Enumerating Branched Orientable Surface Coverings Over a Non-Orientable Surface Ian P

Discrete Mathematics 303 (2005) 42–55 www.elsevier.com/locate/disc

Enumerating branched orientable surface coverings over a non-orientable surface Ian P. Gouldena, Jin Ho Kwakb,1, Jaeun Leec,2 aDepartment of Combinatorics and Optimization, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 bCombinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang 790–784, Korea cMathematics, Yeungnam University, Kyongsan 712–749, Korea

Received 21 November 2002; received in revised form 13 October 2003; accepted 29 October 2003 Available online 10 October 2005

Abstract The isomorphism classes ofseveral types ofgraph coverings ofa graph have been enumerated by many authors [M. Hofmeister, Graph covering projections arising from finite vector space over finite fields, Discrete Math. 143 (1995) 87–97; S. Hong, J.H. Kwak, J. Lee, Regular graph coverings whose covering transformation groups have the isomorphism extention property, Discrete Math. 148 (1996) 85–105; J.H. Kwak, J.H. Chun, J.Lee, Enumeration ofregular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273–285; J.H. Kwak, J. Lee, Isomorphism classes ofgraph bundles, Canad. J. Math. XLII (1990) 747–761; J.H. Kwak, J. Lee, Enumeration ofconnected graph coverings, J. Graph Theory 23 (1996) 105–109]. Recently, Kwak et al [Balanced regular coverings ofa signed graph and regular branched orientable surface coverings over a non-orientable surface, Discrete Math. 275 (2004) 177–193] enumerated the isomorphism classes ofbalanced regular coverings ofa signed graph, as a continuation ofan enumeration work done by Archdeacon et al [Bipartite covering graphs, Discrete Math. 214 (2000) 51–63] the isomorphism classes ofbranched orientable regular surfacecoverings ofa non-orientable surface having a finite abelian covering transformation group. In this paper, we enumerate the isomorphism classes ofconnected balanced (regular or irregular) coverings ofa signed graph and those ofunbranched orientable coverings ofa non-orientable surface,as an answer ofthe question raised by Liskovets [Reductive enumeration under mutually orthogonal group actions, Acta-Appl.

E-mail address: [email protected] (J.H. Kwak). 1 Supported by Com2MaC-KOSEF (R11-1999-054). 2 Supported by KRF(2000-015-DP0037).

Math. 52 (1998) 91–120]. As a consequence ofthese two results, we also enumerate the isomorphism classes ofbranched orientable surfacecoverings ofa non-orientable surface. © 2005 Elsevier B.V. All rights reserved.

Keywords: Graph covering; Signed graph; Surface branched covering; Enumeration

1. Introduction

Let G be a finite connected graph with vertex set V (G) and edge set E(G). Every edge ofa graph G gives rise to a pair ofoppositely directed edges. By e−1 = vu, we mean the reverse edge to a directed edge e = uv. We denote the set ofdirected edges of G by D(G). The neighborhood ofa vertex v ∈ V (G), denoted by N(v), is the set ofdirected edges emanating from v. We use |X| for the cardinality of a set X. The number (G) =|E(G)|− |V (G)|+1 is equal to the number ofindependent cycles in G and it is referred to as the betti number of G. A graph G˜ is called a covering of G with projection p : G˜ → G ifthere is a surjection ˜ p : V(G) → V (G) such that p|N(v)˜ : N(v)˜ → N(v)is a bijection for all vertices v ∈ V (G) and v˜ ∈ p−1(v). We say that the projection p : G˜ → G is an n-fold covering of G if p is ˜ n-to-one. Two coverings pi : Gi → G, i = 1, 2 are said to be isomorphic ifthere exists a ˜ ˜ graph isomorphism : G1 → G2 such that p1 = p2 ◦ . Such a is called a covering isomorphism. Let Sn denote a symmetric group on n elements {1, 2,...,n}.Apermutation voltage assignment (or, voltage assignment)ofG is a function : D(G) → Sn with the property that (e−1) = (e)−1 for each e ∈ D(G). The permutation-derived graph G is defined as follows: V(G ) = V (G) ×{1,...,n} and E(G ) = E(G) ×{1, 2,...,n}, so that an edge (e, i) of G joins a vertex (u, i) to (v, (e)i) for e = uv ∈ D(G) and i = 1, 2,...,n. The first coordinate projection p : G → G is an n-fold covering. Following Gross and Tucker [2], every n-fold covering G ofa graph G can be derived from a voltage assignment which assigns the identity element on the directed edges ofa fixed spanning tree T of G. We call such a reduced. That is, for a covering p : G˜ → G, there exists a reduced voltage assignment of G such that the derived covering p : G → G is isomorphic to ˜ p : G → G. Moreover, for a reduced voltage assignment : D(G) → Sn, the derived graph G is connected ifand only ifthe subgroup of Sn generated by the image ofthe voltage assignment acts transitively on the set {1, 2,...,n}, (see [2]). Such a voltage assignment is said to be transitive. From now on, we assume that every voltage assignment is reduced and transitive. A signed graph is a pair G = (G, ) ofa graph G and a function : E(G) → Z2, Z2 ={1, −1}. We call G the underlying graph of G and the signing of G. An edge e of G is said to be positive (resp. negative) if (e) = 1 (resp. (e) =−1). A signed graph G is balanced ifits vertex set can be partitioned into two partite sets such that a positive edge has the ends in the same partite set, and a negative edge has the ends in different partite sets. Notice that a signed graph G is balanced ifand only ifevery closed walk in G contains even number ofnegative edges, i.e., the value e∈W (e) = 1 for every closed walk W in G. 44 I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55

A graph map h : G → H between two signed graphs is a graph map between their underlying graphs G and H which preserves the signing ofeach edge, i.e., (h(e)) = (e) e ∈ E(G) G˜ G for each . A signed graph ˜ is called a covering ofanother with projection p : G˜ → G p : G˜ → G ◦ p = ˜ p : G˜ → G ˜ if is a covering and . Two coverings ˜ ˜ and p : G → G ofa signed graph G are isomorphic ifthe underlying coverings ˜ p : G˜ → G and p : G˜ → G ofthe graph G are isomorphic. By a method similar to the construction ofa covering graph in [2], one can see that every covering ofa signed graph can be derived froma reduced voltage assignment of p : G˜ → G G. That is, for a covering ˜ , there exists a reduced voltage assignment of G such that the derived covering p : G → G is isomorphic to p˜ : G˜ → G. We deﬁne a signing : E(G ) →{−1, 1} by (ei) = (e) for each ei ∈ E(G ). Then, the covering p : (G ) → G p : G˜ → G is isomorphic to ˜ . In [11], Kwak et al. enumerate the isomorphism classes ofregular balanced coverings of a signed graph and those ofregular bipartite coverings ofa graph [4–7,9]. As an application ofthe results, they also enumerate the isomorphism classes ofregular branched orientable surface coverings of a non-orientable surface having a ﬁnite abelian covering transformation group. It gives a partial answer for the question raised by Liskovets in [13]. In Section 2, we discuss an enumeration method for the isomorphism classes of (regular or not) balanced coverings ofan unbalanced graph. In Section 3, we enumerate the isomorphism classes ofbalanced coverings ofa signed graph. As an application ofour results, we enumerate the isomorphism classes ofbipartite coverings over a non-bipartite graph G and in Section 4, we obtain an enumeration formula for the isomorphism classes of branched or unbranched orientable surface coverings of a non-orientable surface.

2. A characterization of balanced coverings

Let Isoc(G; n) denote the number ofthe isomorphism classes ofconnected n-fold coverings ofa graph G. Similarly, Isoc(G; n) denotes the signed ones. Notice that Isoc(G; n) = B Isoc(G; n) for any signing on a graph G. Let Isoc (G; n) denote the number ofthe isomorphism classes ofconnected balanced n-fold coverings of a signed graph G. Notice that if G is balanced, then every covering of G is also balanced, which gives B Isoc (G; n) = Isoc(G; n) = Isoc(G; n) for any natural number n. We also notice that every odd-fold covering of an unbalanced graph is also unbalanced. Therefore, we restrict our discussion to the even-fold coverings of an unbalanced graph G. We observe that there are some analogous properties between the balancedness and bipartiteness. For example, a graph is bipartite ifand only ifevery cycle has an even length, and analogously a signed graph is balanced ifand only ifevery cycle contains an even number ofnegative edges. Some ofsuch analogous properties are listed in Table 1. From the correspondence in Table 1, one can image that the enumeration method for the bipartite coverings of a graph used in [1] can be applied to derive some enumeration formulae for the balanced coverings ofa signed graph. Let G be an unbalanced graph and let T be a ﬁxed spanning tree of G. A cotree edge e in E(G) − E(T ) is said to be unbalanced (resp., balanced)ifT + e has an unbalanced I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 45

Table 1 A relationship

Graph G Signed graph G

Bipartiteness ←→ balancedness Even cycle ←→ balanced cycle Odd cycle ←→ unbalanced cycle

B U (resp., balanced) cycle. Let ET (G) (resp. ET (G)) denote the set ofall balanced (resp. U unbalanced) cotree edges in E(G) − E(T ). Let U(G, T ) =|ET (G)| and B(G, T ) = B |ET (G)|, so that (G) = B(G, T ) + U(G, T ). For convenience, let P2n be the set ofall permutations in the symmetric group S2n which preserve the parity in {1, 2,...,2n}, and let R2n be the set ofall permutations in S2n which reverse the parity in {1, 2,...,2n}. Then PR2n := P2n ∪ R2n is a subgroup of S2n and 2 |PR2n|=2(n!) . Let : D(G) → S2 ={1, = (12)} be a reduced voltage assignment deﬁned by (e) = 1 B U for each e ∈ ET (G) and (e) = for each e ∈ ET (G). Then the derived double covering graph (G ) is balanced. By a method similar to the proofofTheorem 3.1 in [1], one can show that every balanced covering ofthe unbalanced graph G is a covering ofthe double covering graph (G ) as a signed graph. More precisely, for any balanced p : G˜ → G G q : G˜ → (G ) covering ˜ of there exists a covering map ˜ such that p = p ◦ q. We call (G ) the canonical balanced double covering of G. Notice −1 −1 −1 −1 −1 −1 that p (v) = q ((p ) (v)) = q (v1) ∪ q (v2). Now, by labelling q (v1) ={vi : −1 i = 1, 3,...,2n − 1} and q (v2) ={vi : i = 2, 4,...,2n}, one can have the following theorem.

Theorem 1. Let G be an unbalanced graph and let T be a spanning tree of G. Let p : G˜ → G n ˜ be an even-fold (say,2 ) connected balanced covering. Then there exists a reduced transitive voltage assignment : D(G) → PR2n such that (e) ∈ P2n if e ∈ EB(G ) (e) ∈ R e ∈ EU(G ) p : (G ) → G T , 2n if T , and is isomorphic to p : G˜ → G ˜ .

B Let CT (G, 2n) denote the set ofall reduced transitive voltage assignments : D(G) → B U PR2n such that (e) ∈ P2n if e ∈ ET (G) and (e) ∈ R2n if e ∈ ET (G).

Theorem 2. Let G be an unbalanced graph and let T be a ﬁxed spanning tree of G. CB(G , n) (G ) (G) Let and be two voltage assignments in T 2 . Then and are isomorphic as coverings if and only if there exists a permutation ∈ PR2n such that (e) = ◦ (e) ◦ −1 for all e ∈ D(G − T).

(G ) (G) G G Proof. It is clear that and are isomorphic ifand only if and are isomorphic as coverings. Since and are reduced, by Theorem 2 in [8], G and G are isomorphic as coverings ifand only ifthere exists a permutation ∈ S2n such that 46 I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55

(e) = ◦ (e) ◦ −1 for all e ∈ D(G − T). Since G and G are connected and , B are elements of CT (G, 2n), such an should be an element of PR2n. This completes the proof.

B We observe that CT (G, 2n) can be identiﬁed with the set ofall transitive (G)-tuples ( ,..., , ,..., ) (P )B(G,T ) × (R )U(G,T ) 1 B(G,T ) B(G,T )+1 (G) in 2n 2n . Deﬁne an B PR2n-action on CT (G, 2n) by · ( ,..., , ,..., ) 1 B(G,T ) B(G,T )+1 (G) = ( −1,..., −1, −1,..., −1) 1 B(G,T ) B(G,T )+1 (G) for any ∈ PR2n. From Theorem 2 and the Burnside lemma, we have the following.

Corollary 3. The number of the isomorphism classes of connected balanced 2n-fold coverings of an unbalanced graph G is equal to B 1 Isoc (G, 2n) = |Fix|, 2(n!)2 ∈PR2n B where Fix ={ ∈ CT (G, 2n) : · = }.

3. Enumeration of balanced coverings

B In this section, we shall give an enumerating formula for Isoc (G; 2n). To do this, we start with the following. With a ﬁxed spanning tree T, let CT (G, n) denote the set ofall reduced transitive voltage assignments : D(G) → Sn. For a convenience, we identify this set CT (G, n) with the (G) set ofall transitive (G)-tuples ( 1,..., (G)) in (Sn) , where the transitivity of (G)- tuples means that the subgroup generated by { 1,..., (G)} acts transitively on the set {1, 2,...,n}. Let SA(n) denote the number ofsubgroups ofindex n in a group A and let tm, denote the number oftransitive -tuples in (Sm) . Then, the number ofsubgroups ofindex n in the fundamental group of the graph G, which is a free group F(G) on (G) |C (G, n)|/(n − )! S (n) = t /(n − )!, elements, is equal to T 1 , i.e., F(G) n,(G) 1 (see [12]).

Lemma 4. (1) For any two natural numbers m, 1, we have m−1 m − 1 tm, = (m!) − ((m − j)!) tj,, j − 1 j=1 where the summation over the empty index set is deﬁned to be 0. B (2) Let G be an unbalanced graph. Then |CT (G, 2n)|=n! tn,2(G)−1.

S (n) Proof. (1) comes from the Hall’s recursive formula in [3] for F(G) and the relation S (n) = t /(n − )! F(G) n,(G) 1 . I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 47

S (n) S (2) The number F2(G)−1 can be considered in two ways: ﬁrst, the equality F2(G)−1 (n)=tn,2(G)−1/(n−1)! is known already as in (1). On the other hand, this number is equal to the number ofsubgroups on index n in the fundamental group of the canonical double covering (G ) , because (G ) = 2(G) − 1. Furthermore, those subgroups ofthe fundamental group of (G ) are in one-to-one correspondence to the subgroups ofindex 2 n in the fundamental group of G which correspond to the connected balanced 2n-fold coverings S (n) =|CB(G , n)|/(n − )!n! of G by Theorem 1. Hence, we have the equality F2(G)−1 T 2 1 by noting that the number ofelements in PR2n which ﬁx the symbol 1 is (n − 1)!n!.Now, a comparison ofthe two equalities gives the proof.

Now, we aim to compute |Fix| for an ∈ PR2n. First, note that an has a fixed point ifand only if is a regular permutation, which means by definition that is a product B ofdisjoint 2 n/ cycles oflength . Moreover, a voltage assignment ∈ CT (G, 2n) is (e) Z () fixed by ifand only ifforeach cotree arc e, belongs to the centralizer S2n of in S2n. We recall that is oftype if, in the cycle decomposition of , every cycle has length . For such an , it is well-known that the centralizer Z() ofsuch an in S2n can be represented as the wreath product Z S2n/, where Z is the cyclic group oforder . Notice that each element a ∈ Z S2n/ can be written in the form a =(c1,c2,...,c2n/;ã), c ,...,c ∈ Z a˜ ∈ S ∈ (e) ∈ Z () where 1 2n/ and 2n/. For a Fix, so that S2n for each ˜ e ∈ D(G) − D(T ). For such a ∈ Fix, the function : D(G) → S2n/ define by ˜ (e) = (e) is also a voltage assignment. First, let ∈ P2n and let be a product ofdisjoint 2 n/ cycles oflength for some |2n. Since each ofthese cycles of consists ofonly either odd numbers or even numbers, 2n/ must be even and hence is a divisor of n. Moreover, Z() ∩ P2n = Z (Sn/ {1}), Z()∩R2n=Z(Sn/S2)−Z(Sn/{1}), and Z()∩PR2n=(ZSn/)S2=Z(Sn/S2). P m Let Fix ( ) denote the set ofall reduced voltage assignments : D(G) → Z (Sm S2) B such that (e) ∈ Z (Sm {1}) if e ∈ ET (G), (e) ∈ Z (Sm S2) − Z (Sm {1}) U ˜ B B if e ∈ ET (G), and ∈ CT (G, 2m). Recall that CT (G, 2m) is the set ofall reduced transitive voltage assignments such that the 2m-fold covering G of G is balanced. Let ˜ P ∈ CB(G , n) Fixn/ be the set ofall voltage assignments T 2 which is fixed by . Then, by a method similar to Liskovets [12], we have the following.

Next, let ∈ R2n. Then is a product ofdisjoint 2 n/ cycles oflength . Since each cycle of reverses the parity, = 2 for some divisor of n, i.e., is a product of disjoint n/ cycles oflength 2 for some divisor of n. Moreover, Z() ∩ P2n =2Sn/, Z() ∩ R2n = (Z −2) Sn/. 48 I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55

R m Let Fix ( ) denote the set ofall reduced voltage assignments : D(G) → Z2 Sm B U such that (e) ∈2Sm if e ∈ ET (G), (e) ∈ (Z −2) Sm if e ∈ ET (G), and ˜ ˜ ∈ CT (G; m) such that G is unbalanced. Notice that the number |{ ∈ CT (G; m) : G is balanced}| is0ifm is odd and (m − 1)!tm/2,2(G)−1/(m/2 − 1)! if m is even. Let R ∈ CB(G , n) Fixn/ be the set ofall voltage assignments T 2 which is ﬁxed by . Then, by a slight modiﬁcation ofthe method in Liskovets [12], we have the following.

tm,(G) if misodd, t− = t m,(G) m/2,2(G)−1 tm,(G) − (m − 1)! if miseven. ((m/2) − 1)! B Now, we are ready to estimate the number Isoc (G; 2n).

Theorem 7. Let G be an unbalanced graph. Then  n t B(G ; n) = 1  d2((G)−1)+1 ,2(G)−1 Isoc 2 n d ( − )! 2 |n d|(n/) 1  t− n ((G)− )+ ,(G) + d 1 1  , d ( − 1)! d|(n/),n/(d):odd where is the Möbius function.

Proof. Let be a permutation which can be expressed as a product ofdisjoint 2 n/ cycles of n/ 2 length , and is a divisor of n.Then, the number ofsuch elements in P2n is (n!/(n/)! ) . Let be a permutation which can be expressed as a product ofdisjoint n/ cycles oflength 2 n/ 2 for a divisor of n. Then, the number ofsuch elements in R2n is (n!) /(n/)! .By Corollary 3, we have   2 2 B 1  n! P (n!) R  Isoc (G; 2n) = |Fix n/ |+ |Fix n/ | . (n!)2 (n/)! n/ (n/)! n/ 2 |n |n

Now, by Lemmas 5 and 6, we have the theorem.

B In Table 2, we list the numbers Isoc(G; 2n) and Isoc (G; 2n) for small n and small = (G). I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 49

Table 2 B Two numbers Isoc(G; 2n) and Isoc (G; 2n)

n 12 31 2 3 Isoc 3 26 624 15 14120 371515454 IsocB 1 5 24 1 71 23778

As a direct application ofour results, one can enumerate the isomorphism classes of bipartite 2n-fold connected coverings over a non-bipartite graph G. Let G be a graph and let : E(G) →{1, −1} be a signing deﬁned by (e) =−1 for all e ∈ E(G). Then, G is bipartite ifand only if G is balanced. Hence, we have the following.

B Corollary 8. Let G be a connected non-bipartite graph. Then, for any n, Isoc (G ; n) is equal to the number of the isomorphism classes of bipartite n-fold connected coverings of G.

4. Applications to orientable branched coverings

In this section, with an aid of the enumeration formula for the connected balanced coverings ofa signed graph, we enumerate the isomorphism classes ofbranched orientable surface coverings of a non-orientable surface. It gives an answer for the following question raised by Liskovets in [13]: count unramiﬁed orientable coverings ofa non-orientable surface. A surface means a compact connected 2-manifold S without boundary. A continuous function p : S˜ → S from a surface S˜ onto another S is called a branched covering if S S˜ − p−1(B) p| : there exists a ﬁnite set B in such that the restriction of p to , S˜ −p−1(B) S˜ − p−1(B) → S − B, is a covering projection in a usual sense. The smallest subset B of S which has this property is called the branch set. Two branched coverings p : S˜ → S and q : S˜ → S are isomorphic ifthere exists a homeomorphism h : S˜ → S˜ such that p = q ◦ h. An embedding ofa graph G into a surface S is a homeomorphism i : G → S of G into S. Ifevery component of S − i(G), called a region, is homeomorphic to an open disk, then i : G → S is called a 2-cell embedding. A rotation system for a graph G is an assignment ofa cycle permutation v on the neighborhood N(v)={e ∈ D(G) : ie = v} to each v ∈ V (G).Anembedding scheme ( , ) for a graph G consists ofa rotation scheme and a signing , where : E(G) →{1, −1}. It is well-known that every embedding scheme for a graph G determines a 2-cell embedding of G into an orientable or non-orientable surface, and every 2-cell embedding of G into a surface is determined by such a scheme (see [15,16]). The sign (e) for an edge e depends on the twistedness or untwistedness ofthe collar-neighborhood ofthe edge e in a surface S. 50 I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55

Ifan embedding scheme ( , ) for a graph G determines a 2-cell embedding of G into a surface S, then the orientability of S can be detected by looking at the balancedness of cycles of G. In fact, S is orientable ifand only ifevery cycle of G is balanced. Let i : G → S be a 2-cell embedding with the embedding scheme ( , ), and let :

D(G) → Sn be a voltage assignment. The derived graph G has the derived embedding ( , ) ( ) (e ) = ( (e)) (e ) = (e) scheme , which is deﬁned by vi i v i and i for each ˜ ei ∈ D(G ). Then, the embedding scheme ( , ) determines a 2-cell embedding i :

G → S ofthe derived graph G into a surface, say S . Moreover, if G is connected, then S is connected. It is a fact [2] that the surface S is a branched n-fold covering of the surface S, which is said to be induced by an embedding i : G → S and an voltage assignment : D(G) → Sn.Itisknown[2] that every branched n-fold covering of a surface is isomorphic to a surface branched covering induced by a suitable 2-cell embedding of a graph with suitable voltage assignment on it. Let Nk be a non-orientable surface with k crosscaps, and let Bm denote the graph consisting ofone vertex and m self-loops, say 1,...,m. We call it the bouquet of m circles or simply, a bouquet. As in [7], we construct the standard embedding Bb+k &→ Nk − B ofthe bouquet Bb+k into the non-orientable surface Nk, where b =|B|, as follows: an embedding scheme ( , ) for the standard embedding Bk+b &→ Nk − B is deﬁned by

= ( −1 −1 ... −1 ) v 1 1 2 2 k+b k+b , where v is the vertex ofthe bouquet Bk+b, with the standard signing −1ifs = 1, 2,...,k, (s) = 1ifs = k + 1,k+ 2,...,k+ b.

For example, we consider the standard embedding (B3, )&→ N2 − B with one branch =( −1 −1 −1) ( )=( )= point set B; its embedding scheme is given by v 1 1 2 2 3 3 , 1 2 −1 and (3) = 1. Note that the loops 1 and 2 are embedded as the polygonal boundary in the polygonal representation ofthe Klein bottle N2, which are corresponded to the center lines oftwo Möbius bands attached to the sphere with 2 holes to make 2 crosscaps in the Klein bottle, as in Fig. 1.

l1 l1 -1 l l l3 1 3

v l3 l1 l2 l 1 1 l2 v l -1 2 v l2 l2

Fig. 1. An embedding scheme for (B3, )&→ N2 − B. I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 51

O For a b-subset B ofthe surface Nk, let B ((Bb+k, )&→ Nk − B; 2n) denote the set of voltage assignments : D(Bb+k) → S2n which satisfy the following two conditions:

(a) is transitive and (B , ) is balanced, b+k ( ) = i = k + ,...,k+ b k ( ) ( ) |B| ( ) = (b) i 1 for each 1 and i=1 i i i=1 k+i 1.

Note that condition (a) guarantees that the surface Nk is connected and orientable, and condition (b) does that the set B is the same as the branch set ofthe branched covering p˜ : Nk → Nk. Furthermore, we have the following as in [10]: every voltage O assignment in B ((Bb+k, )&→ Nk − B; 2n) induces a connected branched orientable 2n- fold covering of the surface Nk with branch set B. Conversely, every connected branched orientable 2n-fold covering of Nk with branch set B can be derived from a voltage as- O signment in B ((Bb+k, )&→ Nk − B; 2n). Moreover, for any two voltage assignments O , ∈ B ((Bb+k, )&→ Nk − B; 2n), two branched 2n-fold surface coverings p˜ : Nk → Nk and p˜ : Nk → Nk are isomorphic ifand only iftwo graph coverings p : Bb+k → Bb+k and p : Bb+k → Bb+k are isomorphic. Equivalently, there exists −1 a permutation ∈ S2n such that (i) = (i) for all i ∈ D(Bb+k). It means that O O Isoc (Nk,B; 2n) =|B ((Bb+k, )&→ Nk − B; 2n)/S2n|. Now, by using a method similar to the proofofTheorem 2 in [10], we have the following.

Theorem 9. Let Nk be a non-orientable surface of genus k and let B be a b-subset of Nk. Then, every connected branched orientable covering of Nk must be even-fold, and for any n, the number of the isomorphism classes of connected branched orientable 2n-fold coverings of Nk with branch set B is

b−1 O b O t B Isoc (Nk,B; 2n) = (−1) Isoc (Nk, ∅; 2n) + (−1) Isoc ((Bb+k−t−1, ); 2n), t=0 O where Isoc (Nk, ∅; 2n) is the number of the isomorphism classes of connected unbranched 2n-fold orientable coverings of Nk.

B Since the term Isoc ((Bb+k−t−1, ); 2n) is already computed in Theorem 7, we need to O compute only the term Isoc (Nk, ∅; 2n). O (N ) To simplify this computation, we introduce a new voltage set. Let B2n k be the subset O of B ((Bb+k, )&→ Nk −∅;2n) consisting ofall transitive voltage assignments : D(B ) → PR ( ) ∈ R i = , ,...,k k ( )2 = k 2n such that i 2n for each 1 2 and i=1 i 1. O (N ) n Then, by Theorem 2, B2n k contains all representatives ofconnected orientable 2 -fold coverings of Nk and O (N , ∅; n) =| O ((B , )&→ N −∅; n)/S |=| O (N )/PR | Isoc k 2 B b+k k 2 2n B2n k 2n . O Hence, the number Isoc (Nk, ∅; 2n) ofthe isomorphism classes ofconnected unbranched 2n-fold orientable coverings of Nk is equal to the number oforbits ofthe coordinatewise PR O (N ) conjugacy action ofthe group 2n on B2n k . In order to apply the Burnside lemma in 52 I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 this situation, we compute the number ofﬁxed points of for each element ∈ PR2n.To do this, we need the following lemma. Let Sk(m) denote the number ofsubgroups ofindex m in the fundamental group + 1(Nk, ∗) ofthe non-orientable surface Nk ofgenus k and let Sk(m) denote the number ofsubgroups ofindex m in the fundamental group of Nk whose corresponding covering surfaces are orientable.

Lemma 10 (Mednykh and Pozdnyakova [14]). Let k and m be any two natural numbers. Then m (−1)s+1 Sk(m) = m i i ...i , s 1 2 s i +i +···+i =m s=1 1 2 s i ,i ,...,i 1 2 s 1 where h! k−2 = , h f () ∈Dh () Dh is the set of all irreducible representations of the symmetric group Sh, and f is the degree of the representation , and m , + 0if m is odd Sk (m) = S (k− ) if m is even. 2 1 2

Let ∈ PR2n such that |Fix| = 0. If ∈ P2n, then Z() ∩ P2n = Z (Sn/ {1}) and Z() ∩ R2n = Z (Sn/ S2) − Z (Sn/ {1}) for some |n. ¯ m Let F( ) be the set ofall transitive reduced voltage assignments : D(Bk) → Z() ∩ R ˜ ∈ BO (N ) 2n such that 2n/ k . Then, by a method similar to that used in Section 3, one can see that n/− n/ |Fix|= (d) d 2 1 |F¯ ((/d) )|. d| | O (N )|/(n− )! n!=S+( n)=S (n) Now, by Theorem 3.1 in [14] and the fact that B2n k 1 k 2 2(k−1) , one can have (2n/)(k−1)+1 n n n |F¯ ((/d)n/)|= − ! ! S d 1 2(k−1) . Hence, we have the following.

Lemma 11. Let ∈ P2n. Then |Fix| = 0 if and only if is a regular permutation which is a product of disjoint 2n/ cycles of length for some |n. For such an   (n/)(2k−3)+1 n n n | |= (d) d n/−1 − ! S  ! n/ Fix d 1 2(k−1) , d| where is the Möbius function. I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 53

˜ m Let ∈ R2n. Then Z()∩R2n =(Z2 −2)Sn/. Let F( ) be the set ofall transitive reduced voltage assignments : D(G) → Z2 Sm such that (e) ∈ (Z2 −2) Sm, and ˜ 1 1 ∈ CN (k; m), where CN (k; m) is the set ofall transitive voltage assignments which induce non-orientable coverings. Then, by a method similar to that used in Section 3, one can see that n/− n/ |Fix|= (d)d 1 |F˜ ((/d) )|. d|,d:odd

Now, by a slight modiﬁcation ofthe proofofTheorem 3.1 in [14] and the fact that 1 + |CN (k; m)|=(m − 1)! Sk (n/) − Sk (n/) , one can have (n/)(k−1) n |F˜ ((/d)n/)|= − ! d 1 (kn/)(/d−1) 1 + (−1) n + n × gcd 2, Sk − S . 2 d k Hence, we have the following.

Lemma 12. Let ∈ R2n. Then, |Fix| = 0 if and only if is a regular permutation which is a product of disjoint n/ cycles of length 2 for some |n. For such an , (n/)(k−1) n | |= (d) d n/−1 − ! Fix d 1 d|,d:odd (kn/)((/d)−1) 1 + (−1) n + n × gcd 2, Sk − S , 2 d k where is the Möbius function.

Now, by the Burnside lemma, we have the following.

Theorem 13. Let Nk be a non-orientable surface of genus k. Then, every connected unbranched orientable covering of Nk must be even-fold, and for any n, the number of the isomorphism classes of connected unbranched 2n-fold orientable coverings of Nk is O (N , ∅; n) Isoc k 2 n n 1 2(k−2)+2 (k−2)+1 =  d S (k− )()+ d 2n d 2 1 d |n d|(n/) d|(n/),n/(d):odd k(d−1) 1 + (−1) + × gcd(2,d) Sk() − S () , 2 k

+ where is the Möbius function, and Sk() and Sk () are the numbers mentioned in Lemma 10. 54 I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55

Table 3 O The number Isoc (Nk, ∅; 2n)

nN1 N2 N3 N4

11 1 1 1 20 3 9 39 3 0 3 57 1483 4 0 6 847 354009 5 0 4 15303 208211284

For example, if n is a prime p, then the number ofthe isomorphism classes ofconnected unbranched 2p-fold orientable coverings of Nk is  S ( ) + S ( ) + 2k−2  2(k−1) 2 k 2 2 + k−1( + (− )k) − p = , O (N , ∅; p) = 1 2 1 1 2if2 Isoc k 2 2k−2 2p S2(k−1)(p) + Sk(p) + p +pk−1 − 2 otherwise. In particular, the number ofthe isomorphism classes ofconnected unbranched 2 p-fold O orientable coverings ofthe Klein bottle N2 is Isoc (N2, ∅; 2p) =(p + 3)/2.InTable 3, O we compute the number Isoc (Nk, ∅; 2n) for small n and k.

Acknowledgements

The authors would like to express their gratitude to the referee for valuable comments.

References

[1] D. Archdeacon, J.H. Kwak, J. Lee, M.Y. Sohn, Bipartite covering graphs, Discrete Math. 214 (2000) 51–63. [2] J.L. Gross, T.W. Tucker, Topological Graph Theory, Wiley, New York, 1987. [3] M. Hall Jr., Subgroups offinite index in freegroups, Canadian J. Math. 1 (1949) 187–190. [4] M. Hofmeister, Graph covering projections arising from finite vector space over finite fields, Discrete Math. 143 (1995) 87–97. [5] S. Hong, J.H. Kwak, J. Lee, Regular graph coverings whose covering transformation groups have the isomorphism extention property, Discrete Math. 148 (1996) 85–105. [6] J.H. Kwak, J.-H. Chun, J. Lee, Enumeration ofregular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998) 273–285. [7] J.H. Kwak, S. Kim, J. Lee, Distribution ofregular branched prime-foldcoverings ofsurfaces,Discrete Math. 156 (1996) 141–170. [8] J.H. Kwak, J. Lee, Isomorphism classes ofgraph bundles, Canad. J. Math. XLII (1990) 747–761. [9] J.H. Kwak, J. Lee, Enumeration ofconnected graph coverings, J. Graph Theory 23 (1996) 105–109. [10] J.H. Kwak, J. Lee, A. Mednykh, Enumerating branched surface coverings from unbranched ones, LMS J. Comput. Math. 6 (2003) 89–104. [11] J.H. Kwak, J. Lee, Y. Shin, Balanced regular coverings ofa signed graph and regular branched orientable surface coverings over a non-orientable surface, Discrete Math. 275 (2004) 177–193. [12] V. Liskovets, On the enumeration subgroups ofa freegroup, Dokl. Akad. Nauk. BSSR 15 (1971) 6–9. I.P. Goulden et al. / Discrete Mathematics 303 (2005) 42–55 55

[13] V. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta-Appl. Math. 52 (1998) 91–120. [14] A. Mednykh, G. Pozdnyakova, Number ofnonequivalent coverings over a nonorientable compact surface, Sib. Math. Z. 27 (1983) 123–131. [15] S. Stahl, Generalized embedding schemes, J. Graph Theory 2 (1978) 41–52. [16] A.T. White, Graphs, Groups and Surfaces, North-Holland, New York, 1984.