = b; equivalently a black vertex of egree i is ajacent to i b white leaves. The two ways of phrasing Conition (iii) are equivalent because a half-ege incient to a black vertex has weight - if an only if it belongs to an ege incient to a white leaf. Examples of b-ibranching mobiles are given in Figure 9. The possible eges of a b-ibranching mobile are represente for ifferent values of b in Figure 8. b = b = b b i b i leaf leaf leaf b i {,...,b } Figure 8. The possible eges of b-ibranching mobiles. The white leaves are inicate. Claim 9. Any b-ibranching mobile has excess b. Proof. Let T be a b-ibranching mobile. Let e be the number of eges an β be the number of bus. Let v b an v w be the number of black an white vertices respectively. Let h b an h w be the number of non-bu half-eges incient to black an white vertices, respectively. By efinition, the excess δ of the mobile is δ = h w β. Now, by Conition (iii) on black vertices, one gets (h b +β)/+s = bv b, where S is the sum of weights of the half-eges incient to black vertices. By Conitions (i) an (ii), one gets e(b ) = bv w +S. Eliminating S between these relations gives e(b )+h b +β = b(v b +v w ). Lastly, plugging v b +v w = e+ an e = h b +h w in this relation, one obtains h w β = b. We now come to the main result of this subsection, which is the corresponence between the set C b of bipartite maps an the b-ibranching mobiles. First of all, Theorem 7 allows one to ientify the set C b of bipartite maps with the set of b/(b )-oriente plane maps in Õ. We now consier the image of this subset of Õ by the master bijection Φ. In view of the parameter-corresponence inuce by the master bijection Φ (Theorem ), it is clear that Conitions (i), (ii), (iii) of the b/(b )-orientations correspon respectively to Conitions (i), (ii), (iii) of the b-ibranching mobiles. Thus, by Theorem, the master bijection Φ inuces a bijection between the set of b/(b )-oriente plane maps in Õ an the set of

12 O. BERNARDI AND É. FUSY Figure 9. Bijections for bipartite maps in the cases b = (left) an b = (right). Top: a plane bipartite map of girth b an outer egree b enowe with its suitable b/(b )-orientation.bottom: the associate b-ibranching mobiles. b-ibranching mobiles of excess b. Moreover, by Claim 9 the constraint on the excess is reunant. We conclue:

13 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH Theorem. For any positive integer b, bipartite plane maps of girth b an outer egree b are in bijection with b-ibranching mobiles. Moreover, each inner face of egree i in the map correspons to a black vertex of egree i in the mobile. Figure 9 illustrates the bijection on two examples (b =, b = ). The case b = an its relation with [6] is examine in more etails in Section General case. We now treat the case of general (not necessarily bipartite) maps. In this subsection, is a fixe positive integer. Definition. Let M be a plane map of outer egree having no face of egree less than. A /( )-orientation of M is an amissible Z-biorientation such that every outgoing half-ege has weight, or an (i) each inner ege has weight, (ii) each inner vertex has weight, (iii) each inner face f has egree an weight satisfying eg(f)+w(f) =. Figure 9 shows some /( )-orientations for = an = 5. The cases = an = are represente in Figures 5 an respectively. Definition of /( )-orientations actually generalizes the one given in [8] for -angulations. Note that /( )-orientations of -angulations are in fact N-biorientations since Conition (iii) implies that the weight of every outgoing half-ege is. Theorem. Let M be a plane map of outer egree having no face of egree less than. Then, M amits a /( )-orientation if an only if M has girth. In this case, there exists a unique suitable /( )-orientation of M. Remark. If = b an M is a bipartite plane map of outer egree an girth, then the unique suitable /( )-orientation of M is obtaine from its suitable b/(b )-orientation by oubling the weight of every inner half-ege (since the Z-biorientation obtaine in this way is clearly a suitable /( )-orientation). The proof of Theorem is elaye to Section 8. We now efine the class of mobiles that we will show to be in bijection with C. Definition 4. For a positive integer, a -branching mobile is a Z-mobile such that half-eges incient to black vertices have weight, or an (i) each ege has weight, (ii) each white vertex has weight, (iii) each black vertex v has egree an weight satisfying eg(v)+w(v) =. = = = 4 leaf leaf leaf leaf i i i {,} i i i {,..., } Figure. The possible eges of -branching mobiles. The white leaves are inicate.

14 4 O. BERNARDI AND É. FUSY Figure. The bijection applie to maps in C ( = on the left an = 5 on the right). Top: a plane map of girth an outer egree enowe with its suitable /( )-orientation. Bottom: the associate -branching mobiles. The possible eges of a -branching mobile are represente for ifferent values of in Figure. The following claim can be prove by an argument similar to the one use in Claim 9.

15 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH 5 Claim 5. Any -branching mobile has excess. We now come to the main result of this subsection, which is the corresponence between the set C of plane maps of girth an outer egree an the set of -branching mobiles. By Theorem, the set C can be ientifie with the subset of /( )-oriente plane maps in Õ. Moreover, as in the bipartite case, it is easy to see from Theorem that the master bijection Φ inuces a bijection between the set of /( )-oriente plane maps in Õ an the set of -branching mobiles. We conclue: Theorem 6. For any positive integer, plane maps of girth an outer egree are in bijection with -branching mobiles. Moreover, each inner face of egree i in the map correspons to a black vertex of egree i in the mobile. Figure illustrates the bijection on two examples ( =, = 5). The bijection of Theorem 6 is actually a generalization of the bijection given in [8] for -angulations of girth (for -angulations there are no negative weights). The cases = an = of Theorem 6 are examine in more etails in Section 7, in particular the relation between our bijection in the case = an the bijection escribe by Bouttier, Di Francesco an Guitter in [] (we also show a link with another bijection escribe by the same authors in []). Remark 7. For = bitisclearfromremarkthatthebijectionoftheorem is equal to the specialization of the bijection of Theorem 6 to bipartite maps, up to iviing the weights of the mobiles by two. 5. Bijections for maps with two root-faces In this section we escribe bijections for annular maps. An annular map is of type (p,q) if the outer an inner root-faces have egrees p an q respectively. We enote by A (p,q) the class of annular maps of type (p, q) with non-separating girth at least an separating girth p (in particular A (p,q) = unless q p). In the following we obtain a bijection between A (p,q) an a class of mobiles. Our strategy parallels the one of the previous section, an we start again with the bipartite case which is simpler. In Section 6. we will show that counting results for the classes A (p,q) can be use to enumerate also the annular maps with separating girth smaller than the outer egree. 5.. Bipartite case. In this subsection we fix positive integers b,r,s with r s. We start with the efinition of the Z-biorientations that characterize the bipartite maps in A (r,s) b. Definition 8. Let M be a bipartite annular map of type (r,s) having no face of egree less than b. A b/(b )-orientation of M is an amissible Z-biorientation such that every outgoing half-ege has weight or - an (i) each inner ege has weight b, (ii) each inner vertex has weight b, (iii) each non-root face f has egree an weight satisfying eg(f)/+w(f) = b, (iv) the inner root-face has egree s an weight r s.

16 6 O. BERNARDI AND É. FUSY Figure (top left) shows a b/(b )-orientation for b =. Note that when r = b (outer root-face of egree b) every inner face(incluing the inner root-face) satisfies eg(f)/+w(f) = b, in which case we recoverthe efinition of b/(b )-orientations for plane bipartite maps of outer egree b, as given in Section 4. Theorem 9. Let M be an annular bipartite map of type (r,s). Then M amits a b/(b )-orientation if an only if M is in A (r,s) b. In this case, there exists a unique suitable b/(b )-orientation of M. The proof of Theorem 9 (which extens Theorem 7, corresponing to the case r = b) is elaye to Section 8. We now efine the class of Z-mobiles that we will show to be in bijection with bipartite maps in A (r,s) b. Definition. A b-ibranchingmobileoftype (r,s) is a Z-mobile with a marke black vertex calle special vertex such that half-eges incient to black vertices have weight or an (i) each ege has weight b, (ii) each white vertex has weight b, (iii) each non-special black vertex v has egree an weight satisfying eg(v)/ + w(v) = b; equivalently a non-special black vertex of egree i is ajacent to i b white leaves. (iv) the special vertex v has egree s an weight r s; equivalently v has egree s an is ajacent to s r white leaves. A -ibranching mobile of type (6, 8) is represente in Figure (bottom left). As a straightforwar extension of Claim 9 we obtain: Claim. Any b-ibranching mobile of type (r, s) has excess r. We now come to the main result of this subsection, which is the corresponence between the set A (r,s) b of annular bipartite maps an b-ibranching mobiles of type (r,s). First of all, by Theorem 9 the set A (r,s) b can be ientifie with the subset of b/(b )-oriente annular maps of type (r,s) in Õ. Thus, it remains to show that the master bijection inuces a bijection between this subset an the set of b-ibranching mobiles of type (r, s). In view of the parameter-corresponence of the master bijection Φ (Theorem ), it is clear that Conitions (i), (ii), (iii), (iv) of the b/(b )-orientations correspon respectively to Conitions (i), (ii), (iii), (iv) of the b-ibranching mobiles. Thus, by Theorem, the master bijection Φ inuces a bijection between the set of b/(b )-oriente annular maps of type (r,s) in Õ an the set of b-ibranching mobiles of type (r,s) an excess r. Moreover, by Claim the constraint on the excess is reunant. We conclue: Theorem. Bipartite annular maps in A (r,s) b are in bijection with b-ibranching mobiles of type (r,s). Moreover, each non-root face of egree i in the map correspons to a non-special black vertex of egree i in the mobile. Theorem is illustrate in Figure (left). Observe that the case b = r in Theorem correspons to the bijection of Theorem where an inner face is marke. 5.. General case. We now treat the case of general (not necessarily bipartite) maps. In this subsection we fix positive integers,p,q with p q.

17 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH Figure. Bijection for annular maps. Left: a bipartite annular map in A (r,s) b with {b =,r =,s = 4}enowewith its suitable b/(b )-orientation, an the associate b-ibranching mobile of type (r,s); to be compare with the left part of Figure 9 (b =, one root-face). Right: annular map in A (p,q) with { =,p = 4, q = 5} enowe with its suitable /( )-orientation, an the associate -branching mobile of type (p, q); to be compare with the left part of Figure ( =, one root-face).

18 8 O. BERNARDI AND É. FUSY Definition. Let M be an annular map of type (p,q) having no face of egree less than. A /( )-orientation of M is an amissible Z-biorientation such that every outgoing half-ege has weight, or an (i) each inner ege has weight, (ii) each inner vertex has weight, (iii) each non-root face f has egree an weight satisfying eg(f)+w(f) =, (iv) the inner root-face has egree q an weight p q. Figure (top right) shows a /( )-orientation for =. Note that when p = every inner face satisfies eg(f) + w(f) =, in which case we recover the efinition of /( )-orientations for plane maps of outer egree, as given in Section 4. Theorem 4. Let M be an annular map of type (p,q) having no face of egree less than. Then, M amits a /( )-orientation if an only if M is in A (p,q). In this case, there exists a unique suitable /( )-orientation of M. Remark 5. If = b an M is a bipartite annular map in A (p,q), then the unique suitable /( )-orientation of M is obtaine from its suitable b/(b )-orientation by oubling the weight of every inner half-ege. The proof of Theorem 4 (which extens Theorem ) is elaye to Section 8. Definition 6. A -branching mobile of type (p,q) is a Z-mobile with a marke black vertex calle special vertex such that half-eges incient to black vertices have weight, or an (i) each ege has weight, (ii) each white vertex has weight, (iii) each non-special black vertex v has egree an weight satisfying eg(v) + w(v) =, (iv) the special vertex has egree q an weight p q. The proof of the following claim is similar to the one use for Claim 9. Claim 7. Any -branching mobile of type (p, q) has excess p. We now come to the main result of this subsection, which is the corresponence between the set A (p,q) of annular maps an -branching mobiles of type (p, q). First of all, by Theorem 4, the set A (p,q) can be ientifie with the set of /( )- oriente annular maps of type (p,q) in Õ. Moreover, as in the bipartite case, it is easy to see from Theorem that the master bijection Φ inuces a bijection between this subset of Õ an the set of -branching mobiles of type (p,q). We conclue: Theorem 8. Annular maps in A (p,q) are in bijection with -branching mobiles of type (p,q). Moreover, each non-root face of egree i in the map correspons to a non-special black vertex of egree i in the mobile. Theorem 8 is illustrate in Figure (right). The case = p in Theorem 8 correspons to the bijection of Theorem 6 where an inner face is marke. Remark 9. For = bitisclearfromremark5thatthebijectionoftheorem is equal to the specialization of the bijection of Theorem 8, up to iviing the weights of the mobiles by two.

19 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH 9 6. Counting results In this section we erive the enumerative consequences of the bijections escribe in the previous sections. 6.. Counting maps with one root-face. In this subsection we give, for each positive integer, a system of equations specifying the generating function F of roote maps of girth an outer egree counte accoring to the number of inner face of each egree. We first set some notation. For any integers p,q we enote by [p..q] the set of integers {k Z, p k q}. If G(x) is a (Laurent) formal power series in x, we enote by [x k ]G(x), the coefficient of x k in G(x). For each non-negative integer j we efine the polynomial h j in the variables w,w,... by: () h j (w,w,...) := [t j ] w i w ir. i> ti w i = r i,...,i r> i + +i r=j In other wors, h j is the (polynomial) generating function of integer compositions of j where the variable w i marks the number of parts of size i. Note that h =. Let be a positive integer. By Theorem an Claim 4, counting roote plane maps of girth an outer egree reuces to counting -branching mobiles roote at an expose bu. To carry out the latter task we simply write the generating function equation corresponing to the recursive ecomposition of trees. We call plante -branching mobile a mobile with a angling half-ege that can be obtaine as one of the two connecte components after cutting a -branching mobile M at the mile of an ege. The weight of the angling half-ege h is calle the rootweight, an the vertex incient to h is calle the root-vertex. Recall that the half-eges of a -branching mobiles have weight in [..]. For j in [..], we enote by W j the family of plante -branching mobiles of root-weight j. We enote by W j W j (x,x +,x +...) the generating function of W j, where for k the variable x k marks the black vertices of egree k. We now consier the recursive ecomposition of plante mobiles an translate it into a system of equations characterizing the series W,...,W. Let j be in [.. ], an let T be a plante mobile in W j. Since j >, the root-vertex v of T is white, hence is incient to half-eges having positive weights. Let e,...,e r be the eges incient to v. For all i =...r, let T i be the plante mobile obtaine by cutting the ege e i in the mile (T i is the subtree not containing v), an let α(i) > be the weight of the half-ege of e i incient to v (so that T i is in W α(i) ). Since the white vertex v has weight, one gets the constraint i α(i) = j +. Conversely, any sequence of plante mobiles T,...,T r in W α(),...,w α(r) such that α(i) > an iα(i) = j + gives a plante mobile in W j. Thus for all j in [.. ], W j = W i W ir = h j+ (W,...,W ). r i,...,i r> i + +i r=j+ Note that the special case W = is consistent with our convention h =. Observe also that W = h (W ) = W whenever >. Now let j be in [..], let T be a plante mobile in W j. Since j, the root-vertex v of T is black. If v has egree i, then there is a sequence of i

20 O. BERNARDI AND É. FUSY bus an non-angling half-eges incient to v. Each non-angling half-ege h has weight α {,, },an cutting the ege containing h gives a plante mobile in W α. Lastly, Conition (iii) of -branching mobiles implies that the sum of weights of non-angling half-eges is eg(v) ( j) = j + i. Conversely, any sequence of bus an non-angling half-eges satisfying these conitions gives a plante mobile in W j. Thus for all j in [..], W j = [u j+ ] i x i u i (+W +u W +u ) i, where the summans, W, u W an u = u W in the parenthesis correspon respectively to the bus an non-angling half-eges of weight,, incient to v. We summarize: Theorem. Let be a positive integer, an let F F (x,x +,x +,...) be the generating function of roote maps of girth with outer egree, where each variable x i counts the inner faces of egree i. Then, () F = W j= W j W j, where W =,W,W,...,W are the unique formal power series satisfying: () W j = h j+ (W,...,W ) for all j in [.. ], W j = [u j+ ] i x i u i (+W +u W +u ) i for all j in [..], where the polynomials h j are efine by (). In particular, for any finite set {,+,+,...}, the specialization of F obtaine by setting x i = for all i not in is algebraic (over the fiel of rational function in x i,i ). For =, Theorem gives exactly the system of equations obtaine by Bouttier, Di Francesco an Guitter in []. Observe that for any integer the series W an W are equal, so the number of unknown series is + in these cases. Moreover for the series W is not neee to efine the other series W,W,...,W. Lastly, uner the specialization {x = x, x i = i > } one gets W = W = an W = x( +W ) ; in this case we recover the system of equations given in [8] for the generating function of roote -angulations of girth. Proof. The fact that the solution of the system () is unique is clear. Inee, it is easy to see that the series W,W,...,W have no constant terms, an from this it follows that the coefficients of these series are uniquely etermine by inuction on the total egree. We now prove (). By Theorem 6 an Claim 4 (first assertion) the series F is equal to the generating function of -branching mobiles with a marke expose bu (where x k marks the black vertices of egree k). Moreover by the secon assertion of Claim 4, F is equal to the ifference between the generating function B of -branching mobiles with a marke bu, an the generating function H of -branching mobiles with a marke half-ege incient to a white vertex. Lastly, B = W because -branching mobiles with a marke bu ientify with plante mobiles in W, an H = j= W jw j because -branching mobiles with

21 BIJECTIONS FOR MAPS WITH PRESCRIBED DEGREES AND GIRTH a marke half-ege incient to a white vertex are in bijection (by cutting the ege) with orere pairs (T,T ) of plante -branching mobiles in W j W j for some j in [.. ]. We now explore the simplifications occurring in the bipartite case. Theorem. Let b, an let E b F b (x b,,x b+,,x b+4...) be the generating function of roote bipartite maps of girth b with outer egree b, where each variable x i marks the number of inner faces of egree i. Then, (4) E b = V b b j= V j V b j, where V =,V,...,V b are the unique formal power series satisfying: V j = h j+ (V,...,V b for all j in [..b ], (5) V j = ( ) i x i (+V ) i+j for all j in {b,b}. i j i b Theorem can be obtaine by a irect counting of b-ibranching mobiles(which are simpler than -branching mobiles). However in the proof below we erive Theorem as a consequence of Theorem. Proof. Equations (4) an (5) are obtaine respectively from () an () simply by setting for all integer i, x i+ =, W i = V i, W i+ =. Hence we only nee to prove that the series W i efine by () satisfy for all i, W i+ (x b,,x b+,,...) =. This property hols because one can show that every monomial in the series W i+ (x b,x b+,x b+,...),i Z contains at least one variable x r with r o, by a simple inuction on the total egree of these monomials. 6.. Counting maps with two root-faces. In this subsection we count roote annular maps accoring to the face egrees an accoring to the two girth parameters. For positive integers,e,p,q, we enote by A (p,q),e the class of annular maps of type (p,q) having non-separating girth at least an separating girth at least e. Recall that an annular map is roote if a corner is marke in each of the root-faces. We will now erive an expression for the generating functions G (p,q),e of maps obtaine by rooting the annular maps in A (p,q),e. Theorem. For any positive integers,e,p,q, the series G (p,q),e = G (p,q),e (x,x +,...) counting roote annular maps of type (p,q) with non-separating girth at least an separating girth at least e (where x k marks the number of non-root faces of egree k) is (6) G (p,q),e = p e q e i= j= β(p,i,e)β(q,j,e) i+j p+q (mo ) (+W ) (p+q i j)/ W i+j, p+q i j where the formal power series W,W,...,W are specifie by (), an where p! β(p,i,e) := i! p i e! p i+e!.