BISINGULAR MAPS ON SOME SURFACES

2004,24B( 2) :313-320

.At'at~sPcientia

1t~~JI~1 BISINGULAR MAPS ON SOME SURFACES

1

Li Zhaoxiang ( 4'~# ) Department of Mathematics, Central University for Nationalities, Beijing 100081, China Ren Han ( *# ) Department of Mathematics, East China Normal University, Shanghai 200062, China

Liu Yanpei ( ~IJ!J 1Jit ) Department of Mathematics, Northern Jiaotong University, Beijing 100044, China Abstract A map is bisingular if each edge is either a loop (This paper only considers planar loop) or an isthmus (i.e., on the boundary of the same face). This paper studies the number of rooted bisingular maps on the sphere and the torus, and also presents formulae for such maps with three parameters: the root-valency,the number of isthmus, and the number of planar loops. Key words

Bisingular map, enumerating function, Lagrangian inversion

2000 MR Subject Classification

1

05CIO, 05C30, 05C45

Introduction

A surface is a compact 2-manifold. An (A) orient able (non-orientable) surface of genus 9 is homeomorphic to the sphere with 9 handles (crosscaps) and is denoted by Bg (Ng ) . A map M on (or embedded on) Bg (or Ng ) is a graph drawn on the surface so that each vertex is a point on the surface, each edge {x, y}, x =1= y, is a simple open curve whose endpoints are x and y, each loop incident to a vertex x is a simple closed curve containing x, no edge contains a vertex to which it is not incident, 'and each connected region of the complement of the graph in the surface is homeomorphic to a disc and is called a face. A map is rooted if an edge, a direction along the edge, and a side of the edge are all distinguished. If the root is the oriented edge from u to v, then u is the root-vertex while the face on the oriented side of the edge is defined as the root-face. The concept of rooted map was first introduced by W.Tutte. His series of census papers[15-18] laid a foundation for the theory. Since then, the theory has been developed by several reseavelers such as D.Arques[l], W.Brown[7,8], R.Mullin et al[14] , W.Tutte[19], Walsh et al[20] , E.Bender et al[2-6], Gao[9,lO], and Liu[12,13]. Among them D.Arques, W.Brown, and Walsh et al did some influential works on the (exact) enumeration of nonplanar maps. Since exact counting of nonplanar maps is very difficult, E.Bender et al and Gao did the most extended work in the field of asymptotic evaluation of nonplanar maps. A survey can be found in [5]. 1 Received September 18, 2000; revised April 2, 2003. Supported by fifteenth programming of Central University for Nationalities, NNSFC under Grant No.10271048 and 19831080.

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A map is bisingular if each edge is either a loop (In this paper, we only consider planar loops) or an isthmus (i.e., on the boundary of the same face). In this paper we study the number of rooted bisingular maps (a concept by Liu[12), where a planar bisingular map was defined as a bi-tree) on the sphere and the torus and also we present explicit formulae for such maps with three parameters: the root-valency, the number of isthmuses, and the number of planar loops. An isthmus (or singular edge as some scholar called it[9]) is an edge on the boundary of only one face. Since the removing of an isthmus from a map will result a connected region not homeomorphic a disc, we see that an edge of a planar map is an isthmus if and only if it is a cut-edge. A map is bisingular if all of its edges are either loops or isthmuses. Let C be a circuit (or curve) on a surface E, If ~-C has a connected region homeomorphic to a disc, then C is called trivial (or contractible as some scholars defined it); otherwise it is essential (or noncontractible). A loop is called essential if it is an essential circuit; otherwise it is called planar or trivial. One may easily see that the sphere has no essential circuits. Let Bpi and B be respectively the set of rooted spherical and torus bisingular maps. Their enumerating functions are, respectively fpl(x,y,z) =

L

xm(Mlys(Mlzn(Ml,

MEBpl

f(x,y,z) =

L

xm(Mlys(Mlzn(Ml

MEB

where m(M), s(M) and n(M) are respectively the root valency of M, the number of isthmus of M and the number of planar loops of M.

2

Bisingular Maps on the Sphere

Given two maps M I and M 2 with roots "i = r(Md and r2 = r(M2), respectively, we define M = M I 8 M 2 to be the map obtained by identifying the root-vertex, the root-edge of M is same as those of M I , but the root-face of M is the composition of fr(Md and fr(M2) , where fr(Mi) is the root-face of Mi(i = 1,2). The operation for getting M from M 1 and M 2 is called a Iv-production. Further, for two sets of maps M I and M 2 , the set of maps

is said to be the lv-production of M I and M 2 . The set Bpi may be partitioned into three parts as

where B~I is the vertex-map, B~l = {Mler(M) is an isthmus},

and B;l = {Mler(M) is a loop }.

al: BISINGULAR

Li et

No.2

315

MAPS ON SOME SURFACES

Then, we obtain Lemma 2.1 B~l = Bpl(l) OB pl, where B pl(l) is the set of bisingular maps on the sphere of root-valency 1. Proof For any map M in B~l' the root-edge er(M) is an isthmus. The two submaps determined by er(M) are respectively in Bpl(l) and Bpi. Hence, B~l is a subset of B pl(l) OBpl. Conversely, any map M in Bpl(l) BpI must have its root-edge separable by the definition BpI ~ B~l' of the lv-production of two maps, Bpl(l) Lemma 2.2 Let B~pl) = {M - er(M)IM E B;l}' Then B~pl) = BpI OBpl' Proof This may be verified by the fact that the maps in two regions determined by a loop are the elements in BpI. According to Lemmas 2.1 and 2.2, the contributions of B~l and B~l are resp.

o

B~l

o

:f------+

f;l = hpl(l)fpl,

2 ..f------+ f2pi -BpI

X 2 z f2 pl :

Where h pl (1) is the enumerating function of Bpl(l). Since h pl(l) = xyh pl, h pl = f pl!x=ll

fpl = 1 + xyhpzipl + x2zf;I' 1

(1)

fpl = 1- (2 ). X Zfpl +xyh pl . and with x = 1 we have

hpl =

1

1 - (y + z)h pl

(2)

.

Applying Lagrangian inversionll l , pp17-18] for(l) , we obtain

f

'" 1 Dm-1 { 1 } pI = ~ m! !pl=O (1 - (x 2zfpl + xyhpl))m

L m>l n;:::~-l

m 1 (m+n-1) ( ti ) xm+n-1yn-m+l zm-l n m-1

<::

(3)

Again applying Lagrangian inversion to (2) and (1), we further find that for s ;:::: 1

h;l = fS pi

=

'"

. L.J

'2 n-s20

L ys ( l>s

2l - s - 1 ) (y l- 1

+ Z)l-s ,

s(n+i-1)! x,+n-s ,+s-nzn-shi+s-n n!(n - s)!(i + s - n)! y pl

(4)

(5)

By substituting (4) into (3), we have that Theorem 1 The function fpl satisfies the following equation

x2zf;1

+ (xyhpl -

l)fpl

+1=

0,

hpl = fpzlx=l'

and furthermore, for s ;:::: 1

(

2l- s - 1 ) (y l- 1

+ z)l-s,

(6)

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VoJ.24 Ser.B

ACTA MATHEMATICA SCIENTIA

n-mlm+ (m +nn- 1) (m-l n) 2l-n+m-2) x m+n- 1yn-m+1 (y + Z

L

ipl =

1

171>1,n>rn-l l~n-~+l

x

(

)1-n+m-1 zm-1.

l-1

Hence, the number of rooted planar bisingular maps with k links (i.e., non-loop edges) and j loops is . 1 (2(j+k)) (j+k). ]+k+l j+k k.

In particular, the number of those having 1 edges is

»+ (2l). 1

1

1

Figure 1 (8 distinct rooted maps with two edges in Bpi, the leftmost are 4 maps with 1 link and 1 loop)

Let z = O. Then we have a famous result for trees: Corollary 1[19J The enumerating functions of rooted plane trees are

h;l(y,O) =

L I (2l- -1) yl-s. S

l?s

S

1- 1

Remark Although a planar bisingular map is composed of a tree with several loops on each vertex, one can not easily deduce Theorem 1 from the correspoding results for trees. If y = z, then we have Corollary 2 The enumerating function of rooted planar bisingular maps (with the rootvalency and the number of edges as parameters) is 21- n +m - 1 (n - m

L

ipl(X,y,y) =

ml 2l-n+m-2) x m+n- 1yn-m+1.

m.>l,n>tn-l i~n-~+l

x

(

+ 1)

l-1

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No.2

Each loop of a map M E Bpi separates the plane into two regions each of which uniquely determines a submap of M. Then by the definition of fpl(x,y,z), we have Corollary 3 The enumerating function of rooted planar bisingular maps with exactly k loop-bounded submaps is

~ ok fpII k!

ozk z=o x

(2l - n + m - 2) l-l

3

(

l- n+m - l ) k-m+l

X

m+n-1 l-k+m-1

y

.

Bisingular Maps on Torus

A 'l9-map is a smallest nonseparable bisingular map without vertices of valency 2. A map is rooted if an edge with a direction along its one side is distinguished. Two rooted maps are considered to be the same if there exists an automorphism preserving the rooting. It is easy to see that Lemma 3.1 There are only two different kinds of rooted 'l9-maps on the torus (as shown in Fig.2). Based on the two 'l9-maps in Fig.2 we calculate the numbers of all the rooted bisingular maps on the torus with various kinds of parameters.

(J,

Figure 2

Lemma 3.2

Let P be the set of rooted paths, then its enumerating function is

xy I»> l-y

X2

y2

+ (l-y)2'

Proof The set P may be parted into

where P1 is the set of rooted paths with root-valency 1 and 0 is the lv-production of the maps. Since the enumerating function of P1 is f'!:y, the lemma follows. The set B may be parted into two parts as

where B 1 = {M [e,(M)

is not on an essential circuit};

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ACTA MATHEMATICA SCIENTIA

B2 = {Ml er(M)

Vol.24 Ser.B

is on an essential circuit} .

The set B1 may be parted into two parts as

where

Bf = {MIM E B1 , er(M ) is an isthmus},

Bi = {MIM E B ,er(M) 1

is a planar loop}.

As we have reasoned in Lemma 2.1, we obtain th e following Sf = B(l) Bpi + Bpl(l) OB, where B(1) is th e set of map s in B with root -valency l. As for th e set Bf, the root-edge is a planar loop and separates the map on the toru s into two regions: one determines a planar map in Bpi while the other contains a torus map in B . Since there are two ways of rootin g a planar loop on the torus we have th e next Lemma 3.4 Let B~1) = {M - er(M )IM E Bf}. Then B~1) = BpI OB + B OBpl. Lemmas 3.3 and 3.4 imply th at the contributions of Bt and Bf are, resp., Lemma 3.3

o

f(l)fpl

+ fpl(l)f,

2

2x z f pzf,

(7)

where f(l) and f pl(1) are, respectively, the enumerating functions of B(l ) and B pl (l) . It is clear th at f(l) = x yh, f pl(1) = xyhpl (here h = fl x=l ,h pl = fpzl x=l)' Substituting this into (7) we obt ain

(8) Lemma 3.5

Let Bn s denot e the set of rooted nonseparabl e bisingular maps without loops on the torus. Th en B - B(l ) B(2) ns -

ns

+

ns '

where B~il is the set of rooted singular maps determined by the 'l9-map Bi (which is depicted in Fig.2), i = 1,2 . Proof By the definitions of Bi , we can obtain B~l by replacing each edge of Bi with a rooted path of P( i = 1,2. ) Since th e the procedure is reversable and does not change the bisingular prop erty, the Lemma follows. Lemma 3.6 Th e set B2 may be parted into two parts as

where B~i ) is determined by placing a plan ar bisingular map at each corner around every vert ex of B~il (i = 1,2 .). Proof By the definition of B2 , B2 is th e set of maps with their root- edges on some essential cycles. Since maps in B2 have exactly two essential cycles with only one common path, they are classified into two parts determined by B~~ and B~~) respectively.

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Li et al: BISINGULAR MAPS ON SOME SURFACES

No.2

Now let us begin to calculate fJi), the enumerating function of B~i), i = 1,2. Notice that the maps in B~l} are either with their root-valency 4 or 2, by considering the placement of planar bisingular map at each corner of their vertices, the contribution of B~l) is B2( l )

. .

4Y4f4pi h 4pi - (1- yh;I)2

(1) _

f2

X

+

Y4f2pi h6pi (1- yh;I)3' X

2

(9)

in which fpl is the enumerating function of rooted planar bisingular map and h pl = f pllx=l. Similarly we have the contribution of B~2) as

B2(2)

.

•

(2) _

f2

X

3Y3f3pi h 3pi

- (1 - yh;l)3

X

2

6

Y4f2pi h pi

+ (1 -

(10)

yh;I)4·

According to Lemmas 3.2-3.6, we have that Theorem 2 The function f satisfies the following equation

f = xy(hfpl

+ hpd) + 2x

x2y41;lh~1

+ (1- yh;l)3 +

2

zfpd +

x 3y3f:lh~l (1 - yh;I)3

4Y4f4pi h 4pi (1 - yh;l)2 X

(11)

x2y41;lh~1

+ (1 -

yh;I)4

where h = flx=l. Applying (11), we obtain

L

h(y,z) =

2

j- 1(i

+ 1)(i 2 + 8i + 18)(2i + j + 8)(2l- 2i 3l!(l- 2i - j - 8)!

i>O,j>O

1~2i+T+8

+

L

2

j- 1(i

+ 2)(i + 1)(2i + j + 6)(2l- 2i l!(l-2i-j-6)!

i>O,j>O

j - 9)!

y

j - 7)! i+3( Y

i+4(

y +z

y+z

)1-2i-8

)1-2i-6

.

18!i+J+6

Let z = 0, then we have Corollary 4 The enumerating function of rooted singular maps on torus with the number of edges as parameters is 2j- 1(i·+ 1)(i2 + 8i + 18)(2i + j + 8)(2l - 2i - j - 9)! l-i-4 h(y,O) = 3l!(l - 2i - j - 8)! Y i>O,j>O 1~2i+J+8

+

2j- 1(i + 2)(i

L

+ 1)(2i + j + 6)(2l- 2i -

j - 7)! l-i-3

l!(l - 2i - j - 6)!

i>O,j>O

Y

l2:2 i +T+6

If z = y, then we have Corollary 5 The enumerating function of rooted bisingular maps on torus with the number of edges as parameters is

h(y, y) = i>O,j>O

21-2i+j-9(i + 1)(i 2 + 8i + 18)(2i + j + 8)(2l - 2i - j - 9)! l-i-4 3l!(l - 2i - j - 8)! Y

12:2 i+J+8

+

L

i>O,j>O

1~2i+T+6

21-

2,+j-7(i

+ 2)(i + 1)(2i + j + 6)(2l- 2i l!(l - 2i - j - 6)!

j - 7)!

1-,-3

Y

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References

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Arques D. Relations fonctionelleset denombremant descartes pointees surle tore. J Combin Theory, 1987, 43B: 253-274 Bender E A. Asymptotic methods in enumeration. SIAM Rev, 1974,16: 485-515 Bender E A, Canfield E R, Robinson R W. The enumeration of maps on the torus and the projective plane. Canada Math Bull, 1988, 31: 257-271 Bender E A, Canfield E R. The asymptotic number of rooted maps on a surface. J of Combin Theory, 1986, 43A: 244-257 Bender E A, Richmond L B. A survey of the asymptotic behaviour of maps. J Combin Theory, 1986, 40B: 297-329 Bender E A, Wormald N C. The asymptotic number of rooted nonseparable maps on a given surface. J Combin Theory, 1988, 49A: 370-380 . Brown W G. Enumeration of nonseparable planar maps. Canada J Math, 1963, 15: 526-545 Brown W G. On the number of nonplanar maps. Mem Amer Math Soc, 1966, 65: 1-42 Gao Z C. The number of rooted 2-connected triangular maps on the projective plane. J of Combin Theory, 1991, 53B: 130-142 Gao Z C. The asymptotic number of rooted 2-connected triangular maps on a surface. J Combin Theory, 1992, 54B: 102-112 Goulden I P, Jackson D M. Combinatorial Enumeration. Wiley, 1983. 17-18 Liu Yanpei. Enumerative Theory of Maps. Boston: Kluwer, 1999 Yanpei Liu. On the number of rooted c-nets. J Combin Theory, 1984, 36: 118-123 Mullin R C, Schellenberg P J. The enumeration of c-netsvia quadrangulations. J Combin Theory, 1964, 4: 256-276 Tutte W T. A census of planar triangulations. Canada J Math, 1962, 14: 21-38 Tutte W T. A census of slicings. Canada J Math, 1962, 14: 708-722 Tutte W T. A census of hamiltonian polygons. Canada J Math Soc, 1962,68: 402-417 Tutte W T. A census of planar maps. Canada J Math, 1963, 15: 249-271 Tutte W T. On the enumeration of planar maps. Bull Amer Math Soc, 1968,14: 64-74 Walsh T, Lehman A B. Counting rooted maps by genus I, II. J of Combin Theory, 1972, 13B: 122-141, 192-218