Covalence sequences of planar vertex-homogeneous maps

Transcription

1 Discrete Mathematics 307 (2007) Covalence sequences of planar vertex-homogeneous maps Jana Siagiová a, Mar E. Watins b a Department of Mathematics, Slova University of Technology, Ralinsého 11, Bratislava, Slovaia b Department of Mathematics, Syracuse University, Syracuse, NY, USA Receive 5 October 2005; accepte 13 July 2006 Available online 16 October 2006 Abstract Given a cyclic -tuple of integers at least 3, we consier the class of all 1-ene 3-connecte -valent planar maps such that every vertex manifests this -tuple as the (clocwise or counterclocwise) cyclic orer of covalences of its incient faces. We obtain necessary an/or sufficient conitions for the class to contain a Cayley map, a non-cayley map whose unerlying graph is a Cayley graph, a vertex-transitive graph whose subgroup of orientation-preserving automorphisms acts (or fails to act) vertex-transitively, a non-vertex-transitive map, or no planar map at all Elsevier B.V. All rights reserve. Keywors: Vertex-homogeneous; Vertex-transitive; Cayley map; Cayley graph; Covalence sequence 1. Introuction an preliminaries Two frequently investigate topics in the stuy of infinite planar maps are the geometry of tilings an the symmetries of such maps (see notably the wor of Grünbaum an Shephar [4] an of Graver an Watins [2]). Let us at the outset establish that unless explicitly state otherwise, throughout this article the term map means a 3-connecte, 1-ene, locally finite planar map. Maps may be finite or infinite, an we o not istinguish between whether they are embee in the Eucliean plane or in the hyperbolic plane (though most frequently we will be ealing with the latter). It is well nown that for such a map, any automorphism of its unerlying graph is extenable to a homeomorphism of the plane. In an intuitive sense, the most symmetric planar maps are the regular tessellations. These are the planar tessellations all of whose vertices have the same valence an all of whose faces have the same covalence. For such maps the group of orientation-preserving automorphisms acts regularly on the set of arts (eges with preassigne irection), an hence the regular tessellations are also both vertex- an face-transitive. Moreover, each pair of integers an etermines a unique map [5]. A broaer class of infinite planar graphs are those that are merely ege-transitive; they have been classifie in etail by Graver an Watins [2] in both the 1-ene an the multi-ene cases. Such maps are ege-homogeneous, in the sense that every ege manifests the same pair of values (possibly equal) of its incient vertices an, ually, of its incient faces. Grünbaum an Shephar [5] showe for the 1-ene case that ege-homogeneous maps are uniquely etermine by these pairs of values an that ege-homogeneity implies ege-transitivity. Analogous results are partly true in the 2-ene case [21] but fail in the infinite-ene case [2]. (Combining results of Halin [6] an Jung [8] yiels that the number of ens of an almost-transitive, infinite, locally finite connecte graph is exactly 1, 2, or 2 ℵ 0.) aresses: siagiova@math.s (J. Siagiová), mewatin@syr.eu (M.E. Watins) X/$ - see front matter 2006 Elsevier B.V. All rights reserve. oi: /j.isc

2 600 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) With the goal of measuring the growth of tessellations of the hyperbolic plane, Moran [12] consiere certain classes of maps, or tilings, that are face-homogeneous, meaning that the cyclic sequence of valences of incient vertices is the same about every face. In particular, she investigate in etail the regular tessellations mentione above an all 3-covalent tessellations. The investigation of the present paper concerns the ual situation to that of Moran, namely vertex-homogeneous tessellations, an we classify their symmetry accoring to a schema escribe below. Let us give some precise efinitions. Given a -tuple σ = ( 0, 1,..., 1 ), its reverse is the -tuple σ 1 = ( 1,..., 1, 0 ). Definition 1.1. Let M be a map an let v be a -valent vertex of M. Acovalence sequence at v is a cyclic -tuple ( 0, 1,..., 1 ) of integers 3 that lists in cyclic orer the covalences of the faces incient with v as one procees aroun v in either the clocwise or counterclocwise irection. Thus we mae the ientification ( 0, 1,..., 1 ) ( 1,..., 1, 0 ). If a given cyclic -tuple σ has the property that for every vertex v of M either σ or σ 1 is the clocwise covalence sequence of v, then we say that M is vertex-homogeneous an that M has covalence sequence σ. Notation: Ifσ is a cyclic -tuple, then σ = β t means that σ is forme by catenating t 1 copies of some b-tuple β, where b. For a map M, weletv (M), E(M), F (M), Aut(M), an Aut + (M) enote, respectively, the vertex set, the ege set, the face set, the automorphism group, an the subgroup of orientation-preserving automorphisms of M. We say that M is orientably vertex-transitive if Aut + (M) acts transitively on V(M). Note that the inex [Aut(M) : Aut + (M)] always equals 1 or 2. Following [18], one may efine a Cayley map to be a map such that some subgroup of Aut + (M) acts regularly on V(M). Thus, the unerlying graph of a Cayley map is a Cayley graph (cf. Proposition 1.1), an all vertices have the same covalence sequence in the clocwise irection. (An equivalent formulation of this notion will be given at the beginning of the next section.) Definition 1.2. A vertex-homogeneous map M is at: Level 1: If M is not vertex-transitive. Level 2: If M is vertex-transitive but not orientably vertex-transitive an the unerlying graph of M is not a Cayley graph. Level 3A: If the unerlying graph of M is a Cayley graph but M is not orientably vertex-transitive. Level 3B: If M is orientably vertex-transitive but the unerlying graph of M is not a Cayley graph. Level 4: If M is a Cayley map. Let M(σ) enote the class of all -valent vertex-homogeneous maps M such that the cyclic sequence σ is a covalence sequence at every vertex of M. In these terms we can now state the main thrust of our investigation. General Problem. Given a cyclic sequence σ, etermine the levels (an how many at each level) of the maps in M(σ). To roun out the above classification, we nee Level 0: M(σ) is empty, i.e., σ is not realizable by a map. Asie from Level 0, M(σ) may contain maps at more than one level as well as many ifferent maps at the same level. This article is organize as follows. In Section 2 we characterize the covalence sequences of Cayley maps (Level 4). This result combine with a truncation technique will be use in Section 3 to obtain a corresponing characterization of orientably vertex-transitive maps (Level at least 3B). Some properties of covalence sequences of vertex-transitive maps in general (Level at least 2) are given in Section 4. Covalence sequences of finite vertex-transitive maps are iscusse in Section 5. Section 6 contains a sample of sufficient conitions an necessary conitions for non-realizability (Level 0) of a covalence sequence. Following some Concluing Remars is an Appenix in tabular form completely classifying maps of valences 3 5. This introuctory section conclues with some statements of nown results that will be invoe in the course of our wor.

3 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Proposition 1.1 (Sabiussi [15]). A graph Γ is a Cayley graph of a group G if an only if there is a subgroup of Aut(Γ) isomorphic to G that acts regularly on V(Γ). Proposition 1.2 (Grünbaum an Shephar [5]). There exists an ege-homogeneous (finite or 1-ene) map with valences 1 an 2 an covalences 1 an 2 if an only if 1, 2, 1, 2 are integers 3 an exactly one of the following hols: (1) all of 1, 2, 1, 2 are even; (2) 1 = 2 an at least one of 1, 2 is o; (3) 1 = 2 an at least one of 1, 2 is o; (4) 1 = 2, 1 = 2, an all are o. Such a map is ege-transitive an is uniquely etermine by the parameters 1, 2, 1 an 2. If 1 = 2, then it is vertex-transitive. If 1 = 2, then it is face-transitive. Proposition 1.3 (Richter et al. [14] Theorem 2.2). A -valent map M on an orientable surface is a Cayley map if an only if M is a regular covering space of a single-vertex -valent map M 1 on some orientable surface, or equivalently, M can be obtaine as a lift of M 1 by a suitable voltage assignment on the art set of M 1 in some group. Proposition 1.4 (Richter et al. [14] Theorem 10.1). Let an be positive integers 3 an let M have covalence sequence (). The following are equivalent: (1) M is a Cayley map. (2) The unerlying graph of M is a Cayley graph. (3) The integer has a prime ivisor p such that p. 2. Covalence sequences of Cayley maps (Level 4) We assume that the reaer is familiar with basic notions from the theory of graphs an maps incluing, in particular, (finite or infinite) planar maps an vertex- an ege-transitivity of maps. As Cayley maps are a more recent notion, we review for completeness some of the terminology from [14]. Let G be a group with ientity ι, an let X = (x i i Z ) be a cyclic sequence of generators of G (calle henceforth a cyclic generating sequence) together with an involution τ on Z such that x τ(i) = xi 1 ; the involution τ is calle the istribution of inverses. ACayley map CM(G, X, τ) may then be efine in these terms as a map on some orientable surface whose vertex set is G an whose art set is {(g, i) : g G; i Z }. The reverse of the art (g, i) is the art (gx i, τ(i)). A pair of (mutually) reverse arts forms an ege (or, exceptionally, a semi-ege if x i = ι an τ(i) = i). At each vertex g G the clocwise cyclic orer of arts at g is simply ((g, 0), (g, 1),...,(g, 1)). Thus in a Cayley map, the clocwise cyclic orering of generators emanating from a vertex is the same at each vertex. Left multiplication by elements of G inuces a transitive action of G on itself as a group of orientation-preserving map automorphisms of CM(G, X, τ); i.e., Cayley maps are not only vertex-transitive but also orientably vertex-transitive. This explains why the incomparable Levels 3A an 3B fall strictly between Levels 2 an 4. Cayley maps as presente in the previous paragraph may have loops, multiple eges an semi-eges. This is important from the stanpoint of the theory of regular covering spaces for Cayley maps evelope in [14]. The reaer is referre to [3] for further etails concerning this theory an its equivalent formulation in terms of a voltage assignment from the art set into a group an the corresponing lift. As explaine in etail in [14], Cayley maps are regular covers of single-vertex maps with branch points possibly at angling ens of semi-eges an at face centers but with no branch point at the single vertex. In the sequel, M 1 will always enote a single-vertex map in some orientable surface. From [14] combine with Poincaré s polygon theorem (see [7]) it also follows that planar Cayley maps are regular lifts of such single-vertex maps M 1 where the efining relations for the voltage group are exactly those inuce by the faces of M 1. Single-vertex Cayley maps M 1 of valence can be conveniently escribe by a single involution on Z (as in [14]). First, label the arts of M 1 with elements of Z by assigning 0 arbitrarily to some art an then continue clocwise

4 602 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) as the arts appear aroun the single vertex of M 1. This labeling inuces an involution τ on the set Z such that (1) τ(i) = j if an only if i an j enote mutually reverse arts of the same loop an (2) τ(i) = i if i enotes a semi-ege. The map M 1 associate with τ is enote by M 1 (, τ). We note that M 1 can be escribe equivalently as a Cayley map of the trivial group, in which case the involution τ coincies with the istribution of inverses. From [14] it also follows that the faces of M 1 (, τ) are in one-to-one corresponence with the orbits (or cycles) of the permutation τ + of Z given by τ + (i) = τ(i) + 1. Inee, a face corresponing to an orbit of τ + is forme by corners that appear to the left of the arts that form the orbit. Note that, although both τ an τ + epen upon the selection of the art originally labele 0, their respective orbit structures are inepenent of this selection. For each i Z, let τ + (i) enote the length of the orbit of τ + that contains i. It follows immeiately that the covalence sequence σ τ of M 1 (, τ) is σ τ = ( τ + (0), τ + (1),..., τ + ( 1) ). We now give more etaile information about Cayley maps that cover this single-vertex map M 1 (, τ). Let {x i i Z } be a set of objects, which will eventually become a set of generators of a group G τ. Let O(τ + ) enote the set of cycles of the permutation τ +. To each cycle α = (i 1,...,i s ) in O(τ + ) we formally assign the wor w α = x i1 x i2...x is. Define the group G τ by G τ = x i,i Z w l α = ι, α O(τ + ), where each l α is a positive integer. Then, by [14], all Cayley maps covering M 1 (, τ) can be obtaine as lifts of M 1 with voltages in groups such as G τ. The covalence sequence of the lift of M 1 (, τ) in the group G τ is (l 0 τ + (0),l 1 τ + (1),...,l 1 τ + ( 1) ), where l i = l α if i appears in α. Informally, the covalence of a face corresponing to an orbit α O(τ + ) is multiplie l α times in the lift. (The existence of such lifts follows from Section 8 of [14].) Definition 2.1. A cyclic sequence σ = (c 0,c 1,...,c 1 ) is a τ-multiple of the cyclic sequence σ τ = ( τ + (0), τ + (1),..., τ + ( 1) ) if for each i Z there exists a positive integer l i such that c i = l i τ + (i) for each i Z, an l i = l j if i an j are in the same orbit of τ +. The foregoing analysis irectly implies the following: Theorem 2.1. A cyclic sequence σ of length is the covalence sequence of a Cayley map if an only if there exists an involution τ on the set Z such that σ is a τ-multiple of the cyclic sequence σ τ. Let us apply Theorem 2.1 to the case = 5. The reaer may wish simultaneously to refer to the first two columns of Table 2 in the Appenix. Example 2.1. We characterize all covalence sequences of Cayley maps of valence 5. Up to cyclic translation there are exactly six involutions τ i on Z 5, namely: τ 1 = (0)(1)(2)(3)(4), τ 2 = (12), τ 3 = (13), τ 4 = (12)(34), τ 5 = (13)(24), an τ 6 = (14)(23). The cycle ecompositions of the corresponing permutations τ + i are: τ + 1 = (01234), τ+ 2 = (0134)(2), τ + 3 =(014)(23), τ+ 4 =(013)(2)(4), τ+ 5 =(01432), an τ+ 6 =(01)(24)(3). The covalence sequences σ i=σ τi of the associate single-vertex maps M 1 (5, τ i ) (allowing cyclic shift) are σ 1 =σ 5 =(5, 5, 5, 5, 5), σ 2 =(4, 4, 4, 4, 1), σ 3 =(3, 3, 3, 2, 2), σ 4 = (3, 3, 1, 3, 1) an σ 6 = (2, 2, 2, 1, 2). The six maps M 1 (5, τ i ) are shown in Fig. 1; M 1 (5, τ 5 ) is toroial while the others are in the sphere. By Theorem 2.1, all possible covalence sequences of Cayley maps of valence 5 are τ-multiples of some σ i, namely (5j,5j,5j,5j,5j), (4j,4j,4j,4j,), (3j,3j,3j,2, 2), (3j,3j,,3j,l), an (2j,2j,2, l, 2), where j,,l enote arbitrary positive integers such that all terms are 3. It is significant to compare the above example with Theorem 9.1 of [14], which characterizes only sets of integers comprising all the terms in a covalence sequence of a Cayley map of a given valence. For example, for =5 a feasible

5 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 Fig. 1. One-vertex maps M 1 (5, τ i ). set is {3, 4, 6}. However, this implies neither the multiplicities nor the orer of the terms in a covalence sequence of a corresponing 5-valent Cayley map. It follows from Example 2.1 that, for example, the cyclic sequences (3, 3, 4, 3, 6) an (4, 4, 6, 3, 6) are realizable as Cayley maps while the sequence (3, 3, 3, 4, 6) is not. In fact, M(3, 3, 3, 4, 6) contains no map above Level 1. While Theorem 2.1 characterizes covalence sequences of Cayley maps, we have no analogous result for non-cayley maps whose unerlying graphs are Cayley graphs. However, all such maps of a given valence 3 an given covalence sequence σ can be constructe as follows. It is funamental that in any given planar embeing M of a 3-connecte Cayley graph Cay(G, X), for any element g G, left-multiplication λ g : v gv inuces a map-automorphism that preserves the color of each ege, in the sense that λ g maps the ege [v, vx] to the ege [gv, (gv)x] for all v G an x X. Since λ g λ x λ 1 g = λ gxg 1 is an orientation-preserving map-automorphism if an only if λ x is, we efine the generator x to be orientation-preserving or orientation-reversing accoringly. In particular, ajacent vertices v an vx have ientical clocwise covalence sequences if an only if the generator x is orientation-preserving. With these notions at han we may run through all sets X of generators that are close uner inverses. Taing into account which subsets of X (close uner inverses) consist of orientation-reversing generators, one may trace in a given irection each face bounary of M an recor the (cyclic) sequence of generators create as one moves from ege to ege aroun each face. The cyclic sequences thus note etermine the relations of the Cayley group. Due to vertex-transitivity, it suffices to consier only the faces incient with some common vertex. At this stage it remains to compare the covalence sequences obtaine by this proceure with the sequence σ. In this way we are able to ientify all possible ways perhaps none at all that σ can be realize by a map having a Cayley graph as an unerlying graph. If only orientation-preserving sets of generators are consiere, one obtains realizability of σ by Cayley maps. This metho is use later in the Appenix to complete the first two columns of the tables. If the cyclic sequence σ satisfies σ = σ 1, then we say that σ is reversible.ifis even, then the terms i an i+/2 are in opposite positions. Theorem 2.2. Let σ be a cyclic sequence of length 3. If σ is reversible an contains either at most one o term or exactly two o terms an they are in opposite positions, then M(σ) contains a Cayley map. Proof. Suppose that σ = ( 0, 1,..., 1 ) is reversible. Assume first that is o an, without loss of generality, that i = i for all i Z. Define τ on Z by τ(i) = 1 i for each i, 0 i 1. Then τ + (i) = i an therefore σ τ = (1, 2, 2,...,2). All τ-multiples of σ τ have the form (m 0, 2m 1, 2m 2,...,2m 1 ), where m i = m i. The conclusion now follows from Theorem 2.1. If is even then the argument procees in a similar fashion, where τ on Z is efine by τ(i) = i for the sequence (2) or by τ(i) = 1 i for the sequence (1, 2,...,2) 2. (The corresponing one-vertex maps are shown in Fig. 2.) Theorem 2.3. Let σ = (a 0,b,a 1,b,a 2,b,...,a r 1,b)be a cyclic sequence of length 2r whose terms are 3. (1) If r b, then σ is a covalence sequence of a Cayley map. (2) If there exists a -subset {i 0,i 1,...,i 1 } {0, 1,...,r 1} an a positive integer c such that a i0 = a i1 = =a i 1 = b = c(r + ), then M(σ) contains both a Cayley map an a non-cayley map whose unerlying graph is a Cayley graph. Proof. Let σ have the form (a 0,b,a 1,b,...,a r 1,b). Define τ to be the prouct of 2-cycles of the form (2i, 2i + 1) where 0 i r 1. Then, τ + (2i)=2i +2 an τ + (2i +1)=2i +1, yieling the cyclic sequence σ τ =(1,r,1,r,...,1,r). Part (1) now follows from Theorem 2.1.

6 604 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Fig. 2. One-vertex maps with σ τ = (1, 2, 2,...,2), (2), an (1, 2,...,2) 2. An equivalent way to approach this situation is to consier a one-vertex quotient map with r loops corresponing to a cyclic generating sequence X = (x 0,x0 1,x 1,x1 1,...,x r 1,xr 1 1 ), where each generator x i has orer 3. This interpretation is use to prove (2). If all the generators x i are orientation-preserving, then the single-vertex quotient map consists of r loops with isjoint interiors, an so its covalence sequences is (1,r,1,r,...,1,r). In this case the covalence sequence of the original Cayley map M must be of the form (a 1,cr,a 2,cr,...,a r,cr) for some positive integer c, as shown before Theorem 2.1. If exactly one generator, say x r 1, is orientation-reversing, then the outer face of the (non-orientable) quotient map has covalence r +1 an the covalence sequence of the quotient is (1, r +1, 1, r +1,...,1, r +1, 1, r +1, 1, r +1, r +1, r +1). (This is because, as the faces of the quotient map are trace, whenever the loop labele x r 1 is encountere, it is trace twice, once on each sie.) It follows that in the sequence σ = (a 0,b,a 1,b,a 2,b,...,a r 1,b) we have b = a r 1 = c(r + 1) for some positive integer c. An extension to any -set of orientation-reversing generators is now obvious; each orientation-reversing generator x j augments by 1 the covalence of the quotient s outer face, an a multiple of this covalence r + will also appear at the corresponing position of a j in σ. In this way we construct in M(σ) a non-cayley map whose unerlying graph is a Cayley graph. A Cayley map M M(σ) is obtaine by replacing in the generating sequence X each of the pairs of orientationreversing generators an their respective inverses by a pair of involutions. The consequence for the quotient map is that each loop inuce by an orientation-reversing generator is replace by a pair of consecutive semi-eges. The covalence sequence of the quotient an also that of M will be the same, respectively, as those of a non-cayley map whose unerlying graph is a Cayley graph. Another situation where we can guarantee that the unerlying graph is a Cayley graph is presente in the next theorem. Theorem 2.4. If all terms in a covalence sequence σ of length 3 are even, then M(σ) contains a map whose unerlying graph is a Cayley graph. Proof. Let σ = (2m 0, 2m 1,...,2m 1 ). Let Cay(G, X) enote the Cayley graph where X ={x i : i Z } an G = x i x 2 i = (x i x i+1 ) m i = ι,i Z. Since all the relators in the group G are of even length, so are all wors erivable from the relators (see, for example [9, pp ]). Thus Cay(G, X) is bipartite. We efine an embeing of Cay(G, X) so that the cyclic orering (x 0,x 1,...,x 1 ) of the generators is achieve in the clocwise irection at each vertex of one orbit an in the counterclocwise irection at each vertex of the other orbit. In this way one constructs a map whose unerlying graph is a Cayley graph with covalence sequence σ=(2m 0, 2m 1,...,2m 1 ) embee in an orientable surface. This surface, however, is the plane because all the efining relations of the group correspon to faces, i.e., by Poincaré s funamental polygon theorem [7], there are no non-contractible cycles. The above theorems are illustrations of the profusion of results that might be obtaine by choosing various arrangements of loops an semi-eges of the one-vertex quotient map an ifferent arrangements of orientation-preserving an orientation-reversing generators. We conclue this section with a few cautionary remars. If σ is a covalence sequence of a Cayley map M, then it nee not hol that σ = σ 1. Moreover, even if this equality oes hol, then the reflective symmetry of σ nee not be extenable to an element of Aut(M). All that can be sai in this regar is that if σ is a covalence se-

7 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) b a c a b (1) (2) Fig. 3. Patches an quotients of non-isomorphic Cayley maps with σ = (4, 6, 6, 6). b b a c a Fig. 4. Patch an quotient of embee Cayley graph with σ = (4, 6, 6, 6). quence of a map M, then there exists a map isomorphic to M an at the same level as M whose covalence sequence is σ 1. Finally, a given cyclic sequence may, in general, be realizable by non-isomorphic Cayley maps as well as a non-cayley map whose unerlying graph is a Cayley graph (Level 3A), as illustrate by the following example. Example 2.2. Let G 1 = a,b,c b 2 = c 2 = a 4 = (abc) 2 = ι, with the cyclic orer of generators (a, a 1,b,c), an let G 2 = a,b a 4 = b 6 = (ab) 3 = ι, where the cyclic orer of generators is (a, a 1,b,b 1 ).Fori {1, 2} Fig. 3(i) shows patches of Cayley maps (Level 4) with the same covalence sequence σ = (4, 6, 6, 6) for the groups G i, together with their one-vertex quotients. The maps are not isomorphic. Since they are 1-ene, 3-connecte an planar, their unerlying graphs are also not isomorpic. Let G 3 = a,b,c b 2 = c 2 = a 4 = abcba 1 c = ι. In this case, a an b are orientation-preserving, while c is orientation-reversing. The Cayley graph for G 3 an the generating set {a,a 1,b,c} also has a planar embeing with

8 606 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Fig. 5. Map with σ = ( 0, 1,..., 1 ). Fig. 6. Truncation. the same covalence sequence (4, 6, 6, 6), but the embeing is not a Cayley map. A patch of this map is illustrate in Fig 4. The same figure also shows the two-vertex quotient relative to the orientation-preserving subgroup of G 3. When σ = (4, 6, 6, 6), the class M(σ) also contains non-vertex-transitive maps (see Section 4). 3. Covalence sequences of orientably vertex-transitive maps (Level 3B) Throughout this section we assume that M M(σ) is an orientably vertex-transitive map, where σ=( 0, 1,..., 1 ). A patch of M is illustrate in Fig. 5. Let H = Aut + (M). Then for any vertex u V(M)the stabilizer Stab H (u) Z b, where bt = for some positive integer t. The group H oes not act semi-regularly on the vertex set of M unless M is a Cayley map (which is the case when b = 1). For our purposes, however, we woul lie to wor with a semi-regular action on a map, since such actions are in an elementary way relate to coverings of graphs an maps (see, e.g., [3, Theorem 2.2.2]). The action of H can be mae to be semi-regular if it is consiere to act rather on the vertex set of the truncation T(M)of M, efine as follows. At each vertex u V(M) we cut off one-thir of each of its incient eges an replace u an its incient ege-pieces by a -gon as inicate in Fig. 6. In T(M)all vertices are 3-valent an all the corresponing covalences from M have been ouble, as inicate in the more etaile Fig. 7. The group H, which was orientation-preserving an transitive on V(M)in the original map M, is still orientationpreserving on T(M)but now acts semi-regularly on V(T(M))with t =/b orbits. If we mo out by H, we obtain the finite quotient map T (M)/H in some orientable surface. Its unerlying graph is 3-valent of orer t an one of its faces is boune by a t-circuit, i.e., it is Hamiltonian. The H-orbits of T(M)are the vertices of T (M)/H, an two vertices of T (M)/H are ajacent if an only if some ege of T(M)joins a pair of vertices in their respective orbits (Fig. 8). The truncate map T(M)can be reconstructe by lifting the quotient map T (M)/H with voltages α H on arcs of T (M)/H (see Fig. 9). Let w t enote the face of covalence t, an let the remaining (not necessarily istinct) faces be enote by w 0,w 1,..., w t 1. Note that each face w i,0 i t 1, has even covalence, say 2l i. This is because the eges incient with w i are alternately projections of eges incient with a face that projects onto w t an projections of eges not incient with

9 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Fig. 7. Truncate map. t Fig. 8. Quotient map T(M) α T(M)/H w 0 w t-1 w 1 w t w 2 Fig. 9. Projection of a truncate map. such a face, i.e., eges joining two such faces. For the voltages α wj in the group H aroun the faces w j we obtain the following ientities: (α wt ) b = ι, (α wi ) i/l i = ι, 0 i t 1. (1)

10 608 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Since all the relations efining the voltage group H come only from faces, it follows again from Poincaré s funamental polygon theorem that the lift is a planar map. The foregoing analysis immeiately implies the following proceure for etermining whether a cyclic sequence σ of length is the covalence sequence of an orientably vertex-transitive map M. Proceure. Let a covalence sequence σ = ( 0, 1,..., 1 ) be given. For each positive ivisor t of such that Z /t acts on σ, list all possible 3-valent Hamiltonian maps on t vertices where one face f is boune by a Hamiltonian circuit. For each such map, form the (clocwise or counterclocwise) cyclic sequence (2l 0, 2l 1,...,2l t 1 ) of the covalences that appear aroun the face f. There exist positive integers m 0,m 1,...,m t 1 such that m 0 l 0,m 1 l 1,...,m t 1 l t 1 are t consecutive terms of σ if an only if σ is the covalence sequence of an orientably vertex-transitive map. Realizability of σ via Cayley maps can be chece by running the Proceure for t =. This is because Cayley maps coincie with regular lifts of one-vertex maps (cf. Proposition 1.3). This remar about Cayley maps reveals a notable connection between covalence sequences of orientably vertex-transitive maps an Cayley maps: for t 3 the sequence (l 0,l 1,...,l t 1 ) of half-lengths of the outer faces corresponing to the trivalent map with a t-covalent Hamiltonian face in Fig. 8 is itself a covalence sequence of a t-valent planar Cayley map. This is not so when t 2, because by contracting the Hamiltonian face we obtain a one-vertex graph of valence t 2. Sequences corresponing to t 2 are constant sequences () when t = 1 an alternating sequences (, l) /2 of even length when t = 2. (The latter are exactly the ege-homogeneous maps characterize in Proposition 1.2.) It follows that covalence sequences of orientably vertex-transitive planar maps are also obtainable by catenating copies (possibly just one copy) of a covalence sequence of a Cayley map. We summarize this iscussion as follows. Theorem 3.1. Let σ be a cyclic sequence of length which is neither constant nor alternating. Then σ is the covalence sequence of an orientably vertex-transitive map if an only if σ=α /t for some ivisor t 3 of, where α is a covalence sequence of a t-valent Cayley map. 4. Covalence sequences of vertex-transitive maps (Level 2) The Proceure of the previous section was state only for orientably vertex-transitive maps. In principle, one may exten the Proceure in orer to ientify covalence sequences of all vertex-transitive graphs. As before, one must consier all possible quotients of truncate maps, but now the supporting surface of the quotient may be non-orientable an it may contain bounary components (cf. [19,3, Chapter 6]). Reconstruction of all possible lifts of such quotients is more complicate than in the orientable case, an the corresponing theory is at the time of this writing still uner evelopment [16]. An application of the Proceure to vertex-transitive graphs in general is seen in the following result. Theorem 4.1. For a cyclic sequence σ there exist only finitely many (1-ene, 3-connecte, planar) orientably vertextransitive maps with covalence sequence σ. Proof. Accoring to the Proceure, for any given sequence σ of length we nee only consier all ivisors t of such that Z /t acts on σ. For each such t we construct all the corresponing quotients that are orientable embeings of 3-valent graphs on t vertices such that one of their faces is boune by a Hamiltonian circuit. As for each given sequence of length there are only finitely many such quotients, the number of their lifts in groups efine precisely by the relations (1) is also finite. It follows that the number of planar orientably vertex-transitive maps having covalence sequence σ is finite. Theorem 4.1 is false if the conition of vertex-transitivity is roppe. Counterexamples inclue the covalence sequences σ 1 = (4, 4, 4, 5) an σ 2 = (3, 4, 3, 4, 4). We illustrate this for σ 1 ; the corresponing result for σ 2 can be obtaine using Bilinsi iagrams which are briefly iscusse in Section 6. Example 4.1. In Fig. 10(a) we see a patch of a vertex-transitive realization of the covalence sequence σ = (4, 4, 4, 5). There are strips of 4-gons in it through which one can cut the map into exactly two (infinite) pieces, then translate one

11 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) (a) (b) Fig. 10. Patches of non-isomorphic maps with σ = (4, 4, 4, 5). of the pieces by one 4-gon, an then reattach the two pieces to obtain the map shown in Fig. 10(b). (The otte lines in Fig. 10 inicate the cuts.) The new map will have the same covalence sequence (4, 4, 4, 5) at each vertex, but it is not isomorphic to the original map, an it is not vertex-transitive. Infinite sequences of such cuts with translation can be selecte in uncountably many ways to prouce uncountably many pairwise non-isomorphic maps, all with covalence sequence (4, 4, 4, 5). The existence of infinitely many maps with the valence sequence (4, 4, 4, 6) was mentione by Moran [12, p. 179] in a ifferent context. Lemma 4.1. Let M be a (finite or infinite, planar or non-planar) vertex-transitive map, let A = Aut(M), an let u V(M). (1) If Stab A (u) contains at least one reflection, then M is orientably vertex-transitive. (2) If Stab A (u) contains exactly one reflection, then M is a Cayley map. Proof. (1) Let H = Aut + (M), an suppose that [A : H ]=2. To show that H acts transitively on V(M), let v V(M). By the vertex-transitivity of M, we may choose g A such that g(u) = v.ifg H there is nothing to prove. If g/ H, let r be a reflection in Stab A (u). Then gr(u) = v an gr H. (2) If Stab G (u) contains exactly one reflection then Stab H (u) is trivial. Hence H acts regularly on V(M), an so M is a Cayley map for the group H by Proposition 1.1. Corollary 4.1. Let M be a vertex-transitive map of a prime valence p having covalence sequence σ. Then exactly one of the following hols: (1) M is a Cayley map (Level 4); (2) M is not a Cayley map but its unerlying graph is a Cayley graph (Level 3A), an σ contains at least two ifferent values; (3) M is an orientably vertex-transitive map, its unerlying graph is a Vertex-transitive non-cayley graph (Level 3B), an σ = () p where has no ivisor t such that 2 t p.

12 610 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Proof. Suppose that σ contains at least two ifferent values. Then the stabilizer of a vertex of M is either trivial, which implies that the unerlying graph of M is a Cayley graph (by Proposition 1.1), or the stabilizer contains exactly one reflection, in which case M is a Cayley map by Lemma 4.1(2). Now suppose that σ = (). Then the existence of M is assure by Proposition 1.2. By Lemma 4.1(1), M is orientably vertex-transitive. The rest follows from Proposition 1.4. Example 4.2. Let σ be the cyclic sequence of the form (3, 4, 3, 4, 4). Accoring to Corollary 4.1 any vertex-transitive map in M(σ) woul have to be a Cayley map or a non-cayley map whose unerlying graph is a Cayley graph. But then, since σ is of the form (,,l,,l)an neither 3 = 4 nor 2 l = 3, Table 2 tells us that any map in M(σ) is at most at Level 1. The way to show that M(σ) is not empty for this particular σ (an others of that il) is explaine in Section 6. Corollary 4.2. Let σ be a covalence sequence of a vertex-transitive map M such that a number appears in σ either (1) in exactly one set of consecutive positions but not all positions or (2) exactly twice but not in opposite positions. Then the unerlying graph of M is a Cayley graph. Proof. In both cases the vertex-stabilizers either are trivial or contain at most one reflection, an Lemma 4.1 applies. 5. Finite vertex-transitive maps The class of covalence sequences of infinite, 1-ene, 3-connecte, planar orientably vertex-transitive maps coincies precisely with the class of covalence sequences of finite vertex-transitive maps on orientable surface of positive genus, as we now emonstrate. Theorem 5.1. Let σ be a cyclic sequence of length 3 whose terms are integers 3. Then σ is a covalence sequence of some finite orientably vertex-transitive map on some compact orientable surface of positive genus if an only if σ is covalence sequence of an infinite (3-connecte,1-ene, planar) orientably vertex-transitive map. Proof. Let M be an orientably vertex-transitive map with covalence sequence σ on an orientable surface S γ of genus γ 1. Let the surface S γ be represente in the stanar way by ientifying certain sies of a regular (4γ)-gon (cf., for example [11, p. 10ff.]) with M embee in its interior. We now tessellate the plane (Eucliean if γ = 1, hyperbolic if γ > 1) by congruent copies of this (4γ)-gon again with appropriate sie ientifications. The union of all copies of M in all these polygons inuces a map M. The map M is a regular covering space of M, an all automorphisms of M lift to automorphisms of M. (This was explicitly prove in Proposition 5.2 of [17] for γ = 1 an the metho generalizes to all γ 1.) It follows that M is infinite, planar, orientably vertex-transitive, an 3-connecte an has the covalence sequence σ. Conversely, let M be an infinite (3-connecte, 1-ene, planar) orientably vertex-transitive map with the given covalence sequence σ. We fix a vertex v in M an enote by W the set of all vertices incient with faces incient with v. Let B be the set of all non-ientity automorphisms g Aut + ( M) such that g(v) W. As a consequence of a result in [10] the group Aut + ( M) is resiually finite (see also [14,20]), which means that there exists a normal subgroup H in Aut + ( M) of finite inex such that H B =. Let M = M/H be the quotient map. It follows that M is a finite orientably vertex-transitive map on some orientable surface of positive genus. Let π : M M be the corresponing covering projection. Since H B =, the restriction of π to W is a bijection. This implies that M an M have the same covalence sequence σ. For vertex-transitive maps on non-orientable surfaces we have an analogous result only in one irection. Theorem 5.2. Let σ be a cyclic sequence of length 3 whose terms are integers 3. If σ is a covalence sequence of some finite vertex-transitive map on some compact non-orientable surface of genus at least 2, then σ is covalence sequence of an infinite (3-connecte,1-ene, planar) vertex-transitive map. Proof. As in the proof of Theorem 5.1 we represent a non-orientable surface of genus η 2 bya(2η)-gon with the finite map M embee in its interior, an we tessellate the plane (Eucliean if η = 2, hyperbolic if η > 2) by congruent

13 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) copies of this (2η)-gon. As before, the union of all copies of M in all these polygons inuces a map vertex-transitive, 3-connecte an has the same covalence sequence as M. M which is We note that Theorem 5.2 coul as well be prove using two-fol orientable covers. Theorem 5.3. Let σ be a covalence sequence of some finite orientably vertex-transitive map on some orientable surface of positive genus. Then there exist infinitely many non-isomorphic finite orientably vertex-transitive maps with covalence sequence σ. Proof. Let M be a finite orientably vertex-transitive map on some orientable surface of positive genus. We follow the metho use in the secon part of the proof of Theorem 5.1. First, we lift the map M onto an infinite (1-ene, 3-connecte, planar) orientably vertex-transitive map M with covalence sequence σ. Let B be an arbitrary finite superset of the set B efine in the proof of Theorem 5.1 such that B oes not contain the ientity automorphism. By resiual finiteness, the group Aut + (M) contains a normal subgroup H of finite inex such that B H =. Since there are infinitely many choices for such finite supersets B of B an subgroups H, there also exist infinitely many non-isomorphic quotient maps M/H that are finite, orientably vertex-transitive an have covalence sequence σ. 6. Some non-realizability conitions (Level 0) It is ifficult to establish existence of vertex-homogeneous but not vertex-transitive maps by the algebraic methos of Section 4. A more geometric approach is the use of Bilinsi iagrams, first efine rigorously in [13] but also use in [12] an earlier in [4]. Intuitively, in a Bilinsi iagram of a map, vertices are arraye on concentric circles in the plane aroun some central vertex. Eges appear either as arcs joining pairs of consecutive vertices on the same circle or as segments joining pairs of vertices on consecutive circles. The latter eges partition the annuli between two consecutive circles into faces. This concentric property hols for vertex-homogeneous maps that o not have both low valence an a covalence sequences with consecutive or nearly consecutive small entries (cf. [1]). At the expense of losing some generality, it is safe to assume that the valence is at least 4 an there are no 3-covalent faces, or that the valence is at least 5 an there are no ajacent 3-covalent faces, or that the valence is at least 6. In orer to prove existence of a vertex-homogeneous map with a given covalence sequence it suffices to show that its Bilinsi iagram can be constructe inuctively. In other wors, it suffices to prove that, for any n 1, any patch of a Bilinsi iagram consisting of a set of n concentric circles an their interiors can be extene by ajoining a new corona. We will illustrate this with the following result. (See the cautionary remar following Theorem 2.3, lest the reaer be tempte to say that it follows from Lemma 4.1(2)!) Theorem 6.1. Let σ = (,,...,,l)be a cyclic sequence of length. Assume that (1),,l 4, or (2) 5 an either 4 or l 5, or (3) 6. Then there exists a vertex-homogeneous map with covalence sequence σ. Proof. In orer to show that a patch of a Bilinsi iagram of n coronas can be extene, we observe that in the aitional corona no two l-covalent faces are ajacent, but a -covalent face may be ajacent to either a -covalent or an l-covalent face. It therefore suffices to show that the -covalent an l-covalent faces can always be arrange in such a way (subject to the above restriction) to exten the patch. That the faces may be inee so combine is illustrate in Fig. 11 for valence = 5. These iagrams are naturally generalizable to any valence greater than 5, an a minor moification wors for valence 4 subject to the above limitations (an also for = 3 uner yet more stringent assumptions). This metho was use to obtain the parameters for non-vertex transitive maps of the tables in the Appenix. There seem to be no results in the literature relate to non-realizability of cyclic sequences by vertex-homogeneous maps. We offer here two simple theorems which are also use in the Appenix. Theorem 6.2. Let, l, m be istinct integers 3, l o. Let σ be a cyclic sequence of length 3 containing one or more segments of the forms [, l, m] or [m, l, ]. Assume further that if there is any other instance of the number l elsewhere in σ, then it neither immeiately precees nor immeiately follows or m. Then σ is not realizable as a covalence sequence of a map.

14 612 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) l l [] [l] l l (l) l () Fig. 11. Patch of Bilinsi iagram for σ = (,,,,l). Proof. Suppose that M is a map realizing the covalence sequence σ. Then there exists an l-covalent face f of M the ajacent faces of which have covalences an m alternately aroun f. Since = m, the covalence of f must be even. A similar proof yiels the following result. Theorem 6.3. Let σ be a cyclic sequence of length 3 containing one or more segments of the forms [, l, l, m] or [m, l, l, ], where, l, m are istinct integers 3, an l is o. Assume further that if there is any other instance of the number l elsewhere in σ, then it neither immeiately precees nor immeiately follows or m or l. Then σ is not realizable as a covalence sequence of a map. 7. Concluing remars Some of our classification theorems for covalence sequences of planar Cayley maps an planar orientably vertextransitive maps have an algorithmic flavor. The existence of a simple arithmetic characterization of these covalence sequences that is analogous to the results in [14, Theorems 9.1 an 9.2] for sets of covalences is unliely. A main tool in the present article for gaining information about covalence sequences of planar maps of given level of transitivity is to pass to quotient maps obtaine by moing out by a subgroup H of automorphisms of the original map. As iscusse at the beginning of Section 4, this metho can be use to characterize covalence sequences also in the case when orientation-reversing automorphisms are taen into account. However, while moing out by a subgroup H consisting of the orientation-preserving automorphisms always leaves a quotient map embee in an orientable surface, this nee not be the case when H contains an orientation-reversing automorphism. In fact, in such a case the quotient surface may be non-orientable an may contain bounary components. We efer this much more complex analysis to a forthcoming project. Appenix Tables 1 an 2 in this appenix contain a complete classification of maps of valences 3, 4, an 5. The following abbreviations are use: CM a Cayley map (Level 4); NCMCG a non-cayley map whose unerlying graph is a Cayley graph (Level 3A); VTNCG a vertex-transitive map whose unerlying graph is not a Cayley graph (Levels 3B an 2); NVTM a map that is not vertex-transitive (Level 1); NR a non-realizable sequence (Level 0).

15 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Table 1 Classification for valences 3 an 4 Sequence σ CM NCMCG VTNCG NVTM NR (,,) (,6) = 1 (, 6) = 1 (, 6) = 1 (,,l) 2 4 or 2 2, l (,l,m) 2, l, m At least one of, l, m o (,,,) (,6) = 1 (, 6) = 1 (, 6) = 1 (,,, l) (, 6) = 1 3 or 4 All values or 2, l of, l (,,l,l) 2, l 2, l 2 or 2 l (,l,,l) 2 or 2 l 2, l 2, l (,, l, m) 2, l, m 2, l, m At least one of, l, m o (,l,,m) 2 (4 &2 m) All remaining or 2, l, m values of, l, m (,l,m,n) 2, l, m, n At least one of, l, m, n o Table 2 Classification for valence 5 Sequence σ CM NCMCG VTNCG NVTM NR (,,,,) (,30) = 1 (, 10) = 1 (, 30) = 1 or 6 (,,,, l) (, 6) = 1 4 or 6 All values or 2, l of, l (,,,l,l) (,6) = 1&2 l (,6) = 1&2 l 2 l 2 l (,, l,, l) 3 or ((, 6) = 1&2 l) All values 2, l or 6 of, l (,,, l, m) 2, l, m 2 l, m 2 l or 2 m (,,l,,m) 3 2, l, m or All values (4 &2 l) of, l, m or 6 (,, l, l, m) 2, l, m 2, l, m At least one of, l, m o (, l,, l, m) 2, l, m 2 m 2 m (,, l, m, l) 2, l 2, l, m or 2 or 2 l (4 &2 l) or (2 &4 l) (,, l, m, n) 2, l, m, n 2, l, m, n At least one of, l, m, n o (, l,, m, n) 2, l, m, n or 2, m, n At least one of (4 &2 m, n), m, n o (, l, m, n, p) 2, l, m, n, p At least one of, l, m, n, p o By 2, l, m (respectively, 2, l, m) we mean that all of, l, an m are ivisible (respectively, are not ivisible) by 2. As usual, the symbol (a, b) enotes the greatest common ivisor of a an b. We mae the convention that istinct letters in a covalence sequence are presume to enote istinct values. Entries in the CM an NCMCG columns were obtaine by the proceure escribe in Section 2 before the statement of Theorem 2.4 an also Corollary 4.1(2). The VTNCG column was compile on the basis of Lemma 4.1 an Corollary 4.1(3). Entries in the NVTM column can be obtaine by the metho of Bilinsi iagrams as inicate in Section 6. The last column follows from Theorems 6.2 an 6.3. We note that some of the entries for valences 3 an 4 can also be extracte from [14].

16 614 J. Siagiová, M.E. Watins / Discrete Mathematics 307 (2007) Acnowlegments We woul lie to than R. Bruce Richter, Jozef Sirá n, an Thomas W. Tucer for stimulating iscussions an valuable comments. The first author acnowleges support from the VEGA Grant no. 1/9176/02 an from the APVT Grant no References [1] J.A. Bruce, M.E. Watins, Concentric Bilinsi iagrams, Austral. J. Combin. 30 (2004) [2] J.E. Graver, M.E. Watins, Locally finite, planar, ege-transitive graphs, Mem. Amer. Math. Soc. 126 (601) (1997). [3] J.L. Gross, T.W. Tucer, Topological Graph Theory, Wiley, New Yor, [4] B. Grünbaum, G.C. Shephar, Tilings an Patterns, W. H. Freeman an Co., New Yor, [5] B. Grünbaum, G.C. Shephar, Ege-transitive planar graphs, J. Graph Theory 11 (1987) [6] R. Halin, Automorphisms an enomorphisms of infinite locally finite graphs, Abh. Math. Sem. Univ. Hamburg 39 (1973) [7] B. Iversen, Hyperbolic Geometry, Cambrige University Press, Cambrige, [8] H.A. Jung, A note on fragments of infinite graphs, Combinatorica 1 (1981) [9] W. Leermann, Introuction to Group Theory, Longman, New Yor, [10] I.A. Mal cev, On the faithful representation of infinite groups by matrices, Mat. Sb. 8(50) (1940) (in Russian); Amer. Math. Soc. (2) 45 (1965) 1 18 (Translate English). [11] W.S. Massey, Algebraic Topology: an Introuction, Harcourt, Brace an Worl, Inc., New Yor, [12] J.F. Moran, The growth rate an balance of homogeneous tilings in the hyperbolic plane, Discrete Math. 173 (1997) [13] P. Niemeyer, M.E. Watins, Geoetic fibers an rays in one-ene planar graphs, J. Combin. Theory Ser. B 69 (1997) [14] R.B. Richter, J. Sirá n, R. Jajcay, T.W. Tucer, M.E. Watins, Cayley maps, J. Combin. Theory Ser. B 95 (2005) [15] G. Sabiussi, On a class of fixe point-free graphs, Proc. Amer. Math. Soc. 9 (1958) [16] J. Sirá n, T.W. Tucer, A Theory of Branche an Fole Maps, in preparation. [17] J. Sirá n, T.W. Tucer, M.E. Watins, Realizing finite ege-transitive orientable maps, J. Graph Theory 37 (1) (2001) [18] M. Soviera, J. Siráň, Regular maps from Cayley graphs, Part 1: balance Cayley maps, Discrete Math. 109 (1992) [19] T.W. Tucer, Finite groups acting on surfaces an the genus of a group, J. Combin. Theory Ser. B 34 (1983) [20] A. Vince, Regular combinatorial maps, J. Combin. Theory Ser. B 35 (1983) [21] M.E. Watins, Ege-transitive strips, Discrete Math. 95 (1991)