Limit Laws of Planar Maps with Prescribed Vertex Degrees

Transcription

1 Combnatorcs, Probablty and Computng (206) 00, c 206 Cambrdge Unversty Press OI: 0.07/S Prnted n the Unted Kngdom Lmt Laws of Planar Maps wth Prescrbed Vertex egrees G. C O L L E T, M. R M O T A, L.. K L A U S N E R TU Wen, Insttute of screte Mathematcs and Geometry, Wedner Hauptstraße 8 0, A-040 Wen, Austra gwendal.collet@tuwen.ac.at, drmota@tuwen.ac.at, lukas.d.klausner@tuwen.ac.at We prove a general mult-dmensonal central lmt theorem for the expected number of vertces of a gven degree n the famly of planar maps whose vertex degrees are restrcted to an arbtrary (fnte or nfnte) set of postve ntegers. Our results rely on a classcal bjecton wth mobles (objects exhbtng a tree structure), combned wth refned analytc tools to deal wth the systems of equatons on nfnte varables that arse. We also dscuss possble extensons to maps of hgher genus and to weghted maps.. Introducton and Results In ths paper we study statstcal propertes of planar maps, whch are connected planar graphs, possbly wth loops and multple edges, together wth an embeddng nto the plane. Such objects are frequently used to descrbe topologcal features of geometrc arrangements n two or three spatal dmensons. Thus, the knowledge of the structure and of propertes of typcal objects may turn out to be very useful n the analyss of partcular algorthms that operate on planar maps. We say that a map s rooted f an edge e s dstngushed and orented. It s called the root edge. The frst vertex v of ths orented edge s called the root vertex. The face to the rght of e s called the root face and s usually taken as the outer (or nfnte) face. Smlarly, we call a planar map ponted f just a vertex v s dstngushed. However, we have to be really careful wth the model. In rooted maps the root edge destroys potental symmetres, whch s not the case f we consder ponted maps. The enumeraton of rooted maps s a classcal subject, ntated by Tutte n the 960 s, see [3]. Among many other results, Tutte computed the number M n of rooted maps wth n edges, provng the formula M n = 2(2n)! (n + 2)!n! 3n Partally supported by the Austran Scence Fund FWF, Project SFB F50-02.

2 2 G. Collet, M. rmota and L.. Klausner whch drectly provdes the asymptotc formula M n 2 π n 5/2 2 n. We are manly nterested n planar maps wth degree restrctons. Actually, t turns out that ths knd of asymptotc expanson s qute unversal. Furthermore, there s always a (very general) central lmt theorem for the number of vertces of gven degree. Theorem.. Suppose that s an arbtrary set of postve ntegers but not a subset of {, 2}. Let M be the class of planar rooted maps wth the property that all vertex degrees are n and let M,n denote the number of maps n M wth n edges. Furthermore, f contans only even numbers, then set d = gcd{ : 2 }; set d = otherwse. Then there exst postve constants c and ρ wth M,n c n 5/2 ρ n, n 0 mod d. (.) Furthermore, let X n (d) denote the random varable countng vertces of degree d ( ) n maps n M. Then E(X n (d) ) µ d n for some constant µ d > 0 and for n 0 mod d, and the (possbly nfnte) random vector X n = (X n (d) ) d (n 0 mod d) satsfes a central lmt theorem, that s, n (X n E(X n )), n 0 mod d, (.2) converges weakly to a centered Gaussan random varable Z (n l 2 ). Note that maps where all vertex degrees are or 2 are very easy to characterse and are not really of nterest, and that actually, ther asymptotc propertes are dfferent from the general case. It s therefore natural to assume that s not a subset of {, 2}. Snce we can equvalently consder dual maps, ths knd of problem s the same as the one consderng planar maps wth restrctons on the face valences. Ths means that the same results hold f we replace vertex degree by face valency. For example, f we assume that all face valences equal 4, then we just consder planar quadrangulatons (whch have also been studed by Tutte [3]). In fact, our proofs wll refer just to face valences. Theorem. goes far beyond known results. There are some general results for the Euleran case where all vertex degrees are even. Frst, the asymptotc expanson (.) s known for Euleran maps by Bender and Canfeld [2]. Furthermore, a central lmt theorem of the form (.2) s known for all Euleran maps (wthout degree restrctons) [9]. However, n the non-euleran case there are almost no results of ths knd; there s only a one-dmensonal central lmt theorem for X n (d) for all planar maps [0]. The unform dstrbuton of planar maps accordng to the number of edges s not the only dstrbuton that has been studed. Many probablstc results on planar maps have also be extended to other probablty dstrbutons, based on q-boltzmann maps. Let q = (q, q 2,... ) be a sequence of non-negatve weghts. A q-boltzmann map s a random planar map wth arbtrary vertex degrees, where the probablty of choosng a gven map

3 M s proportonal to Planar Maps wth Prescrbed egrees 3 >0 #vertces of degree n M q. When such a procedure descrbes a well-defned probablty dstrbuton, q s called admssble. (By dualty we could equvalently use weghts of the form #faces of degree n M q.) In [, 2], the authors showed that, under some ntegrablty condtons, random q- Boltzmann maps have the same profle as random unform planar maps. In ths sprt we wll show that (under certan condtons) Theorem. also apples to q-boltzmann maps. Theorem.2. Let q = (q, q 2,... ) be a weght sequence wth q = Θ( α ) for some α 3 (d) 2 and consder correspondng q-boltzmann maps. Furthermore, let Y n denote the random varable countng vertces of degree d. Then E(Y n (d) ) µ d n for some constant µ d > 0 and the nfnte random vector Y n = (Y n (d) ) d satsfes a central lmt theorem. Secton 2 ntroduces planar mobles whch, beng n bjecton wth ponted planar maps, wll reduce our analyss to smpler objects wth a tree structure. Ther asymptotc behavour s derved n Secton 3, frst for the smpler case of bpartte maps (. e. when contans only even ntegers), then for famles of maps wthout constrants on. Sectons 4 and 5 are devoted to the proof of the central lmt theorem usng analytc tools from [8, 9]. Fnally, n Secton 6 we dscuss combnatorcs and asymptotcs of maps on orentable surfaces of hgher genus. The expressons we obtan are much more nvolved than n the planar case, but we obtan smlar analytc results. 2. Mobles Instead of nvestgatng planar maps themselves, we wll follow the prncple presented n [5], whereby ponted planar maps are bjectvely related to a certan class of trees called mobles. (Ther verson of mobles dffers from the defnton orgnally gven n [3]; the equvalence of the two defntons s not shown explctly n [5], but [7] gves a straghtforward proof.) efnton. A moble s a planar tree that s, a map wth a sngle face such that there are two knds of vertces (black and whte), edges only occur as black black edges or black whte edges, and black vertces addtonally have so-called legs attached to them (whch are not consdered edges), whose number equals the number of whte neghbour vertces. A bpartte moble s a moble wthout black black edges. The degree of a black vertex s the number of half-edges plus the number of legs that are attached to t. A moble s called rooted f an edge s dstngushed and orented. The essental observaton s that mobles are n bjecton to ponted planar maps.

4 4 G. Collet, M. rmota and L.. Klausner Theorem 2.. There s a bjecton between mobles that contan at least one black vertex and ponted planar maps, where whte vertces n the moble correspond to nonponted vertces n the equvalent planar map, black vertces correspond to faces of the map, and the degrees of the black vertces correspond to the face valences. Ths bjecton nduces a bjecton on the edge sets such that the number of edges s the same. (Only the ponted vertex of the map has no counterpart.) Smlarly, rooted mobles that contan at least one black vertex are n bjecton to rooted and vertex-ponted planar maps. Fnally, bpartte mobles wth at least two vertces correspond to bpartte maps wth at least two vertces, n the unrooted as well as n the rooted case. Proof. For the proof of the bjecton between mobles and ponted maps we refer to [7], where the bpartte case s also dscussed. It just remans to note that the nduced bjecton on the edges can be drectly used to transfer the root edge together wth ts drecton. 2.. Bpartte Moble Countng We start wth bpartte mobles snce they are more easy to count, n partcular f we consder rooted bpartte mobles, see [7]. Proposton 2.2. Let R = R(t, z, x, x 2,...) be the soluton of the equaton R = tz + z ( ) 2 x 2 R. (2.) Then the generatng functon M = M(t, z, x, x 2,...) of bpartte rooted maps satsfes M t = 2 (R/z t), (2.2) where the varable t corresponds to the number of vertces, z to the number of edges, and x 2,, to the number of faces of valency 2. Proof. Snce rooted mobles can be consdered as ordered rooted trees (whch means that the neghbourng vertces of the root vertex are lnearly ordered and the subtrees rooted at these neghbourng vertces are agan ordered trees), we can descrbe them recursvely. Ths drectly leads to a functonal equaton for R of the form tz R = z x 2 ( 2 ) R whch s apparently the same as (2.). Note that the factor ( ) 2 s precsely the number of ways of groupng legs and edges around a black vertex (of degree 2; one edge s already there). Hence, the generatng functon of rooted mobles that are rooted by a whte vertex s gven by R/z. Snce we have to dscount the moble that conssts just of one (whte) vertex, the generatng functon of rooted mobles that are rooted at a whte vertex and

5 contan at least two vertces s gven by Planar Maps wth Prescrbed egrees 5 R/z t = ( ) 2 x 2 R. (2.3) We now observe that the rght-hand sde of (2.3) s precsely the generatng functon of rooted mobles that are rooted at a black vertex (and contan at least two vertces). Summng up, the generatng functon of bpartte rooted mobles (wth at least two vertces) s gven by 2(R/z t). Fnally, f M denotes the generatng functon of bpartte rooted maps (wth at least two vertces) then M t corresponds to rooted maps where a non-root vertex s ponted (and dscounted). Thus, by Theorem 2. we obtan (2.2). It s clear that Formula (2.2) can be specalsed to count M for any subset of even postve ntegers: It suffces to set x 2 = for 2 and x 2 = 0 otherwse General Moble Countng We now proceed to develop a mechansm for general moble countng that s adapted from [5]. For ths, we wll requre Motzkn paths. A Motzkn path s a path startng at 0 and gong rghtwards for a number of steps; the steps are ether dagonally upwards (+), straght (0) or dagonally downwards ( ). A Motzkn brdge s a Motzkn path from 0 to 0. A Motzkn excurson s a Motzkn brdge whch stays non-negatve. We defne generatng functons n the varables t and u, whch count the number of steps of type 0 and, respectvely. (Explctly countng steps of type s then unnecessary, of course.) The ordnary generatng functons of Motzkn brdges, Motzkn excursons, and Motzkn paths from 0 to + shall be denoted by B(t, u), E(t, u) and B (+) (t, u), respectvely. By decomposng these three types of paths by ther last passage through 0, we arrve at the equatons (compare wth [5]): E = + te + ue 2, B = + (t + 2uE)B, B (+) = EB. In what follows we wll also make use of brdges where the frst step s ether of type 0 or. Clearly, ther generatng functon B s gven by B = tb + ub (+) = B(t + ue). When Motzkn brdges are not constraned to stay non-negatve, they can be seen as am arbtrary arrangement of a gven number of steps +, 0,. It s then possble to obtan explct expressons for ( ) l + 2m B l,m = [t l u m ]B(t, u) =, (2.4) B (+) l,m = [tl u m ]B (+) (t, u) = l, m, m ( l + 2m + l, m, m + B l,m = [t l u m ]B(t, u) = B l,m + B (+) l,m = l + m l + 2m ), (2.5) ( ) l + 2m. (2.6) l, m, m Usng the above, we can now fnally compute relatons for generatng functons of proper classes of mobles. We defne the followng seres, where t corresponds to the number of whte vertces, z to the number of edges, and x,, to the number of black vertces of degree :

6 6 G. Collet, M. rmota and L.. Klausner L(t, z, x, x 2,...) s the seres countng rooted mobles that are rooted at a black vertex and where an addtonal edge s attached to the black vertex. R(t, z, x, x 2,...) s the seres countng rooted mobles that are rooted at a whte vertex and where an addtonal edge s attached to the root vertex. Smlarly to the above we obtan the followng equatons for the generatng functons of mobles and rooted maps. Proposton 2.3. of the equaton Let L = L(t, z, x, x 2,...) and R = R(t, z, x, x 2,...) be the solutons L = z l,m x 2m+l+ B l,m L l R m, R = tz + z l,m and let T = T (t, z, x, x 2,...) be gven by x l+2m+2 B (+) l,m Ll R m+, (2.7) T = + l,m x 2m+l B l,m L l R m, (2.8) where the numbers B l,m, B (+) l,m, and B l,m are gven by (2.4) (2.6). Then the generatng functon M = M(t, z, x, x 2,...) of rooted maps satsfes M = R/z t + T, (2.9) t where the varable t corresponds to the number of vertces, z to the number of edges, and x,, to the number of faces of valency. Proof. The system (2.7) s just a rephrasement of the recursve structure of rooted mobles. Note that the numbers B l,m and B (+) l,m are used to count the number of ways to crcumscrbe a specfc black vertex and consderng whte vertces, black vertces and legs as steps, 0 and +. The generatng functon T gven n (2.8) s then the generatng functon of rooted mobles where the root vertex s black. Fnally, the equaton (2.9) follows from Theorem 2. snce R/z t corresponds to rooted mobles wth at least one black vertex where the root vertex s whte and T corresponds to rooted mobles where the root vertex s black. 3. Asymptotc Enumeraton In ths secton we prove the asymptotc expanson (.). It turns out that t s much easer to start wth bpartte maps. Actually, the bpartte case has already been treated by Bender and Canfeld [2]. However, we apply a slghtly dfferent approach, whch wll then be extended to cover the general case as well the central lmt theorem. 3.. Bpartte Maps Let be a non-empty subset of even postve ntegers dfferent from {2}. Then by Proposton 2.2 the countng problem reduces to the dscusson of the solutons R =

7 Planar Maps wth Prescrbed egrees 7 R (t, z) of the functonal equaton R = tz + z ( ) 2 R (3.) 2 and the generatng functon M (t, z) that satsfes the relaton M t = 2 (R /z t). (3.2) Let d = gcd{ : 2 }. Then for combnatoral reasons t follows that there only exst maps wth n edges for n that are dvsble by d. Ths s reflected by the fact that the equaton (3.) can we rewrtten n the form R = t + ( ) 2 z /d R, (3.3) 2 where we have substtuted R (t, z) = z R(t, z d ). (Recall that we fnally work wth R /z.) Lemma 3.. There exsts an analytc functon ρ(t) wth ρ() > 0 and ρ () 0 that s defned n a neghbourhood of t =, and there exst analytc functons g(t, z), h(t, z) wth h(, ρ()) > 0 that are defned n a neghbourhood of t = and z = ρ() such that the unque soluton R = R (t, z) of the equaton (3.) that s analytc at z = 0 and t = 0 can be represented as R = g(t, z) h(t, z) z ρ(t). (3.4) Furthermore, the values z = ρ(t)e(2πj/d), j {0,,..., d }, are the only sngulartes of the functon z R (t, z) on the dsc z ρ(t), and there exsts an analytc contnuaton of R to the range z < ρ(t) +η, arg(z ρ(t)e(2πj/d)) 0, j {0,,..., d }. Proof. From general theory (see [8, Theorem 2.2]) we know that an equaton of the form R = F (t, z, R), where F s a power seres wth non-negatve coeffcents, has usually a square-root sngularty of the form (3.4). We only have to assume that the functon R F (t, z, R) s nether constant nor a lnear polynomal and that there exst solutons ρ > 0, R 0 > 0 of the system of equatons R 0 = F (, ρ, R 0 ), = F R (, ρ, R 0 ) whch are nsde the range of convergence of F. Furthermore, we have to assume that F z (, ρ, R 0 ) > 0 and F RR (, ρ, R 0 ) > 0 to ensure that (3.4) holds not only for t = but n a neghbourhood of t =, and the condton F t (, ρ, R 0 ) > 0 ensures that ρ () 0. Ths means that n our case we have to deal wth the system of equatons R 0 = ρ + ρ ( ) 2 R 0, = ρ ( ) 2 R0, 2 2

8 8 G. Collet, M. rmota and L.. Klausner or just wth a sngle equaton (after elmnatng ρ) ( ) 2 ( ) R0 =. (3.5) 2 It s clear that (3.5) has a unque postve soluton f s fnte. (We also recall that all, snce 2 has to be postve.) If s nfnte, we have to be more precse. Actually, we wll show that (3.5) has a unque postve soluton R 0 < /4. Ths follows from the fact that ( ) 2 ( ) 4 2 π. Thus, f s nfnte, t follows that the power seres x H(x) = 2 ( )( 2 has radus of convergence /4 and we also have H(x) as x /4 snce each non-zero term satsfes lm ( ) x /4 ( 2 ) x 2 π, whch s unbounded for. Fnally, we set ρ = ( 2 ( ) ) 2 R. 0 It s clear that F z (ρ,, R 0 ) > 0, F RR (ρ,, R 0 ) > 0, and F t (ρ,, R 0 ) > 0. Hence we obtan the representaton (3.4) n a neghbourhood of z = ρ = ρ() and t =. Next, let us dscuss the analytc contnuaton property. If d = gcd{ : 2 } = then t follows from the equaton (3.) that the coeffcents [z n ]R (, z) are postve for n n 0 (for some n 0 ). Consequently [8, Theorem 2.2] (see also [8, Theorem 2.6]) mples that there s an analytc contnuaton to the regon z < ρ(t) + η, arg(z ρ(t)) 0. If d >, then we can frst reduce equaton (3.) to a an equaton (3.3) for the functon R that s gven by R (t, z) = z R(t, z d ). We now apply the above method to ths equaton and obtan correspondng propertes for R. Of course, these propertes drectly translate to R, and we are done. It s now relatvely easy to obtan smlar propertes for M (t, z). Lemma 3.2. The functon M = M (t, z) that s gven by (3.2) has the representaton ( M = g 2 (t, z) + h 2 (t, z) z ) 3/2 (3.6) ρ(t) n a neghbourhood of t = and z = ρ(), where the functons g 2 (t, z), h 2 (t, z) are analytc n a neghbourhood of t = and z = ρ() and we have h 2 (, ρ()) > 0. Furthermore, the values z = ρ(t)e(2πj/d), j {0,,..., d }, are the only sngulartes of the functon z M (t, z) on the dsc z ρ(t), and there exsts an analytc contnuaton of M to the range z < ρ(t) + η, arg(z ρ(t)e(2πj/d)) 0, j {0,,..., d }. Proof. Ths s a drect applcaton of [8, Lemma 2.27]. In partcular t follows that M (, z) has the sngular representaton of the form (3.6) wth a domnant sngularty ( z/ρ()) 3/2 near z = ρ(). The sngular representatons ) x

9 Planar Maps wth Prescrbed egrees 9 are of the same knd near z = ρ()e(2πj/d), j {,..., d }, and we have the analytc contnuaton property. Hence t follows by usual sngularty analyss (see for example [8, Corollary 2.5]) that there exsts a constant c > 0 such that [z n ]M (, z) c n 5/2 ρ() n, n 0 mod d, whch completes the proof of the asymptotc expanson n the bpartte case General Maps We now suppose that contans at least one odd number. It s easy to observe that n ths case we have [z n ]M (, z) > 0 for n n 0 (for some n 0 ), so we do not have to deal wth several sngulartes. By Proposton 2.3 we have to consder the system of equatons for L = L (t, z), R = R (t, z): and also the functon L = z m R = tz + z B 2m,m L 2m R m, (3.7) m T = T (t, z) = + B (+) 2m 2,m L 2m 2 R m+, (3.8) m B 2m,m L 2m R m. Lemma 3.3. There exsts an analytc functon ρ(t) wth ρ() > 0 and ρ () 0 that s defned n a neghbourhood of t =, and there exst analytc functons g(t, z), h(t, z) wth h(, ρ()) > 0 that are defned n a neghbourhood of t = and z = ρ() such that R /z t + T = g(t, z) h(t, z) z ρ(t). (3.9) Furthermore, the value z = ρ(t) s the only sngularty of the functon z R /z t+t on the dsc z ρ(t), and there exsts an analytc contnuaton of R to the range z < ρ(t) + η, arg(z ρ(t)) 0. Proof. The system of equatons (3.7) (3.8) whch we wrte n short-hand notaton as L = F (t, z, L, R ), R = G(t, z, L, R ) s a strongly connected system of two equatons of the form where F and G are power seres wth non-negatve coeffcents. It s known that such a system of equatons has n prncple the same analytc propertes (ncludng the sngular behavour of ts solutons) as a sngle equaton, see [8, Theorem 2.33]; however, we have to be sure that the regons of convergence of F and G are large enough. In partcular, f s fnte, then we have a postve algebrac system and we are done, see []. In the nfnte case we have to argue n a dfferent way. Frst of all, t s clear from the explct solutons of E = E(t, u) = (( t ( t) 2 4u)/(2u) and B = B(t, u) = / ( t) 2 4u that F and G (and all ther dervatves wth respect to L and R ) are certanly convergent f 2 L L R <. On the other hand, t follows smlarly to the bpartte case that the dervatves of F and G are dvergent f

10 0 G. Collet, M. rmota and L.. Klausner L > 0, R > 0, and 2L L 2 + 2R =. To see ths we consder the functon B(t/s, us 2 ) = 2t/s t 2 /s 2 4uw 2 = B l,m s 2m+l t l u m = l,m s m B 2m,m t 2m u m. By sngularty analyss t follows (for t, u > 0) that B 2m,m t 2m u m c /2 h(t, u), m where c > 0 and h = h(t, u) > 0 satsfes the equaton 2t/h t 2 /h 2 4uh 2 = 0. Smlarly, we can consder dervatves of F whch correspond, for example, to sums of the form B 2m,m mt 2m u m c /2 h(t, u). m In partcular, f h(t, u) = (whch s the case f 2t t 2 4u = ), then ths term dverges for. Thus, the dervatves of F and G dverge f L > 0, R > 0, and 2L L 2 + 2R =. In order to determne the sngularty of the system L = F (t, z, L, R ), R = G(t, z, L, R ) we have to fnd postve solutons of L 0, R 0, ρ of the system L 0 = F (, ρ, L 0, R 0 ), R 0 = G(, ρ, L 0, R 0 ), = G L F R F L + G R. (3.0) We do ths n the followng way. Startng wth ρ = 0, we ncrease ρ and solve the frst two equatons to get L 0 = L 0 (ρ), R 0 = R 0 (ρ) tll the thrd equaton s satsfed. (Note that for ρ = 0, the rght-hand sde s 0 and, thus, smaller than.) As long as the rght-hand sde of the thrd equaton s smaller than, t follows from the mplct functon theorem that there s a local analytc contnuaton of the solutons L 0 = L 0 (ρ), R 0 = R 0 (ρ). Furthermore, snce L 0 > 0 and R 0 > 0, we have to be n the regon of convergence of the dervatves of F and G, that s, 2L 0 L R 0 <. From ths t also follows that the solutons L 0 = L 0 (ρ), R 0 = R 0 (ρ) naturally extend to a pont where the rght-hand sde of the thrd equaton equals, so that the above system has a soluton (, ρ, L 0, R 0 ). Of course, at ths pont the dervatves of F and G have to be fnte, whch mples that (, ρ, L 0, R 0 ) les nsde the regon of convergence of F and G. Ths fnally shows that all assumptons of [8, Theorem 2.33] are satsfed. Thus, sngular representaton of type (3.9) and the analytc contnuaton property follow for the functons L = L (t, z), R = R (t, z). Hence, the same knd of propertes follows for T = T (t, z) and consequently also for R /z t + T. Lemma 3.3 shows that we are precsely n the same stuaton as n the bpartte case (actually, t s slghtly easer snce there s only one sngularty on the crcle z = ρ(t)). Hence we mmedately get the same property for M as stated n Lemma 3.2 and consequently the proposed asymptotc expanson (.).

11 Planar Maps wth Prescrbed egrees 4. Central Lmt Theorem for Bpartte Maps Based on ths prevous result, we now extend our analyss to obtan a central lmt theorem. Actually, ths s mmedate f the set s fnte, whereas the nfnte case needs much more care. Let be a non-empty subset of even postve ntegers dfferent from {2}. Then by Proposton 2.2 the generatng functons R = R (t, z, (x 2 ) 2 ) and M = M (t, z, (x 2 ) 2 ) satsfy the equatons R = tz + z 2 x 2 ( 2 ) R and M t = 2 (R /z t). (4.) If s fnte, then the number of varables s fnte, too, and we can apply [8, Theorem 2.33] to obtan a representaton of R of the form z R = g(t, z, (x 2 ) 2 ) h(t, z, (x 2 ) 2 ) ρ(t, (x 2 ) 2 ). (4.2) A proper extenson of the transfer lemma [8, Lemma 2.27] (where the varables x 2 are consdered as addtonal parameters) leads to ( ) 3/2 z M = g 2 (t, z, (x 2 ) 2 ) + h 2 (t, z, (x 2 ) 2 ), (4.3) ρ(t, (x 2 ) 2 ) and fnally [8, Theorem 2.25] mples a multvarate central lmt theorem for the random vector X n = (X n (2) ) 2 of the proposed form. Thus, we just have to concentrate on the nfnte case. Actually, we proceed there n a smlar way; however, we have to take care of nfntely many varables. There s no real problem to derve the same knd of representaton (4.2) and (4.3) f s nfnte. Everythng works n the same way as n the fnte case, we just have to assume that the varables x are unformly close to. And of course we have to use a proper noton of analytcty n nfntely many varables. We only have to apply the functonal analytc extenson of the above cted theorems that are gven n [9]. Moreover, n order to obtan a proper central lmt theorem we need a proper adapton of [9, Theorem 3]. In ths theorem we have also a sngle equaton y = F (z, (x ) I, y) for a generatng functon y = y(z, (x ) I ) that encodes the dstrbuton of a random vector (X n () ) I n the form y = ) y n (E x X() n z n, n where X n () = 0 for > cn (for some constant c > 0) whch also mples that all appearng potentally nfnte products are n fact fnte. (In our case ths s satsfed snce there s no vertex of degree larger than 2n f we have n edges.) As we can see from the proof of [9, Theorem 3], the essental part s to provde tghtness of the nvolved normalsed random vector, and tghtness can be checked wth the help of moment condtons. It s clear that asymptotcs of moments for X n () can be calculated wth the help of dervatves of F, for example EX n () = F x /(ρf z ) n + O(). Ths follows from the fact all nformaton on the asymptotc behavour of the moments s encoded n the dervatves of the sngularty I

12 2 G. Collet, M. rmota and L.. Klausner ρ(t, z, (x ) I ), and by mplct dfferentaton these dervatves relate to dervatves of F. More precsely, [9, Theorem 3] says that the followng condtons are suffcent to deduce tghtness of the normalsed random vector: F x <, Fyx 2 <, F xx <, I I F zx = o(), F zxx = o(), F yyx = o(), F yyxx = o(), F zzx = O(), F zyx = O(), F zyyx = O(), F yyyx = O(), as, where all dervatves are evaluated at (, ρ, () I, y(ρ)). The stuaton s slghtly dfferent n our case snce we have to work wth M nstead of R. However, the only real dfference between R and M s that the crtcal exponents n the sngular representatons (4.2) and (4.3) are dfferent, but the behavour of the sngularty ρ(t, (x ) I ) s precsely the same. Note that after the ntegraton step we can set t =. Now tghtness for the normalsed random vector that s encoded n the functon M follows n the same way as for R, and snce the sngularty ρ(, (x ) I ) s the same, we get precsely the same condtons as n the case of [9, Theorem 3]. Ths means that we just have to check the above condtons hold for F = F (, z, (x 2 ) 2, y) = z + z 2 I x 2 ( 2 ) y, where all dervatves are evaluated at z = ρ, x 2 =, and y = R (ρ) < /4. However, they are trvally satsfed snce ( 2 ) K y < for all K > 0 and for postve real y < /4. 5. Central Lmt Theorem for General Maps We now assume that contans at least one odd number. By Proposton 2.3 we have to consder the system of equatons L = z x B 2m,m L 2m R, m m R = tz + z x m B (+) 2m 2,m L 2m 2 R m+, for the generatng functons L = L (t, z, (x ) ) and R = R (t, z, (x ) ), the generatng functon T = T (t, z, (x ) ) = + x B 2m,m L 2m R m m and fnally the generatng functon M = M (t, z, (x ) ) that satsfes the relaton M t = R /z t + T. Agan, f s fnte, we can proceed as n the bpartte case by applyng [8, Theorem 2.33, Lemma 2.27, and Theorem 2.25] whch mples the proposed central lmt theorem. If s nfnte, we argue n a smlar way as n the bpartte case. The only dfference

13 Planar Maps wth Prescrbed egrees 3 s that we are not startng wth one equaton but wth a system of two equatons that have the (general) form L = F (t, z, (x ), L, R), R = G(t, z, (x ), L, R). Nevertheless, t s possble to reduce two equatons of ths form to a sngle one. The proof of [8, Theorem 2.33] shows that there are no analytc problems snce we have a postve and strongly connected system. We use the frst equaton to obtan an mplct functon soluton f = f(t, z, (x ), r) that satsfes f = F (t, z, (x ), f, r). Then we substtute f for L n the second equaton and arrve at a sngle functonal equaton R = G(t, z, (x ), f(t, z, (x ), R), R) for R = R (t, z, (x ) ). Note that the proof of [8, Theorem 2.33] assures that f s analytc although L and R get sngular. Hence by settng H(t, z, (x ), r) = G(t, z, (x ), f(t, z, (x ), r), r) we obtan a sngle equaton R = H(t, z, (x ), R) for R = R and we can apply the same method as n the bpartte case. Of course, the calculatons get more nvolved. For example, we have where H x = G x + G LF x F L, F L = ρ ( 2m )B 2m,m L 2m 2 0 R0 m, m F x = ρ B 2m,m L 2m 0 R0 m, m G L = ρ ( 2m 2)B (+) 2m 2,m L 2m 3 0 R0 m, m G x = ρ m B (+) 2m 2,m L 2m 2 0 R m 0. From the proof of Lemma 3.3 we already know that 2L 0 L R 0 <, whch mples that m K ( 2m )B 2m,m L 2m 2 0 R0 m < m for all K > 0. Furthermore, we have F L < and G R <. Hence t follows that H x <. In the same way, we can handle the other condtons whch completes the proof of Theorem Weghted Maps In order to cope wth weghted maps we just have to substtute x = q. Then the coeffcent M q,n := [z n ]M(, z, q) s just the weghted sum of all maps wth n edges.

14 4 G. Collet, M. rmota and L.. Klausner Actually, under the condton that q = Θ( α ) wth α 3 2 t follows that M q,n c n 5/2 γ n for some postve constants c, γ. The reason s that we can show (almost n the same way as n Lemma 3.3) that there exst solutons L 0 > 0, R 0 > 0, ρ > 0 wth 2L 0 L R 0 < of the correspondng system (3.0). The smple reason s that the seres α dverges for α 3 2. Ths proves that we have a square-root sngularty for the functons L and R etc. The central lmt theorem can be proved also n the same way as above; we just have to replace x by x q. We leave the detals to the reader Mean and Covarance Recall that we have used [8, Theorem 2.33, Lemma 2.27, and Theorem 2.25] to prove the central lmt theorem. Actually ths method provdes us also expressons for the constants µ d (d ) and a covarance matrx Σ = (σ d,d 2 ) d,d 2 for the lmtng Gaussan random varable Z. In the bpartte case we get ( ) 2j µ 2j = ρ R j 0 and s 2j,2k = µ 2j δ j,k µ 2j µ 2k ( + (j )(k )c) j for some postve constant c and where ρ and R 0 are defned n the proof of Lemma 3.3. In prncple the same procedure also works for non-bpartte maps, however, the expressons are much more nvolved. Therefore we only state the results for the basc case = N. The correspondng constants µ d and σ d,d 2 are gven by and m 0 µ d = A d + 2A d σ d,d 2 = µ d δ d,d µ d µ d (d )(d 2 )A d A d2 + 9 ( ) (Ad + (d )A d )A d2 + (A d2 + (d 2 )A d2 )A d 2 ( ) ( 39Ad + (d )µ d 39Ad2 + (d 2 )µ d2 ) 8 2 (µ d A d2 + µ d2 A d ) + 2 ((d 2 )µ d µ d2 + (d )µ d2 µ d ), where A d = ( ) d 6 d 4 m and A d = ( ) d d 2m, m, m 6 d 4 m. d 2m 2, m, m + m 0 6. Maps of Hgher Genus The bjecton used n Secton 2 reles solely on the orentablty of the surface on whch the maps are embedded. Therefore t can easly be extended to maps of hgher genus, Gregory Mermont has ponted out to the second author a very nce probablstc nterpretaton of these representatons n terms of monotype Galton Watson trees and nfnte sequences of Gaussan random varables.

15 Planar Maps wth Prescrbed egrees 5. e. embedded on an orentable surface of genus g Z>0 (whle planar maps correspond to maps of genus 0). The man dfference les n the fact that the correspondng mobles are no longer trees but rather one-faced maps of hgher genus, whle the other propertes stll hold. However, due to the appearance of cycles n the underlyng structure of mobles, another dffculty arses. Indeed, n the orgnal bjecton, vertces and edges n mobles could carry labels (related to the geodesc dstance n the orgnal map), subject to local constrants. In our settng, the legs actually encode the local varatons of these labels, whch are thus mplct. Local constrants on labels are naturally translated nto local constrants on the number of legs. But the labels have to reman consstent along each cycle of the mobles, whch gves rse to non-local constrants on the repartton of legs. In order to deal wth these addtonal constrants, and to be able to control the degrees of the vertces at the same tme, we wll now use a hybrd formulaton of mobles, carryng both labels and legs. As before, we wll focus on the smpler case of mobles comng from bpartte maps and establsh the drect generalzaton of Theorem.: Theorem 6.. Suppose that {2} s an arbtrary set of postve even ntegers, let M (g) be the class of rooted bpartte maps of genus g wth the property that all vertex degrees are n and let M (g),n denote the number of maps n M(g) wth n edges. Furthermore, set d = gcd{ : 2 }. Then there exst postve constants c (g) and ρ(g) wth M (g),n c(g) n5(g )/2 (ρ (g) ) n, n 0 mod d. (6.) Furthermore, let X n (d) denote the random varable countng vertces of degree d ( ) n maps n M (g). Then E(X(d) n ) µ d n for some constant µ d > 0 and for n 0 mod d, and the (possbly nfnte) random vector X n = (X n (d) ) d (n 0 mod d) satsfes a central lmt theorem. Theorem. can be easly recovered for planar bpartte maps, by settng g = 0. The man dfference les n the exponent 5(g )/2, whch appears to be also unversal for rooted maps of genus g. Hence Theorem 6. s expected to hold for wthout any restrcton. 6.. g-mobles Gven g Z 0, a g-moble s a one-faced map of genus g embedded on the g-torus such that there are two knds of vertces (black and whte), edges only occur as black black edges or black whte edges, and black vertces addtonally have so-called legs attached to them (whch are not consdered edges), whose number equals the number of whte neghbour vertces. Furthermore, for each cycle c of the g-moble, let n, n and n respectvely be the numbers of whte vertces on c, of legs danglng to the left of c and of whte neghbours to the left of c. One has the followng constrant (see Fgure 6.): n = n + n (6.2)

16 6 G. Collet, M. rmota and L.. Klausner n = 7 n = 4 n = 3 n n n = 0 Fgure. An orented cycle n a g-moble and the constrant on ts left (coloured area). Notce that a smlar constrant holds on ts rght, but s necessarly satsfed thanks to the propertes of a g-moble. Fgure 2. A -moble on the torus and ts scheme. The degree of a black vertex s the number of half-edges plus the number of legs that are attached to t. A bpartte g-moble s a g-moble wthout black black edges. A g-moble s called rooted f an edge s dstngushed and orented. Notce that a 0-moble s smply a moble as descrbed n efnton 2. Actually there s a drect analogue of Theorem 2.: g-mobles are n bjecton to ponted maps of genus g (wth precsely the same propertes as stated n Theorem 2.. Ths generalzaton of the bjecton to hgher genus was frst gven n [6] for quadrangulatons and [4] for Euleran maps (from whch we wll explot many deas n ths secton). g-mobles are not as easly decomposed as planar mobles, due to the exstence of cycles. However, they stll exhbt a rather smple structure, based on scheme extracton. The g-scheme (or smply the scheme) of a g-moble s what remans when we apply the followng operatons (see Fgure 2): frst remove all legs, then remove teratvely all vertces of degree and fnally replace any maxmal path of degree-2-vertces by a sngle edge. Once these operatons are performed, the remanng object s stll a one-faced map of genus g, wth black and whte vertces (whte whte edges can now occur), where the vertces have mnmum degree 3. The mportant property s that there s a fnte number of schemes of a gven genus Bpartte g-moble Countng A g-moble can now be unquely decomposed as a scheme where each edge s substtuted by a sequence of elementary cells. By defnton of a g-moble, one needs to track the ncrement of each cell to ensure that the overall cycle constrants are satsfed. An elementary cell s a half-edge connected to a black vertex tself connected to a whte vertex wth a danglng half-edge. The whte vertex has a sequence of black-rooted mobles attached on each sde. For an elementary cell of ncrement, the black vertex

17 Planar Maps wth Prescrbed egrees 7 has k 0 whte-rooted mobles and k legs on ts left, l 0 whte-rooted mobles and l + legs on ts rght, and ts degree s 2(k + l + 2). The generatng seres P := P (t, z, R, (x 2 ), s) of a cell, where s marks the ncrement, s: P (t, z, R, (x 2 ), s) = R2 t k,l 0 l+ x 2(k+l+2) R k+l = k ( )( ) 2k + + 2l + s. k l ependng on the edge end colours, there mght be an addtonal black or whte vertex nserted at the end of the sequence of elementary cells. Ths s reflected by an extra factor n the generatng seres S e := S e (t, z, (x 2 ), s): S (u,v) (t, z, (x 2 ), s) = P stz R ( 2 P ) R 2 stz P f (u, v) = (, ) or (, ), f (u, v) = (, ), f (u, v) = (, ). We can now express the generatng seres Q S := Q S (t, z, (x 2 )) of rooted bpartte g-mobles wth scheme S (for the defnton of a labellng see Appendx B): Q S (t, z, R, (x 2 ); s) = 2 z ( ) [ C R z 2 E z E t V [s ncr(e) ]S (e,e tz +) (l c) labellng e E deg(v) ( ) 2k + l ck+ l ck + R k x 2(deg(v)+ k ). (6.3) v V,..., deg(v) 0 k= k Proposton 6.2. The generatng seres M (g) (g) := M (t, z, (x 2)) for the famly of rooted bpartte maps of genus g, where the vertex degrees belong to, satsfes the relaton: M (g) = 2 Q S (t, z, (x 2 {2 } )). (6.4) t z S scheme of genus g Proof. Ths follows drectly from the bjectons between g-mobles and maps of genus g and equaton (6.3) Asymptotcs of g-mobles We proceed smlarly to [4]. However, for the sake of brevty we wll not work out all techncal detals. For example, we wll take only care of the (local) sngular expanson and restrct ourselves to the case d =. Frst we need proper expanson of the coeffcents of ( P ). Lemma 6.3. We have, as, [s ] P (t, z, R(t, z), (x 2 ); s) = C sgn (t, z)α sgn (t, z) + ( α ± (t, z) δ),

18 8 G. Collet, M. rmota and L.. Klausner where α ± (t, z) c ( z/ρ) /4 and C ± (t, z) c 2 ( z/ρ) /4 for some postve constants c, c 2 and ρ = ρ(t, (x 2 )) Proof. Wth the help of (3.5) t s easy to check that the followng three relatons hold when we evaluate at t close to, x 2, 2, close to, R = R 0 (t, (x 2 )), z = ρ = ρ(t, (x 2 )), and s = : P =, P s = 0, P ss 0, P R 0. Thus we have locally two solutons s = α,2 (t, z) of the equaton ( P (t, z, R(t, z), (x 2 ), s) = that are of the form α,2 (t, z) = c ( z/ρ) /4 + O ( z/ρ) /4). For s wth α (t, z) < s < α 2 (t, z) we also have P < and consequently by Cauchy ntegraton appled to the Laurent seres s P [s ] P (t, z, R(t, z), (x 2 ); s) = ds 2π P (t, z, R(t, z), (x 2 ); s) s +, s =s 0 where α (t, z) < s 0 < α 2 (t, z). Clearly s = α,2 (t, z) are polar sngulartes of /( P ). Thus, f we shft the ntegral to a crcle s = α 2 (t, z) + δ (for some δ > 0) and by collectng the resdue at s = α 2 (t, z), we get, as +, [s ] P (t, z, R(t, z), (x 2 ); s) = C (t, z)α 2 (r, z) + ( α 2 (t, z) + δ), where C 2 (t, z) = /P s (t, z, R(t, z), (x 2 )), α 2 (t, z)) = c 2 ( z/ρ) /4 + O(). Smlarly we obtan the correspondng expanson for. Thus, settng α + (t, z) = α 2 (t, z), α (t, z) = α (t, z), C + (t, z) = C 2 (t, z), and C (t, z) = C (t, z) completes the proof of the lemma. Next we observe that for any d Z>0, and any,..., d Z such that k = d we have d ( ) x 2k + k 2 k +2d R k = ( )( ) 2m + 2d m + d x 2m+2d R m k m + d d,..., d 0 k= m 0 = ( )( ) 2m m x 2m R m d. m d m d Wth the help of these prelmnares we can determne the sngular structure of the generatng functons Q S (t, z, (x 2 )) related to a scheme S. For the sake of brevty we wll only dscuss labelled schemes where all vertces are whte. Thus all edges are whte whte and labels are carred by the whte vertces. Wthout loss of generalty, one can assume that the mnmal label s 0 (by shftng all labels, as only the dfferences matter). In order to handle the sums over all labellngs, defne λ : V [ 0, M ] (where M = card({labels of V }) ), the relatve order of the labels. Labels can then be rewrtten as: λ(v) v V, l v = δ, wth δ Z>0. =

19 Planar Maps wth Prescrbed egrees 9 For a gven edge e = (u, v) of the labelled scheme, one substtutes a sequence of cells of overall ncrement (e) = l v l u + = j A e,jδ j +, where A e,j = λ(u)<j λ(v). Hence, we obtan, after rewrtng, that: Q S (t, z, (x 2 )) z z z z z z z z z z z z c 3 z z E z E t V E z E t V E z E t V E z E t V E z E t V ( R tz ( R tz ) C (l c) labellng e E ) C ( ) C ( R tz tz R 2 ( R tz ( R tz δ,...,δ M >0 e E [s (e) ]S (e,e +) ) E tz R 2 [s j Ae,jδ j ] δ,...,δ M >0 e E ) C ( ) E tzc(t, z) R 2 ) C ( ) E M tzc(t, z) R 2 E t V E C(t, z) E ( α + (t, z)) M ( z ) ( E M)/4 E ρ Q S (t, z, (x 2 ) 2 ) c 3 z z E ( ) P C + (t, z)α + (t, z) j Ae,jδj δ,...,δ M >0 j= ( ) δj α + (t, z) Ae,j j e α +(t, z) Ae,j e α +(t, z) Ae,j The man contrbuton wll then come from cubc schemes wth maxmal M,. e. where all labels are dstnct. Thus E = 6g 3, M = V = 4g 3. ( ) 5g/2+3/2 z ρ(t, (x 2 ) 2 ) Smlar asymptotcs can be derved wth more techncal computatons for the mobles where the scheme also has black vertces. Summng up over all the maxmal cubc schemes of genus g, and after an ntegraton step, we recover the expected sngular behavour M (g) (t, z, (x z 2) 2 ) c 4 z 2 E ( ) 5g/2+5/2 z ρ(t, (x 2 ) 2 ) whch corresponds to the asymptotcs gven n Theorem 6. (when we set t = and x 2 =, 2 ). The central lmt theorem follows as n the planar case by varyng x 2 around. Acknowledgements. The authors are very grateful to Mrelle Bousquet-Melou, Gullaume Chapuy, and Gregory Mermont for ther help and for several valuable remarks. e References [] Cyrl Banderer and Mchael rmota. Formulae and asymptotcs for coeffcents of algebrac functons. Combnatorcs, Probablty and Computng, 24(): 53, 205.

20 20 G. Collet, M. rmota and L.. Klausner [2] E.A. Bender and E.R. Canfeld. Enumeraton of degree restrcted maps on the sphere. In Planar graphs. Workshop held at IMACS from November 8, 99 through November 2, 99, pages 3 6. Provdence, RI: Amercan Mathematcal Socety, 993. [3] J. Boutter, P. Francesco, and E. Gutter. Planar maps as labeled mobles. Electron. J. Combn., ():Research Paper 69, 27, [4] Gullaume Chapuy. Asymptotc enumeraton of constellatons and related famles of maps on orentable surfaces. Combn. Probab. Comput., 8(4):477 56, [5] Gullaume Chapuy, Érc Fusy, Mhyun Kang, and Blyana Sholekova. A complete grammar for decomposng a famly of graphs nto 3-connected components. Electron. J. Combn., 5():Research Paper 48, 39, [6] Gullaume Chapuy, Mchel Marcus, and Glles Schaeffer. A bjecton for rooted maps on orentable surfaces. SIAM Journal on screte Mathematcs, 23(3):587 6, [7] Gwendal Collet and Érc Fusy. A smple formula for the seres of bpartte and quasbpartte maps wth boundares. screte Math. Theor. Comput. Sc., pages , 202. [8] Mchael rmota. Random trees. An nterplay between combnatorcs and probablty. Wen: Sprnger, [9] Mchael rmota, Bernhard Gttenberger, and Johannes F. Morgenbesser. Infnte systems of functonal equatons and Gaussan lmtng dstrbutons. In Proceedng of the 23rd nternatonal meetng on probablstc, combnatoral, and asymptotc methods n the analyss of algorthms (AofA 2), Montreal, Canada, June 8 22, 202, pages Nancy: The Assocaton. screte Mathematcs & Theoretcal Computer Scence (MTCS), 202. [0] Mchael rmota and Konstantnos Panagotou. A central lmt theorem for the number of degree-k vertces n random maps. Algorthmca, 66(4):74 76, 203. [] Jean-Franços Marckert and Grégory Mermont. Invarance prncples for random bpartte planar maps. The Annals of Probablty, 35(5): , [2] Grégory Mermont and Mathlde Wel. Radus and profle of random planar maps wth faces of arbtrary degrees. Electronc Journal of Probablty, 3(4):79 06, [3] W.T. Tutte. A census of planar maps. Can. J. Math., 5:249 27, 963. Appendx A. Schemes of g-mobles A g-moble s not as easly decomposed as a planar one, due to the exstence of cycles. However, t stll exhbts a rather smple structure, based on scheme extracton. The g-scheme (or smply the scheme) of a g-moble s what remans when we apply the followng operatons (see Fgure 2): frst remove all legs, then remove teratvely all vertces of degree and fnally replace any maxmal path of degree-2-vertces by a sngle edge. Once these operatons are performed, the remanng object s stll a one-faced map of genus g, wth black and whte vertces (whte whte edges can now occur), where the vertces have mnmum degree 3. To count g-mobles, one key ngredent s the fact that there s only a fnte number of schemes of a gven genus. Indeed, let d be the number of vertces of degree n a g-scheme: 2)d = k 3( d 2 d = 2(#edges #vertces) = 4g 2. k 3 k 3 The number of vertces (respectvely edges) s then bounded by 4g 2 (respectvely 6g 3), where ths bound s reached for cubc schemes (see an example n Fgure 2).

21 Planar Maps wth Prescrbed egrees Fgure 3. The varatons of labels around a black vertex and along an orented cycle. To recover a proper g-moble from a gven g-scheme, one would have to nsert a sutable planar moble nto each corner of the scheme and to substtute each edge wth some knd of path of planar mobles. Unfortunately, ths cannot be done ndependently: Around each black vertex, the total number of legs n every corner must equal the number of whte neghbours, and around each cycle, (6.2) must hold. In order to make these constrants more transparent, we wll equp schemes wth labels on whte vertces and black corners. Now, when tryng to reconstruct a g-moble from a scheme, one has to ensure that the local varatons are consstent wth the global labellng. To be precse, the label varatons are encoded as follows (see Fgure 3): Around a black vertex of degree d, let (l,..., l d ) be the labels of ts corners read n clockwse order: + f there s a leg between the two correspondng corners, for all, l + l = 0 f there s a black neghbour, f there s a whte neghbour. Along the left sde of an orented cycle, the label decreases by after a whte vertex or when encounterng a whte neghbour and ncreases by when encounterng a leg. The above statements hold for general as well as bpartte mobles. In the followng, we wll only consder bpartte mobles, as they are much easer to decompose. Appendx B. Reconstructon of Bpartte Maps of Genus g In the followng, t wll be convenent to work wth rooted schemes. One can then defne a canoncal labellng and orentaton for each edge of a rooted scheme. An edge e now has an orgn e and an endpont e +. The k corners around a vertex of degree k are clockwsely ordered and denoted by c,..., c k. Gven a scheme S, let V, V, C, C be respectvely the sets of whte and black vertces and of whte and black corners. A labelled scheme (S, (l c ) c V C ) s a par consstng of a scheme S and a labellng on whte vertces and black corners, wth l c 0 for all c. Labellngs are consdered up to translaton, as they wll not affect local varatons. For an edge of S e E S, we assocate a label to each extremty l e, l e+. If an extremty s

22 22 G. Collet, M. rmota and L.. Klausner l + l k j j l + l l k + k j k j + j l + l l k + k j k j + k j + Fgure 4. Steps () (3) of the algorthm. a whte vertex of label l, ts label s l. If the extremty s a black vertex, ts label s the same as the next clockwse corner of the black vertex. Let a doubly-rooted planar moble be a rooted (on a black or whte vertex) planar moble wth a secondary root (also black or whte). These two roots are the extremtes of a path (v,..., v k ). The ncrement of the doubly-rooted moble s then defned as n n n, whch s not necessarly 0, as the path s not a cycle. Smlarly as n [4], we present a non-determnstc algorthm to reconstruct a g-moble: Algorthm. () Choose a labelled g-scheme (S, (l c ) c V C ). (2) For all v V, choose a sequence of non-negatve ntegers ( k ) k deg(v), then attach k planar mobles and k + l ck+ l ck + legs to c k (the k th corner of v). (3) For all e S, replace e by a doubly-rooted moble of ncrement ncr(e) = l e+ { + f e s whte, l e + f e s black. (4) On each whte corner of S, nsert a planar moble. (5) stngush and orent an edge as the root. Proposton B.. Gven g > 0, the algorthm generates each rooted bpartte g-moble whose scheme has k edges n exactly 2k ways. Proof. One can easly see that the obtaned object s ndeed bpartte. Attachng planar mobles and legs added at step (2) n a corner c k creates new corners, such that: The frst carres the same label l ck as c k, and the last carres the label l ck + ( k + l ck+ l ck + ) k = l ck+ +. The next corner should then be labelled (l ck+ + ) = l ck+, due to the next whte neghbour, whch s precsely what we want. In the same fashon, at step (3), a smple countng shows that each edge s replaced by a path such that the labels along t evolve accordng to the scheme labellng. We thus obtan a well-formed rooted bpartte g-moble, wth a secondary root on ts scheme. Snce the frst root destroys all symmetres, there are exactly 2k choces for the secondary root whch would gve the same rooted g-moble.