Vertex Degrees in Planar Maps

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1 Vertex Degrees n Planar Maps Gwendal COLLET, Mchael DRMOTA and Lukas Danel KLAUSNER Insttute of Dscrete Matheatcs and Geoetry Techncal Unversty of Venna 1040 Venna Austra February 19, 2016 Abstract We prove a general ult-densonal central lt theore for the expected nuber of vertces of a gven degree n the faly of planar aps whose vertex degrees are restrcted to an arbtrary (fnte or nfnte) set of postve ntegers D. We also dscuss the possble extenson to aps of hgher genus. 1 Introducton and Results In ths paper we study statstcal propertes of planar aps, whch are connected planar graphs, possbly wth loops and ultple edges, together wth an ebeddng n the plane. Such objects are frequently used to descrbe topologcal features of geoetrc arrangeents n two or three spatal densons. Thus, the knowledge of the structure and of propertes of typcal objects ay turn out to be very useful n the analyss of partcular algorths that operate on planar aps. We say that ap 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. Slarly, we call a planar ap ponted f just a vertex v s dstngushed. However, we have to be really careful wth the odel. In rooted aps the root edge destroys potental syetres, whch s not the case f we consder ponted aps. The enueraton of rooted aps s a classcal subject, ntated by Tutte n the 1960 s, see [11]. Aong any other results, Tutte coputed the nuber M n of rooted aps wth n edges, provng the forula M n = whch drectly provdes the asyptotc forula 2(2n)! (n + 2)!n! 3n M n 2 π n 5/2 12 n. Key words: Analytc cobnatorcs, planar aps, central lt theore Matheatcs Subject Classfcaton: Bjectve and analytc cobnatorcs. Funded by FWF SFB F50 Algorthc and Enueratve Cobnatorcs 1

2 We are anly nterested n planar aps wth degree restrctons. Actually, t turns out that ths knd of asyptotc expanson s qute unversal. Furtherore, there s always a (very general) central lt theore for the nuber of vertces of gven degree. Theore 1. Suppose that D s an arbtrary set of postve ntegers but not a subset of {1, 2}, let M D be the class of planar rooted aps wth the property that all vertex degrees are n D and let M D,n denote the nuber of aps n M D wth n edges. Furtherore, f D contans only even nubers, then set d = gcd{ : 2 D}; set d = 1 otherwse. Then there exst postve constants c D and ρ D wth M D,n c D n 5/2 ρ n D, n 0 od d. (1) Furtherore, let X n (d) denote the rando varable countng vertces of degree d ( D) n aps n M D. Then E(X n (d) ) µ d n for soe constant µ d > 0 and for n 0 od d, and the (possbly nfnte) rando vector X n = (X n (d) ) d D (n 0 od d) satsfes a central lt theore, that s, 1 n (X n E(X n )), n 0 od d, (2) converges weakly to a centered Gaussan rando varable Z (n l 2 ). Note that aps where all vertex degrees are 1 or 2 are very easy to characterze and are not really of nterest, and that actually, ther asyptotc propertes are dfferent fro the general case. It s therefore natural to assue that D s not a subset of {1, 2}. Snce we can equvalently consder dual aps, ths knd of proble s the sae as consderng planar aps wth restrctons on the face valences. Ths eans that the sae results hold f we replace vertex degree by face valency. For exaple, f we assue that all face valences equal 4, then we just consder planar quadrangulatons (whch have also been studed by Tutte [11]). In fact, our proofs wll refer just to face valences. Theore 1 goes far beyond known results. Only n the Euleran case where all vertex degrees are even there are soe general results. Frst, the asyptotc expanson (1) s known for Euleran aps by Bender and Canfeld [2]. Furtherore, a central lt theore of the for (2) s known for all Euleran aps (wthouth degree restrctons) [9]. However, n the non-euleran case there are alost no results of ths knd; there s only a one-densonal central lt theore for X n (d) for all planar aps [10]. Secton 2 ntroduces planar obles whch, beng n bjecton wth ponted planar aps, wll reduce our analyss to spler objects wth a tree structure. Ther asyptotc behavour s derved n Secton 3, frst for the spler case of bpartte aps (.e., when D contans only even ntegers), then for fales of aps wthout constrants on D. Secton 4 s devoted to the proof of the central lt theore usng analytc tools fro [8, 9]. Fnally, n Secton 5 we dscuss the cobnatorcs of aps of hgher genus. The expressons we obtan are uch ore nvolved than n the planar case, but t s expected to lead to slar analytc results. 2 Mobles Instead of nvestgatng planar aps theselves, we wll follow the prncple presented n [5], whereby ponted planar aps are bjectvely related to a certan class of trees called obles. (Ther verson of obles dffer fro the defnton orgnally gven n [3]; the equvalence of the two defntons s not shown explctly n [5], but [7] gves a straghtforward proof.) 2

3 Defnton 1. A oble s a planar tree that s, a ap 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 the (whch are not consdered edges), whose nuber equals the nuber of whte neghbor vertces. A bpartte oble s a oble wthout black black edges. The degree of a black vertex s the nuber of half-edges plus the nuber of legs that are attached to t. A oble s called rooted f an edge s dstngushed and orented. The essental observaton s that obles are n bjecton to ponted planar aps. Theore 2. There s a bjecton between obles that contan at least one black vertex and ponted planar aps, where whte vertces n the oble correspond to non-ponted vertces n the equvalent planar ap, black vertces correspond to faces of the ap, and the degrees of the black vertces correspond to the face valences. Ths bjecton nduces a bjecton on the edge sets so that the nuber of edges s the sae. (Only the ponted vertex of the ap has no counterpart.) Slarly, rooted obles that contan at least one black vertex are n bjecton to rooted and vertexponted planar aps. Fnally, bpartte obles wth at least two vertces correspond to bpartte aps wth at least two vertces, n the unrooted as well as n the rooted case. Proof. For the proof of the bjecton between obles and ponted aps we refer to [7], where the bpartte case s also dscussed. It just reans to note that the nduced bjecton on the edges can be drectly used to transfer the root edge together wth ts drecton. 2.1 Bpartte Moble Countng We start wth bpartte obles snce they are ore easy to count, n partcular f we consder rooted bpartte obles, see [7]. Proposton 1. Let R = R(t, z, x 1, x 2,...) be the soluton of the equaton R = tz + z ( ) 2 1 x 2 R. (3) 1 Then the generatng functon M = M(t, z, x 1, x 2,...) of bpartte rooted aps satsfes M t = 2 (R/z t), (4) where the varable t corresponds to the nuber of vertces, z to the nuber of edges, and x 2, 1, to the nuber of faces of valency 2. Proof. See Appendx. 2.2 General Moble Countng We now proceed to develop a echans for general oble countng that s adapted fro [5]. For ths, we wll requre Motzkn paths. 3

4 Defnton 2. A Motzkn path s a path startng at 0 and gong rghtwards for a nuber of steps; the steps are ether dagonally upwards (+1), straght (0) or dagonally downwards ( 1). A Motzkn brdge s a Motzkn path fro 0 to 0. A Motzkn excurson s a Motzkn brdge whch stays non-negatve. We defne generatng functons n the varable t and u, whch count the nuber of steps of type 0 and 1, respectvely. (Explctly countng steps of type 1 s then unnecessary, of course.) The ordnary generatng functons of Motzkn brdges, Motzkn excursons, and Motzkn paths fro 0 to +1 shall be denoted by B(t, u), E(t, u) and B (+1) (t, u), respectvely. Contnung to follow the presentaton of [5] and decoposng these three types of paths by ther last passage through 0, we arrve at the equatons: E = 1 + te + ue 2, B = 1 + (t + 2uE)B, B (+1) = EB. In what follows we wll also ake use of brdges where the frst step s ether of type 0 or 1. Clearly, ther generatng functon B s gven by B = tb + ub (+1) = B(t + ue). When Motzkn brdges are not constraned to stay non-negatve, they can be seen as a rando arrangeent of a gven nuber of steps +1, 0, 1. It s then possble to obtan explct expressons for ( ) l + 2 B l, = [t l u ]B(t, u) =, (5) l,, ( ) B (+1) l l, = [tl u ]B (+1) (t, u) =, (6) l,, + 1 B l, = [t l u ]B(t, u) = B l 1, + B (+1) l, 1 = l + ( ) l + 2. (7) l + 2 l,, Usng the above, we can now fnally copute relatons for generatng functons of proper classes of obles. We defne the followng seres, where t corresponds to the nuber of whte vertces, z to the nuber of edges, and y, 1, to the nuber of black vertces of degree : L (t, z, y 1, y 2,...) s the seres countng rooted obles that are rooted at a black vertex and where an addtonal edge s attached to the black vertex. L (t, z, y 1, y 2,...) s the seres countng rooted obles that are rooted at a unvalent whte vertex, whch s not counted n the seres. R(t, z, y 1, y 2,...) s the seres countng rooted obles that are rooted at a whte vertes and where an addtonal edge s attached to the root vertex. Slarly to the above we obtan the followng equatons for the generatng functons of obles and rooted aps. 4

5 Proposton 2. Let L = L (t, z, y 1, y 2,...), L = L (t, z, y 1, y 2,...), and R = R(t, z, y 1, y 2,...) be the solutons of the equaton L = z l, and let T = T (t, z, y 1, y 2,...) be gven by y 2+l+1 B l, L l R, L = z y l+2+2 B (+1) l, Ll R, (8) l, R = tz, 1 L T = 1 + l, y 2+l B l, L l R, (9) where the nubers B l,, B (+1) l,, and B l, are gven by (5) (7). Then the generatng functon M = M(t, z, y 1, y 2,...) of rooted aps satsfes M = R/z t + T, (10) t where the varable t corresponds to the nuber of vertces, z to the nuber of edges, and y, 1, to the nuber of faces of valency. Proof. The syste (8) s just a rephraseent of the recursve structure of rooted obles. Note that the nubers B l, and B (+1) l, are used to count the nuber of ways to crcuscrbe a specfc black vertex and consderng whte vertces, black vertces and legs as steps 1, 0 and +1. The generatng functon T gven n (9) s then the generatng functon of rooted obles where the root vertex s black. Fnally, the equaton (10) follows fro Theore 2 snce R/z t corresponds to rooted obles wth at least one black vertex where the root vertex s whte and T corresponds to rooted obles where the root vertex s black. Reark 1. Note that Proposton 1 s a specal case of Proposton 2. We just have to restrct to the ters correspondng to l = 0 snce bpartte obles have no black black edges. In partcular, the seres for L s not needed any ore and second and thrd equaton fro (8) can be easly used to elnate L n order to recover the equaton (3). 3 Asyptotc Enueraton In ths secton we prove the asyptotc expanson (1). It turns out that t s uch easer to start wth bpartte aps. Actually, the bpartte case has been already 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 lt theore. 3.1 Bpartte aps Let D be a non-epty subset of even postve ntegers dfferent fro {2}. Then by Proposton 1 the countng proble reduces to the dscusson of the solutons R D = R D (z, t) of the functonal equaton R D = z + zt ( ) 2 1 RD (11) 2 D 5

6 and the generatng functon M D (z, t) that satsfes the relaton M D t = 2 (R D /z t). (12) Let d = gcd{ : 2 D}. Then for cobnatoral reasons t follows that there only exst aps wth n edges for n that are dvsble by d. Ths s reflected by the fact that the equaton (11) can we rewrtten n the for R = t + ( ) 2 1 z /d R, (13) 2 D where we have substtuted R D (z, t) = z R(z d, t). (Recall that we fnally work wth R D /z.) Lea 1. There exsts an analytc functon ρ(t) wth ρ(1) > 0 and ρ (1) 0 that s defned n a neghborhood of t = 1, and there exst analytc functons g(z, t), h(z, t) wth h(ρ, 1) > 0 that are defned n a neghborhood of z = ρ(1) and z = 1 such that the unque soluton R D = R D (z, t) of the equaton (11) that s analytc at z = 0 and t = 0 can be represented as R D = g(z, t) h(t, z) 1 z ρ(t). (14) Furtherore, the values z = ρ(t)e(2πj/d), j {0, 1,..., d 1}, are the only sngulartes of the functon z R D (z, t) on the dsc z ρ(t), and there exsts an analytc contnuaton of R D to the range z < ρ(t) + η, arg(z ρ(t)e(2πj/d)) 0, j {0, 1,..., d 1}. Proof. See Appendx. It s now relatvely easy to obtan slar propertes for M D (z, t). Lea 2. The functon M = M D (z, t) that s gven by (12) has the representaton ( M D = g 2 (z, t) + h 2 (t, z) 1 z ) 3/2 (15) ρ(t) n a neghborhood of z = ρ(1) and z = 1, where the functons g 2 (z, t), h 2 (z, t) are analytc n a neghborhood of z = ρ(1) and z = 1 and we have h 2 (ρ, 1) > 0. Furtherore, the values z = ρ(t)e(2πj/d), j {0, 1,..., d 1}, are the only sngulartes of the functon z M D (z, t) on the dsc z ρ(t), and there exsts an analytc contnuaton of M D to the range z < ρ(t) + η, arg(z ρ(t)e(2πj/d)) 0, j {0, 1,..., d 1}. Proof. Ths s a drect applcaton of [8, Lea 2.27]. In partcular t follows that M D (z, 1) has the sngular representaton ( M D = g 2 (z, 1) + h 2 (z, 1) 1 z ρ around z = ρ. The sngular representaton are of the sae knd around z = ρe(2πj/d), j {1,..., d 1} and we have the analytc contnuaton property. Hence t follows by usual sngularty analyss (see for exaple [8, Corollary 2.15]) that there exsts a constant c D > 0 such that ) 3/2 [z n ]M D (z, 1) c D n 5/2 ρ n, n 0 od d, whch copletes the proof of the asyptotc expanson n the bpartte case. 6

7 3.2 General Maps We now suppose that D contans at least one odd nuber. It s easy to observe that n ths case we have [z n ]M D (z, 1) > 0 for n n 0 (for soe n 0 ) so that we do not have to deals wth several sngulartes. By Proposton 2 we have to consder the syste of equatons for L,D = L,D (z, t), L,D = L,D (z, t), R D = R D (z, t): L,D = z D L,D = z D tz R D =, 1 L,D B 2 1, L 2 1,D R D, B (+1) 2 2, L 2 2,D R D, (16) and also the functon T D = T D (z, t) = 1 + D B 2, L 2,D R D. Lea 3. There exsts an analytc functon ρ(t) wth ρ(1) > 0 and ρ (1) 0 that s defned n a neghborhood of t = 1, and there exst analytc functons g(z, t), h(z, t) wth h(ρ, 1) > 0 that are defned n a neghborhood of z = ρ(1) and z = 1 such that R D /z t + T D = g(z, t) h(t, z) 1 z ρ(t). (17) Furtherore, the values z = ρ(t) s the only sngularty of the functon z R D /z t + T D on the dsc z ρ(t), and there exsts an analytc contnuaton of R D to the range z < ρ(t) + η, arg(z ρ(t)). Proof. See Appendx. Lea 3 shows that we are precsely n the sae 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 edately get the sae property for M D as stated n Lea 2 and consequently the proposed asyptotc expanson (1). 4 Central Lt Theore for bpartte aps Based on ths prevous result, we now extend oru analyss to obtan a central lt theore. Actually, ths s edate f the set D s fnte, whereas the nfnte case needs uch ore care. Let D be a non-epty subset of even postve ntegers dfferent fro {2}. Then by Proposton 1 the generatng functons R D = R D (z, t, (x 2 ) 2 D ) and M D = M D (z, t, (x 2 ) 2 D ) satsfy the equatons R D = z + zt ( ) 2 1 x 2 RD (18) 2 D and M D t = 2 (R D /z t). (19) 7

8 If D s fnte, then the nuber of varables s fnte, too, and we can apply [8, Theore 2.33] to obtan a representaton of R D of the for z R D = g(z, t, (x 2 ) 2 D ) h(t, z, (x 2 ) 2 D ) 1 ρ(t, (x 2 ) 2 D ), (20) a proper extenson of the transfer lea [8, Lea 2.27] (where the varables x 2 are consdered as addtonal paraeters) leads to ( ) 3/2 z M D = g 2 (z, t, (x 2 ) 2 D ) + h 2 (t, z, (x 2 ) 2 D ) 1, (21) ρ(t, (x 2 ) 2 D ) and fnally [8, Theore 2.25] ples a ultvarate central lt theore for the rando vector X n = (X n (2) ) 2 D of the proposed for. Thus, we just have to concentrate on the nfnte case. Actually, we proceed there n a slar way, however, we have to take care of nfntely any varables. There s no real proble to derve the sae knd of representaton (20) and (21) f D s nfnte. Everythng works n the sae way as n the fnte case, we just have to assue that the varables x are unforly bounded. And of course we have to use a proper noton of analytcty n nfntely any varables. We only have to apply the functonal analytc extenson of the above cted theores that are gven n [9]. Moreover, n order to obtan a proper central lt theore we need a proper adapton of [9, Theore 3]. In ths theore 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 rando vector (X n () ) I n the for ) y = n y n ( E I where X n () = 0 for > cn (for soe constant c > 0) whch also ples 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 n f we have n edges.) As we see fro the proof of [9, Theore 3], the essental part s to provde tghtness of the nvolved noralzed rando vector, and tghtness can be checked wth the help of oent condtons. It s clear that asyptotcs of oents for X n () can be calculated wth the help of dervatves of F, for exaple EX n () = F x /(ρf z ) n + O(1). Ths follows fro the fact all nforaton on the asyptotc behavor of the oents s encoded n the dervatves of the sngularty ρ(z, (x ) I ) and by plct dfferentaton these dervatves relate to dervatves of F. More precsely, [9, Theore 3] says that the followng condtons are suffcent to deduce tghtness of the noralzed rando vector: F x <, I x X() n z n, Fyx 2 <, F x x <, I I F zx = o(1), F zx x = o(1), F yyx = o(1), F yyx x = o(1), F zzx = O(1), F zyx = O(1), F zyyx = O(1), F yyyx = O(1), ( ), where all dervatves are evaluated at (ρ, y(ρ), (1) I ). The stuaton s slghtly dfferent n our case snce we have to work wth M D nstead of R D. However, the only real dfference between R D and M D s that the crtcal exponent n the sngular representatons (20) and (21) are dfferent, but the behavor of the sngularty ρ(z, t, (x ) I ) s precsely the sae. Note that after the ntegraton step we can set t = 1. Now tghtness for the noralzed rando vector that s 8

9 encoded n the functon M D follows n the sae way as for R D. And snce the sngularty ρ(z, 1, (x ) I ) s the sae, we get precsely the sae condtons as n the case of [9, Theore 3]. Ths eans that we just have to check the above condtons appled to F = F (z, (x 2 ) 2 D, y) = z + z ( ) 2 1 x 2 y, 2 D where all dervatves are evaluated at z = ρ, x 2 = 1, and y = R D (ρ) < 1/4. However, they are trvally satsfed snce ( ) 2 1 K y < 1 for all K > 0 and for postve real y < 1/4. Reark 2. As stated n Theore 1, the results and ethods extend to the general case as well. The an dea s to reduce the (postve strongly connected) syste of two equatons (16) to a sngle functonal equaton, by applyng [8, Theore 3]. A ore detaled proof s provded n the Appendx. 5 Maps of Hgher Genus The bjecton used n Secton 2 reles solely on the orentablty of the surface on whch the aps are ebedded. Therefore t can easly be extended to aps of hgher genus,.e., ebedded on a surface of genus g > 0 (whle planar aps correspond to aps of genus 0). The an dfference les n the fact that the correspondng obles are no longer trees but rather one-faced aps of hgher genus, whle the other propertes stll hold. However, due to the apparton of cycles n the underlyng structure of obles, another dffculty arses. Indeed, n the orgnal bjecton, vertces and edges n obles could carry labels (related to the geodesc dstance n the orgnal ap), subject to local constrants. In our settng, the legs actually encode the local varatons of these labels, whch are thus plct. Local constrants on labels are naturally translated nto local constrants on the nuber of legs. But the labels have to rean consstent along each cycle of the obles, 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 sae te, we wll now use a hybrd forulaton of obles, carryng both labels and legs. As before, we wll focus on the spler case of obles cong fro bpartte aps. 5.1 g-mobles Defnton 3. Gven g Z 0, a g-oble s a one-faced ap of genus g ebedded 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 the (whch are not consdered edges), whose nuber equals the nuber of whte neghbor vertces. Furtherore, for each cycle c of the g-oble, let n, n and n respectvely be the nubers 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 5.1): n = n + n (22) The degree of a black vertex s the nuber of half-edges plus the nuber of legs that are attached to t. A bpartte g-oble s a g-oble wthout black black edges. A g-oble s called rooted f an edge s 9

10 n = 7 n = 4 n = 3 n n n = 0 Fgure 1: An orented cycle n a g-oble and the constrant on ts left (colored area). Notce that a slar constrant holds on ts rght, but s necessarly satsfed thanks to the propertes of a g-oble. Fgure 2: A 1-oble on the torus and ts schee. dstngushed and orented. Notce that a 0-oble s sply a oble as descrbed n Defnton 1. Theore 3. Gven g 0, there s a bjecton between g-obles that contan at least one black vertex and ponted aps of genus g, where whte vertces n the oble correspond to non-ponted vertces n the equvalent ap, black vertces correspond to faces of the ap, and the degrees of the black vertces correspond to the face valences. Ths bjecton nduces a bjecton on the edge sets so that the nuber of edges s the sae. (Only the ponted vertex of the ap has no counterpart.) Slarly, rooted g-obles that contan at least one black vertex are n bjecton to rooted and vertexponted aps of genus g. Proof. Ths generalzaton of the bjecton to hgher genus was frst gven n [6] for quadrangulatons and [4] for Euleran aps, fro whch we wll explot any deas n the present secton. 5.2 Schees of g-moble g-mobles are not as easly decoposed as planar obles, due to the exstence of cycles. However, they stll exhbt a rather sple structure, based on schee extracton. The g-schee (or sply the schee) of a g-oble s what reans when we apply the followng operatons (see Fgure 2): frst reove all legs, then reove teratvely all vertces of degree 1 and fnally replace any axal path of degree-2-vertces by a sngle edge. Once these operatons are perfored, the reanng object s stll a one-faced ap of genus g, wth black and whte vertces (whte whte edges can now occur), where the vertces have nu degree 3. To count g-obles, one key ngredent s the fact that there s only a fnte nuber of schees of a gven genus. Indeed, let d be the nuber of degree vertces of a g-schee: 2)d = k 3( d 2 d = 2(#edges #vertces) = 4g 2. k 3 k 3 10

11 Fgure 3: The varatons of labels around a black vertex and along an orented cycle. The nuber of vertces (respectvely edges) s then bounded by 4g 2 (respectvely 6g 3), where ths bound s reached for cubc schees (see an exaple n Fgure 2). To recover a proper g-oble fro a gven g-schee, one would have to nsert a sutable planar oble nto each corner of the schee and to substtute each edge wth soe knd of path of planar obles. Unfortunately, ths cannot be done ndependently: Around each black vertex, the total nuber of legs n every corner ust equal the nuber of whte neghbors, and around each cycle, (22) ust hold. In order to ake these constrants ore transparent, we wll equp schees wth labels on whte vertces and black corners. Now, when tryng to reconstruct a g-oble fro a schee, 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 1,..., l d ) be the labels of ts corners read n clockwse order: +1 f there s a leg between the two correspondng corners,, l +1 l = 0 f there s a black neghbor, 1 f there s a whte neghbor. Along the left sde of an orented cycle, the label decreases by 1 after a whte vertex or when encounterng a whte neghbor and ncreases by 1 when encounterng a leg. The above stateents hold for general as well as bpartte obles. In the followng, we wll only consder bpartte obles, as they are uch easer to decopose. 5.3 Reconstructon of Bpartte Maps of Genus g In the followng, t wll be convenent to work wth rooted schees. One can then defne a canoncal labellng and orentaton for each edge of a rooted schee. 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 1,..., c k. Gven a schee S, let V, V, C, C be respectvely the sets of whte and black vertces and of whte and black corners. A labelled schee (S, (l c ) c V C ) s a par consstng of a schee 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 e E S, an edge of S, we assocate a label to each extrety l e, l e+. If 11

12 l + 1 l k j j l +1 l j k j + 1 l k +1 k j l +1 l j k j + 1 l k +1 k k 1 j + 1 Fgure 4: Steps (2) (4) of the algorth. an extrety s a whte vertex of label l, ts label s l. If the extrety s a black vertex, ts label s the sae as the next clockwse corner of the black vertex. Let a doubly-rooted planar oble be a rooted (on a black or whte vertex) planar oble wth a secondary root (also black or whte). These two roots are the extretes of a path (v 1,..., v k ). The ncreent of the doubly-rooted oble s then defned as n n n, whch s not necessarly 0, as the path s not a cycle. Slarly as n [4], we present a non-deternstc algorth to reconstruct a g-oble: Algorth. (1) Choose a labelled schee of genus g (S, (l c ) c V C ). (2) v V, choose a sequence of non-negatve ntegers ( k ) 1 k deg(v), then attach k planar obles and k + l ck+1 l ck + 1 legs to c k (the k th corner of v). { +1 f e s whte, (3) e S, replace e by a doubly-rooted oble of ncreent ncr(e) = l e+ l e + (4) On each whte corner of S, nsert a planar oble. (5) Dstngush and orent an edge as the root. 1 f e s black. Proposton 3. Gven g > 0, the algorth generates each rooted bpartte g-oble whose schee has k edges n exactly 2k ways. Proof. One can easly see that the obtaned object s ndeed bpartte. Attachng planar obles and legs added at step (2) n a corner c k create new corners, such that: The frst carres the sae label l ck as c k, and the last carres the label l ck + ( k + l ck+1 l ck + 1) k = l ck The next corner should then be labelled (l ck+1 + 1) 1 = l ck+1, due to the next whte neghbor, whch s precsely what we want. In the sae fashon, at step (3), a sple countng shows that each edge s replaced by a path such that the labels along t evolve accordng to the schee labellng. We thus obtan a well-fored rooted bpartte g-oble, wth a secondary root on ts schee. Snce the frst root destroys all syetres, there are exactly 2k choces for the secondary root, whch would gve the sae rooted g-oble. 5.4 g-moble Countng A doubly-rooted bpartte planar oble can be decoposed along a sequence of eleentary cells forng the path between ts two roots. Its ncreent s sply the su of the ncreents of ts cells. 12

13 Defnton 4. An eleentary 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 obles attached on each sde. The black vertex has j 0 legs and k 0 whte-rooted obles on ts left, l 0 and k+l j+2 legs on ts rght, and ts degree s 2(k + l + 1). The ncreent of the cell s then k j 1. The generatng seres P := P (t, (x 2 ), z, s) of a cell, where s arks the ncreent, s: P (t, (x 2 ), z, s) = z2 R 2 ( )( j + k k + l j + 2 )s k j 1 x 2(k+l+1) R k+l = z2 R 2 P t j l st j,k,l 0 The generatng seres S := S(t, (x 2 ), z, s) of a doubly-rooted oble depends on the color of ts roots (u, v): 1 f (u, v) = (, ) or (, ), 1 P z S (u,v) (t, (x 2 ), z, s) = P f (u, v) = (, ), 1 P f (u, v) = (, ). zr 2 st(1 P ) We can now express the generatng seres R S schee S: := R S (t, (x 2 ), z) of rooted bpartte g-obles wth R S (t, (x 2 ), z) = 2 z ( ) C 1 R z 2 E z E t V zt deg(v) ( 2k + l l ) ck+1 c k + 1 x 2 k +2deg(v) [s ncr(e) ]S (e,e + ) (23) (l c) labellng v V 1,..., deg(v) 0 k=1 k Proposton 4. The generatng seres M (g) (g) D := M D (t, (x 2), z) for the faly of rooted bpartte aps of genus g, where the vertex degrees belong to D, satsfes the relaton: e E M (g) D t = 2 z S schee of genus g R S (t, (x 2 1 {2 D} ), z) (24) Proof. It follows drectly fro Theore 3 and Equaton (23). 6 Concluson Theore 1 confrs the exstence of a unversal behavour of planar aps. The asyptotcs (wth exponent 5/2) and ths central lt theore for the expected nuber of vertces of a gven degree are beleved to hold for any reasonable faly of aps. It has also been shown n [6, 4] that a slar phenoeno occurs for aps of hgher genus: The generatng seres of several fales (quadrangulatons, general and Euleran aps) of genus g exhbt the sae asyptotc exponent 5g/2 5/2. The expresson obtaned n Secton 5 needs to be properly studed n order to obtan an asyptotc expanson. It refnes prevous results by controllng the degree of each vertex n the correspondng ap. 13

14 References [1] Cyrl Banderer and Mchael Drota, Forulae and asyptotcs for coeffcents of algebrac functons, Cobnatorcs, Probablty and Coputng 24 (2015), no. 1, [2] E.A. Bender and E.R. Canfeld, Enueraton of degree restrcted aps on the sphere., Planar graphs. Workshop held at DIMACS fro Noveber 18, 1991 through Noveber 21, 1991, Provdence, RI: Aercan Matheatcal Socety, 1993, pp [3] J. Boutter, P. D Francesco, and E. Gutter, Planar aps as labeled obles, Electron. J. Cobn. 11 (2004), no. 1, Research Paper 69, 27. [4] Gullaue Chapuy, Asyptotc enueraton of constellatons and related fales of aps on orentable surfaces, Cobn. Probab. Coput. 18 (2009), no. 4, [5] Gullaue Chapuy, Érc Fusy, Mhyun Kang, and Blyana Sholekova, A coplete graar for decoposng a faly of graphs nto 3-connected coponents, Electron. J. Cobn. 15 (2008), no. 1, Research Paper 148, 39. [6] Gullaue Chapuy, Mchel Marcus, and Glles Schaeffer, A bjecton for rooted aps on orentable surfaces, SIAM Journal on Dscrete Matheatcs 23 (2009), no. 3, [7] Gwendal Collet and Érc Fusy, A sple forula for the seres of bpartte and quas-bpartte aps wth boundares, Dscrete Math. Theor. Coput. Sc. (2012), [8] Mchael Drota, Rando trees. An nterplay between cobnatorcs and probablty., Wen: Sprnger, [9] Mchael Drota, Bernhard Gttenberger, and Johannes F. Morgenbesser, Infnte systes of functonal equatons and Gaussan ltng dstrbutons., Proceedng of the 23rd nternatonal eetng on probablstc, cobnatoral, and asyptotc ethods n the analyss of algorths (AofA 12), Montreal, Canada, June 18 22, 2012, Nancy: The Assocaton. Dscrete Matheatcs & Theoretcal Coputer Scence (DMTCS), 2012, pp [10] Mchael Drota and Konstantnos Panagotou, A central lt theore for the nuber of degree-k vertces n rando aps., Algorthca 66 (2013), no. 4, [11] W.T. Tutte, A census of planar aps., Can. J. Math. 15 (1963), A Proof of Proposton 1 Let R = R(t, z, x 1, x 2,...) be the generatng functon of rooted bpartte obles, where the root vertex s whte and where we addtonally attach a planted edge to the (whte) root vertex next to the root edge (for exaple, n counterclockwse order). The varable t corresponds to the nuber of whte vertces, z to the nuber of edges, and x 2, 1, to the nuber of black vertces of degree 2. Snce rooted obles can be consdered as ordered rooted trees (whch eans that the neghborng vertces of the root vertex are lnearly ordered and the subtrees rooted at these neghborng vertces are 14

15 agan ordered trees) we can descrbe the recursvely. Ths leads drectly to a functonal equaton for R of the for tz R = 1 z 1 x ( 2 1 ) 2 R 1 whch s apparently the sae as (3). Note that the factor ( ) 2 1 s precsely the nuber of ways of groupng legs and 1 edges around a black vertex (of degree 2; one edge s already there). Hence, the generatng functon of rooted obles that are rooted by a whte vertex s gven by R/z. Snce we have to dscount the oble that conssts just of one (whte) vertex, the generatng functon of rooted obles that are rooted at a whte vertex and contan at least two vertces s gven by R/z t = ( ) 2 1 x 2 R. (25) 1 We now observe that the rght hand sde of (25) s precsely the generatng functon of rooted obles that are rooted at a black vertex (and contan at least two vertces). Sung up, the generatng functon of bpartte rooted obles (wth at least two vertces) s gven by 2(R/z t). Fnally, f M denotes the generatng functon of bpartte rooted aps (wth at least two vertces) then M t corresponds to rooted aps, where a non-root vertex s ponted (and dscounted). Thus, by Theore 2 we obtan (4). B Proof of Lea 1 Fro general theory (see [8, Theore 2.21]) we know that an equaton of the for R = F (z, t, R), where F s a power seres wth non-negatve coeffcents, has usually a square-root sngularty of the for (14). We only have to assue that the functon R F (z, t, R) s nether constant nor a lnear polynoal and that there exst solutons ρ > 0, R 0 > 0 of the syste of equatons R 0 = F (ρ, 1, R 0 ), 1 = F R (ρ, 1, R 0 ), whch are nsde the range of convergence of F. Furtherore, we have to assue that F z (ρ, 1, R 0 ) > 0 and F RR (ρ, 1, R 0 ) > 0 to ensure that (14) holds not only for t = 1 but n a neghborhood of t = 1, and the condton F t (ρ, 1, R 0 ) > 0 ensures that ρ (1) 0. Ths eans that n our case we have to deal wth the syste of equatons ( ( R 0 = ρ + ρ 2 D ) R 0, 1 = ρ 2 D or (after elnatng ρ) wth the equaton ( ) 2 1 R0 = 1 + ( ) 2 1 R0 2 D 2 D ) R 1 0, whch we can rewrte to ( ) 2 1 ( 1) R0 = 1. (26) 2 D 15

16 It s clear that (26) has a unque postve soluton f D s fnte. (We also recall that all 1, snce 2 has to be postve.) If D s nfnte, we have to be ore precse. Actually, we wll show that (26) has a unque postve soluton R 0 < 1/4. Ths follows fro the fact that ( ) 2 1 ( 1) 4 2 π. Thus, f D s nfnte, t follows that the power seres x H(x) = 2 D( ( ) 2 1 1) x has radus of convergence 1/4 and we also have H(x) as x 1/4 snce each non-zero ter satsfes ( ) 2 1 l ( 1) x x 1/4 2 π, whch s unbounded for. Fnally, we set ρ = ( 2 D ( ) ) 2 1 R It s clear that F z (ρ, 1, R 0 ) > 0, F RR (ρ, 1, R 0 ) > 0, and F t (ρ, 1, R 0 ) > 0. Hence we obtan the representaton (14) n a neghborhood of z = ρ = ρ(1) and t = 1. Next, let us dscuss the analytc contnuaton property. If d = gcd{ : 2 D} = 1 then t follows fro the equaton (11) that the coeffcents [z n ]R D (z, 1) are postve for n n 0 (for soe n 0 ). Consequently [8, Theore 2.21] (see also [8, Theore 2.16]) ples that there s an analytc contnuaton to the regon z < ρ(t) + η, arg(z ρ(t)) 0. If d > 1, then we can frst reduce equaton (11) to a an equaton (13) for the functon R that s gven by R D (z, t) = z R(z d, t). We now apply the above ethod to ths equaton and obtan correspondng propertes for R. Of course, these propertes drectly translate to R D, and we are done. C Proof of Lea 3 It s convenent to reduce the nuber of equatons. If we substtute the second equaton of (16) for L,D nto the thrd one and ultply wth the denonator, we obtan the equvalent syste y B 2 1, L 2 1,D RD, L,D = z D R D = zt + z D y B (+1) 2 2, L2 2,D R +1 D. Ths s a strongly connected syste of two equatons of the for L,D = F (z, t, L,D, R D ), R D = G(z, t, L,D, R D ), where F and G are power seres wth non-negatve coeffcents. It s known that such a syste of equatons has n prncple the sae analytc propertes (ncludng the sngular behavor of ts solutons) as a sngle equaton, see [8, Theore 2.33]; however, we have to be sure that the regons of convergence of F and G are large enough. In partcular, f D s fnte, then we have a postve algebrac syste and we are done, see [1]. In the nfnte case we have to argue n a dfferent way. Frst of all, t s clear fro the explct solutons of E = E(t, u) = ((1 t (1 t) 2 4u)/(2u) and B = B(t, u) = 1/ (1 t) 2 4u that F and G (and all ther dervatves wth respect to L,D and R D ) are certanly convergent f 2 L,D L,D

17 2 R D < 1. On the other hand, t follows slarly to the bpartte case that the dervatves of F and G are dvergent f L,D > 0, R D > 0, and 2L,D L 2,D + 2R D = 1. To see ths we consder the functon B(t/s, us 2 ) = 1 1 2t/s t 2 /s 2 4uw 2 = l, B l, s 2 l t l u = By sngularty analyss t follows (for t, u > 0) that B 2, t 2 u c 1/2 h(t, u), s B 2, t 2 u. where c > 0 and h = h(t, u) > 0 satsfy the equaton 1 2t/h t 2 /h 2 4uh 2 = 0. Slarly, we can consder dervatves of F whch correspond, for exaple, to sus of the for B 2, t 2 u c 1/2 h(t, u). In partcular, f h(t, u) = 1 (whch s the case f 2t t 2 4u = 1), then ths ter dverges for. Thus, the dervatves of F and G dverge f L,D > 0, R D > 0, and 2L,D L 2,D + 2R D = 1. In order to deterne the sngularty of the syste L,D = F (z, t, L,D, R D ), R D = G(z, t, L,D, R D ) we have to fnd postve solutons of L 0, R 0, ρ of the syste L 0 = F (ρ, 1, L 0, R 0 ), R 0 = G(ρ, 1, L 0, R 0 ), 1 = G L,D F RD 1 F L,D + G RD. 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 0 and, thus, saller than 1.) As long as the rght-hand sde of the thrd equaton s saller than 1, t follows fro the plct functon theore that there s a local analytc contnuaton of the solutons L 0 = L 0 (ρ), R 0 = R 0 (ρ). Furtherore, 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 < 1. Fro 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 1, so that the above syste has a soluton (ρ, L 0, R 0 ). Of course, at ths pont the dervatves of F and G have to be fnte, whch ples that (ρ, L 0, R 0 ) les nsde the regon of convergence of F and G. Ths fnally shows that all assuptons of [8, Theore 2.33] are satsfed. Thus, sngular representaton of type (17) and the analytc contnuaton propery follow for the functons L,D = L,D (z, t), L,D = L,D (z, t), R D = R D (z, t). Hence, the sae knd of propertes follows for T D = T D (z, t) and consequently also for R D /z t + T D. D Central Lt Theore for General Maps We now assue that D contans at least one odd nuber. By Proposton 2 (and by the applyng the sae elnaton procedure as n the proof of Lea 3) we have to consder the syste of equatons y B 2 1, L 2 1,D RD, L,D = z D R D = zt + z D y B (+1) 2 2, L2 2,D R +1 D, 17

18 for the generatng functons L,D = L,D (z, t, (y ) D ) and R D = R D (z, t, (y ) D ), the generatng functon T D = T D (z, t, (y ) D ) = 1 + y B 2, L 2,D R D D and fnally the generatng functon M D = M D (z, t, (y ) D ) that satsfes the relaton M D = R D /z t + T D. t Agan, f D s fnte, we can proceed as n the bpartte case by applyng [8, Theore 2.33, Lea 2.27, and Theore 2.25] whch ples the proposed central lt theore. If D s nfnte, we argue n a slar way as n the bpartte case. The only dfference s that we are not startng wth one equaton but wth a syste of two equatons that have the (general) for L = F (z, t, (y ) D, L, R), R = G(z, t, (y ) D, L, R). Nevertheless, t s possble to reduce two equatons of ths for to a sngle one. The proof of [8, Theore 2.33] shows that there are no analytc probles snce we have a postve and strongly connected syste. We use the frst equaton to obtan an plct functon soluton f = f(z, t, (y ) D, r) that satsfes f = F (z, t, (y ) D, f, r). Then we substtute f for L n the second equaton and arrve at a sngle functonal equaton R = G(z, t, (y ) D, f(z, t, (y ) D, R), R) for R = R D (z, t, (y ) D ). Note that the proof of [8, Theore 2.33] assures that f s analytc although L and R get sngular. Hence by settng H(z, t, (y ) D, r) = G(z, t, (y ) D, f(z, t, (y ) D, r), r) we obtan a sngle equaton R = H(z, t, (y ) D, R) for R = R D and we can apply the sae ethod as n the bpartte case. Of course, the calculatons get ore nvolved. For exaple, we have where H y = G y + G LF y 1 F L, F L = ρ (2 1)B 2 1, L R0, D F y = ρ G L = ρ D G y = ρ B 2 1, L R 0, (2 2)B (+1) 2 1, L2 3 0 R0, B (+1) 2 2, L2 2 0 R 0. Fro the proof of Lea 3 we already know that 2L 0 L R 0 < 1, whch ples that K (2 1)B 2 1, L0 2 2 R0 < 1 for all K > 0. Furtherore, we have F L < 1 and G R < 1. Hence t follows that H y <. D In the sae way, we can handle the other condtons whch copletes the proof of Theore 1. 18