ARS MATHEMATICA CONTEMPORANEA Volume 15, Number 2, Fall/Winter 2018, Pages 267-542 Covered by: Mathematical Reviews zbMATH (formerly Zentralblatt MATH) COBISS SCOPUS Science Citation Index-Expanded (SCIE) Web of Science ISI Alerting Service Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES) The University of Primorska The Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The Slovenian Discrete and Applied Mathematics Society The publication is partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications. ARS MATHEMATICA CONTEMPORANEA Blacklists vs. Whitelists In the age of open access (OA) it is sometimes hard to distinguish between honest, genuine research journals and fake, predatory journals run by predatory publishers whose only interest is to make money and who are ready to publish anything under the APC (article processing charges) business model. There exist several blacklists of publishers and journals with such unethical practices. There are well-known cases where the whole editorial board of a prominent journal resigns over an unethical and greedy policy of a publisher and creates another, unblemished journal. It is not uncommon for individual mathematicians and other scientists to boycott certain publishers for the same reason. Not publishing a paper in a predatory journal is certainly a legitimate choice for any researcher. However, maintaining a public blacklist is more dangerous. It may be challenged in court and may result in heavy penalties for the author of such a list. We think that the solution lies in whitelists in which learned societies and trustworthy individuals can endorse high-quality OA journals that are free both for readers and authors. In a sense both MathSciNet and zbMATH form such whitelists. To a certain extent even the Web of Knowledge represents a whitelist. A journal that does not appear on these lists is either not a mathematical journal, is too young or has some ethical issues. Several journals, including ours, declare that they follow the EMS Code of Practice. Unfortunately, no one really checks whether this is indeed the case. The Ethics Committee should look at such journals and confirm their claims when appropriate. This explicit addition to the whitelist would be of great importance for any emerging good journal. It would also serve authors when faced with the problem of choosing a venue for their publication. Klavdija Kutnar, Dragan Marušic and Tomaž Pisanski Editors in Chief Contents Calculating genus polynomials via string operations and matrices Jonathan L. Gross, Imran F. Khan, Toufik Mansour, Thomas W. Tucker . . 267 Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane John Kalliongis, Ryo Ohashi.........................297 A combinatorial problem and numerical semigroups Aureliano M. Robles Pérez, José Carlos Rosales...............323 Tilings of hyperbolic (2 x n)-board with colored squares and dominoes Takao Komatsu, László Németh, László Szalay...............337 Mirrors of reflections of regular maps Adnan Melekoglu...............................347 The thickness of Ki)n,n and K2,n,n Xia Guo, Yan Yang..............................355 Touching perfect matchings and halving lines Micha A. Perles, Horst Martini, Yaakov S. Kupitz..............375 Characterizing all graphs with 2-exceptional edges Drago Bokal, Jesús Leaños..........................383 Isomorphisms of generalized Cayley graphs Xu Yang, Weijun Liu, Lihua Feng......................407 On constructing expander families of G-graphs Mohamad Badaoui, Alain Bretto, David Ellison, Bassam Mourad......425 The Hosoya polynomial of double weighted graphs Tina Novak, Darja Rupnik Poklukar, Janez Zerovnik ............441 On Jacobian group and complexity of I-graph I(n, k, l) through Chebyshev polynomials Ilya A. Mednykh ............................... 467 The isolated-pentagon rule and nice substructures in fullerenes Hao Li, Heping Zhang ............................487 Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime Yan-Quan Feng, Da-Wei Yang, Jin-Xin Zhou................499 Wonderful symmetric varieties and Schubert polynomials Mahir Bilen Can, Michael Joyce, Benjamin Wyser .............. 523 Volume 15, Number 2, Fall/Winter 2018, Pages 267-542 ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 267-295 https://doi.org/10.26493/1855-3974.939.77d (Also available at http://amc-journal.eu) Calculating genus polynomials via string operations and matrices Jonathan L. Gross * Dept. of Computer Science, Columbia University, New York, NY 10027, USA Imran F. Khan PUCIT, University of the Punjab, Lahore 54000, Pakistan Toufik Mansour Department of Mathematics, University of Haifa, 3498838 Haifa, Israel Thomas W. Tuckerf Dept. of Mathematics, Colgate University, Hamilton, NY 13346, USA Received 21 September 2015, accepted 27 November 2017, published online 20 June 2018 To calculate the genus polynomials for a recursively specifiable sequence of graphs, the set of cellular imbeddings in oriented surfaces for each of the graphs is usually partitioned into imbedding-types. The effects of a recursively applied graph operation t on each imbedding-type are represented by a production matrix. When the operation t amounts to constructing the next member of the sequence by attaching a copy of a fixed graph H to the previous member, Stahl called the resulting sequence of graphs an H-linear family. We demonstrate herein how representing the imbedding types by strings and the operation t by string operations enables us to automate the calculation of the production matrices, a task requiring time proportional to the square of the number of imbedding-types. Keywords: Graph imbedding, genus polynomial, production matrix, transfer matrix method. Math. Subj. Class.: 05A15, 05A20, 05C10 * J. L. Gross is supported by Simons Foundation Grant #315001. tT. W.Tucker is supported by Simons Foundation Grant #317689. E-mail addresses: gross@cs.columbia.edu (Jonathan L. Gross), imran.farid@pucit.edu.pk (Imran F. Khan), tmansour@univ.haifa.ac.il (Toufik Mansour), ttucker@colgate.edu (Thomas W. Tucker) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ 1 Introduction The genus polynomial of a graph G is defined to be the generating function rG (z) = £ gi(G)zi, i> 0 where gi(G) counts the cellular imbeddings of G in the closed oriented surface Si of genus i. Following their introduction by [12] in 1987, and starting with the work of [6], the genus polynomials for a recursively constructed sequence of graphs have most frequently been calculated, as in [8, 9, 13], by partitioning the imbeddings according to the cyclic orderings of occurrences of root-vertices on the face-boundary walks (abbr. fb-walks) of the imbeddings. In this paper, we describe how to expedite such calculations. 1.1 Rotation systems All our graphs come with a labeling of the edges. All graph imbeddings in this paper are assumed to be cellular, that is, each component of the complement of the imbedded graph is homeomorphic to the interior of the unit disk. All surfaces are assumed to be closed and oriented. To describe the imbeddings of a graph G, we assign + and - orientations to the edges, including self-loops. Then any imbedding defines, for each vertex, a cyclic order of the signed edge-ends initiating at that vertex, which is called the rotation at that vertex. The rotations collectively form a rotation system (e.g., see [19]), which acts as a permutation p on the oriented edge set. If A is the involution that reverses the orientation of each edge, then the face boundary walks of the imbedding are the orbits of the permutation pA. A rotation system for a graph has also been called a "ribbon graph" or a "fat graph", especially in the context of algebraic geometry, Riemann surfaces, and the theory of dessins ([3, 20, 24]). We use the Euler polyhedral formula |V |-| E | + IF | = 2 - 27(S) to compute the genus y(S) of the imbedding surface S. Two imbeddings b1,b2: G ^ S determine the same rotation system if and only if there is a homeomorphism of the surface S taking i\(G) to 12(G) that acts as the identity isomorphism on the graph G (i.e., respects the labeling of edges). Accordingly, there is a bijection from the set of imbeddings of G to the set of rotation systems. A problem in calculating genus polynomials is that the number of possible cyclic orderings of the edge-ends incident at a d-valent vertex is (d — 1)!. Thus, the number of imbeddings of a graph G is the product n(dv — 1)!, taken over all vertices v of G, where dv is the valence of v. It is well-known [30] that the problem of calculating the minimum genus of a graph is NP-hard, even when the graph is 3-regular. It follows that calculating the genus polynomial is at least that hard. For example, the number of rotation systems for the complete graph K7 is (5!)7 « 3.6 x 1014, and the genus polynomial for K7 has only recently been computed [2]. Table 1 gives the list of coefficients. J. L. Gross et al.: Calculating genus polynomials via string operations and matrices 269 Table 1: Genus distribution of the complete graph K7. 1 9i 0 0 1 240 2 3,396,960 3 3,746,107,320 4 594,836,922,960 5 20,761,712,301,960 6 158,500,382,165,280 7 178,457,399,105,280 1.2 Context Genus polynomials for recursively specified families of graphs have been computed mostly within a general paradigm in which the recursive operation occurs in the vicinity on a small number of vertices or edges designated as roots. The set of all imbeddings of each graph in the family is partitioned into what we now call imbedding-types, according to incidence of the fb-walks on the roots, a technique for calculating genus polynomials that was introduced by [6]. This basic paradigm is exemplified by [8, 13] for root-vertices, and by [25, 26] for root-edges. This paper integrates several embellishments of the basic paradigm: • The genus polynomial for a graph is partitioned into a pgd-vector, with one coordinate for each imbedding type, such that each coordinate is a polynomial that gives the number of oriented imbeddings of that imbedding type in every orientable surface. • The recursively applied topological operation is represented by a production system, as developed by Gross, Khan, and Poshni, in a series of papers [8, 13, 25, 26], that transforms the pgd-vector for a given graph into the pgd-vector for the graph resulting from an application of the recursive operation used to specify the graph family. In those papers, the productions were calculated with the aid of a multiplicity of drawings of rotation projections. • The representation of production systems by matrices was introduced by Stahl [27], for application to pgd-vectors of some graphs in what he called H-linear families. Such matrices are now called production matrices, and the graph sequences are now called H-linear sequences, or simply linear sequences. Stahl used what he called permutation-partition pairs to derive production matrices. • The representation of imbedding-types by strings of root-vertices, as developed by Gross [11]. • Using string operations directly to calculate the production matrices, as suggested subsequently by Mohar [23]. The general idea of a linear sequence is that a copy of a graph H is attached to each graph in the sequence to form the next graph in the sequence. It is necessary to attach each copy of H in the same way, as described precisely by [4]. 1.3 Outline of this paper Our main focus in this paper is the calculation of production matrices. Since the size of the matrix increases with the number of imbedding types, and since the number of imbedding-types grows exponentially with the number of roots and with the valences of the roots, most of the calculations of genus polynomials have been for sequences of graphs with at most two roots and valences no larger than 4. The string notation by which we concisely represent imbedding types allows us to automate the bookkeeping used in tracking the way imbedding types are changed by the addition of paths between root vertices. The advantages of this system are many. It allows us to derive in a few lines (see Subsection 4.3) the computation of production matrices that formerly involved many figures [10] or detailed paper-and-pencil applications of what Stahl [28] called the "Walkup reduction" for permutation-partition pairs. String notation facilitates the computer calculation of production matrices whose derivation would be un-feasibly tedious by hand (see the 12 x 12 matrix in Section 5). Finally, it reveals ways of combining different imbedding types to get smaller matrices (see Subsection 5.1). Following a review in Section 2 of the representation of imbedding-types by strings, Section 3 introduces the representation of topological and vertex-labeling operations on imbeddings by string operations. Section 3 also introduces the concept of grouping two or more i-types into a "super-type". As an illustration of how the string operations are used in calculations of genus distributions, Section 4 applies these representations to two previously published examples, one of which (the iterated claw) we have adapted here to give a detailed example of grouping. Also, we explain in Section 4 how our use of productions to calculate pgd-vectors is interpretable as an embellishment of the transfer matrix method, along the lines described by [29]. Section 5 explores issues related to computation. It uses the theory developed in the previous sections to calculate genus polynomials for a vertex-amalgamation path of copies of K4 and for an edge-amalgamated path of copies of K4. Without string operations, both derivations would be long and tedious. We used two computational aids while preparing this paper. • The computational system Maple®. • A computer program, based on string operations, that calculates production matrices. Such kinds of aids are what we have in mind in various comments here, rather than a state-of-the-art computer. Section 5 includes an additional example of the grouping of i-types into a super-type. In Section 6, we use Burnside's Lemma to derive a formula for the maximum number of imbedding types for a graph with two roots of any possible combination of valences. We generalize the formula to more that two roots. From the rapid growth rate of the number of imbedding-types, as valences and the number of roots of the graphs at issue increases, it becomes clear that programmable computation tools are a virtual necessity when seeking to calculate genus polynomials. 2 Representing imbedding-types by strings In this section, we develop a system of notation that uses strings of root-labels, so that representing the addition of an edge to a graph becomes a simple matter of applying a few string-processing rules. 2.1 Face-boundary-walks We assign labels 0,1,2,... to the roots of a graph G. Given an imbedding of G, we represent a face as a string of roots, in the order they are encountered in a traversal of its fb-walk following the orientation of the surface. If an fb-walk does not contain any roots, we call its string empty. Two strings are equivalent representations of an fb-walk if one is a cyclic shift of the other. We denote an entire equivalence class of strings by putting a representative string of labels inside parentheses. The canonical representative for the equivalence class of fb-walks is the one with minimum lexicographic order with respect to the labels 0,1,... . Remark 2.1. Vertices that are not roots do not appear in the string representing a face. Accordingly, the appearance of consecutive labels... 12 ... within a string would not imply that there is an edge between vertices 1 and 2. Also, since any labeled vertex may appear more than once around an fb-walk, the corresponding cyclic list of root-labels is not a permutation. 2.2 Imbedding types The collection of non-empty strings for all the fb-walks of an oriented imbedding of a rooted graph G is called an imbedding-type of G (abbr. i-type). The collection of all imbedding types over all imbeddings of G is called the full collection of imbedding types for G. In order to compare imbedding types for the same rooted graph, we usually use the shortlex order [31] on canonical representatives to make a list of fb-walks (rather than a set): shorter faces are listed before longer ones, and if two faces have the same length, the one with shortlexically least canonical representative is listed first. We call such a list the canonical form for the i-type. Example 2.1. Figure 1 shows an imbedding of K4 in the sphere with roots 0,1,2, and 3. If the "interior" fb-walks are oriented counterclockwise (which forces the "exterior" fb-walk to appear as clockwise, from the perspective of vertex 0), then the i-type (in canonical form) is Notice that each face is represented by its canonical form (cyclic shift with least lexicographic order) and that the faces are listed in shortlex order. Since for this example every vertex is a root, it follows that two consecutive vertices (with respect to cyclic order) in the (012)(023)(031)(132). Figure 1: An imbedding of K4 in S0. representation of a face actually does represent a directed edge. For any two roots i and j, the directed edge ij appears exactly once. If i = 0 and j = 1, we could suppress the labels 2,3 to obtain the i-type (01)(0)(01)(1) = (0)(1)(01)(01) for the imbedding of Figure 1. If the only root is 0, then the imbedding type would be (0)(0)(0). Notice in the last imbedding type, the number of strings is less than the number of faces, because the fb-walk (132) contains no instances of vertex 0, and we do not list empty faces. If we reverse the orientation of the sphere and have all four vertices 0,1,2,3 as roots, then the i-type in canonical form would be (021)(032)(013)(123) = (013)(021)(032)(123). Observe that the shortlex order for the faces differs from the previous orientation. However, the i-type for roots 0, 1 is the same as before, as is the i-type for root 0, when labels 1, 2, and 3 are suppressed. Example 2.2. Considering all 24 rotation systems for K4, we get the following census of i-types for roots 0,1, given in shortlex order: • 2 of i-type (0)(1)(01)(01) • 2 of i-type (0)(01011) • 2 of i-type (1)(00101) • 2 of i-type (01)(0011) • 8 of i-type (01)(0101) Notice that since there is only one edge 01, only one of the substrings 01 in an i-type, for example (01)(0101), comes from an edge. The other juxtapositions of 0 and 1 come from suppressing incidences of the roots 2 and 3. We conclude that {(0)(1)(01)(01), (0)(01011), (1)(00101), (01)(0011), (01)(0101)} is a full set of i-types for K4 with roots 0 and 1. In Section 6 of this paper, we shall see that the maximum number of i-types for a pair of 3-valent roots is 38. Remark 2.2. We observe that within the string representation of any i-type, each root-vertex appears as many times as its valence. If there is an edge between roots i and j , then both ij and ji must appear at least once in every i-type. On the other hand, as we have noted, the appearance of ij in a string does not imply that there is an edge between i and j. Remark 2.3. Suppose that G has no multi-edges or self-loops, and suppose that every vertex is a root. Then each rotation system for the graph G determines a unique i-type, since each i-type determines a rotation system for the dual graph. In this circumstance, the number of i-types would be the same as the number of rotation systems. At the opposite extreme, the set of imbeddings for a tree with one root-vertex has only one i-type. Remark 2.4. When there are multi-edges or loops and every vertex is a root, it happens that different rotation systems can determine the same i-type. For example, the bouquet Bn has only one vertex 0 and has n loops at that vertex. Then an i-type is simply a partition of 2n into k parts, where k is the opposite parity of n (k is the number of faces, so the Euler characteristic 1 - n + k must be even). Thus, the number of i-types for imbeddings of Bn with k faces is at most the Stirling subset number {2" } (i.e., the Stirling number of the second kind), where k and n have opposite parities. 2.3 String notational conventions We adopt two notational conventions for strings: • The concatenation of a string S with a string T is denoted by ST. • The reverse string for a string S is denoted by S-1. We emphasize that SS-1 is not the empty string, but rather the concatenation of S with its reverse (which forms a palindrome). This notation does satisfy the relations (ST)-1 = T-1S-1 and (S-1)-1 = S as if in a group, even though our strings are not permutations (since roots can repeat), and even though they do not form a group. 2.4 Pgd-vectors Given an i-type t, we write its partial genus polynomial in the form J2aizi where ai is the number of type-t imbeddings of G of genus i. If the i-types are listed in shortlex order, then we can associate the set of partitioned genus polynomials for G with a column vector whose rth coordinate is the partial genus polynomial for the rth i-type. This is called apgd-vector for the graph G. For instance, the partitioned genus distribution for the complete graph K4 given by Example 2.2 corresponds to the vector [2 2z 2z 2z 8z]T where the superscript T denotes the transpose. 3 Operations on imbedding-types In this section, we describe how a path-adding operation affects the i-types. We also describe the relabeling of root-vertices, and the suppression of some root-labels, which are used, for instance, when there are no more paths to be added at a root-vertex. 3.1 Adding a path within a face and between faces Let G be a rooted graph and let iUj be a path whose endpoints i,j are roots of G but all other vertices of U are not in G. If U is empty, we have simply the edge ij. The effect of adding iUj into a face with fb-walk (iSjT) is given by the following operation: (iSjT) + iUj ^ (iSjU-1)(iUjT). (3.1) In calculations, we may denote the right-hand side by AddiUj [iSjT]. If the i-type in which the fb-walk (iSjT) occurs is of the form (iSjT)W1W2 ...Wk, which includes other fb-walks, then applying Operation (3.1) to that i-type yields the i-type (iSjU-1)(iUjT )WiW2 ...Wk. That is, the other fb-walks of the i-type are simply recopied. The effect of adding the path iUj between two faces (iS) and (jT) is given by this operation: [(iS), (jT)] + iUj ^ z(iSiUjTjU-1). (3.2) The right-hand side may be expressed as Add^ [(iS), (jT)]. When applying Operation (3.2) to an i-type with fb-walks (iS) and (jT), any other fb-walks of the i-type are simply recopied, the same as for Operation (3.1). The multiplier z indicates that the genus of the imbedding rises by 1 when a handle is added to the surface. For the circumstance in which the faces (iS) and (jT) lie within (disjoint) imbeddings i and i' of separate graphs G and G', the effect of joining the imbeddings by adding the path iUj between the two faces (iS) and (jT) is given by this operation: [(iS), (jT)] + iUj ^ (iSiUjTjU-1). (3.3) The non-presence of the multiplier z signifies the fact that the genus of the surface in which the resulting graph is imbedded is simply the sum of the genera of the imbeddings i and i'. Example 3.1. Consider an imbedding of the 4-cycle 0213 in the sphere. There are two faces, one with fb-walk (0213) and the other with fb-walk (0312). Thus, the initial i-type is (0213)(0312). There are four ways to add a path 0451 to such an imbedding, one within the face (0213), one within the face (0312) and two between the faces (0213) and (0312). Figure 2 shows the four possible ways to add the path 0U1 and the resulting i-type for each. (0213)(0312) 2 -, 0 1 (02154)(04513)(0312) 2 -, 0 1 (i) (0213)(03154)(04512) z(02130451203154) z(03120451302154) 2 2 4 (ii) 0 2 0 3 (iv) Figure 2: Adding the path 0451 to an imbedding of a 4-cycle in the sphere. 1 1 1 (i) Inserting path 0451 into the face (0213) yields the imbedding type (02154)(04513)(0312), as per Operation (3.1). We now have three faces. Root-vertices 0 and 1 now have valence 3, so they now appear three times in this representation of the i-type. (ii) Inserting the path 0451 instead into the face (0312) yields i-type (0213)(03154)(04512). (iii) If we join the two faces, from endpoint 0 inside the face (0213), to endpoint 2 inside the face (0312), then the resulting string expression is z(02130451203154). (iv) If we add the path 0451 with edge-end 0 now inside the face (0312) and edge-end 1 inside the face (0213), we get the string expression z(02154031204513). It follows that the net result of adding the path 0451 to the i-type (0213)(0312) is the following linear combination of i-types taken over the ring Z[z] of polynomials with integer coefficients: (02154)(04513)(0312) + (0213)(03154)(04512) + z(0213045120354) + z(03120451302154). Remark 3.1. The path ii for adding a self-loop is simply a special case. As a variation on Operation (3.1), we have (iS) + ii ^ (i)(iSi) As a variation on Operation (3.2), we have [(iS), (iT)] ^ z(iSiiTi) Remark 3.2. If a graph already has an edge ij, then adding the path P = ij creates a multiple adjacency. 3.2 Suppressing roots and relabeling roots Given a subset of roots {i, j,... }, the root-suppression operator Supi j acts to suppress every occurrence of the root-labels i, j,... within an i-type t. For example, SuPi,2 [(1)(12)(0212)(0231303)] = (0)(03303). Observe that we delete empty pairs of parentheses as a final step in suppressing roots. Example 3.1, continued. Suppressing roots 2 and 3 as well as any roots along U transforms the i-type (021U-1)(0U 13)(0312) into the i-type (01)(01)(01). Similarly, Sup12U [z(021U-1 )(0U 13)(0312)] = z(010101). Moreover, when root-suppression is applied to a linear combination of i-types, it can reduce the number of terms. For instance, Sup2,3U [(021U-1)(0U 13)(0312) + (0213)(031U-1)(0U 12) + z(02130U 1203U-1) + z(03120U 13021U-1)] = 2(01)(01)(01) + 2z(010101). We can also relabel roots, by using the root-relabeling operator. Suppose that the label i appears in i-type t and label j does not. Then Labj [t] is the i-type obtained by replacing in t all occurrences of i by j. Thus, Lab24[(1)(2)(22)(1323)] = (1)(4)(44)(1343). We denote by Lab^y,... [t] the result of relabeling i by i', j by j' etc. 3.3 Reversing orientation If the orientation of a graph imbedding is reversed, the effect on i-types is as follows: • the cyclic order of each fb-walk is reversed; • the genus of the imbedding stays the same. We call this the i-type reversal operator. Given an i-type t, we denote by t-1 the i-type in which each fb-walk string is reversed. Note that if (ST) is an fb-walk within i-type t, then the corresponding fb-walk in t-1 is (T-1S-1), for which a cyclic shift gives (S-1T-1). On the other hand, the i-type (R-1S-1T-1) is not a cyclic shift of the i-type (RST)-1 = (T-1S-1R-1). Proposition 3.3. The i-type reversal operator commutes with the operators Add, Sup, and Lab. Proof. Clearly, we can reverse lists either before of after suppressing or relabeling vertices, and the result is the same. Using Rule (3.1) for adding a path within a face, we have Addp [(iSjT )]-1 = [(SP-1)(PT)] = (T-1P-1)(S-1P) and (3.4) AddP [(iSjT )-1] = AddP [iT-1jS-1] = (T-1P-1)(S-1P) (3.5) Using Rule (3.2) for adding an edge between two faces, we have Addp[(iS), (jT)]-1 = z(PTP-1S)-1 = z(S-1PT-1P-1) and (3.6) Addp[(iS)-1, (jT)-1] = Addp[(iS-1), (jT-1)] = z(PT-1P-1S-1) (3.7) □ 3.4 Combining i-types into super-types As we have observed, the number of i-types grows exponentially with the valence and the number of roots, so any way of reducing the number of i-types is welcome. For example, in building a graph by path-addition, we can always group an i-type with its reverse, since i-type reversal commutes with edge path-adding. Indeed, root-suppression is also a way of grouping many i-types together. Suppose that the rooted graph H is obtained from the rooted graph G by a sequence Op of the following kinds of operations: path-additions, root-suppression, and root-relabeling. Let T be the full collection of i-types for G, and let S be the full collection of i-types for H, both in shortlex order. Then for any i-type t G T, we see that The expression Op (t) is a linear combination of elements of S, with coefficients taken from the ring Z[z] of polynomials in z. We represent Op, therefore, as a matrix M whose columns are labeled by i-types in S, and whose rows are labeled by i-types in T, where Ms,t is the coefficient of i-type s in the expression Op(t). Let P and Q be partitions of S and T, respectively. Suppose that we order the i-types within S and the i-types within T so that the i-types within each cell of P and within each cell of Q are contiguous in the respective orderings, inducing a partitioning of the production matrix M into blocks that satisfy this criterion: Within each block, the column sums are the same. (This requirement applies also to the blocks that span only a single row of the matrix M, which implies that the entries in such a row are identical.) Then we call the partitions P and Q compatible with M. Moreover, we call each part of P and Q a super-type for the operation Op. We can then condense the matrix M to a smaller one whose columns are indexed by P and rows by Q, and whose entries are the constant column sum of the block of M determined by the respective parts. We have already encountered super-types in two contexts: type-reversal and root-suppression. For type-reversal, we partition a full collection of i-types into parts by grouping together an i-type and its reverse. Since type-reversal commutes with path-adding, root-suppression, and root-relabeling, it is compatible with any sequence of those operations. We can also view root-suppression Supi j itself as creating super-types. In this case, we have S = T. The parts of P are just singletons; i-types s, t are in the same part of Q if and only if Supi j ... (s) = Supi j ... (t). Notice in this case, the matrix M is just the identity matrix and each block is a part of a single column of M. The condensed matrix has a single 1 in each column. Another way to create super-types is to exploit any symmetry between roots. With H, G, S, T as before, suppose there is a graph automorphism f of H that permutes the roots of H and G. Then f also induces a permutation of the S and T. We can then use orbits of that permutation as super-types. Grouping types into super-types by graph automorphisms and reversal is illustrated particularly well in the family of iterated claws in Subsection 4.3, where 12 i-types are reduced to three super-types. For now we consider an example that provides a clear illustration of the theory underlying the reduction. Example 3.2. Suppose that G = K4, as in Example 2.2, with roots 0 and 1, and that the graph H is obtained from G by the operation of adding a second edge between 0 and 1. Since there is an automorphism of the graph G interchanging 0 and 1, we have the partition given in Table 2 for the full set T of i-types of the graph G, under the partition Q (induced by this automorphism), with the parts of Q indicated by square brackets. Table 2: Partitioning the i-types for (K4, {0,1}). T _T /Q (0)(1)(01)(01) (0)(1)(01)(01) (0)(01011) [(0)(01011), (1)(00101)] (1)(00101) (01)(0011) (01)(0011) (01)(0101) (01)(0101) We can construct the full set S of 13 i-types for the graph H, by adding the path 01 to the i-types in T for the graph G. In Table 3, we again use square brackets to enclose the parts of the partition P. Table 3: Partitioning the i-types for (K4 +01, {0,1}). Vh(z) Figure 5: Functor from the category of graphs and string operations to the category of ring modules and matrices with integer polynomial coefficients. T 4.4 Polynomial matrix and transfer matrix methods There are models in the physical sciences where the computational process uses polynomial matrix entries, like our production matrices. Some such models in chemistry were explored in [21, 22], which uses the terminology polynomial matrix method. This method was adapted by [1] for application to matching polynomials of polygraphs. As described by [7], the transfer matrix method for various mathematical contexts concerns the transformation of a given problem into a matter of counting walks in a digraph. We observe that if A is the adjacency matrix of a digraph, then the ij entry of the matrix Ak counts the numbers of paths from vertex v to vertex vj. A generalization of this problem (see [29]) is concerned with a digraph in which the arc from vertex i to vertex j, for all i and j, is labeled with the element m^j of a commutative ring, with M = (m^ ). Instead of counting the paths of length k, we are calculating the sum of the products of all length-k paths from v to vj. Of course, the ij entry of the matrix Mk gives this sum for vj and Vj. In [5] and [23], the matrix M is called a "transfer matrix". When calculating pgd-vectors for a graph sequence {Gn : n = 0,1,...} that is specified by recursive application of a topological operation t, we take the imbedding types as vertices of the digraph. We label the arc from type-« to type-j by the coefficient of type-j in the production for type-«. 5 Machine computation of production matrices In this section, we give two examples of linear sequences whose production matrices have been calculated with the aid of a computer program. It should be clear that calculating these production matrices by hand would be daunting. Heretofore, such calculations have been done mostly by hand, which has limited us to calculating the genus polynomials only for relatively few graph families. As a consequence, we have very little data to study deep issues, such as the log-concavity conjecture, that the genus distribution of every graph is a log-concave polynomial (see [18, 16]). 5.1 Vertex-amalgamation path of copies of K4 We define the graph T\ to be the complete graph on four vertices, with a single root, labeled 0. The graph Tn is obtained from Tn-1 by vertex-amalgamating a new copy of K4 to Tn-1. The graphs T2 and T3 are illustrated in Figure 6. Figure 6: The graphs T2 and T3. Following the paradigm of [13], we could obtain Tn from Tn-1 by vertex-amalgamating a doubly rooted copy of K4 to a singly rooted copy of Tn-1. However, whereas a pair of 2-valent root-vertices involves at most 10 i-types, it can be seen in Table 5 that for two 3-valent root-vertices, the number of i-types could be as large as 38. Moreover, the potential number of productions for amalgamating two graphs with 38 i-types could be as large as 382 = 1444. In what follows, we see that using the string-operation paradigm enables us to reduce the number of i-types from 38 to 3. The topological operation of vertex-amalgamating an additional copy of K4 to the rooted graph (Tn-1,0) can be represented by the following sequence of string operations. Procedure 5.1. Add the next copy of K4 by vertex-amalgamation. Add0i230 Addo2 Add 13 SuPo,1,3 Lab2o (5.1) (5.2) (5.3) (5.4) (5.5) We see that the i-types for a graph with a single 3-valent root-vertex named 0 are (0)(0)(0) (0)(00) (000) More generally, the number of i-types for a graph with a single k-valent root-vertex equals at most the number of partitions of the integer k. Nonetheless, even though only three productions would be needed, deriving them with pencil-and-paper calculations would be tedious work. Just for a start, there are 12 ways to insert the path 01230 into an imbedding of Tn-1, two ways between each of the three pairs of distinct corners at root-vertex 0 and two ways at each corner. The total number of imbeddings of Tn that are consistent with each imbedding of Tn-1 is 480. Theorem 5.1. The pdg-vector of the graph Tn is Mn V1 is [2 12z 2z] and the production matrix is -i V1; where the initial pgd-vector Mt (z) = 96z + 18 80 z + 30 60 " 48z2 + 156z 220z 360z 144z2 + 18z 120z2+ 30z 60z (5.6) Proof. The initial pgd-vector V1 for (K4,0) and the production matrix are best calculated by a computer program. □ 5.2 Edge-amalgamation path of copies of K4 Here we define T1 to be the complete graph K4 with a single root-edge 01. The graph Tn is obtained from Tn by edge-amalgamating a copy of K4. The new root-edge is the edge in the new copy that is independent of the edge amalgamated to the previous root-edge. The graphs T2 and T3 are illustrated in Figure 7. 'V v 0 0 Figure 7: The graphs T2 and T3. The topological operation of extending the graph Tn-1 by edge-amalgamating an additional copy of K4 can be represented by the following sequence of string operations. Procedure 5.2. Add the next copy of K4 by edge-amalgamation. Addo23i Addo3 Add i2 Supo,i Lab 20,31 (5.7) (5.8) (5.9) (5.10) (5.11) 1 1 We determine that the i-types for the graphs Tn are as follows, grouped by classes under the automorphism interchanging 0 and 1 and listed in shortlex order: 1. (0)(1)(01)(01) 7. (01)(0011) 2. (0)(1)(0011) 8. (01)(0101) 3. (0)(01)(011), (1)(01)(001) 9. (001)(011) 4. (0)(00111), (1)(00011) 10. (000111) 5. (0)(01011), (1)(00101) 11. (001011), (001101) 6. (01)(01)(01) 12. (010101) Each imbedding of Tn-i in each of these 12 super-types has 576 possible extensions to an imbedding of Tn. Theorem 5.2. The pdg-vector of the graph Tn is M' matrix is i-i (z)V(z), where the production 4 18 8 36 40 6 20 22 12 72 80 84 8z 0 16z 0 0 24z 32z 32z 32z 0 0 0 64z 96z 96z 96z 96z 96z 128z 128z 128z 0 0 0 48z2 32z2 32z2 0 0 48z2 0 0 0 0 0 0 8z 36z 16z 72z 80z 12z 40z 44z 24z 144z 160z 168z 60z 56z 72z 48z 48z 60z 64z 64z 96z 0 0 0 4z2 + 4z 48z2 + 18z 64z2 + 8z 36z 40z 72z2 + 6z 20z 22z 12z 72z 80z 84z 16z 72z 32z 144z 128z 24z 80z 72z 48z 288z 256z 240z 104z2 48z2 64z2 0 0 72z2 0 0 0 0 0 0 32z3 0 0 0 0 0 0 0 0 0 0 0 64z2 96z2 96z2 96z2 96z2 96z2 128z2 128z2 128z2 0 0 0 60z2 56z2 72z2 48z2 48z2 60z2 64z2 64z2 96z2 0 0 0 The initial graph (Ti, 0) has the pgd-vector V(z) = [2 0 0 0 4z 0 2z 8z 0 0 0 0]T. Proof. The initial pgd-vector and the production matrix were calculated by our computer program. □ If follows that T2 8 + 376z 16z + 320z2 128z+ 1664z2 96z2 16z + 752z2 120 z + 832z2 584z2 + 8z 32z + 1248z2 208z2 64z3 128z2 + 1664z3 120z2 + 832z3 and Ta 32 + 5040z + 119552z2 + 207616z3 64z + 9216z2 + 111872z3 512z + 56064z2 + 612864z3 384z2 + 28416z3 + 103424z4 64z + 10080z2 + 239104z3 + 415232z4 480z + 43200z2 + 365568z3 5872z2 + 32z + 176256z3 + 389376z4 128z + 19136z2 + 414464z3 + 644096z4 832z2 + 56704z3 + 181760z4 256z3 + 12032z4 512z2 + 56064z3 + 612864z4 480z2 + 43200z3 + 365568z4 6 Enumerating possible imbedding types Various previously published genus polynomial calculations have involved recursive constructions of families of graphs with two 2-valent root-vertices, for which ten i-types are sufficient. As we progress toward more general results, most especially in regard to the LCGD conjecture, we are encountering recursive graph constructions for which we use arbitrarily many vertex roots, of arbitrary degrees. In this section, we first use Burnside's Lemma to calculate the number of i-types that can occur for two 2-valent roots. Then we generalize to obtain lower and upper bounds on the number of i-types for arbitrarily many root-vertices or arbitrary valences. Interestingly, our method provides a formula for calculating the number of possible cyclic partitions of a multi-set. Thus, it is a generalization of Stirling numbers of the first kind. 6.1 Two 2-valent roots Early papers on genus polynomial calculations via pgd-vectors used ten mnemonics for the i-types for graphs with two 2-valent roots. The following table lists the ten mnemonics and their corresponding type-names: dd0 dd' dd'' ds0 ds' (0)(0)(1)(1) (0)(01)(1) (01)(01) (0)(0)(11) (0)(011) sd0 sd' ss0 ss1 ss2 (00)(1)(1) (001)(1) (00)(11) (0101) (0011) An ad hoc examination confirms that the ten type-names contain all the possible partitions of the multi-set {0,0,1,1} into cyclic cells. We now undertake a reconfirmation of this calculation of ten possible i-types, using Burnside's Lemma. Our set of objects is the set of disjoint cycle decompositions of the 24 permutations in the symmetric group S4, with domain {0,1,2, 3}. Our permutation group on them has the permutations e (identity) (0 2) (1 3) (0 2)(1 3) (6.1) where we regard the numbers 2 and 3 as second copies of the numbers 0 and 1, respectively. Under the action of this permutation group, the orbit of the permutation (0 1)(2)(3) is (0)(1)(2 3) (0)(3)(1 2) (1)(2)(0 3) (2)(3)(0 1) This orbit corresponds to the imbedding-type (0)(1)(01). The identity permutation e fixes all 24 disjoint cycle representations of S4. The permutation (0 2) fixes the subgroup of disjoint cycle representations in which both 0 and 2 are fixed or transposed, whose cardinality is 4. The permutation (1 3) fixes the same subgroup of cardinality 4. The permutation (0 2)(1 3) fixes that same subgroup, plus the set (0 1)(2 3) (0 3)(1 2) (0 1 2 3) (0 3 2 1) for a total of 8 fixed points. Applying Burnside's Lemma, we divide the sum of the sizes of the fixed-point sets by the cardinality of the permutation group (6.1) to obtain 24 + 4 + 4 + 8 = 40 = 4 = 4 = as the maximum number of i-types for two 2-valent roots. J. L. Gross et al.: Calculating genus polynomials via string operations and matrices 289 6.2 Two roots, 2-valent and 3-valent Suppose that root 0 is 2-valent and root 1 is 3-valent. Then there are 18 imbedding-types, as in Table 4. Table 4: Table of the i-types for two roots, one 2-valent and one 3-valent. structure imbedding types 1l (0)(0)(1)(1)(1) 13 2 (0)(0)(1)(11) (0)(1)(1)(01) (1)(1)(1)(00) 1 22 (0)(01)(11) (1)(00)(11) (1)(01)(01) 12 3 (0)(0)(111) (0)(1)(011) (1)(1)(001) 23 (00)(111) (01)(011) (11)(001) 14 (0)(0111) (1)(0011) (1)(0101) 5 (00111) (01011) The action of the permutation group £{0,2} x £{1,3,4} on the elements of £{0,1,2,3,4} has the cycle index -2 [il + 4t3t2 + 3tit2 + 2Î2Î3] . We now consider the number of fixed points for each of the four permutation types. Type t\. The identity permutation fixes all 120 elements of £{0,1,2,3,4}. Type t\t2. Each permutation of structure tft2 fixes 12 elements of £{0,1,2,3,4}. For instance, (0 2) fixes each of the six elements with the 1-cycles (0) and (2) and each of the six with the 2-cycle (02), for a total of 12. The sum of the sized of the fixed-point sets of the four permutations of structure if t2 is 48. Type t\t2,. Each permutation of structure 11i2 fixes 8 elements of £{0,1,2,3,4}. For instance, (0 2)(1 3) fixes both of the elements with the 1-cycles (0), (2), and (4), both with the 2-cycle (02) and the 1-cycle (4), and also the four elements (0 1)(2 3), (0 3)(1 2), (0 1 2 3), and (0 3 2 1) for a total of 8. The sum of the sized of the fixed-point sets of the four permutations of structure 11i2 is 24. Type t]t3. Each permutation of structure t2t3 fixes 6 elements of £{0,1,2,3,4}. In particular, (0 )(2)(134) fixes Z{0,2} x Z{1,3,4}, as does (0)(2)(1 4 3). Together, they make a contribution of 12 to the sum of the sizes of the fixed point sets. Type t2t3. These two permutations each fix the same 6 elements of £{0,1,2,3,4} as in the preceding case, for a net contribution of 12. Applying Burnside's Lemma, we infer that the number of orbits is 120 + 48 + 24+ 12+ 12 216 12 12 6.3 Several roots of arbitrary degrees We now calculate lower and upper bounds on the number of i-types. Theorem 6.1. For a class of graphs with roots 0,1,..., k — 1 of respective degrees do, di,..., dk-i, the number of i-types is at least (dp + dj +-----+ dfc_i)i do!di! • • • dfc_i! (6.2) Proof. In addition to their respective primary names 0,1,..., k — 1, each root j has dj — 1 aliases chosen from among the numbers k, k +1, .. ., do + di +----+dfc_i with no two different primary names having any aliases in common. Accordingly, our set of objects is the set of disjoint cycle representations of the symmetric group , where K = d0 + di + • • • + dk-i. The permutation group that acts on them is isomorphic to X ^di X • • • X Sdfc-1 Since the identity permutation fixes all the cycle forms of , the sum of the sizes of the sets of fixed points is at least K!. The cardinality of the permutation group is di!d2! • • • dk!. Thus, by Burnside's Lemma, a lower bound on the number of i-types is given by (6.2). □ Theorem 6.2. For a class of graphs with roots 0 and 1, of respective degrees a and b, the number ofi-types is at most EnkckCk! E E E 1 c k=i yi,Pi + qi = Oi (iPi 2P2 ...aPa )ePa (i'i 2'2 ...j9b )ePb EI j=i iP%Pi^Y\j = i j9j qj'! where the sumYl,c is over all partitions 1ci 2C2 • • • nCn G Pn and Pn is the set of all partitions of the number n. Proof. The action of the permutation group ^{i,3,4,...,a+i} X ^{2,a+2,a+3,...,a+j} on the elements of S{i 2,...,n}, where n = a + b, has the cycle index Ca,j = E E na=i ^ nj=i j (ipi 2P2.*• aPa)ePa (i'l2'2 ...jqb}epb n®=i iPipi! nj=ijqj qj! where Pm is the set of all partitions of m. The number of fixed points for a permutation of cycle type 1ci 2C2 • • • nCn is given by i!b!Ca,j(1Cl2c2 ••• ncn) n kckcfc!, k=i where Ca,b(1ci 2C2 • • • nCn ) is the coefficient of 111122 • • • tr in the polynomial Ca,b. Thus, each permutation of structure t^1t22 • • • tn fixes n kck ck ! y y y -—h-. 11 k ^ ^ ^ IT iPi v !Fib i q a ! Vi,pi + qi = ci (1P1 2P2 ...aPa )£Pa (1«1 292 ...b«b ) £ Pb I li=1 i Vi! i i j=1 J aJ ! k=1 elements of £{1j2i...jn}. Applying Burnside's Lemma, we conclude that the number of orbits is given by 1 ^a^nk Ck ! E E E na iPi ^b Jqj ! k = 1 Vi,pi+qi=ci (1P1 2P2 ...aPa )£Pa (191 292 ... b9 b ) £ Pbl ii=1 1 Pï-Hj^J yj- i!b! which equals 1 IT k Ck' ^ ' ^ ' no I pr b -q. | ' c fc=1 Vi,pi + qi = ci (1P1 2P2 ...apa )ePa (1*1 2*2 ...bqb !j=1 % %Pi'l i j = 1 j ° Qj ' where the sum J2c is over all partitions 1C12C2 • • • nCn e Pn. Applying our formula for a,b < 10, we obtain Table 5. Table 5: The maximum number of i-types for two root-vertices, of valences a and b. □ a\b 1 2 3 4 5 6 7 8 9 10 1 2 4 7 12 19 30 45 67 97 139 2 4 10 18 34 56 94 146 228 340 506 3 7 18 38 74 133 233 385 623 977 1501 4 12 34 74 158 297 550 951 1614 2627 4202 5 19 56 133 297 602 1166 2133 3775 6437 10692 6 30 94 233 550 1166 2382 4551 8424 14953 25835 7 45 146 385 951 2133 4551 9142 17639 32680 58659 8 67 228 623 1614 3775 8424 17639 35492 68356 127443 9 97 340 977 2627 6437 14953 32680 68356 136936 264747 10 139 506 1501 4202 10692 25835 58659 127443 264747 530404 Theorem 6.3. The formula corresponding to that of Theorem 6.2 for m roots of degrees (a1 ,a2,..., am) is given by En k°kCk! E Vi,P1i+P2i +-----+Pdi = Ci E 1 c k=1 Vd=1,2, nmun a= 1 iPd< v^ where the sum^c is over all partitions 1C12C2 • • • nCn G Pn. Proof. This proof uses the same arguments as for Theorem 6.2. □ Using the formula from Theorem 6.3 for the calculations, we present in Table 6 the maximum number of imbedding-types for triply rooted graphs with root-vertices of va- lences 1 < i,j,k < 5. 1Pd1 2Pd2 ...ad £Pad Table 6: The maximum number of imbedding-types for three roots, of valences i, j, k for i = 1, 2,3,4,5. i = 1 i=2 i=3 i=4 i=5 j\k 1 2 3 4 5 1 6 14 28 52 90 2 14 38 84 170 316 3 28 84 206 450 899 4 52 170 450 1058 2254 5 90 316 899 2254 5110 j\k 1 2 3 45 1 14 38 84 170 316 2 38 120 290 644 1284 3 84 290 788 1886 4074 4 170 644 1886 4868 11214 5 316 1284 4074 11214 27556 j\k 1 2 3 45 1 28 84 206 450 899 2 84 290 788 1886 4074 3 206 788 2370 6146 14302 4 450 1886 6146 17170 42696 5 899 4074 14302 42696 112966 j\k 1 2 3 45 1 52 170 450 1058 2254 2 170 644 1886 4868 11214 3 450 1886 6146 17170 42696 4 1058 4868 17170 51630 137070 5 2254 11214 42696 137070 387146 j\k 1 2 3 45 1 90 316 899 2254 5110 2 316 1284 4074 11214 27556 3 899 4074 14302 42696 112966 4 2254 11214 42696 137070 387146 5 5110 27556 112966 387146 1161498 7 Conclusions We have focused here primarily on the computational aspects involved in applying string operations toward the determination of genus polynomials of graphs. We recognize the following two immediate benefits of the string-operations paradigm: 1. It enables us to reduce the number of partial genus polynomials (one for each imbed-ding-type) into which a genus polynomial must be partitioned. 2. The imbedding-types, the production matrix, and the partial genus polynomials (which are the coordinates of a pgd-vector) can be calculated by a computer program, which enables us to generate a much larger set of experimental data. Beyond using string operations in new calculations of enumerative results on graph imbeddings, some new theoretical insights may arise from them. One may reasonably consider how the paradigm of string operations relates to the log-concavity conjecture, that every genus polynomial is log-concave (see [16,18]). We observe that using Theorem 4.7.2 of [29] could give generating functions for the individual entries of a power of a production matrix. In a sequel [15], we regard a linear family of graphs as a Markov process is which the states are i-types and a slightly modified form of the production matrix is the transition matrix. We explore the properties of such Markov processes. The methods described here seem amenable to extension. Suppose that instead of a fixed production matrix M(z) for a graph sequence {Gn : n = 0,1,...}, with pgd-vectors Vn(z) we had a sequence of production matrices Mn(z), such that Recursion (4.1) was generalized to Mn(z)Vn(z) = Vn+l(z), and Equation (4.2) to Vn(z) = Mn-l(z)Mn-2(z) • • • Mq(z)Vq(z). A tractable recursion or a closed formula for Mn(z) would enable us to calculate the pgd-vector Vn(z) reasonably rapidly. Of course, such a sequence of production matrices corresponds to a non-stationary Markov process. References [1] D. Babic, A. Graovac, B. Mohar and T. Pisanski, The matching polynomial of a polygraph, Discrete Appl. Math 15 (1986), 11-24, doi:10.1016/0166-218x(86)90014-4. [2] S. Beyer, M. Chimani, I. Hedtke and M. Kotrbcik, A practical method for the minimum genus of a graph: models and experiments, in: A. V. Goldberg and A. S. Kulikov (eds.), Experimental Algorithms, Springer, volume 9685 of Lecture Notes in Computer Science, pp. 75-88, 2016, doi:10.1007/978-3-319-38851-9_6, proceedings of the 15th International Symposium (SEA 2016) held in St. Petersburg, June 5-8, 2016. [3] B. Bollobas and O. M. Riordan, A polynomial invariant of graphs on orientable surfaces, Proc. London Math. Soc. 83 (2001), 513-531, doi:10.1112/plms/83.3.513. [4] Y. Chen, J. L. Gross, T. Mansour and T. W. Tucker, Recurrences for the genus polynomials of linear sequences of graphs, 2016, manuscript, 26 pages. [5] T. Y. Chow and J. West, Forbidden subsequences and Chebyshev polynomials, Discrete Math. 204 (1999), 119-128, doi:10.1016/s0012-365x(98)00384-7. [6] M. L. Furst, J. L. Gross and R. Statman, Genus distributions for two classes of graphs, J. Comb. Theory Ser. B 46 (1989), 22-36, doi:10.1016/0095-8956(89)90004-x. [7] I. M. Gessel and R. P. Stanley, Algebraic enumeration, in: R. L. Graham, M. Grotschel and L. Lovasz (eds.), Handbook of Combinatorics, Volume II, Elsevier, Amsterdam & MIT Press, Cambridge, Massachusetts, pp. 1021-1061, 1995. [8] J. L. Gross, Genus distribution of graph amalgamations: self-pasting at root-vertices, Australas. J. Comb. 49 (2011), 19-38, https://ajc.maths.uq.edu.au/pdf/4 9/ajc_v4 9_ p019.pdf. [9] J. L. Gross, Genus distributions of cubic outerplanar graphs, J. Graph Algorithms Appl. 15 (2011), 295-316, doi:10.7155/jgaa.00227. [10] J. L. Gross, Embeddings of cubic Halin graphs: genus distributions, Ars Math. Contemp. 6 (2013), 37-56, doi:10.26493/1855-3974.217.440. [11] J. L. Gross, Embeddings of graphs of fixed treewidth and bounded degree, Ars Math. Contemp. 7 (2014), 379-403, doi:10.26493/1855-3974.366.dd1. [12] J. L. Gross and M. L. Furst, Hierarchy for imbedding-distribution invariants of a graph, J. Graph Theory 11 (1987), 205-220, doi:10.1002/jgt.3190110211. [13] J. L. Gross, I. F. Khan and M. I. Poshni, Genus distribution of graph amalgamations: pasting at root-vertices, Ars Combin. 94 (2010), 33-53. [14] J. L. Gross, I. F. Khan and M. I. Poshni, Genus distributions for iterated claws, Electron. J. Combin. 21 (2014), #P1.12, http://www.combinatorics.org/ojs/index.php/ eljc/article/view/v21i1p12. [15] J. L. Gross, T. Mansour and T. W. Tucker, Markovian analysis of production matrices for genus polynomials, in preparation. [16] J. L. Gross, T. Mansour, T. W. Tucker and D. G. L. Wang, Log-concavity of combinations of sequences and applications to genus distributions, SIAM J. Discrete Math. 29 (2015), 10021029, doi:10.1137/140978867. [17] J. L. Gross, T. Mansour, T. W. Tucker and D. G. L. Wang, Iterated claws have real-rooted genus polynomials, Ars Math. Contemp. 10 (2016), 255-268, doi:10.26493/1855-3974.538.86e. [18] J. L. Gross, D. P. Robbins and T. W. Tucker, Genus distributions for bouquets of circles, J. Comb. Theory Ser. B 47 (1989), 292-306, doi:10.1016/0095-8956(89)90030-0. [19] J. L. Gross and T. W. Tucker, Topological Graph Theory, Dover Publications, Mineola, New York, 2001, reprint of the 1987 original [Wiley, New York] with a new preface and supplementary bibliography. [20] G. A. Jones and J. Wolfart, Dessins d'enfants on Riemann surfaces, Springer Monographs in Mathematics, Springer, Cham, 2016, doi:10.1007/978-3-319-24711-3. [21] M. V. Kaulgud and V. H. Chitgopkar, Polynomial matrix-method for calculation of n-electron energies for linear conjugated polymers, J. Chem. Soc. Faraday Trans. II 73 (1977), 13851395, doi:10.1039/f29777301385. [22] M. V. Kaulgud and V. H. Chitgopkar, Polynomial matrix method for the calculation of charge densities and bond orders in linear conjugated n-electron systems, J. Chem. Soc. Faraday Trans. II74 (1978), 951-957, doi:10.1039/f29787400951. [23] B. Mohar, Genus distribution of path-like and ring-like graphs, oral presentation at SIAM DM'12 at Halifax, Nova Scotia, June 2012. [24] M. Mulase and M. Penkava, Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over Q*, Asian J. Math. 2 (1998), 875-919, doi:10.4310/ajm.1998.v2. n4.a11. [25] M. I. Poshni, I. F. Khan and J. L. Gross, Genus distributions of graphs under edge-amalgamations, Ars Math. Contemp. 3 (2010), 69-86, doi:10.26493/1855-3974.110.6b6. [26] M. I. Poshni, I. F. Khan and J. L. Gross, Genus distributions of graphs under self-edge-amalgamations, Ars Math. Contemp. 5 (2012), 127-148, doi:10.26493/1855-3974.166.63e. [27] S. Stahl, Permutation-partition pairs III: Embedding distributions of linear families of graphs, J. Comb. Theory Ser. B 52 (1991), 191-218, doi:10.1016/0095-8956(91)90062-0. [28] S. Stahl, On the zeros of some genus polynomials, Canad. J. Math. 49 (1997), 617-640, doi: 10.4153/cjm-1997-029-5. [29] R. P. Stanley, Enumerative combinatorics, Volume I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, California, 1986, doi:10.1007/978-1-4615-9763-6. [30] C. Thomassen, The genus problem for cubic graphs, J. Comb. Theory Ser. B 69 (1997), 52-58, doi:10.1006/jctb.1996.1721. [31] Wikipedia contributors, Shortlex order — Wikipedia, The Free Encyclopedia, 2015, https: //en.wikipedia.org/wiki/Shortlex_order. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 297-321 https://doi.org/10.26493/1855-3974.806.c9d (Also available at http://amc-journal.eu) Finite actions on the 2-sphere, the projective plane and I-bundles over the projective plane John Kalliongis Department of Mathematics and Statistics, Saint Louis University, 220 North Grand Boulevard, Saint Louis, MO 63103 Ryo Ohashi Department of Mathematics and Computer Science, King's College, 133 North River Street, Wilkes-Barre, PA 18711 Received 5 February 2015, accepted 6 February 2018, published online 25 June 2018 In this paper, we consider the finite groups which act on the 2-sphere S2 and the projective plane P2, and show how to visualize these actions which are explicitly defined. We obtain their quotient types by distinguishing a fundamental domain for each action and identifying its boundary. If G is an action on P2, then G is isomorphic to one of the following groups: S4, A5, A4, Zm or Dih(Zm). For each group, there is only one equivalence class (conjugation), and G leaves an orientation reversing loop invariant if and only if G is isomorphic to either Zm or Dih(Zm). Using these preliminary results, we classify and enumerate the finite groups, up to equivalence, which act on P2 x I and the twisted I-bundle over P2. As an example, if m > 2 is an even integer and m/2 is odd, there are three equivalence classes of orientation reversing Dih(Zm)-actions on the twisted I-bundle over P2. However if m/2 is even, then there are two equivalence classes. Keywords: Achiral symmetry, chiral symmetry, equivalence of actions, finite group action, isometry, orbifold, symmetry. Math. Subj. Class.: 57S25, 05E18, 57M60, 57R18, 58D19, 57M20 1 Introduction The finite orientation preserving groups which act effectively on S2 are known. (See for example Gross and Tucker [5] and Zimmermann [9].) They are the octahedral symmetric group S4, the dodecahedral/icosahedral alternating group A5, the tetrahedral alternating E-mail addresses: kalliongisje@slu.edu (John Kalliongis), ryoohashi@kings.edu (Ryo Ohashi) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ group A4, the cyclic group Zm or the dihedral group Dih(Zm). Using this classification, the actions on the projective plane P2 are also known as folklore, and one can easily compute them by this theorem of Singerman [7] and Tucker [8]. Theorem. Let F be a closed non-orientable surface and let p: F ^ F be the orientable double cover with covering translation t: F ^ F. Then any finite group G acting on F, lifts to an orientation preserving action of G on F that commutes with t. Moreover, the action of G on F is determined by the action of G x (t) on F. If t: S2 ^ S2 is the covering translation such that S2 / (t) = P2, one checks that any rotation of S2 commutes with t. Therefore since these groups consists of rotations, it follows that the orientation preserving actions on S2 project to P2, giving the following corollary. Corollary. Any finite group acting on P2 is isomorphic to one of the following groups: S4, As, A4, Zm or Dih(Zm). A finite G-action on a manifold M is a monomorphism p: G ^ Homeo(M), where G is a finite group, and Homeo(M) is the group of homeomorphisms of M. Two actions p1 and p2 are equivalent if there exists a homeomorphism h of M such that hp1 (G)h-1 = p2 (G). For an action p, the quotient space M/p is an orbifold which is referred to as the quotient type of the action. In this paper, we describe how to visualize the finite groups which act on the 2-sphere S2 and the projective plane P2, and show how to obtain their quotient types. Our approach, for the groups which are not cyclic or dihedral, is to view these groups as subgroups of the symmetric group Sn for an appropriate n, tiling the 2-sphere with appropriate polygons with n vertices for each group, and explicitly defining each action. As for the cyclic and dihedral groups, we use spherical coordinates to precisely describe their actions on S2. For all these groups, we can easily identify an explicit fundamental region for each action and see its quotient type, which is obtained by identifying the boundary of the fundamental region. In this way, it is easy to see the actions on S2, P2 and their quotient types. This part of the paper may be considered expository, and we obtain the following theorem where the description of these quotient types may be found in Figure 1. Theorem 7.1. Let p: G ^ Homeo(P2) be a finite group action on P2. Then G is isomorphic to one of the following groups: S4, A5, A4, Zm or Dih(Zm). The orbifold quotient P2/p is an orbifold homeomorphic to one of the following orbifolds: Oh, Ih, Tv, Zm, S2m, Dm or Dim. There is only one equivalence class for each group. (1) G ~ S4 if and only if P2/p = Oh. (2) G ~ As if and only if P2/p = Ih. (3) G ~ A4 if and only if P2/p = Tv. (4) G ~ Zm and m is even if and only if P2/p = Zm. (5) G ~ Zm and m is odd if and only if P2/p = S2m. (6) G ~ Dih(Zm) and m odd if and only if P2/p = Dm. (7) G ~ Dih(Zm) and m even if and only if P2/p = D^j m This approach relates to topics in topological graph theory found in Gross and Tucker [5]. There, graphs are embedded on surfaces and finite groups act on these spaces with quotient spaces, branch covering maps and branch points, relating to orbifold covering maps and cone points. Using the above result, we classify in Theorem 7.4 the finite group actions, up to equivalence, on P2 x I for I = [0,1]. If G is an action on P2 x I, then G is isomorphic to one of the following groups: S4, S4 x Z2, As, As x Z2, A4, A4 x Z2, Zm, Zm x Z2, Dih(Zm) or Dih(Zm) x Z2. We indicate the number of equivalence classes for each group in Theorem 7.4. If W is the twisted I-bundle over the projective plane P2, then we obtain the following results: Corollary 8.12. Let <: G ^ Homeo(W) be a finite orientation preserving G-action on W. Then G is isomorphic to one of the following groups: S4, A5, A4, Zm or Dih(Zm). The orbifold quotient for each action is a twisted I-bundle orbifold over the following 2-orbifolds: Oh (for S4), Ih (for As), Tv (for A4), Zh (for Zm and m even), S2m (for Zm and m odd), Dvm (for Dih(Zm) and m odd) and Dh (for Dih(Zm) and m even). There is one equivalence class for each quotient type. Theorem 9.4. Let <: G ^ Homeo(W) be an orientation reversing G-action. Then G is isomorphic to one of the following groups: S4, Zm with m even, Dih(Zm), S4 x Z2, As x Z2, A4 x Z2, Zm x Z2 or Dih(Zm) x Z2. (1) If G is either S4, S4 x Z2, As x Z2, A4 x Z2, Zm x Z2 with m even or Dih(Zm) with m odd, there is only one equivalence class. (2) If G is Zm with m > 2 even and m/2 odd, then there are two equivalence classes of Zm = Zm/2 x Z2-actions on W. (3) If G is Zm with either m/2 even or m = 2, then there is only one equivalence class. (4) If G is Dih(Zm) with m > 2 and m/2 even, there are two equivalence classes of Dih(Zm)-actions on W. (5) If G is Dih(Zm) with m > 2 and m/2 odd, there are three equivalence classes of Dih(Zm)-actions on W. (6) If G is Dih(Zm) x Z2 with m even, there is only one equivalence class. (7) If G is Dih(Zm) x Z2 with m odd, then Dih(Zm) x Z2 ~ Dih(Z2m) and there are three equivalence classes of Dih(Z2m)-actions on W. We list all the closed 2-orbifolds with positive Euler number, of which there are 14. (See Figure 1.) In referring to these orbifolds, we use Schonflies notation found in Coxeter and Moser [1], and Dunbar [3]. There are five orientable 2-orbifolds with positive Euler number which have as their underlying space a 2-sphere with the cone points indicated in the notation. They are E(2, 2, n) = Dn, E(2, 3, 3) = T, E(2, 3, 4) = O, E(2, 3, 5) = I, and £(n,Z) = Cn,;. These double cover the following nine non-orientable 2-orbifolds where the double lines are reflector lines. The superscripts h and v stand for horizontal and vertical reflections in their orientable double covers. Except for E(n, Z) = Cn,; where the cone points are at the north and south poles, all the cone points are located on the equator. In this article, where appropriate and depending on the context, we use the same symbol to denote the quotient space and the group acting on S2. For example, O = E(2, 3,4) and O also denote the octahedral group. Here is a brief outline of the paper. We consider each of these orbifolds in Sections 2 through 6, and give model maps which we consider as standard actions, to obtain each quotient type. Summarizing we give the main results for finite actions on P2 and P2 x I in Section 7. Sections 8 and 9 are devoted to classifying the finite actions on the twisted I-bundle W over P2. The authors wish to thank the referees for many helpful comments and suggestions. 2 Chiral octahedral symmetry O and achiral octahedral symmetry Oh We describe the groups O = S4 and Oh = S4 x Z2 acting on the 2-sphere S2, and show how O acts on the projective plane P2. We view S2 as an octahedron which has eight triangles (faces): A125, A145, A126, A146, A235, A236, A345 and A346. (See Figure 2.) Consider elements of S6 where a = (1,2)(3,4)(5,6) and b = (1, 2, 5)(3,4,6). The two elements act on the octahedron. We can see that a is a 180° rotation about the axis passing through the midpoint of edges 1,2 and 3,4. On the other hand, b is a 120° rotation about the axis passing through the barycenter of A125 and A346 respectively. Further, ab = (2, 6,4,5) where ab is a 90° rotation about the axis passing through vertices 1 and 3. As a result, the two elements a and b generate a group isomorphic to S4, and we denote this group by O = (a,b | a2 = b3 = (ab)4 = 1), the octahedral group. Next, we use E to denote the quotient space of S2 by O, and we will find a fundamental region for E on S2. We first claim that A125 will tile the whole octahedron S2 by the action of O. Observe that the action by a sends A125 to A216. Further, b2 (ab)b-2 = (1,4, 3,2) is a 90° rotation about the axis passing through vertices 5 and 6, which shows our claim. Note that the number of fundamental regions for E on S2 must be 24 as the number is the order of the octahedral group O = S4. Since the S2 currently has eight faces, we will have to triangulate them further. Our approach is that we will add one more vertex on the barycenter on each triangle. For instance, one of the triangulations on A125 is shown in the Figure 2. We now show that A12y becomes a fundamental region for E. Since a rotational axis of b passes the vertex y, the barycenter of A125, one can see that b permutes those three triangles A12y, A51y and A25y. In the meantime, edges 1,xi and 2, xi are identified 5 5 2 2 xi 1 6 (a) Front of S2. 6 (b) Back of S2. (c) Triangle 125. Figure 2: S2 as an octahedron. by a. Likewise, edges 1, y and 2, y are identified by b. Points 1, x1 and y will be cone points under the action. Each has an order 4, 2 and 3 respectively. Consequently, we obtain £ = S2/O = £(2,3,4). In order to obtain Oh, we consider an action i = (1,3)(2,4)(5, 6) which is the antipodal map on S2. Notice that the antipodal map commutes with the elements in O, hence it induces the reflection map on S2/O = £(2, 3,4). Now, we choose a triangle whose vertices are 1, x1 and y. Apply (ab)2b(ab)i on the triangle gives us the triangle with vertices 2, x1 and y. Notice that segments 1x1 and 2x1; 1y and 2y have been identified under O-action and the segment x1y has been fixed under the map (ab)2b(ab)i. This argument shows that A1x1y is a fundamental region for O x (i)-action on S2. The vertices of A1x1y become corner reflectors, and the edges minus the vertices become the reflector lines. As a result, S2/[O x (i)] = Oh, where O x (i) = S4 x Z2 = n1 (Oh). We remark that S2/(i) = P2 is the projective plane. Since the antipodal map i commutes with O, the octahedral action on S2 induces the action generated by a and b on P2, which is isomorphic to the octahedral group O. As aresult, we also obtain P2/(a, b) = Oh. We will now describe the octahedral action O on P2. The left diagram in Figure 3 illustrates a fundamental region on S2 used to obtain P2 under the antipodal map i = (1,3)(2,4)(5, 6). For any arc x, y, z, we let [x, y, z] be its projection in P2. The arc 1, 2, 3 (or 3,4,1 etc) on S2 projects to an orientation reversing loop [172,3] on P2. The generator a maps the loop [172,3] onto [2714] = [2TT][174] = [2,1][3, 2], which traces the same loop as [1,2,3] starting at a different point. Thus a leaves the loop [1, 2,3] invariant and restricted to this loop is a rotation. On the other hand, the map b maps the loop [1, 2,3] onto [2, 5,4] whose image is shown as a bold line in the middle _2 _ _ diagram in Figure 3 above. Moreover, b maps the loop [1,2,3] onto the loop [1, 3, 5]. Thus the Z3-action generated by b does not leave the orientation reversing loop [1,2,3] on P2 invariant. However, it is important to emphasize that this does not imply the Z3-action leaves no orientation reversing loops invariant. In fact, we can find another orientation reversing loop on P2 which is left invariant under the map b. It can be found by looking at the octahedron S2 which double covers P2. Consider the circle on S2 which contains the vertices consisting of the midpoints of 4, 5, 5,3, 3, 2, 2, 6, 6,1 and 1,4. One can check that this circle is left invariant under b and the covering translation i, hence it projects to an orientation reversing loop on P2 left invariant under b. It follows that the entire S4 -action on P2 does not leave any orientation reversing loop invariant. Lemma 2.1. Let Z2 be a subgroup of n1(Oh) such that P2 ^ Oh is the covering corresponding to Z2. Then Z2 = (i). Proof. We will show that there is only one element of order two in Oh = S4 x Z2 acting on S2 which is fixed point free and orientation reversing, and that element is i. Since the elements in S4 and Z2 commute, we will first look at all elements of order two in S4. In this group, there are nine such elements. Six of them are a rotation of 180° where their rotational axes are on midpoints of edges. For example, one rotational axis passes the midpoint of 1,4 and 2,3. Another one passes the midpoint of 2, 5 and 4,6. Notice that all six types of these rotations are conjugate in S4. Moreover, there are three types of 90° rotations, call them r1, r2 and r3, where r1 = (1, 2,3,4), r2 = (1, 6,3,5) and r3 = (2, 5,4,6) respectively. Clearly, they generate three kinds of 180° rotations which are conjugate in S4. As a result, S4 has two conjugacy classes of order two elements, and we will choose a and (ab)2 from the group to represent each class. There is an easy way to verify if two elements in Sn are conjugate for n G N by checking their cycle types. (See [2, Chapter 4].) Now, we compose them with the antipodal map i to obtain ai = (1,4)(2,3) and (ab)2i = (1,3). Since both maps have a fixed point, if P2 ^ Oh is the covering corresponding to any Z2, then Z2 = (i). □ Proposition 2.2. Let <: G ^ Homeo(P2) be a finite action such that P2/< is homeomor-phic to Oh. Then G ~ S4 and < is conjugate to the standard action S4 = (a, b). Moreover, no orientation reversing loop is left invariant by the G-action. Proof. Let v: P2 ^ P2/ (a, b) and vv : P2 ^ P2/< be the orbifold covering maps. By assumption there exists a homeomorphism h: P2/(a, b) ^ P2/<. By Lemma 2.1, the Z2 subgroup of ni(P2/<) giving rise to a covering P2 ^ P2/< is unique. Hence h lifts to a homeomorphism h: P2 ^ P2 and we obtain the following commutative diagram: P2 h ) P2 P2/(a, b) —4 P2/< This implies that G ~ S4 and h conjugates < to the standard action (a, 6). □ 3 Chiral dodecahedral/icosahedral symmetry I and achiral dodeca-hedral/icosahedral symmetry Ih We describe the groups I = A5 and Ih = I x Z2 = A5 x Z2 acting on the 2-sphere S2, and show how A5 acts on P2. We view S2 as a dodecahedron consisting of 12 pentagons as shown in the first two figures from the left in Figure 4. We also consider two elements a and b in S20 where a = (1,2)(3, 7)(4,13)(5,8)(6,14)(9,12)(10,19)(11, 20)(15,18)(16,17) and the element b = (2, 5,7)(3,6,13)(4,12, 8)(9,11,19)(10,18,14)(15,17,20). The two elements act on the dodecahedron S2, and we can see that a is a 180° rotation about the axis passing through the midpoint of edges 1,2 and 16,17. On the other hand, b is a 120° rotation about the axis passing through the vertices 1 and 16. Moreover, ab-1 is a 72° rotation about the axis passing through the barycenter of the pentagon whose vertices are 1, 2, 3, 4, 5 and 16, 17,18, 19, 20 respectively since ab-1 = (1,2,3,4, 5)(6, 7,8,9,10)(11,12,13,14,15) (16,17,18,19, 20). Consequently, a and b generate a group isomorphic to I = A5 written by I = (a, b | a2 = b3 = (ab-1)5 = 1). We use E to denote the quotient space of S2 by I, and we will look for a fundamental region for E on S2. We will first observe that one of the pentagons consists of vertices 1, 2, 3, 4 and 5 tiles the remaining pentagons on S2. This pentagon is sent to the pentagon with the vertices 1, 5, 6, 12, 7 by b. Then, ab-1 permutes the remaining pentagons in the front of S2. On the other hand, (ab-1)2 sends the vertices 1, 5, 6,12, 7 to the vertices 4, 3, 9,15, 10. Then, the map a sends them to the vertices 13, 7, 12, 18, 19 on the back of S2. At this stage, one can see that ab-1 permutes all pentagons on the back of S2 except the one on the center whose vertices are 16, 17, 18, 19, 20. However, it can be obtained by applying the map b-1(ab-1)2 on the vertices 4, 3, 9, 15,10. 8 8 14 14' (a) Front of S2. (b) Back of S2. (c) Triangulated face. 5 2 Figure 4: S2 as a dodecahedron. Next, we will add a vertex on the barycenter of the pentagon 1, 2, 3, 4, 5 (see Figure 4), which we denote by y. We also add vertices x (1 < i < 5) on this pentagon. We can see that A12y tiles the remaining triangles on the pentagon 1, 2, 3, 4 and 5 (see Figure 4) by the map ab-1. By the argument above, this proves that A12y is a fundamental region for E. Now the edges 1,x1 and 2, x1 are identified by a G I. Likewise, ab-1 G I identifies edges 1, y and 2, y. The vertices 1, x1 and y are fixed by the elements b, a and ab-1 respectively. Thus, the vertices project to the cone points on E of orders 3, 2 and 5 respectively. Consequently, E = S2/I = E(2, 3, 5). In order to obtain Ih, we consider an antipodal map on S2 defined by i = (1,16)(2,17) (3,18)(4,19)(5,20)(6,14)(7,15)(8,11)(9,12)(10,13). It is easy to check that the antipodal map commutes with the elements of I, hence I x Z2 = (a, b, i | a2 = b3 = (ab-1)5 = i2 = 1, [a, i] = [b, i] = 1). As a result, i induces the reflection map on S2/I = E(2,3, 5). We choose a triangle whose vertices are 1, x1 and y on the fundamental region A12y. When the map a(ab-1)3b-1(ab)i is applied on A1x1y, its image is A4x3y. Since A4x3y is identified to A2x1y by (ab-1)-2 G I, this shows the antipodal map i on S2 induces the reflection map on E(2,3,5) at its equator line. Consequently, S2/[1 x (i)] = Ih, where I x (i) = A5 x Z2. The quotient space S2/[1 x (i)] is a mirrored disk where cone points of order 2, 3 and 5 are on the mirror. Note that S2/ (i) = P2 is the projective plane and the map i commutes with I. Thus, the icosahedral action on S2 induces the action generated by a and b on P2, which is isomorphic to the icosahedral group I. Hence, we obtain P2/ (a, b) = Ih. We will now describe the I = A5-action on P2, and show that it is unique up to conjugation. The front and back of S2 in Figure 4 describe a fundamental region used to obtain P2 = S2/(i) where i is the antipodal map on S2. Note that the boundary of each region in the diagram is left invariant and the interior of each region is exchanged under i. The arc 7,13, 8,14,9,15 (or 15,10,11,6,12,7 etc) projects to an orientation reversing loop [7,13, 8,14, 9,15] on P2. The map ab-1 leaves the outer most loop containing the arc invariant up to the covering translation i. Thus, the induced map ab-1 in P2 leaves this orientation reversing loop invariant. On one hand, a leaves the circle containing vertices 1, 2, 3, 9,15,16,17,18,12, 7, 1 in S2 invariant which double covers an orientation reversing loop on P2. Note that a leaves this orientation reversing loop invariant. However, the orientation reversing loops [7,13, 8,14,9,15] and [3,4,5, 6,12,18] = [3,4,5,6,9, 3] in P2 are exchanged by a. Finally, b will induce a map b on P2. One can see this since three orientation reversing loops in P2, namely [7,13, 8,14, 9,15], [2,3,4,10,11,17] = [2,3,4,10,8, 2] and [5,6,12,18,4,5] = [5, 6, 9,3,4, 5], are permuted under b. Note that although we can find an orientation reversing loop left invariant under b, no common orientation reversing loop exists which is left invariant by both a and b since the two maps generate an A5-action on P2. Lemma 3.1. Let Z2 be a subgroup of n1(1h) such that P2 ^ Ih is the covering corresponding to Z2. Then Z2 = (i). Proof. We claim that there is only one element of order two in Ih = A5 x Z2 acting on S2 which is fixed point free and orientation reversing up to a conjugacy. Notice that all elements in Ih have the form of al bmin for some l, m, n G Z where a, b G A5 and i G Z2. Since a corresponding covering space must be regular, the group generated by al bmin must be a normal subgroup in Ih. In particular, albm generates a normal subgroup of A5 which is impossible unless l = m = 0. Therefore, a covering space of the orbifold Ih corresponding to a Z2 subgroup in n1(/h) = A5 x Z2 is S2/(i) = P2. Therefore, an A5-action on P2 with quotient type Ih is unique up to conjugacy. □ Proposition 3.2. Let f: G ^ Homeo(P2) be a finite action such that P2/f is homeomor-phic to Ih. Then G ~ A5 and f is conjugate to the standard action I = (a, b). Moreover, no orientation reversing loop is left invariant by the G-action. Proof. The proof is similar to that of Proposition 2.2 and uses the above Lemma 3.1. □ 4 Chiral tetrahedral symmetry T and pyritohedral symmetry Tv We consider the groups T = A4 and Tv = T x Z2 = A4 x Z2 acting on the 2-sphere S2 and describe how T acts on the projective plane. 4 4 4 1 2 9 1 1 13 3 3 14 2 2 14 3 Figure 5: S2 as a tetrahedron. We view S2 as a tetrahedron which has four faces: A124, A134, A234 and A123 (see Figure 5). We add a total of 14 vertices on the faces to triangulate the tetrahedron, and A123 in the Figure 5 illustrates a "bottom" of the tetrahedron. Consider elements of S14 where a = (1,2, 3)(5, 6,7)(9,14,13)(10,12,11) and b = (1,2)(3,4)(5, 8)(6, 7)(10,13)(11,14). These two generators act on the tetrahedron S2. For instance, a is a 120° rotation about the axis passing throught the vertices 4 and 8; and b is a 180° rotation about the axis passing throught the vertices 9 and 12. It is easy to see that ab = (1, 3,4)(5,8,6)(9,14,10)(11,13,12) is a 120° rotation about the axis passing through the vertices 2 and 7. Hence, the two elements a and b generate a group isomorphic to A4, and we call this group by T = (a, b | a3 = b2 = (ab)3 = 1), which is the tetrahedral group. Let E be the quotient space of S2 by the group T. We will observe that the face A123 on the "bottom" of this tetrahedron tiles the rest of its faces. To understand this, we look at the map b which sends from A123 to A124. Then, the map a permutes A124 by 120° each time to tile the whole tetrahedron. However, this argument shows that we may choose A128 for a fundamental region for E since the map a permutes within the three triangles A128, A238 and A318 on the "bottom" face A123 of S2. Notice that b, which has the order two, fixes the vertex 9. Hence, it becomes an exceptional point of order two. Further, b identifies edges 1, 9 and 2, 9. On the other hand, a fixes the vertex 8, hence this vertex becomes a cone point of order three. Also, a identifies edges 1,8 and 2,8. Moreover, the map ab, which has an order three, fixes the vertex 2 to obtain an additional cone point of order three. Consequently, E = S2/T =E(2,3,3). Next, we will discuss how to obtain Tv. An antipodal map defined by i = (1, 6)(2, 7) (3,5)(4,8)(9,12)(10,13)(11,14) on S2 commutes with the elements in T, hence we have T x Z2 = (a, b, i | a3 = b2 = (ab)3 = i2 = 1, [a, i] = [b, i] = 1) and i induces a map on S2/T = E(2,3,3). However, it requires some work to analyze what map i induces on the orbifold E(2,3, 3). First, let x and y be the mid-point of the edge 1,8 and 2, 8 respectively. Since the points x and y are identified in E(2, 3, 3) by a G T, we may view the union of x, 9 and 9, y as the vertical equator line on E(2,3,3). Notice that the induced map i on E(2,3,3) fixes all points on the vertical equator line. It can be checked by observing that a-1(ab)-1(a2b)i fixes the points on the line x, 9; and (ab)(a2b)i fixes the points on the line 9, y. Secondly, we will show that the induced map i on E(2, 3, 3) is a reflection on the vertical equator line x, 9 U 9,y. To see this, consider A189 lying on our fundamental region A128. Apply the map a(ab)2i on A189 gives us A819 which is a reflection on x, 9. Likewise, A289 is reflected on 9, y by the map a(ab)2bi to get A829. As a result, the induced map i on E(2, 3, 3) is a reflection at the vertical equator line on the orbifold. By the argument above, S2/[T x (i)j = Tv, where T x (i) = A4 x Z2 and the quotient space S2/[T x (i)] is a mirrored disk containing a corner reflector containing one exceptional points of order 2 and 3 on it and one exceptional point of order 3 in its interior. Recall that the antipodal map i commutes with T on S2, hence a, b e T induce maps a and b on S2/(i) = P2. Moreover, P2/(a, b) = S2/[T x (i)] = Tv, where (a, b) is isomorphic to T. We will now describe the T = A4-action on P2 and show that it is unique up to conju-gacy. 2 14 11 3 V r/ /A 4 _____> 2 14 11 3 >To\ /12\ 4 2 14 11 3 r// yTff\ /12N. 4 Figure 6: Fundamental region on tetrahedron. The left diagram in Figure 6 above illustrates a fundamental region on S2 used to obtain P2 under the antipodal map i, where S2 is viewed as in Figure 5 and S2/ (i) = P2. This can be seen by observing the circle containing vertices, 5, 2, 14, 3, 7,11 is left invariant by i, and the vertices 4, 10, 6, 12 are sent to 8, 13, 1, 9 respectively. The projective plane is obtained by identifying the opposite side in this polygon. Recall a and b are generators of the tetrahedral group T = A4 operating on S2. Furthermore i e T = A4. Thus T induces an action on P2 and the elements a, b e T induce maps a, b on P2. Notice that the generator a maps the loop [5,2,14, 3] = [5~2][2,14][14, 3] in P2 onto [6T3][37T3][T37T] = [6T3][5;i0][106] = [6,3,5,10,6] = [5,10,6, 3], and a2 maps this loop onto [2,6,12,7]. Each image is expressed as a bold line in the Figure 6 above. Thus, the map a does not leave this orientation reversing loop invariant in P2. Likewise, ab and Jab)2 map the loop [5, 2,14, 3] = [2,14,3,7] onto [2,10,4,7] and [2,6,12, 7] respectively. Furthermore, b maps the loop [5,2,14, 3] onto [11,4,6,14]. The loop consists of vertices 2, 6, 12, 7,1 and 9 on the tetrahedron S2 is left invariant by the map b e T and the covering translation i, hence the arc having vertices 2, 6,12 and 7 projects to an orientation reversing loop on P2. There is no orientation reversing loop in P2 which is left invariant by both a and b. Lemma 4.1. Let Z2 be a subgroup of ni(Tv) such that'. sponding to Z2. Then Z2 = (i). ^ Tv is the covering corre- Proof. We will show that the orbifold Tv has only one P2 covering space up to a conjugacy. 22 Notice that A4 has three elements of order two. These elements are b, aba 1 and a2ba which are all equivalent. Thus, A4 x Z2 has two conjugacy classes of order two elements 5 5 5 7 7 7 which reverse orientation, namely bi and i. Since bi = (1,7)(2, 6)(3, 8)(4,5)(9,12) fixing vertices 10, 11, 13 and 14, we have a desired conclusion. □ Proposition 4.2. Let p: G ^ Homeo(P2) be a finite action such that P2/p = Tv. Then G ~ A4 and p is conjugate to the standard action generated by (a, b). Moreover, no orientation reversing loop is left invariant by the G-action. Proof. The proof follows as in Proposition 2.2 and uses Lemma 4.1. □ We remark that [4] contains excellent figures to show us how each element in A4 acts on a tetrahedron. 5 Achiral tetrahedral symmetry Th In Section 1, we have seen the O = S4-action on S2 where S2/O is £(2, 3,4), which is an orientable orbifold. In this section, we will investigate another O = S4-action on S2. However, the resulting quotient space S2/O = Th will be non-orientable this time. More specifically, it will be a mirrored disk which contains two cone points of order three and one cone point of order two on the mirror. Note that we will triangulate S2 as shown in Section 4 which is a tetrahedron. First, we will begin by providing generators to define a group isomorphic to S4. Consider two elements a = (1, 2)(6, 7)(10,11)(13,14) and b = (2,4, 3)(5, 7, 8)(9,11,13) (10,12,14) in Si4. We can see that a is a reflection on the circle containing vertices 4, 5, 9, 8, 3 and 12 in S2. On the other hand, b is a 120° rotation about the axis passing through vertices 1 and 6. It is easy to check ab = (1,2,4, 3)(5, 6,7,8)(9,10,12,13)(11,14). Although ab reverses an orientation, it is called improper rotation. As a result, S4 = (a,b | a2 = b3 = (ab)4 = 1). Secondly, A4 is an index two subgroup of S4 and the subgroup can be expressed by using the two generators for S4. In order to get a presentation for A4, consider (ab)2 = (1,4)(2,3)(5, 7)(6, 8)(9,12)(10,13) which is a 180° rotation about the axis passing through vertices 11 and 14. Then, b(ab)2 = (1, 3,4)(5,8, 6)(9,14,10)(11,13,12) is a 120° rotation about the axis passing through vertices 2 and 7. Consequently, we obtain a desired subgroup A4 = (b, (ab)2 | [(ab)2]2 = b3 = [b(ab)2]3 = 1). Thirdly, we will look for a fundamental region for S2/A4. It is easy to compute that the map b permutes A134, A123 and A142. Further, b(ab)2 maps from A123 to A432. Thus, we will look at A134. However, A137 tiles A134 using the element b(ab)2. Then, the vertices 1 and 7 become order 3 cone points since they are fixed by b and b(ab)2 respectively. Thus, we may choose A137 for our fundamental region. Notice that the vertex 11 is fixed under (ab)2, which becomes the order 2 cone point, and it is identified to the vertex 13 by b(ab)2 € A4. Now, b(ab)2 identifies 17 and 3/7; b(ab)2b-1 identifies 1713 and 3713. As a result, the quotient space S2/A4 is indeed E(2, 3, 3). Finally, we will discuss how to obtain the orbifold Th. Recall the map a € S4 reflects on the circle containing vertices 4, 5, 9, 8, 3 and 12 in S2. We compose this map by a covering translation (ab)-1[(ab)2b]2(ab) = (1,2,3)(5, 6, 7)(9,14,13)(10,12,11) € A4, which is a 120° rotation about the axis passing through vertices 4 and 8. Then, (ab)-1[(ab)2b]2(ab)a sends the triangle containing vertices 1, 13 and 7 to the triangle containing vertices 3, 7 and 13. Notice that 1, 7 and 3,7 are identified in E(2,3,3). Likewise, 1,13, and 3,13 are identified. Thus, the circle containing vertices 1, 7, 13 becomes the line of reflection under the map induced by a on E(2,3, 3). Consequently, we obtain S2/S4 = Th. Unlike the previous orbifolds, Th is not covered by a projective plane. Notice that ni(Th) = S4 contains six elements of order two which are orientation reversing. All of them are a reflection at a plane whose intersection with the tetrahedron is a triangle containing either vertices 2, 11, 3; vertices 1, 12, 2; vertices 3, 10, 1; vertices 4, 9, 3; vertices 4,13, 2; or vertices 4,14,1. Clearly, none of them give a fixed point free action on the tetrahedron S2, and hence this yields the following lemma. Lemma 5.1. The orbifold Th is not covered by a projective plane. 6 Cyclic and dihedral actions We describe the cyclic and dihedral actions on S2 and the projective plane P2. In describing these actions, it is convenient to use spherical coordinates. Therefore for any point (x, y, z) € S2, we let x = sin ^ • cos 0, y = sin ^ • sin 0 and z = cos We begin by defining a rotation of order m on S2 as follows: r(x,y, z) = (sin^ • cos(0 + m), sin^ • sin(0 + m), cos Note that r fixes only the points (0,0,1) and (0,0, -1). A spinning map s which rotates through an angle of n about the y-axis is defined by s(x, y, z) = (—x, y, -z). In terms of the spherical coordinate system, the map is defined by s(x, y, z) = (sin(^ + n) • cos(-0), sin(^ + n) • sin(-0), cos(^ + n)). One can check that s o r o s-1 = r-1, and therefore (r, s) generates a dihedral group Dih(Zm) acting on S2. Finally we define the antipodal map i on S2 by i(x, y, z) = (—x, -y, -z). In terms of the spherical coordinate system, i(x, y, z) = (sin(^ + n) • cos 0, sin(^ + n) • sin 0, cos(^ + n)). We have S2/ (i) = P2. Observe that i o s o i-1 = s and i o r o i-1 = r. Hence i commutes with r and s which implies the following lemma: Lemma 6.1. The maps r and s induce homeomorphisms r and s on P2 respectively. Let k(x,y,z) = (sin^• cos(0 + ^), sin^• sin(0 + ^), cos. A computation shows that k o s o k-1 = r o s, k o r o k-1 = r and k o i = i o k. This implies that the induced map s on P2 conjugates s to s o s and commutes with s. Notice that we can express the three maps above in terms of a PL-category. Let m € N. We assume that vertices from 1 to 2m are located on the equator line of S2. The vertices 2m +1 and 2m + 2 are on the poles. As a result, we obtain 4m many faces (triangles) from these vertices on S2. If m > 1 is odd, then the rotation r is expressed by r = (1,3,..., 2m-1)(2,4,..., 2m) whose order is m. On one hand, if m is even, then r = (1, 2, 3,..., 2m) whose order is 2m. In each case, the vertices 2m + 1 and 2m + 2 are fixed under r since they are the north and the south poles. The spinning map for m > 1 passing through the y-axis is defined by s = (2, 2m)(3, 2m - 1) • • • (m, m + 2)(2m + 1, 2m + 2). The vertices 1 and m + 1 are fixed under s. The antipodal map for m > 1 is defined by i = (1, m + 1)(2, m + 2) • • • (m, 2m)(2m + 1, 2m + 2). For the case when r has order two, we place vertices 1 to 4 on the equator of S2 and vertices 5 and 6 are on the poles (see the Figure 2 in Section 2). Then r = (1,3)(2,4), s = (2,4)(5,6), and i = (1,3)(2,4)(5, 6). Define a map j = (1,6,3,5). A computation shows that j o s o j _1 = r, j o r o j _1 = s and j o i o j _1 = i. Therefore j is conjugate to j on P2. Summarizing we have the following lemma: Lemma 6.2. There exists a homeomorphism k on P2 which conjugates s to j o j and commutes with r. When j has order two, there exists a homeomorphism j on P2 which conjugates rs to s and s to rs. 6.1 Quotient types S(0, m, m), S(2, 2, m), D^ and D^ The space E(0, m, m) is an orbifold whose underlying space is a 2-sphere with two cone points each of order m. Similarly E(2, 2, m) is an orbifold whose underlying space is a 2-sphere with three cone points, two of order 2 and one of order m. The orbifold Dm is a mirrored disk containing a cone point of order m and 2 on the mirror and its interior respectively. The orbifold Dm is a mirrored disk with three cone points on the mirror, one of order m and two of order 2. Observe that we obtain S2/(r) = E(0, m, m), which double covers S2/(r, s) = E(2,2, m). Since i commutes with r and s, we have Dih(Zm) x Z2 = [(r) o_1 (s)] x (i) acting on S2. Now r and s acting on S2 induce a Dih(Zm)-action on S2/(i) = P2. Furthermore, i operating on S2 induces an orientation reversing involution i on E(2,2, m), and we have E(2,2, m)/(i) = P2/(r, s) = S2/(Dih(Zm) x Z2). Thus the fundamental group of the quotient space P2/(r, s) is Dih(Zm) x Z2 = [(r) o_1 (s)] x (i). Let p: S2 ^ S2/(r, s) = E(2, 2, m) be the orbifold covering map and note that p(0,0,1) = p(0,0, -1) is the cone point of order m. Since i(0,0,1) = (0,0, -1) and s(0,0, -1) = (0,0,1), it follows that_i(p(0,0,1)) = p(0,0,1), and thus i fixes the cone point of order m in E(2,2, m). Hence i is a reflection. If m is odd, rk(0,1,0) = (0, -1,0) for any k. Thus p(0,1,0) and p(0, -1,0) are the two distinct cone points of order 2 in E(2,2, m). If m is even, then p(0,1,0) = p(0, -1,0) is a cone point of order 2 since rm (0,1,0) = (0, -1,0). We will consider the cases m odd and m even separately. Suppose m is odd. Then since i(0,1,0) = (0, -1,0), it follows that i(p(0,1,0)) = p(0, -1,0) and thus i exchanges the two cone points of order 2. Since i fixes the cone point of order m, it follows that P2/(r, s) = Dm. The order two elements in Dih(Zm) x Z2 are: i, s, rjsi. One can check that rjsi(x, y, z) = (sin^ • cos(-0 + ), sin^ • sin(-0 + ), cos . By choosing ^ = 0 or n, the map fixes the points (0,0, ±1) on S2. Note that E(0, 2, 2) = S2/(s) is not a regular covering space of Dm since (s) is not a normal subgroup of n1(Dm) = Dih(Zm) x Z2. Thus i is the only orientation reversing element which is fixed-point free. This implies that when m is odd, n1(Dm) has a unique normal Z2 subgroup generated by a fixed-point free orientation reversing element, and the covering of Dm corresponding to this subgroup is P2. Next we suppose m is even and show how to obtain Dm. Write m = 2n and observe that the rotation r of order 2n on S2 is defined as follows: r(x, y, z) = (sin ^ • cos(0 + n), sin ^ • sin(0 + n), cos . Note that r fixes only the points (0,0, ±1), and since rns(1,0,0) = (1,0,0) it follows that p(1,0,0) is one of the cone points of order 2 in E(2, 2,2n). Consider the point (sin(f) • cos(n), sin(f) • sin(21), cos(f)). We see that rn+1 s fixes (sin(72) • cos(n), sin(f) • sin(n), cos(f)), and so it follows that the point p((sin(n) • cos(), sin(f) • sin(2i), cos(f))) is the other cone point of order 2 in E(2, 2, rn). Define a reflection /: S2 ^ S2 by /(x,y, z) = (sin(-^) • cos(—0 + ), sin(-^) • sin(—0 + ), cos(—. A calculation shows that /s/-1 = rs and /r/-1 = r-1. Thus we have Dih(Z2n) o Z2 = [(r) o-1 (s)j o (/) acting on S2 and an induced map f acting on E(2, 2, m) = S2/(r, s). A further computation shows that /(1,0,0) = /(sin(n) • cos(0), sin(|) • sin(0), cos(|)) = (sin(-n) • cos(), sin(-n) • sin(21), cos(-n)). Applying rn to this element, we see that rn(sin(-n) • cos(), sin() • sin(n), cos(-n)) = (sin(2) • cos(), sin(n) • sin(n), cos(f)). Hence the induced map / exchanges the two cone points of order two. In addition, consider a set F C S2 defined by F = {(sinf • cos(4n + n), sinf • sin(41 + n), cosf) | f G R}. Notice that /(sin f • cos( 4n + n), sin f • sin( 41 + n), cos f) = (sin(—f •cos(-4n - n + ), sin(—f •sin(-4n - n + )>cos(-^)) = ( - sin f • cos(+ n), - sin f • sin(41 + n), cos(f)) = rm (sin f • cos(41 + n), sin f • sin(41 + n), cos f). Therefore, p(F) = fixj!} in E(2, 2, m) where p denotes the covering map. Consequently, / is a reflection exchanging the cone points of order 2. Thus E(2,2, m)/(f) = D2n, and n1(D2n) = Dih(Z2n) o Z2 = [(r) o-1 (s)] o (/) where /s/-1 = rs and /r/-1 = r-1. The elements of order two are: rn, rks, rk/ (for any integer k = 0,1,..., 2n — 1). The only orientation reversing elements of order two are rk/, and they all fix the points (0,0,1) and (0,0, —1). Thus there is no orbifold covering P2 ^ D2n. We summarize the above in the following theorem. Theorem 6.3. Let f: G ^ Homeo(P2) be a finite action such that P2/f = Dm. Then m is odd, G ~ Dih(Zm) and f is conjugate to the standard action generated by (f, f). Proof. By the above m is odd. Let v: P2 ^ Dm = P2/ (f, f) be the covering map corresponding to the standard action. For the action f: G ^ Homeo(P2) with P2/f = Dm, let p: P2 ^ P2/f be the covering map and h: Dm ^ P2/f be a homeomorphism. By the above the subgroup p* (n1(P2)) in n1(P2/f) is unique. Thus h lifts to a homeomorphism h of P2 such that hv = ph. This implies that the two actions are conjugate by h. □ Suppose m is even. Since rm (0,1,0) = (0,-1,0) and ¿(0,1,0) = (0,-1,0), it follows that i fixes the cone point p(0,1,0). Since i also fixes the cone point of order m, we have that i is a reflection leaving each cone point fixed and P2/ (r, s) = . The order two elements in Dih(Zm) x Z2 are: i, s, rjsi or rm¿. Since rjsi(0,0,1) = (0,0,1), we only need to consider rm i(x, y, z) = ( sin(^ + n) • cos(0 + n), sin(^ + n) • sin(0 + n), cos(^ + n)). Letting ^ = n and 0 = 0, we see that the point (1,0,0) is fixed by rm i. Thus has a unique P2 covering up to conjugation. This implies that when m is even, ) has a unique normal Z2 subgroup generated by a fixed-point free orientation reversing element, and the covering of corresponding to this subgroup is P2. We now suppose m is odd and show how to obtain . Define a reflection /0 : S2 ^ S2 by /0(x, y, z) = (sin(-• cos(—0), sin(—• sin(-0), cos(—= (—x, y, z). One can check that 10s1-1 = s and /0 r/-1 = r-1. Hence Dih(Zm)oZ2 = [(r)o_1(s)]o(/0) acting on S2 and an induced map "0 acting on E(2,2, m) = S2/(r, s). Clearly /0 fixes the points (0,1,0) and (0, —1,0). Recall p(0,1,0) = (0, —1,0). Hence the induced map /"0 on E(2, 2, m) is a reflection which fixes each cone point. Thus E(2, 2, m)/("0) = and ni(Dm) = Dih(Zm) o Z2 = [(r) o_i (s)] o (/0). The elements of order two are: s, rk s, rk/0 (any integer k = 0,1,..., m — 1), and s/0. The only orientation reversing elements of order two are rk/0 and s/0, but they all have fix-points. Thus there is no orbifold covering P2 ^ . We summarize the above in the following theorem whose proof is similar to Theorem 6.3. Theorem 6.4. Let p: G ^ Homeo(P2) be a finite action such that P2/p = . Then m is even, G — Dih(Zm) and p is conjugate to the standard action generated by (", "). 6.2 Quotient types S2m and Z^ We use S2m and Z^ to denote a projective plane that has one cone point of order m and a mirrored disk containing an order m cone point in its interior respectively. The orbifold Zq denotes a mirrored disk without an exceptional point within its interior, and if m = 1, then S2(1) = P2. Recall (r) x (i) = Zm x Z2 acts on S2. Hence, the involutions in this group are either i, r ^ or ir ^ for an even number m. If m is even, then irtt(x, y, z) = (sin^ • cos0, sin^ • sin0, — cos. The fixed point set of this map is the circle at the equator on S2 and occurs when ^ = 2. Thus, S2/(irm) = S2/Z2 = Zq. Furthermore, r on S2 induces a rotation r on Zq fixing a point not on the mirror, and inducing an action r acts on P2 = S2/ (i). In the meantime, i on S2 induces a reflection i on E(0, m, m) = S2/(r) since rm ( — 1,0,0) = (1,0,0) and i(1,0,0) = ( —1,0,0). As a result, we obtain Z^ = ZQV(r) = P2/(r) = S(0,m,m)/(i) for m is even. Note that n1(Z^) — Zm x Z2 is generated by r and i, where i is the only fixed-point free orientation reversing element. This implies that when m is even, n1(Z^) has a unique normal Z2 subgroup generated by a fixed-point free orientation reversing element, and the covering of Zh corresponding to this subgroup is P2. We now show how to obtain Zh when m is odd. Let p be a homeomorphism of S2 defined by p(x, y, z) = (sin ^ • cos 0, sin ^ • sin 0, — cos . A computation shows that p and r commute. We obtain an orbifold covering map S2 ^ E(0, m, m) = S2 / (r) with p inducing a reflection p on E(0, m, m). The quotient space E(0, m, m)/(p) = Zh and n1(Zm) — Zm x Z2 is generated by r and p. Since m is odd, the only element of order 2 in n1 (Zh) is p which has fixed points. Thus there is no orbifold covering P2 ^ Zh when m is odd. Consequently, the following theorem is obtained: Theorem 6.5. Let <: G ^ Homeo(P2) be a finite action. If P2/< = Zh, then m is even, G — Zh and < is conjugate to the standard action generated by (r). If m is odd, we again have r inducing a map r on P2 = S2/ (i), and one can check that the induced map i on E(0, m, m) = S2/(r) is the antipodal map. Consequently, we obtain S2h = P2/(r) = E(0, m, m)/(l). Furthermore n^S2™) — Zh x Z2 is generated by r and i, where the only order two fixed-point free orientation reversing element is i. Hence when m is odd, ^1(S2m) has a unique normal Z2 subgroup generated by a fixed-point free orientation reversing element, and the covering of S2h corresponding to this subgroup is P2. To obtain S2h when m is even, we write m = 2n and define a homeomorphism h of S2 by h(x, y, z) = (sin($ + n) • cos(0 + 2n), sin($ + n) • sin(0 + 2n), cos($ + n)). Observe that h is a composition of the antipodal map and a rotation through n/2n, and h generates a Z2(2n)-action on S2 and S2/(h2) = E(0,2n, 2n). It follows that the induced map h on E(0,2n, 2n) is the antipodal map and E(0, 2n, 2n)/(h) = S2(2n). Furthermore n1(S2(2n() — Z2(2n) is generated by h. The only element of order 2 is h2n, and h2n(x, y, z) = (sin $ • cos(0 + n), sin $ • sin(0 + n), cos $) has fixed-points. Thus there is no orbifold covering P2 ^ S2h when m is even. Summarizing these results we obtain the following theorem: Theorem 6.6. Let <: G ^ Homeo(P2) be a finite action. If P2/< = S2™, then m is odd, G — Zh and < is conjugate to the standard action generated by (r). 6.3 Nonexistence of quotient type Cm m The orbifold Ch is a mirrored disk with two cone points on the mirror of order m. We will show that the orbifold Ch h is obtained by some covering translations on S2. Recall the reflection map on the yz-plane defined on R3 by Z0(x, y, z) = (—x, y, z) and the rotation r(x, y, z) = (sin $ • cos(0 + h), sin $ • sin(0 + h), cos $). It is easy to check that Dih(Zh) = (r) o_1 (Z0). Then, we obtain E(0, m, m) = S2/(r) and the reflection on S2 induces a reflection Z0 on E(0, m, m). As a result, Ch = E(0, m, m)/(Z0) where ^1(CmJm) = Dih(Zh). The order two elements in ^(Ch ) are rjZ0 for 0 < j < m, or r m for m even. A calculation shows that rjZo(x,y,z) = (sin(-$) • cos(-0 + h), sin(-$) • sin(-0 + h), cos(-$)), which has fixed points at (0,0, ±1) G S2 when $ = 0 or n. Since Z0 and rm when m is even, have fixed points, Ch is not covered by P2. We therefore have shown the following proposition: Proposition 6.7. The projective plane does not cover CV 7 Finite group actions on P2 and P2 x I In this section, we summarize the above results and classify the finite group actions on P2 and P2 x I. Theorem 7.1. Let f: G ^ Homeo(P2) be a finite group action on P2. Then G is isomorphic to one of the following groups: S4, A5, A4, Zm or Dih(Zm). The orbifold quotient P2/f is an orbifold homeomorphic to one of the following orbifolds: Oh, Ih, Tv, , S2m, Dm or Dm. There is only one equivalence class for each group. (1) G ~ S4 if and only if P2/f = Oh. (2) G ~ A5 if and only if P2/f = Ih (3) G ~ A4 if and only if P2/f = Tv. (4) G ~ Zm and m is even if and only if P2/f = Zm. (5) G ~ Zm and m is odd if and only if P2/f = S2m. (6) G ~ Dih(Zm) and m odd if and only if P2/f = Dm. (7) G ~ Dih(Zm) and m even if and only if P2/f = Dm. Proof. Let f: G ^ Homeo(P2) be a finite group action. Then P2 /f is a non-orientable 2-orbifold with positive euler number x(P2 /f). The non-orientable good orbifolds (orbifolds which have manifolds for their universal covering spaces) with positive euler numbers are the following: Cm,m, S2m, Zm, Dm, Dm, Th, Oh, Ih and Tv. The result then follows by the above. □ Theorem 7.2. Lei f: G ^ Homeo(P2) be a finite group action. The action f (G) does not leave any orientation reversing loop in P2 invariant if and only if G is isomorphic to S4, A5 or A4. Furthermore, f is equivalent to one ofthese standard actions. Proof. This follows from Sections 2 through 6. □ Theorem 7.3. Let f: G ^ Homeo(P2 x I) be a finite group action. If for every g G G f(g)(P2 x {0}) = P2 x {0}, then G is isomorphic to one of the following groups: S4, A5, A4, Zm or Dih(Zm). Furthermore, there is only one equivalence class for each group which is represented by one of the standard actions. Proof. By the comment following Theorem 8.1 in [6], we may conjugate f (G) so that it is a product action. This implies that there is a G-action f 1: G ^ Homeo(P2) such that for any g G G, we have f (g)(z, t) = (f1 (g)(z),t). By Theorem 7.1, there exists a homeomorphism k of P2 such that kf 1(G)k-1 is one of the standard actions (1) through (7) listed there. Conjugating this action further by k x id proves the result. □ Theorem 7.4. Let f: G ^ Homeo(P2 x I) be a finite group action. Then G is isomorphic to one of the following groups: S4, S4 x Z2, A5, A5 x Z2, A4, A4 x Z2, Zm, Zm x Z2, Dih(Zm) or Dih(Zm) x Z2. (1) If G is isomorphic to S4, then there are two equivalence classes. (2) If G is isomorphic to either S4 x Z2, A5, A5 x Z2, A4, A4 x Z2, Zm x Z2 or Dih(Zm) x Z2, then there is one equivalence class. (3) Suppose G is isomorphic to Zm. If m is odd, then there is one equivalence class. If m is even, then there are two equivalence classes. (4) Suppose G ~ Dih(Zm). If m = 2 or if m is odd, then there are two equivalence classes. If m > 2 and m is even, there are 3 equivalence classes. Proof. Again by [6], we may assume <: G ^ Homeo(P2 x I) is a product action. Thus there exists a homomorphism <1: G ^ Homeo(P2) such that for any g G G, <(g)(z,t) = (<1(g)(z),t) or <(g)(z,t) = (<1(g)(z), 1 - t). There exists a Z2-action on P2 x I generated by a map R defined by R(z, t) = (z, 1 - t). Suppose first that <1: G ^ Homeo(P2) is not one-to-one. This implies there exists an element g0 = 1 G G such that <1(g0)(z) = z for all z G P2, and so <(g0)(z,t) = (z, 1 -1) or <(g0) = R. Since R commutes with (<1(g)(z), t) and (<1(g)(z), 1 -1), it follows that g0 commutes with every element of G. Let H = {g G G | <(g)(z,t) = (<1(g)(z), t)}. If <(g)(z,t) = (<1(g)(z), 1 - t), then <(g2)(z,t) = (<1(g2)(z),t) showing g2 G H. It follows that H is an index two normal subgroup of G, and G = H x Z2 where Z2 = (g0). Furthermore <1|H: H ^ Homeo(P2) is one-to-one. By Theorem 7.1, H is isomorphic to S4, As, A4, Zm or Dih(Zm) and conjugate to one of the standard actions. As in Theorem 7.3, we may conjugate <|H: H ^ Homeo(P2 x I) by a homeomorphism k x id to a standard action, proving the result in this case. Suppose <1: G ^ Homeo(P2) is one-to-one, and hence is a G-action. Note that in this case R G <(G). By Theorem 7.1, G is isomorphic to S4, A5, A4, Zm or Dih(Zm) and conjugate to one of the standard actions. Thus as above by conjugating <(G), we may assume that <1 is one of the standard actions. Suppose G = S4 and <1(G) = (a, b a = b3 = (ab)4 = 1). Let A and B be actions on P2 x I defined by A(z, t) = (a(z), t) and B(z, t) = (b(z),t). If B o R g <(G), then (B o R)3 = R g <(G), and this would imply that <1: G ^ Homeo(P2) is not one-to-one. Thus B G <(G) and we either have A g <(G) or A o R g <(G). Consequently there are two possibilities <(G) = (A, B) or <(G) = (A o R, B), both isomorphic to S4. They are not conjugate since the quotient space (P2 x I)/(A, B) has two boundary components while the quotient space (P2 x 1)/(AoR,B) has only one boundary component. Suppose G = A5 and let <1(G) = (a, b | a2 = b3 = (ab-1)5 = 1). As above we obtain actions A and B on P2 x I defined by A(z, t) = (a(z), t) and B(z, t) = (b(z), t). We see as in the previous case that B o R G <(g). Furthermore since (AoRoB-1)5 = R, it follows that Ao R G <(G). Thus <(G) = (A, B) with only one equivalence class. The proof is similar for A4. If G ~ Zm, then when m is odd the action is conjugate to (r x id); and when m is even the action is conjugate to either (r x id) or (r x id) o R. We now suppose <1 (G) = Dih(Zm) = (a, a). We first suppose m is even. The possible groups for <(G) are: H1 = ((a x id), ( a x id)), H2 = ((ax id)oR, ( ax id)), H3 = ((ax id), ( sx id) oR), H4 = ((ax id) o R, ( axid)o R). Clearly, H1 is not conjugate to any of the other groups since no element of H1 exchanges the boundary components of P2 x I. The element of order two ( a x id) in H2 does not exchange boundary components, however every element of order two in H3 exchanges boundary components showing H2 is not conjugate to H3. Similarly, the element (a x id) of order m in H3 cannot be conjugate to (a x id) o R in H4, showing H3 and H4 are not conjugate. Notice H4 = ((a x id) o R, ( a x id) o R) = ((a x id) o R, (a a x id)). Using Lemma 6.2, it follows that H2 is conjugate to H4, showing there are three equivalence classes when m > 2. When m = 2, Lemma 6.2 also shows that H2 and H3 are conjugate, and so we have only two equivalence classes in this case. When m is odd, the only two possibilities are ((a x id), ( a x id)) and ((a x id), ( a x id) o R). □ 8 Finite actions on twisted I-bundle over P2 For S2 x I, define a fixed-point free orientation preserving involution a: S2 x I ^ S2 x I by a(z, t) = (i(z), 1 - t). The manifold S2 x I/(a) = W is a twisted I-bundle over the one-sided projective plane P2. Let v: S2 x I ^ W be the covering map and note that v(S2 x {1/2}) = P2 is a one-sided projective plane. The levels of W are v(S2 x {t}),and a homeomorphism h of W is level preserving if h(v(S2 x {t})) = v(S2 x {t}). We may view W as the set of equivalence classes {[z, t] | (z, t) is equivalent to (i(z), 1 - t)}. Let Homeo( W, P2) be the group of homeomorphisms which leave the projective plane P2 invariant. Denote by Centp(a) the subgroup of the centralizer of a which leaves S2 x {1 / 2 } invariant and preserves the sides of S2 x{ 1 / 2 }. Every homeomorphism which leaves P2 invariant lifts to two homeomorphisms of S2 x I, one of which preserves the sides of S2 x {1/2} while the other doesn't. Thus for any homeomorphism f G Homeo(W, P2) there is a unique lift f G Centp(a), and we obtain an isomorphism L: Homeo(W, P2) ^ Centp(a). Note that since v|S2x{0}: S2 x {0} ^ dW is a homeomorphism, it follows that f is orientation preserving if and only if f is orientation preserving. We obtain the following proposition. Proposition 8.1. L: Homeo(W, P2) ^ Centp(a) is an isomorphism. There exists a map R: Homeo(W, P2) ^ Homeo(P2) defined by restricting any homeomorphism to P2. Proposition 8.2. Let : G ^ Homeo(W, P2) be an effective orientation preserving enaction. Then the restriction R^>: G ^ Homeo(P2) is an effective G-action. Proof. Let 92 = L o : G ^ Centp(a) be an orientation preserving G-action on S2 x I. Suppose there exists an element g G G such that Ry>(g) = id, and thus = id or i. Since 2 (G). Then there exists a homeomorphism k of W isotopic to the identity such that (G)k-1 = ¥>2(G). Proof. Let R^1 = 2 by y^-1«^, we may assume on W. Proof. By [7] and [8], there exists an action p: G ^ Cent+(i) where Cent+(i) consists of orientation preserving elements in the centralizer Cent(i) of i in Homeo(S2). Define an action 0: G ^ Centp(a) by 0(g)(x,t) = (p(g)(z), t). Then L-1B: G ^ Homeo(W, P2) is the extension. □ Let E (P2, G) be the set of equivalence classes of effective G-actions on P2, and let E+(W, G) be the set of equivalence classes of effective orientation preserving G-actions on W. Denote by E+ ((W, P2), G) the subset of E+ (W, G) which have a representative that leaves a one-sided projective plane invariant. Proposition 8.8. Let p: G ^ Homeo(W) be a finite action on W. Then there exists a one-sided projective plane P such that p(g)(P) = P for all g G G. Proof. Let Homeo(S2 x I, S2 x {0}) be the group of homeomorphisms which leave S2 x {0} invariant. There exists an injection L0: Homeo(W) ^ Homeo(S2 x I, S2 x {0}) n Cent(a) defined by lifting any homeomorphism to a homeomorphism of S2 x I leaving S2 x {0} invariant. Letting L0p = p: G ^ Homeo(S2 x I, S2 x {0}) n Cent(a), we obtain a G x Z2 action on S2 x I where the Z2-action is generated by a, which projects to the p-action on W. This action is equivalent to a product action by [1], and thus there is an G x Z2-invariant 2-sphere S in int(S2 x I). Furthermore, a(S) = S and v(S) is a p(G)-invariant projective plane in W. □ Corollary 8.9. E+((W, P2), G) = E+(W, G). Proposition 8.10. Let P be a one-sided projective plane in W. Then there exists a homeo-morphism k of W, isotopic to the identity, such that k(P) = P2. Proof. Isotope P to intersect P2 in simple closed curves. We may assume the number of curves in P n P2 is minimal. We will show that the number of simple closed curves in P n P2 is one. Note first that P n P2 = 0, for otherwise P C W - P2 which is isomorphic to S2 x [0,1), and this is impossible. If the number of intersections of P n P2 exceeds one, and hence the number of simple closed curves in P exceeds one, then there is a simple closed curve S G P n P2 which bounds a disk A in P. We may assume A is innermost in P, in the sense that int(A) n P2 — 0. Since S bounds a disk in WV, it follows that S is an orientation preserving loop in P2, and thus bounds a disk D in P2. Now D U A is a separating 2-sphere. If D U A bounds a ball, then we may isotope P to eliminate S. We therefore assume D U A does not bound a ball, and is therefore parallel to the sphere boundary dW = v(S2 x {0}). Lift P to an a-invariant 2-sphere S in S2 x I, let A1 and A2 be the two lifts of A in S, and let D1 and D2 be the two lifts of D in S2 x {1/2}. Denote dAi by Si. We may assume D1U A1 C S2 x [0,1 /2] and D2 U A2 C S2 x [1 /2,1]. Furthermore, there is an a-invariant simple closed curve 7 G S n S2 x {1/2}, separating S1 and S2. Note that (S2 x {1/2} - int(D1)) is a disk in S2 x {1/2} whose boundary is the boundary of the disk A1 in S. Now D1 U A1 is parallel to S2 x {0}, which implies that A1 U (S2 x {1/2} - int(D1)) bounds a ball in S2 x [0,1/2]. Thus we may construct an a-equivariant isotopy, relative to 7, which eliminates the intersections S1 and S2. Projecting this isotopy to W eliminates the S-intersection of P n P2. Thus we have shown that S n S2 x{1/2} is a single simple closed curve 7 which projects to a non-contractable simple closed curve 7 in P n P2. By an argument similar to the one above, there is an a-equivariant isotopy, relative to 7, which isotopes S to S2 x {1/2}. Projecting this isotopy to W, we obtain an isotopy taking P to P2. □ Theorem 8.11. The map r: E(P2, G) ^ E+((W, P2), G) defined by extending G-actions from P2 to W is a bijection. Proof. Let [(] G E(P2, G). By Proposition 8.7, ( can be extended to a G-action (p on W. Define r([(]) = [(]. Suppose - is a G-action on P2 such that [-] = [( G E(P2, G). Then there exists a homeomorphism h of P2 such that h((G)h-1 = -0(G). Lift h to an orientation preserving homeomorphism k of S2 and note that ik = ki. Extend k to a homeomorphism p by p(x, t) = (k(x),t). Letting h = L-1p. we see that h is an extension of h. Since R(h(p(G)h-1) = h((G)h-1 = -(G) = R(-p(G)), it follows by Proposition 8.4 that [(p = [-], and thus r is well defined. Let [S] g E((W, P2),G). Thus there is a one-sided projective plane P such that S(g)(P) = P for all g G G. By Proposition 8.10, there exists a homeomorphism of W taking P to P2. This implies that we may choose a representative S' in [S] such that S'(g)(P2) = P2 for all g G G. By Proposition 8.2, the restriction RS' is an effective G-action on P2 and therefore represents an element in E(P2, G). Let r([RS']) = [RS']. Since R(RS') = RS', it follows by Proposition 8.4 that [RS7] = [S'], and thus r([RS']) = [S'] showing r is a surjection. To show r is one-to-one, suppose that [(], [0] G E (P2, G) are such that their level preserving extensions [(p] = [p] in E+((W, P2), G). Now W/( and W/0 are homeomorphic twisted I-bundle orbifolds over one of the following 2-orbifolds: Oh, Ih, Tv, , S2m, Dm or Dm. Since by Theorem 7.1, there is only one equivalence class for each action on P2 which determines a unique quotient type, it follows that [y] = [0] showing r is one-to-one. □ Corollary 8.12. Let y: G ^ Homeo(W) be a finite orientation preserving G-action on W. Then G is isomorphic to one of the following groups: S4, A5, A4, Zm or Dih(Zm). The orbifold quotient for each action is a twisted I-bundle orbifold over the following 2-orbifolds: Oh (for S4), Ih (for A5), Tv (for A4), (for Zm and m even), S2m (for Zm and m odd), Dm (for Dih(Zm) and m odd) and Dm (for Dih(Zm) and m even). There is one equivalence class for each quotient type. 9 Orientation reversing finite actions on twisted I-bundle over P2 Recall that W = {[z, t] | (z,t) is equivalent to (i(z), 1 - t)} with P2 = {[z, 1/2] G W}, and L : Homeo(W, P2) ^ Centp(a) is an isomorphism. Let / be a homeomorphism of P2, and let / be a lift of /1 to S2. We remark that / commutes with i. A homeomorphism / : W ^ W is a product homeomorphism if /[z,t] = [f1(z),t]. Note that /|P2 = /1. Let Homeo(S2 x I, S2 x {1/2} ) be the group of homeomorphisms of S2 x I which leave S2 x {1/2} invariant. Define the map R: Homeo(S2 x I, S2 x {1/2}) ^ Homeo(S2) by restricting any homeomorphism to S2 x {1/2}. Lemma 9.1. Let f : G ^ Homeo(W, P2) be an effective G-action and let f = Lf : G ^ Centp(a) C Homeo(S2 x I, S2 x {1/2}). Then Rf : G ^ Homeo(S2) is an effective G-action. Proof. Suppose there exists an element g G G such that i?f(g) — ¿d|§2x{1/2}. Since -Rf(g) does not reverse the sides of S2 x {1/2}, it follows that f(g) = id. □ Remark 9.2. Note that the involution p of W defined by p[z, t] = [i(z), t] = [z, 1 - t], has the property that Rp = id|P2, but RLp(z, t) = (i(z), t) and thus does not restrict to the identity on S2 x {1/2}. Theorem 9.3. Let f : G ^ Homeo(W) be an effective G-action. Then f is conjugate to a product action on W. Proof. By Propositions 8.8 and 8.10, we may assume f (g)(P2) = P2 for every g G G. Let f = Lf : G ^ Centp(a). By Lemma 9.1, Rf : G ^ Homeo(S2) is an effective G-action which commutes with i. Define an action 6: G ^ Centp (a) C Homeo(S2 x I, S2 x{1/2}) by $(g) = Rf(g) x id. Thus $(g)|s2x{1/2} = ^(g)|§2x{1/2} for any g G G. Projecting this action to W, we obtain an effective product action 6 : G ^ Homeo(W, P2). We now use the proof in Proposition 8.4 to construct a homeomorphism h which commutes with a and conjugates $(G) to f(G). The homeomorphism h projects to a homeomorphism of W which conjugates 6(G) to f (G), thus completing the proof. □ We will now define the standard actions S4 x Z2, A4 x Z2, A4 x Z2, Zm x Z2 or Dih(Zm) x Z2 on W. Consider first the group S4 = (a, b | a2 = b3 = (ab)4 = 1} acting on S2 commuting with i, and its projection S4 = (a, b | a2 = b3 = (ab)4 = 1} to P2. Define the product maps A, B : W ^ W by A[z,t] = [a(z),t] and B[z,t] = [b(z),t]. Note that (A, B, p} = S4 x Z2. The other standard group actions on W are defined in a similar fashion. Theorem 9.4. Let f: G ^ Homeo(W) be an orientation reversing G-action. Then G is isomorphic to one of the following groups: S4, Zm with m even, Dih(Zm), S4 x Z2, A5 x Z2, A4 x Z2, Zm x Z2 or Dih(Zm) x Z2. (1) If G is either S4, S4 x Z2, A5 x Z2, A4 x Z2, Zm x Z2 with m even or Dih(Zm) with m odd, there is only one equivalence class. (2) If G is Zm with m > 2 even and m/2 odd, then there are two equivalence classes of Zm = Zm/2 x Z2-actions on W. (3) If G is Zm with either m/2 even or m = 2, then there is only one equivalence class. (4) If G is Dih(Zm) with m > 2 and m/2 even, there are two equivalence classes of Dih(Zm)-actions on W. (5) If G is Dih(Zm) with m > 2 and m/2 odd, there are three equivalence classes of Dih(Zm)-actions on W. (6) If G is Dih(Zm) x Z2 with m even, there is only one equivalence class. (7) If G is Dih(Zm) x Z2 with m odd, then Dih(Zm) x Z2 ~ Dih(Z2m) and there are three equivalence classes of Dih(Z2m)-actions on W. Proof. Let f: G ^ Homeo( W) be an effective orientation reversing G-action. We may assume by Theorem 9.3, that there exists G-actions f 1: G ^ Homeo(S2) and f 1: G ^ Homeo(P2), such that f(g)[z,t] = [ 2, or to (Rp, S) when m = 2. If m > 2, then any generator of Zm in (Rp, S) is an odd power of Rp relatively prime to m, and thus orientation reversing. On the other hand, any generator of Zm in (R, Sp) is orientation preserving. Hence these groups cannot be conjugate. This implies that if m/2 is even, there are two equivalence classes of Dih(Zm)-actions. We note that when m/2 is odd and not equal to one, there are three equivalence classes of Dih(Zm)-actions. They are (Rp) 0-1 (S), (R) 0-1 (Sp) and (R2p) 0-1 (S). The last group has an element (R2p)m/2 = p restricting to the identity on P2, and the group may be viewed as ((R2) o-1 (S)) x (p) = Dih(Zm/2) x Z2.This group was identified in the second paragraph of this proof when we assumed f 1: G ^ Homeo(P2) was not an effective G-action. The proof is completed by noting that if G is isomorphic to Dih(Zm) x Z2 and m is odd, then Dih(Zm) x Z2 is isomorphic to Dih(Z2m), and this case has already been dealt with. □ References [1] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, New York, 4th edition, 1980, doi:10.1007/978-3-662-21943-0. [2] D. S. Dummit and R. M. Foote, Abstract Algebra, John Wiley & Sons, Hoboken, New Jersey, 3rd edition, 2004. [3] W. D. Dunbar, Geometric orbifolds, Rev. Mat. Univ. Complut. Madrid 1 (1988), 67-99, http: //www.mat.ucm.es/serv/revmat/vol1-123/vol1-12 3f.html. [4] J. A. Gallian, Contemporary Abstract Algebra, Houghton Mifflin, 5th edition, 2002. [5] J. L. Gross and T. W. Tucker, Topological Graph Theory, Dover Publications, Mineola, New York, 2001. [6] W. H. Meeks, III and P. Scott, Finite group actions on 3-manifolds, Invent. Math. 86 (1986), 287-346, doi:10.1007/bf01389073. [7] D. Singerman, Automorphisms of compact non-orientable Riemann surfaces, Glasgow Math. J. 12 (1971), 50-59, doi:10.1017/s0017089500001142. [8] T. W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Comb. Theory Ser. B 34 (1983), 82-98, doi:10.1016/0095-8956(83)90009-6. [9] B. P. Zimmermann, On finite groups acting on spheres and finite subgroups of orthogonal groups, Sib. Elektron. Mat. Izv. 9 (2012), 1-12, http://semr.math.nsc.ru/v9/p1-12.pdf. /^creative ^commor ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 323-336 https://doi.org/10.26493/1855-3974.989.d15 (Also available at http://amc-journal.eu) A combinatorial problem and numerical semigroups* Aureliano M. Robles Perez Departamento de Matemática Aplicada, Universidad de Granada, 18071-Granada, Spain Jose Carlos Rosales Departamento de Algebra, Universidad de Granada, 18071-Granada, Spain Received 3 December 2015, accepted 2 March 2018, published online 25 June 2018 Abstract Let a = (ai,... ,an) and b = (bi,... ,bn) be two n-tuples of positive integers, let X be a set of positive integers, and let g be a positive integer. In this work we show an algorithmic process in order to compute all the sets C of positive integers that fulfill the following conditions: 1. The cardinality of C is equal to g; 2. If x,y e N \ {0} and x + y e C, then C n {x, y} = 0; 3. If x e C and ^ e N \ {0} for some i e {1,..., n}, then ^ e C; 4. X n C = 0. Keywords: Combinatorial problems, numerical semigroups, Frobenius varieties, Frobenius pseudo-varieties. Math. Subj. Class.: 11B75, 05A99, 20M14 *Both authors are supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andalucía Grant Number FQM-343. The second author is also partially supported by Junta de Andalucía/Feder Grant Number FQM-5849. The authors would like to thank the referee for several comments and suggestions that led to the improvement of this paper. E-mail ¡addresses: arobles@ugr.es (Aureliano M. Robles Perez), jrosales@ugr.es (Jose Carlos Rosales) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ 1 Introduction Let Z and N be the sets of integers and non-negative integers, respectively. Let us suppose that we want to compute a set C c N of six elements such that the following conditions are fulfilled. (C1) If x, y are positive integers such that x + y G C, then C n {x, y} = 0. (C2) If x G C and x - 4 is a positive integer, then x - 4 G C. (C3) If x G C and X-1 is a positive integer, then X-1 G C. (C4) 5 G C. The purpose of this work will be to give an answer to this type of combinatorial problems. Let a = (ai,..., an) and b = (6i,...,bn) be two n-tuples (with n > 1) of positive integers, let X be a non-empty subset of N \ {0}, and let g be a positive integer. Let us denote by P(a, b, X, g) the problem of computing all the subsets C of N \ {0} that fulfill the following conditions. (P1) The cardinality of C is equal to g. (P2) If x, y G N \ {0} and x + y G C, then C n {x, y} = 0. (P3) If x G C and ^ G N \ {0} for some i G {1,..., n}, then ^ G C. (P4) X n C = 0. With the previous notation, we observe that the problem proposed at the beginning is just P((1, 2), (4,1), {5}, 6). A numerical semigroup (see [6]) is a submonoid S of (N, +) such that N \ S is finite. The cardinality of N \ S is the so-called genus of S and is denoted by g(S). It is easy to see that C is a solution of P(a, b, X, g) if and only if S = N \ C is a numerical semigroup that fulfills the following conditions. (51) g(S) = g. (52) If s G S \ {0}, then as + b G Sn (where as + b = (ais + bi,..., ans + bn)). (53) X C S. Let us denote by N(a, b, X) the set {S | S is a numerical semigroup, X C S, and as + b G Sn for all s G S \ {0} }. With this notation, the solutions of P(a, b, X, g) are the elements of the set {N \ S | S G N(a, b, X) and g(S) = g}. Let S be a numerical semigroup. The Frobenius number of S (see [2]), denoted by F(S), is the maximum integer that does not belong to S. A Frobenius variety (see [5]) is a non-empty family of numerical semigroups V that fulfills the following conditions. (V1) If S, T e V, then S n T e V. (V2) If S e V and S = N, then S U {F(S)} e V. Before to show the structure of this paper, let us see some remarks in order to delimit the problem P(a, b, X, g) inside the theory of numerical semigroups. Remark 1.1. If we only impose condition (C1) (equivalently, condition (P2)), then we are considering the family S of all numerical semigroups that, obviously, is a Frobenius variety. Moreover, in [6] it is shown how to arrange the elements of S in a tree with root N. Remark 1.2. Keeping in mind only conditions (C1) and (C4) (equivalently, conditions (P2) and (P4) or condition (S3)), we have the family S(X) of all numerical semigroups containing X. Once again, it is not difficult to check that S(X) is a Frobenius variety. In addition, following the ideas of this paper, it will be clear how we can arrange the elements of S(X) in a tree with root N. Remark 1.3. Now let us consider only conditions (C1), (C2) and (C3) (equivalently, conditions (P2) and (P3) or condition (S2)). In this case we get families of numerical semigroups satisfying a set non-homogeneous patterns (see [1]). This case is related to the results of [1], where the concept of m-variety makes possible to arrange the elements of certain families in trees with root Sm = {0, m, = {0} U {n e N | n > m}. Precisely, m-varieties are examples of Frobenius pseudo-varieties (see [4]). In Section 2 we will see that N(a, b, X) is a Frobenius variety. In addition, we will show that such a variety is finite if and only if gcd(X U {bi,..., bn}) = 1 (where, as usual, gcd(A) is the greatest common divisor of the elements in A). Let us denote by M(a, b, X) the intersection of all the elements in N(a, b, X). Observe that M(a, b, X) is always a submonoid of (N, +). In addition, we will prove that M(a, b, X) is a numerical semigroup if and only if N(a, b, X) has finitely many elements. In Section 2 we will show that P(a, b, X, g) has a solution if and only if the cardinality of N \ M(a, b, X) is greater than or equal to g. Moreover, we will give an algorithm in order to compute M(a, b, X). Therefore, we will have an algorithmic process to decide whether P(a, b, X, g) has a solution. In Section 3, with the help of some results from [5], we will arrange the elements of N(a, b, X) in a tree with root N. Moreover, we will characterize the children of a vertex in such a tree and, consequently, will have a recursive procedure in order to build N(a, b, X). Accordingly, we will have an algorithmic process to compute all the elements of N(a, b, X) with a fixed genus and, in particular, an algorithm to compute all the solutions of P(a, b, X, g). Finally, using the concept of Frobenius pseudo-variety, we will state and solve a generalization of P(a, b, X, g) in Section 4. 2 (a, b)-monoids In this work, unless stated otherwise, a = (a1,..., an) and b = (b1,..., bn) denote two n-tuples of positive integers. If X C N, then N(a, b, X) is the set {S | S is a numerical semigroup, X C S, and as + b e Sn for all s e S \ {0} }, with as + b = (ais + bi,. .., ans + bn). Proposition 2.1. N(a, b, X) is a Frobenius variety. Proof. It is clear that N G N(a, b, X) and, therefore, N(a, b, X) = 0. It is also clear that, if S,T G N(a, b, X), then S n T G N(a,b,X). Now, let S G N(a,b,X) such that S = N. In order to show that S U {F(s)} G N(a, b, X), it is enough to see that aF(S) + b G (S U {F(S)})n. Observe that, if i G {1,..., n}, then a,F(S) + b > F(S) and, therefore, a,F(S) + b, G SU{F(S)}. Consequently, aF(S) + b G (Su{f(S)})". □ We will say that M is an (a, b)-monoid if M is a submonoid of (N, +) fulfilling that am + b G Mn for all m G M \ {0}. Proposition 2.2. Let X be a non-empty subset of N. Then M is an (a, b)-monoid that contains X if and only if there exists J C N (a, b, X) such that M = p| Se j S. Proof. The sufficient condition is trivial. For the necessary one, let Mk = M U {k, for all k G N (nhere {k, = {n G N | n > k}). Then it is clear that Mfc G N(a, b, X) and that M = f|fceN Mfc. □ Let us observe that, if we denote by M(a, b, X) = p|s£n(a b X) S, then M(a, b, X) is the smallest (a, b)-monoid containing X. Theorem 2.3. Let X be a non-empty subset of N \ {0} and let g be a positive integer. Then the problem P(a, b, X, g) has a solution if and only if the cardinality of N \ M (a, b, X) is greater than or equal to g. Proof. (Necessity.) If C is a solution of P(a, b, X, g) and we take S = N \ C, then S G N(a, b, X) and g(S) = g. Since M(a, b, X) C S, we have that N \ S C N \ M(a, b, X) and, thereby, the cardinality of N \ M(a, b, X) is greater than or equal to g. (Sufficiency.) Let us suppose that N \ M(a, b, X) = {ci < • • • < cg < • • • }. If we take S = M(a, b, X) U {cg + 1, then it is clear that S G N(a, b, X) and g(S) = g. Therefore, C = N \ S is a solution of P(a, b, X, g). □ Let us observe that, if P(a, b, X, g) has a solution and, in addition, we have computed M(a, b, X), then the proof of the sufficient condition in the previous theorem gives us a method to compute a solution of P(a, b, X, g). If X is a non-empty subset of N, then we denote by (X} the submomoid of (N, +) generated by X, that is, (X} = {Aixi +-----+ Afcxfc | k G N \ {0}, xi,..., xfc G X, and Ai,..., Afc G N}. If M = (X}, then we will say that M is generated by X or, equivalently, that X is a system of generators of M. The next result is well known (see, for instance, [6]). Lemma 2.4. If X C N, then (X} is a numerical semigroup if and only if gcd(X) = 1. We know that M(a, b, X) is a submonoid of (N, +). From the following proposition we will get that, if X C N \ {0}, then M(a, b, X) is a numerical semigroup if and only if gcd(X U {b i,..., bn }) = 1. Proposition 2.5. If X C N \ {0}, then gcd(M(a, b, X)) = gcd(X U {bi,..., bn}). Proof. Let d = gcd(M(a, b, X)) and d' = gcd(X U {&i,..., bn}). In order to prove the proposition, we will show that d' | d and d | d'. (As usual, if p, q are positive integers, then p | q means that p divides q.) First of all, it is clear that ({d'}} is an (a, b)-monoid containing X. Thus, M(a, b, X) C ({d'}} and, consequently, d' | d. Now, let us take x G X. Then {x, a1x + b1,..., anx + bn} C M(a, b, X) and gcd{x, a1x + b1,..., anx + bn} = gcd{x, b1,..., bn}. Therefore, we have that X U (UUtex + bi | x G X}) C M(a, b, X) and gcd (X U (UUKx + b< | x G X})) = gcd(X U {b1,..., bn}) = d'. Accordingly, d | d'. □ In the next result we show when the Frobenius variety N(a, b, X) is finite. Theorem 2.6. Let X be a subset of N \ {0}. Then the following conditions are equivalent. 1. N (a, b, X) is finite. 2. M(a, b, X) is a numerical semigroup. 3. gcd(X U{b1,...,b„}) = 1. Proof. The equivalence between conditions 2 and 3 is a consequence of Lemma 2.4 and Proposition 2.5. Now, let us see the equivalence between conditions 1 and 2. (1. ^ 2.) It is enough to observe that the finite intersection of numerical semigroups is another numerical semigroup. (2. ^ 1.) If S G N(a, b, X), then M(a,b,X) C S. Thus, S = M(a,b,X) U Y with Y C N \ M(a, b, X). Since N \ M(a, b, X) is finite, we conclude that N(a, b, X) is finite. □ Our next aim in this section will be to give an algorithm in order to compute M(a, b, X). For this is fundamental the following result. Proposition 2.7. Let M be a submonoid of (N, +) generated by X C N \ {0}. Then M is an (a, b)-monoid if and only if ax + b G Mn for all x G X. Proof. The necessary condition is trivial. For the sufficiency, let m G M \ {0}. Then there exist x1,..., xt G X such that m = x1 + • • • + xt. If t = 1, then m = x1 and am + b = ax1 + b G Mn. If t > 2, then am + b = a(x1 + • • • + xt) + b = a(x1 + • • • + xt-1) + axt + b. Since a(x1 + • • • + xt-1), axt + b G Mn, we finish the proof. □ The above proposition will be useful in order to determine whether a submonoid M of (N, +) is or is not an (a, b)-monoid. Let us see an example. Example 2.8. S = ({4, 5,11}} is an ((1, 2), (4,1))-monoid because (1,2)4 + (4,1) = (8,9) G S2, (1,2)5 + (4,1) = (9,11) G S2, and (1, 2)11 + (4,1) = (15,23) G S2. Nevertheless, T = ({5,7, 9}} is not an )(1, 2), (4,1))-monoid because (1,2)5 + (4,1) = (9,11) G T2 (observe that 11 G T). With the help of Proposition 2.7, it would be possible to give an algorithm in order to compute M(a, b, X). However, we are going to postpone such an algorithm because, as we will see now, we can focus on case in which gcd(X U {^,... ,bn}) = 1, and thus simplify the computations. We say that an integer d divides an n-tupla of integers c = (c-1,... ,cn) if d | q for all i e {1,..., n}. In such a case, we denote by d = (cd",..., cf). If A C Z and k e Z, then kA = {ka | a e A}. Finally, if A C Z, d e Z, and d | a for all a e A, then A = {d I a € A}. Lemma 2.9. Let M be an (a, b)-monoid such that M = {0}. If gcd(M) = d, then 1. d divides b; 2. if dd e N \ {0} and dd | d, then M is an (a, Jj) -monoid; 3. if k e N \ {0}, then kM is an (a, kb)-monoid. Proof. 1. If we take X = M \ {0}, and having in mind that M(a, b, X) is the smallest (a, b)-monoid containing X, then this item is a consequence of Proposition 2.5. 2. It is clear that M is a submonoid of (N, +). In addition, if x e M \ {0}, then d'x e M \ {0} and, therefore, ad'x + b e Mn. Consequently, ax + jj e Mf = (M)n. 3. It is clear that kM is a submonoid of (N, +). Now, arguing as in the previous item, if x e kM \ {0}, then f e M \ {0} and, therefore, af + b e Mn. Consequently, ax + kb e k ■ Mn = (kM)n. □ The next result says us that, in order to compute M (a,b,X), (t is sufficient to ca}culate d = gcd (X U{b1,...,bn}) and M (a, j, f). Observe that gcd (X U { ^,..., bf)) = 1 and, therefore, M(a, j, is a numerical semigroup. d'd/ & \ d I _ 1 i c o r> i lm d Proposition 2.10. Let X be a subset of N \ {0}. If gcd(X U {b1,..., bn}) = d, then b X • d7 d . M(a, b, X) = d ■ M(a, d, . Proof. From item 3 of Lemma 2.9, we have that d ■ M(a, d, X) is an (a, b)-monoid containing X. Therefore, M(a, b,X) C d ■ M(a, d, X). On the other hand, from Proposition 2.5 and item 2 of Lemma 2.9, we deduce that M(adb'X} is an (a, d)-monoid containing ^. Consequently, M (a, d, =X) C M(°db'X), that is, d- M(a, d, xX)c M(a,b,X). □ We are now ready to show the announced algorithm. Algorithm 2.11. INPUT: A non(-empty finite set of) positive integers X such that gcd (X U{bi,...,bn}) = 1. Output: M(a,b,X). (1) A = 0 and G = X. (2) If G \ A = 0, then return (G) and stop the algorithm. (3) m = min(G \ A). (4) H = {ajm + bj | i e {1,... ,n} and a4m + bj e (G^. (5) If H = 0, then go to (7). (6) G = G U H. (7) A = A U {m} and go to (2). In order to justify the performance of this algorithm, let us observe that, if the algorithm stops, then it returns (G) such that ag + b e (G)n for all g e G. Therefore, by applying Proposition 2.7, we have that (G) is an (a, b)-monoid. In addition, by construction, it is clear that G must be a subset of every (a, b)-monoid which contains X. Thus, (G) is the smallest (a, b)-monoid containing X. Consequently, in order to justify the algorithm, it will be enough to see that the algorithm stops. In fact, when we arrive to step (7) at the first time, we have that gcd(G) = 1 and, thereby, (G) is a numerical semigroup. Therefore, N \ (G) is finite and we can go to the step (6) only in a finite number of times. Let us illustrate the performance of Algorithm 2.11 with two examples. In the first one M(a, b, X) is a numerical semigroup. Example 2.12. We are going to compute M = M((1, 2), (4,1), {5}). • A = 0 and G = {5}. • m = 5, H = {9,11}, G = {5,9,11}, and A = {5}. • m = 9, H = {13}, G = {5, 9,11,13}, and A = {5, 9}. • m = 11, H = 0, G = {5,9,11,13}, and A = {5,9,11}. • m = 13, H = {17}, G = {5, 9,11,13,17}, and A = {5, 9,11,13}. • m = 17, H = 0, G = {5,9,11,13,17}, and A = {5,9,11,13,17}. Therefore, M = ({5,9,11,13,17}). Going back to the problem P((1,2), (4,1), {5}, 6) of the introduction, we have that, since N \ M = {1,2, 3,4,6,7,8,12} has cardinality equal to 8, then Theorem 2.3 asserts that the proposed problem has a solution. Moreover, the solutions will be some subsets, with cardinality equal to 6, of {1,2,3,4, 6,7,8,12}. In addition, by the proof of the sufficiency of Theorem 2.3, we know that {1, 2, 3,4,6, 7} is a solution of such a problem. Let us see now an example in which M(a, b, X) is not a numerical semigroup. Example 2.13. Let us see that P((2, 3), (4,2), {6, 8}, 9) has a solution. For that, we begin with the computation of M((2,3), (4,2), {6,8}). By applying Paoposition 2.10, since gcd({6,8,4, 2}) = 2, we get that M((2, 3), (4, 2), {6, 8}) =2 • M((2, 3), (2,1), {3,4}). Now, from Algorithm 2.11, M((2, 3), (2,1), {3,4}) = ({3,4}). Therefore, M((2, 3), (4, 2), {6, 8}) = ({6, 8}) = {0, 6, 8,12,14,16,...}. Since N\M((2, 3), (4,2), {6, 8}) has infinitely many elements, its cardinality is greater than or equal to 9 and, consequently, Theorem 2.3 assures that P((2,3), (4, 2), {6,8}, 9) has a solution. Moreover, by the proof of the sufficiency of Theorem 2.3, we have that {1, 2,3,4,5, 7, 9,10,11} is a solution. Remark 2.14. If we suppose, for a moment, that X = 0 (in a sense, we are removing condition (P4) in P(a, b, X, g) such as is observed in Remark 1.2), then it is obvious that Sk = {0, k, for all k e N, are numerical semigroups that belong to N(a, b, X) independently of the chosen n-tuples a, b. Thus M(a, b, X) = {0}, that is, the submonoid of (N, +) generated by X = 0. Remark 2.15. Now, let us suppose that a, b are 0-tuples, that is, we remove condition (P3) in P(a, b, X, g) (see Remark 1.3). In this case, it is straightforward to show that M(a, b, X) is just the monoid generated by X. 3 The tree associated to N(a, b, X) A graph G is a pair (V, E), where • V is a non-empty set whose elements are called vertices of G, • E is a subset of {(v, w) G V x V | v = w} whose elements are called edges of G. A path (of length n) connecting the vertices x and y of G is a sequence of different edges of the form (v0, vi), (vi, v2),..., (vn_i, vn) such that v0 = x and vn = y. We say that a graph G is a tree if there exists a vertex r (known as the root of G) such that, for every other vertex x of G, there exists a unique path connecting x and r. If (x, y) is an edge of the tree, then we say that x is a child of y. We define the graph G(N(a, b, X)) in the following way. • N(a, b, X) is the set of vertices of G(N(a, b, X)); • (S,S') G N (a, b, X) xN (a, b, X) isanedgeof G(N(a,b,X)) if S' = S U{F(S)}. By Proposition 2.1 and [5, Theorem 27], we have that G(N(a, b, X)) is a tree with root N. Our first purpose in this section will be to establish what are the children of a vertex in such a tree. For this we need to introduce some concepts. Let S be a numerical semigroup and let G be a system of generators of S. We say that G is a minimal system of generators of S if S = (Y} for all Y C G. It is well known (see [6]) that every numerical semigroup admits a unique minimal system of generators and that, in addition, such a system is finite. Observe that, if we denote by msg(S) the minimal system of generators of S, then msg(S) = (S \ {0}) \ ((S \ {0}) + (S \ {0})). On the other hand, we have (see [6]) that, if S is a numerical semigroup and s G S, then S \ {s} is another numerical semigroup if and only if s G msg(S). An immediate consequence of [5, Proposition 24, Theorem 27] is the next result. Theorem 3.1. The graph G (N (a, b, X)) is a tree with root N. Moreover, the set of children of a vertex S G N (a, b, X) is {S \ {m} | m G msg(S), m > F(S), and S \ {m} G N(a, b, X)}. In the next result we will show the conditions that must satisfy m G msg(S) in order to have S \ {m} G N(a, b, X). Proposition 3.2. Let S G N(a, b, X) and let m G msg(S). Then S \ {m} G N(a, b, X) if and only if G S \ {0} for all i G {1,... , n} and m G X. Proof. (Necessity.) Since X C S \ {m}, we have that m G X. Let us suppose that there exists i G {1,..., n} such that G S \ {0}. Since = m, we have that G S \ {0, m} and that aj (m__bi) + a = m G S \ {m}. Therefore, S \ {m} G N(a, b, X). (Sufficiency.) If S \ {m} G N(a, b, X), then there exists s G S \ {0, m} and there exists i G {1,..., n} such that a4s + bj G S \ {m}. Since S G N(a, b, X), we know that a4s + bj G S. Therefore, ajs + bj = m and, consequently, m_bi = s G S \ {0}. □ Our next purpose will be to build recurrently G(N(a, b, X)) from its root and joining each vertex with its children by means of edges. In order to make easy that construction, we will study the relation between the minimal system of generators of a numerical semigroup S and the minimal system of generators of S \ {m}, where m is a minimal generator of S greater than F(S). First of all, it is clear to observe that, if S is minimally generated by {m, m + 1,..., 2m - 1} (that is, S = {0, m, then S \ {m} = {0, m + 1, is minimally generated by {m + 1, m + 2,..., 2m + 1}. In other case we can apply the following result, which is [3, Corollary 18]. Proposition 3.3. Let S be a numerical semigroup with minimal system of generators msg(S) = {ni < n2 < • • • < np}. If i G {2,... ,p} and nj > F(S), then i{ni,..., np} \ {nj}, if there exists j G {2,..., i — 1} such that nj + ni — nj G S; ({ni,..., np} \ {nj}) U {nj + ni}, in other case. Let us illustrate the previous results with an example. Example 3.4. By Proposition 2.7, it is easy to see that S = ({5,7,8, 9,11}} belongs to N((1,2), (4,1), {5}). On the other hand, from Theorem 3.1, we know that the set of children of S in the tree G(N((1,2), (4,1), {5})) is {S \ {m} | m G msg(S), m > F(S), and S \ {m} G N((1, 2), (4,1), {5})}. Since F(S) = 6, we have that {m G msg(S) | m > F(S)} = {7,8, 9,11}. Furthermore, by Proposition 3.2, we know that S \ {m} G N((1,2), (4,1), {5}) if and on(y if m G {5} {nd {m - 4, } n (S \ {0}) = 0. Thus, since that {7 - 4, n (S \ {0}) = {8 - 4,^ } n (S \ {0}) = 0, {9 - 4, } n (S \ {0}) = 0, and {11 - 4, ^ } n (S \ {0}) = 0, we conclude that S = ({5, 7,8,9,11}} has two children. Namely, they are ({5, 7, 8, 9,11}}\{7} = ({5, 8, 9,11,12}} and ({5, 7, 8, 9,11}}\{8} = ({5, 7, 9,11,13}}, where we have applied Proposition 3.3. Following the idea of the previous example, we can build G(N((1, 2), (4,1), {5})) in a recurrent way starting from its root, that is, from N = ({1}} (see Figure 1). Let us observe that, since gcd({5} U {4,1}) = 1 and by Theorem 2.6, then we know that N((1,2), (4,1), {5}) is a finite Frobenius variety and, thereby, we have been able of building it completely in a finite number of steps. Let us also observe that, if S g N(a, b, X), then g(S) is equal to the length of the path connecting S with N in the tree G((N(a, b, X)). Therefore, in order to build the elements of N(a, b, X) with a fixed genus g, we only need to build the elements of N(a, b, X) which are connected to N through a path of length less than or equal to g. Consequently, we have an algorithmic process to compute all the solutions of )he problem P(a, b, X, g). For instance, in the tree G(N((1, 2), (4,1), {5})), the numerical semigroups which are connected to N through apath of length 6 are ({5,8,9,11,12}}, ({5,7,9,11,13}}, and ({5, 6,9,13}}. Therefore, the problem proposed in the introduction has three solutions. Namely, N\({5, 8, 9,11,12}} = {1, 2, 3,4,6, 7}, N\({5,7,9,11,13}} = {1,2,3,4, 6,8}, and N \ ({5,6, 9,11,13}} = {1, 2, 3,4,7,8}. We finish with an example in which N(a, b, X) is an infinite Frobenius variety. Example 3.5. Let us compute all the solutions of P((2,3), (4,2), {6,8}, 4). First of all, from Example 2.13, we know that M((2, 3), (4, 2), {6, 8}) = ({6, 8}} and, by Theorem 2.3, that the problem has a solution. Moreover, since gcd({6,8} U {4,2}) = 2 = 1, ({1}) ({2, 3}) ({3,4, 5}) ({2, 5}) ({3, 5, 7}) ({5, 6, 7, 8, 9}) ({4, 5, 7}) ({4, 5, 6}) ({5, 7, 8, 9, 11}) ({5, 6, 8, 9}) ({5, 6, 7, 9}) ({4, 5,11}) ({5, 8, 9, 11, 12}) ({5, 7, 9, 11, 13}) ({5, 6, 9, 13}) ({5, 9, 11, 12, 13}) ({5, 9, 11, 13, 17}) Figure 1: Tree associated to the finite Frobenius variety N((1, 2), (4,1), {5}). we have that N((2,3), (4,2), {6,8}) is a infinite Frobenius variety. However, in a finite number of steps, we can compute the elements of G(N((2,3), (4, 2), {6,8})) which are connected to N through a path of length 4, such as we show in Figure 2. ({1}) = N ({2, 3}) ({4, 5, 6, 7}) ({3, 5, 7}) ({3,4}) ({2, 7}) ({5, 6, 7, 8, 9}) ({4, 6, 7, 9}) ({4, 5, 6}) ({3, 7, 8}) ({3, 5}) ({2, 9}) Figure 2: Five first levels of the tree associated to the infinite Frobenius variety N((2, 3), (4, 2), {6, 8}). Therefore, the sets N \ ({5,6, 7, 8,9}) = {1,2, 3,4}, N \ ({4, 6,7,9}) = {1, 2, 3,5}, N \ ({4, 5, 6}) = {1, 2, 3, 7}, N \ ({3, 7, 8}) = {1, 2,4, 5}, N \ ({3, 5}) = {1, 2,4, 7}, and N \ ({2,9}) = {1,3,5, 7} are the (six) solutions of P((2,3), (4, 2), {6,8}, 4). Remark 3.6. Let us observe that, in the construction of the trees, we can assume that X = 0 or that a, b are 0-tuples (see Remarks 2.14 and 2.15). Then, we obtain all the possible solutions in each case. In particular, if we consider jointly such assumptions, then we get the tree associated to the full family of numerical semigroups (see Remark 1.1). 4 A generalization of the problem Along this section r and g are non-negative integers, a = (ai,..., an) and b = (bi,..., bn) are n-tuples of positive integers, and X is a non-empty subset of {r+1, We will denote by Pr (a, b, X, g) the (generalised) problem of computing all the subsets C of {r + 1, that fulfill the following conditions. (GP1) The cardinality of C is equal to g. (GP2) If x, y G {r + 1, and x + y G C, then C n {x, y} = 0. (GP3) If x G C and ^ G {r + 1, for some i G {1,..., n}, then ^ G C. (GP4) X n C = 0. Let us observe that Po(a, b, X, g) = P(a, b, X, g). It is clear that a set C is a solution of Pr (a, b, X, g) if and only if S = {0, r + 1, \ C is a numerical semigroup that fulfills the following conditions. (GS1) g(S) = r + g. (GS2) If s G S \ {0}, then as + b G Sn. (GS3) X Ç S. Let us denote by Nr (a, b, X) the set of all numerical semigroups which are subsets of {0, r + 1, and satisfy the conditions (GS2) and (GS3). Let us observe that, with this notation, the solutions of Pr (a, b, X, g) are the elements of the set {{0, r+, 1 \ S | S G N(a, b, X) and g(S) = r + g}. Moreover, Nr (a,b,X) = {S G N(a,b,X) | S Ç {0,r + 1, The following proposition is analogous to Theorem 2.3. Proposition 4.1. Let us take Mr (a, b, X) = p|seN(a b x) S. Then Pr (a, b, X, g) has a solution if and only if the cardinality of N \ Mr (a, b, X ) is greater than or equal to g + r. Proof. (Necessity.) If C is a solution of Pr (a, b, X, g), then S = {0, r + 1, \ C belongs to Nr (a, b, X) and g(S) = g + r. Since Mr (a, b, X) Ç S, then we conclude that the cardinality of N \ Mr (a, b, X) is greater than or equal to g + r. (Sufficiency.) If {0, r + 1, \ Mr (a, b, X) = {ci < • • • < cs < • • • } and S = Mr(a, b, X) U {cg + 1, then it is easy to see that S G Nr (a, b, X) and g(S) = g + r. Therefore, C = {0, r +1, \ S is a solution of Pr(a, b, X, g). □ Observe that the cardinality of N \ Mr (a, b, X) is greater than or equal to g + r if and only if the cardinality of {0, r + 1, \ Mr (a, b, X) is greater than or equal to g. Proposition 4.2. Mr (a, b, X ) = M (a, b, X ). Proof. Since Nr (a, b, X) C N(a, b, X), then we have that M K b, X ) = fl SeN (a,b,X) S C fl SeN (a,b,X) S = Mr (a b, X ). Let us now see the other inclusion. Since {0, r + 1, —} G N(a, b, X) and N(a, b, X) is a Frobenius variety, we have that, if S G N(a, b, X), then S n {0, r +1, —} G N(a, b, X). In addition, S n{0,r + 1, —} C {0,r +1, —} and, thus, S n{0,r + 1, —} G Nr (a, b,X). In this way, R = {S n {0, r +1, —} | S G N"(a, b, X)} C Nr(a, b, X). Consequently, Mr K b, X) = DseNr(a,b,x) S CDseR S = flseN(a,b,x) S = M(a, b, X). □ As an immediate consequence of Proposition 4.2 and Proposition 2.10, we have the next result. Corollary 4.3. If gcd (X U {bi,..., b„}) = d, then Mr(a, b, X) = d ■ M(a, f, X). Let us observe that, as a consequence fo the previous corollary, we can use Algorithm 2.11 in order to compute Mr (a, b, X). The following result is the analogous to Theorem 2.6 for the current problem. Corollary 4.4. The following conditions are equivalent. 1. Nr (a, b, X) is finite. 2. Mr (a, b, X) is a numerical semigroup. 3. gcd(X U{bi,...,bn}) = 1. Proof. The equivalence between conditions 2 and 3 is a consequence of Proposition 4.2 and Theorem 2.6. Now, let us see the equivalence between conditions 1 and 2. (1. ^ 2.) It is enough to observe that the finite intersection of numerical semigroups is another numerical semigroup. (2. ^ 1.) If S G Nr (a, b, X), then Mr (a, b, X) C S. Thus, S = Mr (a, b, X) U Y for some Y C N \ Mr (a, b, X). Since N \ Mr(a, b, X) is finite, then we can conclude that Nr (a, b,X) is finite. □ Let us illustrate the previous results with several examples. Example 4.5. Let us see that Pr((1, 2), (4,1), {5}, 6) has a solution if and only if r G {0,1,2}. Since {5} C {r + 1, -}, then r G {0,1, 2, 3,4}. By Proposition 4.2 and Example 2.12, we have that M = Mr ((1, 2), (4,1), {5}, 6) = ({5,9,11,13,17}). Since N \ M = {1, 2,3,4,6, 7, 8,12} has cardinality equal to 8, by applying Proposition 4.1, we easily deduce that Pr ((1, 2), (4,1), {5}, 6) has a solution if and only if r G {0,1,2}. Example 4.6. If r G {0,1,2,3,4, 5}, then Nr ((2, 3), (4, 2), {6, 8}) is an infinite set. In fact, this is an immediate consequence of Corollary 4.4 and that gcd({4,2, 6, 8}) = 2 = 1. Example 4.7. Let us compute P3((2, 3), (4,2), {6, 8}, 9). By Proposition 4.2 and Example 2.13, we have that M3((2, 3), (4, 2), {6, 8}) = ({6, 8}). Now, if we apply the construction given in the sufficiency of Proposition 4.1, we have that {4,5,7, 9,10,11,13,15,17} is a solution. If G is a tree and u, v are two vertices of G such that there exists a path between them, then we will say that u is a descendant of v. The next result has an easy proof. Proposition 4.8. Nr (a, b, X) is the set of all descendants of {0, r + 1, in the tree G(N(a, b,X)). A Frobenius pseudo-variety (see [4]) is anon-empty family P of numerical semigroups that fulfills the following conditions. (PV1) P has a maximum element max(P) (with respect to the inclusion order). (PV2) If S, T G P, then S n T G P. (PV3) If S G P and S = max(P), then S U F(S) G P. As an immediate consequence of Proposition 4.8 and the comment above to Example 7 in [4], we have the following result. Proposition 4.9. Nr (a, b, X ) is a Frobenius pseudo-variety. Let us observe that, if r > 1, then max(Nr (a, b, X)) = {0, r +1, = N. Therefore, by applying [4, Proposition 1], we have that Nr (a, b, X) is not a Frobenius variety. Now, let us notice that the subgraph, of a (ree, which is formed by a vertex and all its descendants is also a tree. We will denote by G(Nr (a, b, X)) the subtree of G(N(a, b, X)) formed by {0, r +1, and all its descendants. Example 4.10. The root of G(N3((1, 2), (4,1), {5})) is {0,4, = ({4,5, 6, 7}). Thus, from Example 3.4, we have that such a tree is given by Figure 3. ({4, 5, 6, 7}) ({5, 6, 7, 8, 9}) ({4, 5, 7}) ({4, 5, 6}) ({5, 7, 8, 9,11}) ({5, 6, 8, 9}) ({5, 6, 7, 9}) ({4, 5,11}) ({5, 8, 9,11,12}) ({5, 7, 9,11,13}) ({5, 6, 9,13}) ({5, 9,11,12,13}) ({5, 9,11,13,17}) Figure 3: Tree associated to the Frobenius pseudo-variety N3((1, 2), (4,1), {5}). Let us observe that Nr (a, b, X) is the set of vertices in G (NT (a, b, X)), and that (S, S') G Nr (a, b, X) x Nr (a, b, X) is an edge of G(N (a, b, X)) if and only if S' = S U {F(S)}. It is also clear that, if S g Nr (a, b, X), then the set formed by the children of S in Nr (a, b, X ) is the same that the set formed by the children of S in N (a, b, X ). In this way, by applying Theorem 3.1, we have the next result. Proposition 4.11. The graph G(Nr (a, b, X)) is a tree with root {0, r +1, Moreover, the set of children of a vertex S in G(Nr (a, b, X )) is {S \ {m} | m G msg(S ), m > F(S), and S \ {m} G N (a, b, X )}. Now, let us notice that, by using Propositions 3.2 and 3.3, we can compute the children of any vertex S in G(Nr (a, b, X)) and, consequently, we have an algorithmic process to recurrently build the elements of Nr (a, b, X). We finish this section with an illustrative example about the above comment. Example 4.12. Let us compute all the solutions of P3 ((2,3), (4,2), {6,8}, 4). In order to do this, we have to determine the vertices of G(Ns((2,3), (4, 2), {6,8})) which are connected to {0,4, = ({4,5,6, 7}} through a path of length 4. Let us observe that, if A is the set of vertices (of a tree) which are connected to the root through a path of length k, then the set formed by all vertices that are children of some vertex of A is just the set of vertices which are connected to the root through a path of length k + 1. Thus, if we denote by Aj the set formed by the vertices of G(N3((2, 3), (4,2), {6, 8})) which are connected to ({4, 5,6,7}} through a path of length i, then (by applying Propositions 4.11, 3.2, and 3.3) we obtain recurrently the following sets. • Ao = {({4, 5, 6, 7}}} • Ai = {({5, 6, 7, 8, 9}}, ({4, 6, 7, 9}}, ({4, 5, 6}}} • A2 = {({6, 7, 8, 9,10,11}}, ({5, 6, 8, 9}}, ({5, 6, 7, 8}}, ({4, 6, 9,11}}, ({4, 6, 7}}} • A3 = {({6, 8, 9,10,11,13}}, ({6, 7, 8,10,11}}, ({6, 7, 8, 9,11}}, ({6, 7, 8, 9,10}}, ({5, 6, 8}}, ({4, 6,11,13}}, ({4, 6, 9}}} • A4 = {({6, 8,10,11,13,15}}, ({6, 8, 9,11,13}}, ({6, 8, 9,10,13}}, ({6, 8, 9,10,11}}, ({6, 7, 8,11}}, ({6, 7, 8,10}}, ({6, 7, 8, 9}}, ({4, 6,13,15}}, ({4, 6,11}}} Therefore, the set of solutions of P3 ((2,3), (4,2), {6,8}, 4) is {({4, 5, 6, 7}} \ S | S € A4} = {{4, 5, 7, 9}, {4, 5, 7,10}, {4, 5, 7,11}, {4, 5, 7,13}, {4, 5, 9,10}, {4, 5, 9,11}, {4, 5,10,11}, {5, 7, 9,11}, {5, 7, 9,13}}. References [1] M. Bras Amor6s, P. A. Garcia Sanchez and A. Vico Oton, Nonhomogeneous patterns on numerical semigroups, Internat. J. Algebra Comput. 23 (2013), 1469-1483, doi:10.1142/ s0218196713500306. [2] J. L. Ramirez Alfonsin, The Diophantine Frobenius Problem, volume 30 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2005, doi:10. 1093/acprof:oso/9780198568209.001.0001. [3] A. M. Robles Perez and J. C. Rosales, The numerical semigroup of phrases' lengths in a simple alphabet, Sci. World J. 2013 (2013), 459024 (9 pages), doi:10.1155/2013/459024. [4] A. M. Robles Perez and J. C. Rosales, Frobenius pseudo-varieties in numerical semigroups, Ann. Mat. PuraAppl. 194 (2015), 275-287, doi:10.1007/s10231-013-0375-1. [5] J. C. Rosales, Families of numerical semigroups closed under finite intersections and for the Frobenius number, Houston J. Math. 34 (2008), 339-348. [6] J. C. Rosales and P. A. Garcia Sanchez, Numerical Semigroups, volume 20 of Developments in Mathematics, Springer, New York, 2009, doi:10.1007/978-1-4419-0160-6. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 337-346 https://doi.org/10.26493/1855-3974.1470.e79 (Also available at http://amc-journal.eu) Tilings of hyperbolic (2 x n) -board with colored squares and dominoes Takao Komatsu School of Mathematics and Statistics, Wuhan University, Wuhan, China Laszlo Nemeth University ofSopron, Institute ofMathematics, Hungary Laszlo Szalay University J. Selye, Department of Mathematics and Informatics, Slovakia Received 23 August 2017, accepted 20 December 2017, published online 26 June 2018 Several articles deal with tilings with squares and dominoes of the well-known regular square mosaic in Euclidean plane, but not any with the hyperbolic regular square mosaics. In this article, we examine the tiling problem with colored squares and dominoes of one type of the possible hyperbolic generalization of (2 x n) -board. Keywords: Tiling, domino, hyperbolic mosaic, Fibonacci numbers, combinatorial identity. Math. Subj. Class.: 05A19, 05B45, 11B37, 11B39, 52C20 1 Introduction In the hyperbolic plane there exist infinite types of regular mosaics, they are denoted by Schlafli's symbol {p, q}, where the positive integers p and q have the property (p - 2)(q -2) > 4, see [5]. If p = 4 they are the regular square mosaics and each vertex of the mosaic is surrounded by q squares. Note that if p = q = 4 we obtain the Euclidean square mosaic. Now we define the (2 x n)-board on mosaic {4, q}, where q > 4. First we take a square Si with vertices A0, Ai, B1,B0 according to Figure 1, and later to Figures 2 and 3. As the second step we consider the square S2, which has a common edge A1B1 with S1. The two new vertices are A2,B2. Similarly, we define the squares S3,... ,Sn, their newly constructed vertices are Aj and Bj (3 < i < n), respectively. The union of Sj (1 < i < n) E-mail addresses: komatsu@whu.edu.cn (Takao Komatsu), nemeth.laszlo@uni-sopron.hu (Laszlo Nemeth), laszlo.szalay.sopron@gmail.com (Laszlo Szalay) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ forms the first level of the board. It is depicted with yellow colors in Figures 1-3. (On the left-hand side of Figure 2 the mosaic {4,5} and the (2 x 4)-board are illustrated in Poincare disk model and on the right-hand side there is a "schematic" (2 x 4)-board from the mosaic.) The second level of the board is formed by the squares of the mosaic having at least one vertex from the set {Ai, A2,..., An} and not from {B\, B2,..., Bn, An+1}, where the last point is the appropriate point of the virtually joined square Sn+1 (A0 is not in the first set, see Figure 3). These are the light blue squares in the figures. In the first level, independently from q there are n squares, while the second level contains n(q - 3) squares (see Figure 3). Let rn be the number of the different tilings with (1 x 1)-squares and (1 x 2)-dominoes (two squares with a common edge) of a (2 x n)-board of mosaic {4, q}. It is known that the tilings of a (1 x n)-board on the Euclidean square mosaic can be counted by the Fibonacci numbers [2, 4]. In fact, rn = fn, where {fn}^=0 is the shifted Fibonacci sequence (Fn = fn-1, where Fn is the n-th Fibonacci number, A000045 in OEIS [12]), so that fn = fn-1 + fn-2 (n > 2) holds with initial values f0 = f1 = 1 (and f-1 = 0). 5 ! ^ 2 5 3 S 4 A4 A 5 B0 B1 B0 B1 B2 B0 B1 B2 B3 B0 B1 B2 B3 B4 Figure 1: (2 x 4)-board on Euclidean mosaic {4,4}. Figure 2: (2 x 4)-board on hyperbolic mosaic {4,5}. McQuistan and Lichtman [9] (generalizations in [6]) studied the tilings in case of the Euclidean square mosaic {4,4} and they proved that rn satisfies the identity rn = 3rn-1 + rn-2 - rn-3 (1.1) for n > 3 with initial values r0 = 1, r1 = 2 and r2 = 7 (A030186 in [12]). Figure 3: (2 x 1)-board and (2 x 4)-board on hyperbolic mosaic {4, q} (q > 5). In the work [3], the generalized Fibonacci number un, where Un = aUn-1 + bun-2, (n > 2) (1.2) with initial values u0 = 1, ui = a (and u-i = 0), is interpreted as the number of ways to tile a (1 x n)-board using a colors of squares and b colors of dominoes. Obviously, if a = b =1 then un = fn. Belbachir and Belkhir proved a couple of general combinatorial identities related to un in [1]. Let Rn be the number of tilings of (2 x n)-board of mosaic {4, q} using a colors of squares and b colors of dominoes. When q = 4 Katz and Stenson [7] showed the recurrence rule Rn = (a2 + 2b)Rn-i + a2b Rn-2 - b3Rn-3, (n > 3) (1.3) with initial values R0 = 1, R1 = a2 + b and R2 = a4 + 4a2 b + 2b2. In this article, we examine the tilings of (2 x n)-board on mosaic {4, q} (q > 4) with colored squares and dominoes in a general way and we obtain the following main theorem. Theorem 1.1. Assume q > 4. The sequence {Rn}^=0 can be described by the fourth order linear homogeneous recurrence relation Rn = «q Rn-1 + Pq Rn-2 + Yq Rn-3 - b2(q-2)Rn-4, (n > 4) (1.4) where (explicit formulas later) aq+2 = aaq+i + baq, (1.5) ftq+3 = (a2 + b)ftq+2 + b(a2 + b)ftq+1 - b3ftq, (1.6) Yq+2 = -abYq+1 + b3Yq (1.7) with initial values «4 = a2 + b, «5 = a(a2 + 3b), = 2b(a2 + b), = b(a2 + b)(a2 + 2b), = b(a6 + 6a4b + 10a2b2 +2b3), Y4 = b2(a2 - b), y5 = — ab3(a2 + b), moreover Ro = 1, R1 = uq-2, R2 = u^-2 + abuq-4uq-3 + bu^-3 + b2u^-4, R3 = (u2 2 + 2abuq-4uq-3 + 2bu^-3 + 2b2u;j-4 )uq-2 + b2(uq-3 uq-4 + (a2 + b)uq-4uq-5 + auq-4)uq-3 + ab3u2-4uq-5. If a = b =1, then Theorem 1.1 leads to the following corollary. Recall that fn = Fn+1 (shifted Fibonacci numbers). Corollary 1.2. The sequence |r„}™=0 can be given by the fourth order linear homogeneous recurrence relation rn = 2fq-3 rn-1 + (5/2-4 + (-1)q-1) fn-2 + 2(-1)qfq-5 rn-3 - r„-4, (n > 4) (1.8) with initial values ro = 1, ri = fq-2, r2 = 7fq2-4 + 7fq-4fq-5 + 2/2-5 and r3 = 22fq3-4 + 36fq2-4fq-5 + 19fq-4fq2-5 + 3fq3-5. Observe, that if q = 4, then (1.4) returns with (1.3) (compute the sum of Rn and bRn-1). Similarly, the extension of (1.1) is (1.8). 2 Tilings on mosaic {4, q} We can see that our tiling exercise of the hyperbolic (2 x 1)-board on the mosaic {4, q} (q > 5) is the same as the tiling exercise of the Euclidean (1 x (q - 2)) -board. So R1 = Uq-2 and r1 = fq-2 (Figure 3). Before the discussion of the main result, we define the break-ability of a tiling. A tiling of a (2 x n)-board is breakable in position i fo( 1 < i < n - 1, if this tiling is a concatenation of the tilings of a (2 x i)-subboard and a (2 x (n - i)) -subboard. Clearly, the number of colored tilings of such a board is RiRn-i. A tiling is unbreakable in position i in three different ways: if a domino covers the last square of the first subboard and the first square of the second subboard either in the first or the second level, or on both levels (see Figure 4). a r^ i & \ c 01 i n 01 i n 01 i n 01 i n Figure 4: Breakable and unbreakable tilings in position i when q = 7. Now, we define three subboards. Let Ai, Bi and Ci be the subboards of (2 x i)-board (1 < i < n), respectively, where the last square from second level, the last square from first level and the last squares from both levels are deleted from (2 x i)-board. In Figure 4 these subboards are illustrated. Let Ai, Bi and Ci denote the number of different colored tilings of Ai, Bi and Ci, respectively. 2.1 Proof of Theorem 1.1 and Corollary 1.2 Our proof is based on the connections among (2 x n)-board, An, Bn and Cn subboards. We can easily give the number of tilings if n =1. They are R1 = uq-2, A1 = uq-4, B1 = uq-3 and C1 = uq-4. Moreover let R0 = 1, A0 = B0 = C0 = 0. Generally, if n > 2, then Figure 5 shows the recurrence connections of the subboards. For example, let us see the first )ow. We can build a full (2 x n)-board by four different ways from the full (2 x (n - 1))-board or from the subboards An-1, Bn-1 and Cn-1. If we join a suitable (2 x 1)-board to a (2 x (n - 1))-board, then the coefficient uq-2 is obvious in case of the breakable tilings in position n - 1. When we complete An-1 to a full (2 x n)-board, we have a domino in the second level with b different colors, and we put a square onto the first level with a colors. (If we replace the laid down domino in the second level with two squares, then these tilings would be a part of the first case when we completed the (2 x (n — 1))-board.) The rest part can be tiled freely. Consequently, the coefficient of An-1 is abuq-4 and these are unbreakable tilings in position n — 1. Now, let us complete Bn-1 and Cn-1 to be full (2 x n)-board with a domino in the first level or with two dominoes, one is in the first level and the other in the second level, respectively. The rest parts can be tiled freely. We obtain buq-3 and b2uq-4 new (unbreakable in position n — 1) tilings. Summarising the result of the first row of Figure 5 we have the first equation of the system of recurrence equations (2.1). The determinations of the other rows can be explained similarly. We mention, that, for example, in the fourth row Bn-1 does not appear, because when we complete it to Cn we do not have new tiling type, the tilings are in the first tiling types in the same row. (The yellow square would be in the grey (2 x (n — 1)) -board - see the last row in Figure 5.) Hence the recurrence equations for n > 1 satisfy the system Figure 5: Base of recurrence connections of the subboards. Rn = Uq-2Rn-l + abUq-^An-l + bUq-^Bn-l + b Uq-4 Cn-l A n = Uq-^Rn-l + abUq-5An-1 + bUq-^Bn-l + b Uq-5 Cn-l (2.1) Bn = Uq-sRn-l + bUq-^An-l Cn = Uq-^Rn-l + bUq-^An-l. coefficients of (2.1) is M are Ro = 1, Ao = Bo = Co Uq- 2 ab Uq-4 b Uq -3 b2 Uq- 4\ Uq- 3 ab Uq-5 b Uq -4 b2 Uq- 5 Uq- 3 b Uq-4 0 0 Uq- 4 b Uq-5 0 0 ) 0. The matrix of the As usual, the characteristic equation of M provides the recurrence relation for {Rn} (and {An}, {Bn}, {Cn}; see the proof in [10]. The computation was made by the help of software Maple.) Thus we have Rn '■q Rn-l + Pq Rn-2 + Yq Rn-3 + ÔqRn-4 > 4) (2.2) where (with some calculation using (1.2)) aq = abuq-5 + uq-2, ftq = b(b2u2q_5 - aUq-5Uq-2 + 2bu2_4 + OMq_quq_3 + u2_3), Yq = -bq (buq_5uq-q - 2uq-Au2q_3 + auq-5uq_3 + u2q_4uq-q), Sq = -bA(u2q_zu2q_3 - 2uq_5u2q_4uq_3 + uAq_4). Moreover, we obtain the initial values of the recurrence for n = 1,2, 3 from system (2.1). They are R1 = uq_2, R2 = u2q_2 + abuq_quq_3 + bu2q_3 + b2u2q_q and R3 = (u2q_2 + abuq_quq_3 + buq_3 + b2 uq_q)uq_2 + (abuq_2 uq_q + a2b2uq_quq_5 + b2V,q_3V,q_q + b3 uq_quq_5)uq_3 + (buq_2uq_3 + ab2uq_q)uq_3 + (b2uq_2uq_q + ab3uq_quq_5)uq_q. In the next part, we prove that relations (1.5)-(1.7) hold. Firstly, we insert aq+2, aq+i and aq into (1.5) to have abuq_3 + uq = a(abuq_q + uq_l) + b(abuq_5 + uq_2 ). (2.3) Apply (1.2) consecutively with n = q,q - 1,... as follows. First plug uq into the equation (2.3), then substitute uq_\ in the new equation, and so an. Finally, when n = q - 3, we find that (2.3) is an identity, so (1.5) holds. If q = 4 and q = 5, then aq provides the initial values. The proofs of (1.6) and (1.7) go similarly. Finally, we show that Sq = -b2(q_2). For q = 4 we immediately obtain Sq = -bq(u2_5uq_3 - 2uq_5u2_quq_3 + uq_q) = -b2 2. Then we consider the recurrence relation (q > 4) xq+1 = b2xq. (2.4) Some calculations show that both expressions (Sq and -b2(q_2)) satisfies recursion (2.4), which implies the equality. We express the values by uq_q and uq_5 by using relation (1.2). Thus we have aq = (a2 + b)uq_q + 2abuq_5, Pq = (2a2 + 2b)bu2q_q + (-a3 + 2ab)buq_quq_5 + (-a2b + 2b2)buq_5, Yq = (a2 - b)b2u3q_q - (a3 - 3ab)b2uq_quq_5 - (3a2b - b2)b2uq_quq_5 - 2abqu3q_5, sq = -b2(q_2). As F2 - FnFn_i - F2_1 = (-l)n_1, if a = b =1, then we obtain aq — 2fq-4 + 2fq-5 — 2fq-3 Pq — 4f2q-q + fq-fq-5 + f— = + (-l)q-1, Yq — 2f2-qfq-5 - 2fq-qf2-5 - fq-5 — 2(-l)q fq-5, 5q — -1. Now the initial values R lead to the initial values ri (i — 1,2,3). 2.2 Unbreakable tilings In this subsection we determine the number of unbreakable tilings. Let rn (and Rn) be the number of different unbreakable tilings with (colored) squares and dominoes of (2 x n)-boardof {4, q}. Moreover, let Ai, Bi and Ci denote the number of the different unbreakable colored tilings of Ai, Bi and Ci, respectively. Theorem 2.1. The sequence {Rn} can be described by the binary recurrence relation Rn = abuq-sRn-i + b2 {uq-4 + buq-5) Rn-2, (n > 3) where the initial values are R\ = uq-q and Rq = abuq-3uq-4 + bUq-3 + b2u^-4. Proof. The proof is similar to the proof of the first theorem. By deleting the breakable tilings from Figure 5 (the second column) we gain the system of recurrence sequences (n > 2) Rn = abuq-4An-1 + buq-0,Bri-l + b uq-4 3) with coefficients linked to Fibonacci numbers, where the initial values are rl = fq-q and C = 2fq-4fq-q + (-1)q-\ 3 Some identities In the sequel, we give certain identities related to the sequences {Rn} and {Rn}. The proofs are based on the tilings, not on the recursive formulae. Identity 3.1. If n > 1, then nl Rn ^ ^ Ri-B-n-i- i=0 Proof. Let us consider the breakable colored tilings in position i (0 < i < n) of (2 x n)-board, where the tilings on the right (2 x (n - i)) -subboard are unbreakable (see Figure 6). The number of this tilings is RiRn-i. If i = 0, then the tilings are unbreakable on the whole (2 x n)-board. Clearly, when i goes from 1 to n - 1, we have different tilings and we consider all of them exhaustedly. □ Figure 6: Breakable tilings in position i in case of Identity 3.1. An equivalent form of Identity 3.1 is Identity 3.2. If n > 1, then n Rn ^ ^ Rn- i-^i-i=l The next statement gives another rule of summation. Identity 3.3. If m > 1 and n > 1, then n m Rn+m RnRm + ^ ^ ^ ^ Rn-iRm-jRi+j • i=1 j = 1 Proof. Let us consider a (2 x (n+m)) -board as the concatenation of (2 x n)-board and (2 x m)-board (in other words, tilings are breakable in position n). First we take the breakable tilings in position n, their cardinality is Rn Rm. Then we examine the unbreakable tilings in this position. We cover the position n by i + j long unbreakable tilings from position n - i to n + j. They give the rest tilings. Figure 7 illustrates these two cases. □ 0 n R Rm 0 n n+m 0 n-i n+j n+m Figure 7: Tilings in case of Identity 3.3. Identity 3.3 admits the following remarkable specific cases by the choice of m = 1, m = (k — 1)n and n = n — k, m = n + k, respectively. Identity 3.4. If n > 1, then Rn+l = RnRl + ^^ Rn-iRi+1- Identity 3.5. If n > 1 and k > 2, then (k-l)n Rkn = RnR(k-1)n + ^^ ^^ Rn-iR n-iR(k- 1)n-j Ri+j i= 1 j = 1 Identity 3.6. If n > k > 0 then n — kn+k R2n = Rn—k Rn+k + ^^ ^^ Rn—k—iRn+k—j Hi+j . i=1 j = 1 Finally, we give an identity about the product of two arbitrary terms of the sequence {Rn}. Identity 3.7. If n, m > 1, then n— 1 m— 1 RnRm ^ ^ ^ ^ RiRj Rn—iRm—j . i=0 j=0 Proof. Consider a (2 x (n + m)) -board as a concatenation of (2 x n)-board and (2 x m)-board. The result is derived in a direct manner from the number of the breakable tilings in position n. See Figure 8. □ 0 in n+j n+m Figure 8: Tilings in case of Identity 3.7. 4 Conclusion and future work In this article, we introduced a generalization of the square boards on the hyperbolic regular square mosaics and examined the combinatorial properties of tilings on these mosaics with colored squares and dominoes. As there are the infinite number of regular mosaics in the hyperbolic plane we hope that the examinations of the combinatorial properties of other tilings give some useful results. Moreover, we are informed on two additional timely articles about hyperbolic space tilings [8, 11]. References [1] H. Belbachir and A. Belkhir, Tiling approach to obtain identities for generalized Fibonacci and Lucas numbers, Ann. Math. Inform. 41 (2013), 13-17, http://ami.ektf.hu/uploads/ papers/finalpdf/AMI_41_from13to17.pdf. [2] A. T. Benjamin, S. S. Plott and J. A. Sellers, Tiling proofs of recent sum identities involving Pell numbers, Ann. Comb. 12 (2008), 271-278, doi:10.1007/s00026-008-0350-5. [3] A. T. Benjamin and J. J. Quinn, The Fibonacci numbers—exposed more discretely, Math. Mag. 76 (2003), 182-192, doi:10.2307/3219319. [4] A. T. Benjamin and J. J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof, volume 27 of The Dolciani Mathematical Expositions, Mathematical Association of America, Washington, DC, 2003, http://www.jstor.org/stable/10.4169/j. ctt6wpwjh. [5] H. S. M. Coxeter, Regular honeycombs in hyperbolic space, in: J. C. H. Gerretsen and J. de Groot (eds.), Proceedings of the International Congress of Mathematicians 1954, Volume III, North-Holland, Amsterdam, 1956 pp. 155-169, proceedings of the International Congress of Mathematicians 1954 held at Amsterdam, September 2-9, 1954. [6] R. Kahkeshani, The tilings of a (2 x n) -board and some new combinatorial identities, J. Integer Seq. 20 (2017), Article 17.5.4, https://cs.uwaterloo.ca/journals/JIS/ VOL20/Kahkeshani/kahk3.html. [7] M. Katz and C. Stenson, Tiling a (2 x n)-board with squares and dominoes, J. Integer Seq. 12 (2009), Article 9.2.2, https://cs.uwaterloo.ca/journals/JIS/VOL12/ Stenson/stenson8.html. [8] Z. Lucic, E. Molnar and N. Vasiljevic, An algorithm for classification of fundamental polygons for a plane discontinuous group, in: M. D. E. Conder, A. Deza, and A. Ivic Weiss (eds.), Discrete Geometry and Symmetry, Springer, Cham, volume 234 of Springer Proceedings in Mathematics & Statistics, pp. 257-278, 2018, doi:10.1007/978-3-319-78434-2_14, dedicated to Karoly Bezdek and Egon Schulte on the occasion of their 60th birthdays. [9] R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys. 11 (1970), 3095-3099, doi:10.1063/1.1665098. [10] L. Nemeth and L. Szalay, Power sums in hyperbolic Pascal triangles, An. St. Univ. Ovidius Constanta, Ser. Mat. 26 (2018), 189-203, doi:10.2478/auom-2018-0012. [11] I. Prok, On maximal homogeneous 3-geometries—A polyhedron algorithm for space tilings, Universe 4 (2018), Article 49, doi:10.3390/universe4030049. [12] N. J. A. Sloane (ed.), The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 347-354 https://doi.org/10.26493/1855-3974.1459.11d (Also available at http://amc-journal.eu) Mirrors of reflections of regular maps Adnan Melekoglu * Department of Mathematics, Faculty of Arts and Sciences, Adnan Menderes University, 09010 Aydin, Turkey Received 4 August 2017, accepted 8 October 2017, published online 27 June 2018 Abstract A regular map M is an embedding of a finite connected graph into a compact surface S such that its automorphism group Aut+(M) acts transitively on the directed edges. A reflection of M fixes a number of simple closed geodesics on S, which are called mirrors. In this paper, we prove two theorems which enable us to calculate the total number of mirrors fixed by the reflections of a regular map and the lengths of these mirrors. Furthermore, by applying these theorems to Hurwitz maps, we obtain some interesting results. In particular, we find an upper bound for the number of mirrors on Hurwitz surfaces. Keywords: Riemann surface, regular map, Hurwitz map, reflection, mirror. Math. Subj. Class.: 05C10, 30F10 1 Introduction Let S be a compact Riemann surface of genus g. It is known that S can be expressed in the form U/A, where U is the Riemann sphere E, the Euclidean plane C, or the hyperbolic plane H, depending on whether g is 0, 1 or > 1, respectively, and A is a discrete group of isometries of U. A conformal or anti-conformal homeomorphism f: S ^ S is called an automorphism of S. If S admits an anti-conformal involution r: S ^ S, then it is called symmetric and r is called a symmetry of S. The fixed-point set of r is either empty, or consists of disjoint simple closed geodesics on S. These geodesics are called the mirrors of r and their number cannot exceed g +1 by a classical theorem of Harnack [10]. All automorphisms of S form a group under composition and it is denoted by Aut±(S). The subgroup of Aut±(S) consisting of orientation-preserving automorphisms is denoted by *The author would like to thank the referees for their valuable comments and suggestions. The author would also like to thank David Singerman for suggesting the alternative proof of Theorem 3.1 described in Remark 3.4. E-mail address: amelekoglu@adu.edu.tr (Adnan Melekoglu) Aut+(S). ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ Let T be a triangle in U, with angles n/2, n/m and n/n, where m and n are integers greater than one and m + n is greater than, equal to or less than 1 depending on whether U is E, C or H, respectively. Such a triangle is said to be a (2, m, n)-triangle. Let r be the group generated by the rotations about the corners of T. Then it is called the ordinary triangle group r[2, m, n] and it has a presentation (x, y, z | x2 = ym = zn = xyz = 1}. If r is the group generated by the reflections in the sides of T, then it is called the extended triangle group r(2, m, n), which has a presentation (a, b, c | a2 = b2 = c2 = (ab)2 = (bc)m = (ca)n = 1}. A map M on S is an embedding of a finite connected graph G into S such that the interior of each face (a component of S \ G) is homeomorphic to an open disc. The genus of M is defined to be the genus of S. A directed edge of M is called a dart and M is said to be of type {m, n} if every face of M has m sides and n darts meet at every vertex. An automorphism of S that leaves M invariant and preserves incidence is called an automorphism of M. All automorphisms of M form a group under composition and this group is denoted by Aut± (M). The subgroup of Aut± (M) consisting of orientation-preserving automorphisms is denoted by Aut+(M). If Aut+(M) is transitive on the darts, then M is called regular. It is clear that if M is regular, then the number of darts is equal to |Aut+(M)| and M has |Aut+(M)|/2 edges, |Aut+(M)|/m faces and |Aut+(M)|/n vertices. If M is a regular map of type {m, n} and S = U/A is the underlying Riemann surface, then by [12], A is normal in the ordinary triangle group r[2, m, n]. If A is also normal in the extended triangle group r(2, m, n), then M is called reflexible. In that case M admits an anti-conformal involution r, which is a symmetry of S with fixed-points, called a reflection of M. In this paper, we prove two theorems which enable us to calculate the total number of mirrors fixed by the reflections of a regular map and the lengths of these mirrors. Furthermore, we use these theorems to obtain an upper bound for the total number of mirrors in Hurwitz maps. Throughout this paper, we assume that the maps we deal with are regular and reflexible. 2 Patterns and mirror automorphisms Let M be a regular map of type {m, n} on a compact Riemann surface S of genus g. By joining the centers of the faces of M to the midpoints of the neighboring edges and vertices by geodesic arcs, we can divide S into |Aut±(M)| (2, m, n)-triangles. If T is one of these triangles, then the group Aut±(M) can be generated by the reflections in the sides of T and it has a presentation of the form (A, B, C | A2 = B2 = C2 = (AB)2 = (BC)m = (CA)n = • • • = 1). (2.1) Similarly, the group Aut+(M) can be generated by the rotations about the corners of T and it has a presentation of the form (X, Y,Z | X2 = Ym = Zn = XYZ = • • • = 1). (2.2) Note that if g = 0, then the groups Aut±(M) and Aut+(M) are finite and the explicitly listed relations in (2.1) and (2.2) give presentations for these groups, respectively. If g > 1, then these presentations must contain at least one more relation. Following [7], we label the vertices, edge-centers and face-centers of M with 0, 1 and 2, respectively. They are called the geometric points of M. As an automorphism preserves the geometric points, it follows that a mirror of a reflection of M passes through some geometric points of M and these geometric points form a periodic sequence. Since S is compact, this sequence is finite and it is called the pattern of the mirror. As an example, consider the icosahedral map on the sphere, which has type {3,5}. Each reflection of this map fixes a mirror with pattern 010212010212, which is abbreviated as (010212)2; see [7]. Each repeated part of a pattern is called a link, and the number of links is called the link index. So in this example, 010212 is a link and the link index is 2. In [15], it has been shown that the pattern of a mirror is always obtained from one of the six links 01, 02, 12, 0102, 0212, 010212, and there cannot be more than three mirrors with different patterns on the same Riemann surface. (See Figures 1 and 2, which represent regular maps admitting two and three different patterns, respectively.) The following theorem expresses this idea and it can be deduced from [15]. Theorem 2.1. Let M be a regular map of type {m, n} on a compact Riemann surface S and let M be a mirror of a reflection of M. Then: (i) If m and n are odd, then M has pattern of the form (010212)^; (ii) If m is even and n is odd, then M has pattern of the form (0102)£l or (12)^2 ; (iii) If m is odd and n is even, then M has pattern of the form (0212)£l or (01)^2 ; (iv) If m and n are even, then M has pattern of the form (01)£l, (02)^2 or (02)^3. Here ¿, i\, t2 and ¿3 are positive integers, which depend only on M, not on M. Furthermore, ¿is in different lines need not be equal. Note that all the patterns listed in each part of Theorem 2.1 do occur. For example, in part (ii) the surface S contains two classes of mirrors such that the mirrors in different classes have different patterns, namely (0102)£l and (12)^2. The same argument applies to all parts of Theorem 2. 1 . Now let M be a regular map on a compact Riemann surface S and let M be a mirror of a reflection of M. Suppose that ¿ is the link index of the pattern of M .If ¿ > 2, then there exist two orientation-preserving automorphisms of M of order ¿, which fix M setwise and have no fixed points on M. They rotate M in opposite directions and cyclically permute the links of the pattern of M. These automorphisms are inverses of each other and they are called the mirror automorphisms of M. Note that if ¿ = 2, then M has a unique mirror automorphism. If ¿ = 1, then we assume that the mirror automorphism of M is the identity. Associated to each pattern, there is a conjugacy class of mirror automorphisms such that the order of each mirror automorphism in this conjugacy class is equal to the link index of the pattern; see [15, Lemma 1]. In Table 1, for each pattern, a representative mirror automorphism is displayed in terms of the generators of Aut+(M) in (2.2). Note that in the table, for each pattern, only one link is displayed. See [15] for details. Table 1: Patterns and mirror automorphisms. Case Link Mirror automorphism 1 01 Zn X 2 02 Yt Zn 3 12 Ym X 4 0102 Z "+1 YZ n+i Y T 5 0212 zn Y ZY 6 010212 Y T+1 ZYT+1 Zn+ YZn+i 3 Number of mirrors From now on, ||M|| will denote the total number of mirrors fixed by the reflections of a regular map M. Theorem 3.1. Let M be a regular map of type {m, n} on a compact Riemann surface, and let t, ti, t2,t3 be as in Theorem 2.1. Then: (i) If m and n are odd, then (ii) If m and n have different parities, then (iii) If m and n are even, then |Aut+(M)| 2£ ' |Aut+(M)| (X + 1 V 2 ( £1 + £2 '' Aut+(M)| (1 + X + 1 ^ 2 (£, + £o + £„ >• Proof. (i) By Theorem 2.1, every mirror of a reflection of M has pattern (010212) . It is clear that each of these mirrors contains t edges of M. Since M has |Aut^(M)| edges, we find that ||M|| = |Aut+(M)|. (ii) Suppose that m is even and n is odd. It follows from Theorem 2.1 that the pattern of a mirror of a reflection of M is either (0102) 1 or (12)*2. It is known that has |Autm(M)| faces and a mirror with pattern (0102)£l passes through the centers of ti faces of M. Also, the number of mirrors with pattern (0102)£l passing through the center of a face F of M is m/2. (See Figure 1, where m = 6 and n = 3. The dashed lines denote the mirrors that have pattern (0102)£l and pass through the center of F.) Therefore, there are |Aut+ (M)| 1 m _ |Aut+(M)| m ti "2 = 2ti mirrors with pattern (0102)£l. A similar argument shows that there are |Aut+ (M)| 1 m |Aut+(M)| ¿2 2 2^2 mirrors with pattern (12)^2. As a result, we find that IM|| |Aut+(M)| 1 1 k + T2 m \ F ; Figure 1: Mirrors with pattern (0102) 1, passing through a face-center. The case where m is odd and n is even is similar. (iii) In this case, by Theorem 2.1, the pattern of a mirror is either (01)£l, (12)^2 or (02)^3. We know that M has |Autvertices and a mirror with pattern (01)£l passes through £1 vertices of M. Moreover, the number of mirrors with pattern (01)£l passing through a vertex of M is n/2. (See Figure 2, where m = n = 4. The dashed lines denote the mirrors that have pattern (01)£l and pass through a vertex v of M.) Thus, there are |Aut+(M)| 1 n |Aut+(M)| l1 2 'H1 mirrors with pattern (01)£l. Similar arguments show that there are |Aut2^2M)| mirrors with pattern (12) 2 and IIM = JAut+' - |Aut+(M)| 2^3 mirrors with pattern (02) 3. Consequently, we find that ( 1 + 1 + 1 ) ( ¿1 + Î2 ^ )- □ n V Figure 2: Mirrors with pattern (01)£l, passing through a vertex. Note that if M is a reflexible regular map and if we are given a presentation for Aut+(M) as in (2.2), then we can easily determine the link indices by using Table 1 and MAGMA [1]. This is because the link indices are the orders of the mirror automorphisms (see [15, Lemma 1]), and the latter are explicitly known (see Table 1). Then by using Theorem 3.1 we can easily calculate ||M||. Example 3.2. Let S be the Riemann surface of genus 7 admitting 504 conformal automorphisms. This surface is known as the Fricke-Macbeath surface; see [8, 14]. It is known that S underlies a regular map M of type {3,7}, which is called the Fricke-Macbeath map. It follows from [15] that Aut+(M) has a presentation (X, Y, Z | X2 = Y3 = Z7 = XYZ = (Y2ZY2Z4YZ4)2 = 1), and Y2ZY2Z4YZ4 is a mirror automorphism. Since this automorphism has order 2, by Theorem 2.1 every mirror on S has pattern (010212)2. Thus, by using Theorem 3.1 we find that ||M|| = 5r = 126. Remark 3.3. Let M be a regular map and M* be its dual. Since the reflections of M and M* coincide, the mirrors of M* are the same as those of M. So ||M|| = ||M* ||. Remark 3.4. Let M be a regular map on a compact Riemann surface S and let M be a mirror of a reflection of M. If £ is the link index corresponding to the pattern of M, then the stabilizer of M in Aut+(M) is the dihedral group D^. Here D^ is generated by a mirror automorphism of M and an involution fixing two antipodal points of M. Since Aut+ (M) is transitive on the mirrors with the same pattern, the orbit of M consists of the mirrors on S which have the same pattern as M. So by the Orbit-Stabilizer theorem, we find that there are | Aut+ (M) |/2£ mirrors in the orbit of M. By Theorem 2.1, there are at most three orbits, and their sizes can be determined in the same way. Therefore, we obtain an alternative proof of Theorem 3.1. 4 Lengths of mirrors Let M be a regular map of type {m, n} on a compact Riemann surface S of genus g and let M be a mirror of a reflection of M. As pointed out in Section 2, S can be divided into |Aut±(M)| (2, m, n)-triangles and M is a combination of the sides of (2, m, n)-triangles. Let a, b and c be the lengths of the sides of a (2, m, n)-triangle as indicated in Figure 3. If g > 1, then by using sine and cosine rules for hyperbolic triangles, we can calculate a, b and c. So the length of M can be calculated as described below. 0 a 1 2 b Figure 3: A (2, m, n)-triangle with side lengths a, b, c. Let m and n be odd. Then M will have pattern of the form (010212)^. Now every link corresponds to a segment of M, which has length 2(a + b + c). Thus, M has length 2^(a + b + c). Clearly, every mirror has the same length as M in this case. If m and n have different parities, then there are two classes of mirrors on S. If m and n are both even, then there are three classes of mirrors on S. In both cases the mirrors in each class have the same length and pattern. In each case, the lengths of the mirrors can be calculated in the same way. Note that if g = 0 or 1, then it is not difficult to find the length of the mirrors explicitly. So we have the following result: Theorem 4.1. Let M be a regular map of type {m, n} on a compact Riemann surface S and let the lengths of the sides of a (2, m, n)-triangle be a, b and c as indicated in Figure 3. Then the lengths of the mirrors of the reflections of M can be determined by the formulae in Table 2, where i, i1, i2 and i3 are the link indices and ¿is in different lines need not be equal. Table 2: Lengths of mirrors. Case Pattern Length of mirror m and n are odd (010212)^ 2i(a + b + c) m odd n even (01)£l 2iia m odd n even (0212)^2 2i2(b + c) m even n odd (12)£l 2i1b m even n odd (0102)^2 2i2(a + c) m and n are even (01)£i 2i1a m and n are even (12)£2 2i2b m and n are even (02)*> 2i3c 5 Application to Hurwitz maps By a classical theorem of Hurwitz [11], a compact Riemann surface of genus g > 1 has at most 84(g - 1) conformal automorphisms. Any such surface S = H/A is called a Hurwitz surface, and in that case Aut+(S) is called a Hurwitz group. It is known that if S is a Hurwitz surface, then A is normal in the ordinary triangle group r[2,3,7]. Thus, every Hurwitz surface underlies a regular map of type {3,7}, which is called a Hurwitz map. Furthermore, Aut+ (M) is isomorphic to Aut+ (S) and has a presentation of the form (X,Y,Z | X2 = Y3 = Z7 = XYZ = •• • = 1). It has been shown by [13] that the upper bound in Hurwitz's theorem is attained for infinitely many values of the genus g. Thus, there exist infinitely many Hurwitz maps and surfaces. See [2, 3, 4, 5, 13] for further details. Theorem 5.1. Let M be a Hurwitz map of genus g and let S be the underlying surface. Then ||M|| < 21(g — 1), where equality holds if and only if S is the Fricke-Macbeath surface. Proof. Let i be the link index of M. By Theorem 3.1, we find that = |Aut+(M)| = 84(g — 1) = 42(g — 1) 1 1 2i 2i i ' It follows from [15, Theorem 5] that i > 2 and hence ||M|| < 21(g — 1) and that equality holds if and only if S is the Fricke-Macbeath surface. See also [9, Theorem 4.1]. □ It follows from Theorem 5.1 that if M is a Hurwitz map of genus g with link index t, then ||M|| is bounded above by 21(g - 1). When this upper bound is attained, t = 2 and the underlying surface is the Fricke-Macbeath surface. However, ||M|| cannot have a lower bound in terms of g. This follows from the theorem below, which was given in [6]. Theorem 5.2. For every positive integer n, there exist Hurwitz maps with link indices 2n and 3n. In particular, the link index of a Hurwitz map can be any even positive integer Let L be the sum of the lengths of the sides a (2,3,7)-triangle. Then by using the sine and cosine rules for hyperbolic triangles we find that L ~ 1.4490747226. It follows from Theorem 4.1 that the length of a mirror on a Hurwitz surface is 2tL, where t is the link index. Also, the minimum possible length of a mirror on a Hurwitz surface is 4L ~ 5.7962988904, and in that case the underlying surface is the Fricke-Macbeath surface; see [15, Theorem 6]. However, by Theorem 5.2, there is no upper bound on the lengths of mirrors on Hurwitz surfaces. References [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235-265, doi:10.1006/jsco.1996.0125. [2] M. Conder, The genus of compact Riemann surfaces with maximal automorphism group, J. Algebra 108 (1987), 204-247, doi:10.1016/0021-8693(87)90135-9. [3] M. Conder, Maximal automorphism groups of symmetric Riemann surfaces with small genus, J. Algebra 114 (1988), 16-28, doi:10.1016/0021-8693(88)90208-6. [4] M. Conder, Hurwitz groups: a brief survey, Bull. Amer. Math. Soc. (N. S.) 23 (1990), 359-370, doi:10.1090/s0273-0979-1990-15933-6. [5] M. Conder, An update on Hurwitz groups, Groups Complex. Cryptol. 2 (2010), 35-49, doi: 10.1515/gcc.2010.002. [6] M. Conder and A. Melekoglu, Link indices of Hurwitz maps, J. Algebra 490 (2017), 568-580, doi:10.1016/j.jalgebra.2017.08.001. [7] H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York, 3rd edition, 1973. [8] R. Fricke, Ueber eine einfache Gruppe von 504 Operationen, Math. Ann. 52 (1899), 321-339, doi:10.1007/bf01476163. [9] G. Gromadzki, On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces, J. PureAppl. Algebra 121 (1997), 253-269, doi:10.1016/s0022-4049(96)00068-0. [10] A. Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), 189-198, doi:10.1007/bf01442458. [11] A. Hurwitz, Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1892), 403-442, doi:10.1007/bf01443420. [12] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (1978), 273-307, doi:10.1112/plms/s3-37.2.273. [13] A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90-96, doi:10.1017/s2040618500034365. [14] A. M. Macbeath, On a curve of genus 7, Proc. London Math. Soc. 15 (1965), 527-542, doi: 10.1112/plms/s3-15.1.527. [15] A. Melekoglu and D. Singerman, The structure of mirrors on regular maps on Platonic surfaces, Geom. Dedicata 181 (2016), 239-256, doi:10.1007/s10711-015-0122-4. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 355-373 https://doi.org/10.26493/1855-3974.1223.b26 (Also available at http://amc-journal.eu) The thickness of Ki,n?n and K2, * Xia Guo School of Mathematics, Tianjin University, Tianjin, P.R.China Yan Yang t School ofMathematics, Tianjin University, Tianjin, P.R.China Received 26 October 2016, accepted 12 June 2018, published online 9 July 2018 The thickness of a graph G is the minimum number of planar subgraphs whose union is G. In this paper, we obtain the thickness of complete 3-partite graph Kl,n,n, K2,n,n and complete 4-partite graph Kljl n n. Keywords: Thickness, complete 3-partite graph, complete 4-partite graph. Math. Subj. Class.: 05C10 1 Introduction The thickness 6(G) of a graph G is the minimum number of planar subgraphs whose union is G. It was first defined by W. T. Tutte [7] in 1963, then a few authors obtained the thickness of hypercubes [5], complete graphs [1, 2, 8] and complete bipartite graphs [3]. Naturally, people wonder about the thickness of the complete multipartite graphs. A complete k-partite graph is a graph whose vertex set can be partitioned into k parts, such that every edge has its ends in different parts and every two vertices in different parts are adjacent. Let KPlP2..Pk denote a complete k-partite graph in which the ith part contains p (1 < i < k) vertices. For the complete 3-partite graph, Poranen proved 6(Kn,n,n) < \nl in [6], then Yang [10] gave a new upper bound for 6(Kn,n,n), i.e., 6(Kn,n,n) < \n+11 + 1 and obtained 6(K„i„i„) = [], when n = 3 (mod 6). And also Yang [9] gave the thickness number of Kl,mn(l < m < n) when l + m < 5 and showed that 6(Ki,m,n) = \1 when l + m is even and n > l(l + m - 2)2; or l + m is odd and n > (l + m - 2)(l + m - 1). In this paper, we obtain the thickness of complete 3-partite graph K1n n and K2 n n, and we also deduce the thickness of complete 4-partite graph K1i1n n from that of K2 n n. *Supported by the National Natural Science Foundation of China under Grant No. 11401430. 1 Corresponding author. E-mail addresses: guoxia@tju.edu.cn (Xia Guo), yanyang@tju.edu.cn (Yan Yang) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ 2 The thickness of K1,n,n In [3], Beineke, Harary and Moon gave the thickness of complete bipartite graphs Km for most value of m and n, and their theorem implies the following result immediately. Lemma 2.1 ([3]). The thickness of the complete bipartite graph Kn ' n + 2" is 0(Kn,n) 4 In [4], Chen and Yin gave a planar decomposition of the complete bipartite graph K4p,4p with p + 1 planar subgraphs. Figure 1 shows their planar decomposition of K4p,4p, in which {u1,..., u4p} = U and {v1,..., v4p} = V are the 2-partite vertex sets of it. Based on their decomposition, we give a planar decomposition of K2,n,n with p + 1 subgraphs when n = 0 or 3 (mod 4) and prove the following lemma. (a) The graph Gr (1 < r < p). ul u2 vl v2 u4p-1 u4p v4p-1 v4p (b) The graph Gv+1. Figure 1: A planar decomposition of K4p,4p. Lemma 2.2. The thickness of the complete 3-partite graph K1,n,n and K2,n,n is ' n + 2 " 0(Ki,n,n) = 0(K2,n,n) = 4 when n = 0 or 3 (mod 4). Proof. Let the vertex partition of K2,n,n be (X,U,V), where X = {x1,x2}, U = {ui,..., un} and V = {vi,..., v„|. When n = 0 (mod 4), let n = 4p (p > 1). Let {Gi,..., Gp+1} be the planar decomposition of Kn,n constructed by Chen and Yin in [4]. As shown in Figure 1, the graph Gp+1 consists of n paths of length one. We put all the n paths in a row, place vertex x1 on one side of the row and the vertex x2 on the other side of the row, join both x1 and x2 to all vertices in Gp+1. Then we get a planar graph, denote it by Gp+1. It is easy to see that {G1,..., Gp, Gp+1} is a planar decomposition of K2,n,n. Therefore, we have 0(K2,n,n) < p +1. Since Kn,n c K1n n c K2,n,n, combining it with Lemma 2.1, we have p + 1 = 0(Kn,n) < 0(K1,n,n) < 0(K2,n,n) < P +1, that is, 0(K^n,n) = 0(K2,n,n) = p +1 when n = 0 (mod 4). When n = 3 (mod 4), then n = 4p + 3 (p > 0). When p = 0, from [9], we have 0(^,3,3) = 0(K2,3,3) = 2. Whenp > 1, since Kn,n c K1,n,n c K2,n,n C K2 according to Lemma 2.1 and 0(K2,4p,4p) = p + 1, we have p + 2 = 0(Kn,n) < 0(K1,n,n) < 0(K2,n,n) < 0(K2,n+1,n+1) = p + 2. Then, we get 0(K1,n,n) = 0(K2,n,n) = p + 2 when n = 3 (mod 4). Summarizing the above, the lemma is obtained. □ Lemma 2.3. There exists a planar decomposition of the complete 3-partite graph K1,4p+2,4p+2 (p > 0) with p + 1 subgraphs. Proof. Suppose the vertex partition of the complete 3-partite graph K1,n,n is (X, U, V), where X = {x}, U = {u1,..., un} and V = {v1,..., vn}. When n = 4p + 2, we will construct a planar decomposition of K1,4p+2,4p+2 with p + 1 planar subgraphs to complete the proof. Our construction is based on the planar decomposition {G1, G2,..., Gp+1} of K4p 4p given in [4], as shown in Figure 1 and the reader is referred to [4] for more details about this decomposition. For convenience, we denote the vertex set Uf=1i=r{u4i-3, u4i-2}^p=1,i=r{u4i-1,u4i}^P=1,i=r{v4i-3,v4i-1} and |JP=Wr{v4i-2,v4i} by U1, U2, V1r and V2r respectively. We also label some faces of Gr (1 < r < p), as indicated in Figure 1, for example, the face 1 is bounded by v4r-3u4rvju4r_1 in which vj is some vertex from Vf. In the following, for 1 < r < p + 1, by adding vertices x, u4p+1, u4p+2, v4p+1, v4p+2 and some edges to Gr, and deleting some edges from Gr such edges will be added to the graph Gp+1, we will get a new planar graph Gr such that {G?1,..., Gp+1} is a planar decomposition of K1,4p+2,4p+2. Because v4r_3 and v4r-1 in Gr (1 < r < p) is joined by 2p - 2 edge-disjoint paths of length two that we call parallel paths, we can change the order of these parallel paths without changing the planarity of Gr. For the same reason, we can do changes like this for parallel paths between M4r-1 and u4r, v4r_2 and v4r, u4r_3 and u4r_2. We call this change by parallel paths modification for simplicity. All the subscripts of vertices are taken modulo 4p, except that of v4p+1, v4p+2, w4p+1 and u4p+2 (the vertices we added to Gr). Case 1. When p is even and p > 2. (a) The construction for Gr , 1 < r < p, and r is odd. Step 1: Place the vertex x in the face 1 of Gr, delete edges v4r_3u4r and u4rv4r_1 from Gr. Do parallel paths modification, such that w4r+6 € U[, v4r+1 G Vf and u4r, v4r-3, v4r-2, v4r-1 are incident with a common face which the vertex x is in. Join x to «4r-3, «4r-1, «4r, ^4r-3, V4r-2, ^4r-1 and «4r+6, V4r + 1. Step 2: Do parallel paths modification, such that w4r+11, u4r+12 € U2 are incident with a common face. Place the vertex v4p+1 in the face, and join it to both w4r+11 and w4r+12. Step 3: Do parallel paths modification, such that w4r+7, u4r+8 € UJ are incident with a common face. Place the vertex v4p+2 in the face, and join it to both w4r+7 and w4r+8. Step 4: Do parallel paths modification, such that v4r+10, v4r+12 € V2r are incident with a common face. Place the vertex w4p+1 in the face, and join it to both v4r+10 and v4r+12. Step 5: Do parallel paths modification, such that v4r+6, v4r+8 € V2r are incident with a common face. Place the vertex w4p+2 in the face, and join it to both v4r+6 and v4r+8. (b) The construction for Gr, 1 < r < p, and r is even. Step 1: Place the vertex x in the face 3 of Gr, delete edges v4rw4r-3 and w4r-3v4r_2 from Gr. Do parallel paths modification, such that w4r+7 € U2, v4r+4 € V2r and w4r-3, u4r-2, u4r, v4r-2, v4r-1, v4r are incident with a common face which the vertex x is in. Join x to U4r—3, W4r_2, U4r, V4r_2, V4r — 1, V4r and U4r+7, V4r+4. Step 2: Do parallel paths modifications, such that w4r+5,w4r+6 € U[, u4r+1,w4r+2 € U[, v4r+5, v4r+7 € Vf, v4r+1, v4r+3 € Vf are incident with a common face, respectively. Join v4p+1 to both w4r+5 and u4r+6, join v4p+2 to both w4r+1 and u4r+2, join w4p+1 to both v4r+5 and v4r+7, join u4p+2 to both v4r+1 and v4r+3. Table 1 shows how we add edges to Gr (1 < r < p) in Case 1. The first column lists the edges we added, the second and third column lists the subscript of vertices, and we also indicate the vertex set which they belong to in brackets. Table 1: The edges we add to Gr (1 < r < p) in Case 1. subscript \ case edge r is odd r is even XUj 4r — 3, 4r — 1, 4r 4r + 6 (Uf) 4r — 3,4r — 2,4r 4r + 7 (UJ ) xvj 4r — 3, 4r — 2, 4r — 1 4r + 1 (Vf ) 4r — 2,4r — 1,4r 4r + 4 (V2r) V4p+lUj 4r + 11, 4r + 12 (UJ) 4r + 5,4r + 6 (UJ) V4p+2Uj 4r + 7, 4r + 8 (U2r) 4r + 1,4r + 2 (UJ) U4p+1Vj 4r + 10, 4r + 12 ( V2r) 4r + 5, 4r + 7 (Vf ) U4p+2 Vj 4r + 6, 4r + 8(V2r) 4r +1, 4r + 3 (Vf ) (c) The construction for Gp+1. From the construction in (a) and (b), the subscript set of u that xuj is an edge in Gr for some r G {1,... ,p} is {4r — 3, 4r — 1,4r, 4r + 6 (mod 4p) | 1 < r < p, and r is odd} U {4r — 3, 4r — 2,4r, 4r + 7 (mod 4p) | 1 < r < p, and r is even} = {1,... ,p}. The subscript set of uj that v4p+1uj is an edge in Gr for some r G {1,...,p} is {4r + 11,4r +12 (mod 4p) | 1 < r < p, and r is odd} U {4r + 5,4r + 6 (mod 4p) | 1 < r < p, and r is even} = {4r — 3, 4r — 2,4r — 1, 4r | 1 < r < p, and r is even}. Using the same procedure, we can list all the edges incident with x, v4p+1, v4p+2, u4p+1 and u4p+2 in Gr (1 < r < p), so we can also list the edges that are incident with x, v4p+1, v4p+2, u4p+1 in K1j4p+2j4p+2 but not in any Gr (1 < r < p). Table 2 shows the edges that belong to K1j4p+2j4p+2 but not to any Gr, 1 < r < p, in which the the fourth and fifth rows list the edges deleted form Gr (1 < r < p) in step one of (a) and (b), and the sixth row lists the edges of Gp+1. The Gp+1 is the graph consists of the edges in Table 2, Figure 2 shows Gp+1 is a planar graph. Table 2: The edges of Gp+1 in Case 1. edges subscript xvip+1,xuip+1,vip+1uj ,'Uip+1Vj j = 4r - 3, 4r - 2, 4r - 1, 4r, 4p + 2 (r = 1, 3,..., p - 1) XV4p+2,XU4p+2,V4p+2Uj ,U4p+2 Vj j = 4r - 3,4r - 2,4r - 1,4r, 4p + 1 (r = 2,4,... ,p) V4r — 3U4r, U4rV4r-i r = 1, 3,..., p - 1 V4rU4r-3, U4r-3V4r-2 r = 2, 4,... ,p Uj Vj j = 1,..., 4p + 2 u4p + 1 v4p + 2 Figure 2: The graph Gp+1 in Case 1. A planar decomposition {G?1,..., G?p+1} of K1j4p+2j4p+2 is obtained as above in this case. In Figure 3, we draw the planar decomposition of K11818, it is the smallest example for the Case 1. We denote vertex u4 and v by i and i' respectively in this figure. Case 2. When p is odd and p > 3. The process is similar to that in Case 1. (a) The construction for Gr, 1 < r < p, and r is odd. Step 1: Place the vertex x in the face 1 of Gr, delete edges v4r-3u4r and u4rv4r-1 from Gr, for 1 < r < p, and delete v2u1 from G1 additionally. (e) The graph G5. Figure 3: A planar decomposition of K11818. For 1 < r < p, do parallel paths modification to Gr, such that u4r+6 € U[, v4r+1 G V and u4r-3, u4r-1, u4r, v4r-3, v4r-2, v4r-1 are incident with a common face which the vertex x is in. Join x to u4r-3, u4r-1, u4r, v4r-3, v4r-2, v4r-1 and u4r+6, v4r+1. Similarly, in G1, join x to u1, u3, u4, v1, v2, v3, v4 and u10 G U/, v5 G V1. In Gp, join X to M4p-3,M4p-1,U4p,V4p_3,V4p_2,V4p_1 and «2 G Up. Step 2: For 1 < r < p, do parallel paths modification to Gr, such that u4r+11, u4r+12 G U2, w4r+7,w4r+8 G Ur, v4r+10, v4r+12 G V2r and v4r+6,v4r+8 G V2r are incident with a common face, respectively. Join v4p+1 to both w4r+11 and w4r+12, join v4p+2 to both u4r+7 and u4r+8, join w4p+1 to both v4r+10 and v4r+12, join w4p+2 to both v4r+6 and V4r+8. Similarly, in Gp, join v4p+1 to u5, u6 G Uf, join v4p+2 to «7, u8 G Uf, join w4p+1 to V6, V8 G V2p, join «4f+2 to V5, V7 G V1p. (b) The construction for Gr, 1 < r < p, and r is even. Step 1: Place the vertex x in the face 3 of Gr, delete edges v4ru4r-3 and u4r-3v4r-2 from Gr, 1 < r < p - 1. Do parallel paths modification to Gr, 1 < r < p-1, such that u4r+7 G U2, v4r+4 G V2r and u4r-3, u4r-2, u4r, v4r-2, v4r-1, v4r are incident with a common face which the vertex x is in. Join x to u4r-3, u4r-2, u4r, v4r-2, v4r-1, v4r and u4r+7, v4r+4. Similarly, in Gp-1,join x to u4p-7, u4p-6, u4p-4, v4p-6, v4p-5, v4p-4 and u7 G Up 1, v4p G V2p 1. Step 2: Do parallel paths modifications, such that u4r+5,«4r+6 G U[, «4r+1, «4r+2 G U[, v4r+5,v4r+7 G Vf, v4r+1, v4r+3 G vr are incident with a common face, respectively. Join v4p+ 1 to both u4r+5 and u4r+6, join v4p+2 to both u4r+1 and u4r+2, join u4p+ 1 to both v4r+5 and v4r+7, join u4p+2 to both v4r+ 1 and v4r+3. Table 3 shows how we add edges to Gr (1 < r < p) in Case 2. (c) The construction for Gp+1. With a similar argument to that in Case 1, we can list the edges that belong to K1,4p+2,4p+2 but not to any Gr, 1 < r < p, in this case, as shown in Table 4. Then Gp+1 is the graph that consists of the edges in Table 4. Figure 4 shows Gp+1 is a planar graph. Therefore, {G1,..., Gp+1} is a planar decomposition of K14p+2 4p+2 in this case. Figure 4: The graph Gp+1 in Case 2. Table 3: The edges we add to Gr (1 < r < p) in Case 2. subscript \ case r is odd r is even edge XUj 4r - 3, 4r - 1, 4r 4r + 6, r = p (Uf) 2, r = p (Uf) 4r - 3, 4r - 2, 4r 4r + 7, r = p - 1 (U2r) 7, r = p - 1 (U2r) XVj 4r - 3, 4r - 2, 4r - 1 4, 5, r = 1 4r + 1, r = 1,p (V) 4r - 2, 4r - 1, 4r 4r + 4(V2r) V4p+lUj 4r +11,4r + 12, r = p (U2r) 5, 6, r = p (Uf) 4r + 5,4r + 6 (Uf) V4p+2Uj 4r + 7,4r + 8 (U2r) 4r + 1,4r + 2 (Uf) U4p+lVj 4r + 10,4r + 12, r = p (V2r) 6, 8, r = p (V2r) 4r + 5, 4r + 7 (Vxr) U4p+2Vj 4r + 6,4r + 8, r = p (V2r) 5, 7, r = p (Vf) 4r + 1, 4r + 3 (Vxr) Table 4: The edges of Gp+\ in Case 2. edges subscript XV4p+ l, V4p+ l Uj j = 4r - 3,4r - 2,4r - 1,4r, 7, 8,4p +2 (r = 3, 5, 7,... ,p) XU4p+ i ,U4p+ i Vj j = 4r - 3,4r - 2,4r - 1,4r, 5, 7,4p + 2 (r = 3, 5, 7, . . . , p) XV4p+2, V4p+2Uj j = 4r - 3,4r - 2,4r - 1,4r, 5, 6,4p + 1 (r = 1,4, 6, 8,... ,p - 1) XU4p+2,U4p+2Vj j = 4r - 3,4r - 2,4r - 1,4r, 6, 8,4p + 1 (r = 1,4, 6, 8,... ,p - 1) U 1V2, V4f_3U4r, U4rV4r_ i r = 1, 3,... ,p V4rU4r—3 , U4r_3V4r_2 r = 2,4,... ,p - 1 Uj Vj j = 1,... ,4p + 2 Case 3. When p < 3. When p = 0, K1j2j2 is a planar graph. When p = 1,2,3, we give a planar decomposition for K1j6,6, K1j10j10 and K1j14j14 with 2, 3 and 4 subgraphs respectively, as shown in Figure 5, Figure 6 and Figure 7. Figure 5: A planar decomposition of K16 6. Figure 6: A planar decomposition of K1j10j10. Lemma follows from Cases 1, 2 and 3. □ v4p + 1 X2 Figure 8: The graph Gp+1 in Case 1. Theorem 2.4. The thickness of the complete 3-partite graph Kiin,n is ' n + 2 ~ 4 Proof. When n = 4p, 4p + 3, the theorem follows from Lemma 2.2. When n = 4p + 1, n = 4p +2, from Lemma 2.3, we have 0(K1,4p+2,4p+2) < p +1. Since 0(^4^,4^) = p +1 and Kp,4p c #1,4^+1,4^+1 c #1,4^+2,4^+2, we obtain p +1 < 0(^1,4^+1,4^+1) < 0(#1,4p+2,4p+2) < p +1. Therefore, 0(KMp+1,4p+1) = 0(#1,4p+2,4p+2) = p + 1. Summarizing the above, the theorem is obtained. □ 3 The thickness of K2,n,n Lemma 3.1. There exists a planar decomposition of the complete 3-partite graph K2,4p+1,4p+1 (p > 0) with p + 1 subgraphs. Proof. Let (X, U, V) be the vertex partition of the complete 3-partite graph K2,n,n, in which X = {x1, x2}, U = {u1,..., un} and V = {v1,..., vn}. When n = 4p + 1, we will construct a planar decomposition of K2,4p+1,4p+1 with p +1 planar subgraphs. The construction is analogous to that in Lemma 2.3. Let {G1, G2,..., Gp+1} be a planar decomposition of K4p,4p given in [4]. In the following, for 1 < r < p + 1, by adding vertices x1, x2, u4p+1, v4p+1 to Gr, deleting some edges from Gr and adding some edges to Gr, we will get a new planar graph Gr such that {G?1,..., G?p+1} is a planar decomposition of K2,4p+1,4p+1. All the subscripts of vertices are taken modulo 4p, except that of u4p+1 and v4p+1 (the vertices we added to Gr). Case 1. When p is even and p > 2. (a) The construction for Gr , 1 < r < p. Step 1: When r is odd, place the vertex x1, x2 and u4p+1 in the face 1, 2 and 5 of Gr respectively. Delete edges v4r-3w4r and M4r-1v4r-2 from Gr. When r is even, place the vertex x1, x2 and u4p+1 in the face 3, 4 and 5 of Gr, respectively. Delete edge v4rw4r-3 and M4r-2v4r-1 from Gr. Step 2: Do parallel paths modifications, then join x1, x2, u4p+1 and v4p+1 to some Wj and vj, as shown in Table 5. (b) The construction for 3. (a) The construction for Gr, 1 < r < p. Step 1: When r is odd, place the vertex x^ x2 and u4p+1 in the face 1, 2 and 5 of Gr respectively. Delete edges v4r-3u4r and u4r-1v4r_2 from Gr. When r is even, place the vertex x1, x2 and u4p+1 in the face 3, 4 and 5 of Gr, respectively. Delete edge v4ru4r-3 and u4r_2v4r-1 from Gr. Step 2: Do parallel paths modifications, then join x1, x2, u4p+1 and v4p+1 to some uj and vj, as shown in Table 7. Table 7: The edges we add to Gr (1 < r < p) in Case 2. subscript \ case edge r is odd r is even Xl Uj 4r - 1, 4r 4r + 5, r = p (U[) 1, r = p (U2) 4r - 3, 4r - 2 4r + 8, r = p - 1 (U22) 8, r = p - 1 (U2) X1 Vj 4r - 3, 4r - 1 4r + 1,r = p (Vl2 ) 4r - 2, 4r 4r + 4 (V2) X2 Uj 4r - 1, 4r 4r + 3, r = p (U22) 8, r = p (U2) 4r - 3, 4r - 2 4r + 2 (Ul2) x2 vj 4r - 2, 4r 4r + 7,r = p (Vl2) 3, r = p (Vl2) 4r - 3, 4r - 1 4r + 6, r = p - 1 (V22) 6, r = p - 1 (V22) U4p+lVj 4r - 2,4r - 1 V4p+lUj 4r + 4,4r + 8, r = p (U22) 4, r = p (U2) 4r - 11,4r - 7 (Uf ) (b) The construction for Gp+1. We list the edges that belong to K2j4p+1j4p+1 but not to any Gr, 1 < r < p, as shown in Table 8. Then Gp+1 is the graph that consists of the edges in Table 8. Figure 10 shows Gp+1 is a planar graph. Therefore, |G?1,..., Gp+1} is a planar decomposition of K2j4p+1j4p+1 in this case. Case 3. When p < 3. When p = 0, K2j1j1 is a planar graph. When p = 1,2, 3, we give a planar decomposition for K2j5j5, K2j9j9 and K21313 with 2, 3 and 4 subgraphs respectively, as shown in Figure 11, Figure 12 and Figure 13. Summarizing Cases 1, 2 and 3, the lemma follows. □ Table 8: The edges of Gp+1 in Case 2. edges subscript Xl Uj j = 2,4r + 3,4r + 6,4p + 1 (r = 1, 3,... ,p - 2) Xi Vj j = 2, 4,4r + 3, 4r + 6,4p + 1 (r = 1, 3,... ,p - 2) X2 Uj j = 1, 2, 9,4r, 4r + 1,4p + 1 (r = 4,... ,p - 1) x2 vj j = 1, 8, 9,4r, 4r + 1,4p + 1 (r = 4,... ,p - 1) U4p+lVj j = 4r - 3,4r (r = 1, 2,...,p) V4p+lUj j = 4r - 2,4r - 1,4p - 7 (r = 1, 2,...,p) V4r-3U4r, V4r-2U4r_1 r = 1, 3,... ,p U4r-3V4r, U4r-2V4r-1 r = 2,4,... ,p - 1 Uj Vj j = 1,... , 4p + 1 Figure 10: The graph Gp+1 in Case 2. Figure 11: A planar decomposition K2,5,5. Figure 13: A planar decomposition of K21313. Theorem 3.2. The thickness of the complete 3-partite graph K2i„in is ~ n + 3 - 0(K2,n,n) = 4 Proof. When n = 4p, 4p + 3, from Lemma 2.2, the theorem holds. When n = 4p +1, from Lemma 3.1, we have 0(K2i4p+1j4p+1) < p +1. Since 0(K4p,4p) = p + 1 and K^p c K2,4p+i,4p+i, we have p + 1 = 0(K4p,4p) < 0(K2,4p+1,4p+i) < P +1. Therefore, 0(K2,4p+i,4p+i) = P +1. When n = 4p + 2, since K4p+3,4p+3 c K2j4p+2j4p+2, from Lemma 2.1, we have p + 2 = 0(K4p+3,4p+3) < 0(K2j4p+2j4p+2). On the other hand, it is easy to see 0(K2,4p+2,4p+2) < 0(K2,4p+1,4p+i) + 1 = p + 2, so we have 0(K2,4p+2,4p+2) = p + 2. Summarizing the above, the theorem is obtained. □ 4 The thickness of K1,1,n,n Theorem 4.1. The thickness of the complete 4-partite graph K1i1in,n is "n + 3- 9(Ki,i,n,n) = 4 Proof. When n = 4p + 1, we can get a planar decomposition for Ki i 4p+i 4p+i from that of K2j4p+ij4p+i as follows. (1) When p = 0, Kiiijiji is a planar graph, 0(Kiiijiji) = 1. When p = 1,2 and 3, we join the vertex x1 to x2 in the last planar subgraph in the planar decomposition for K2i5,5, K2j9j9 and K2,i3,i3 which was shown in Figure 11, 12 and 13. Then we get the planar decomposition for Ki i 5 5, Ki i 9 9 and Ki i i3 i3 with 2, 3 and 4 planar subgraphs respectively. (2) When p > 4, we join the vertex x1 to x2 in Gp+1 in the planar decomposition for K2j4p+ij4p+i which was constructed in Lemma 3.1. The Gp+1 is shown in Figure 8 or 10 according to p is even or odd. Because x1 and x2 lie on the boundary of the same face, we will get a planar graph by adding edge xix2 to Gp+1. Then a planar decomposition for Kiiij4p+ij4p+i with p + 1 planar subgraphs can be obtained. Summarizing (1) and (2), we have Kiiij4p+ij4p+i < p +1. On the other hand, from Lemma 2.1, we have 0(K4p+1j4p+1) = p +1. Due to K4p+1,4p+1 c Ki , i, 4p,4p c Ki ,i,4p+i,4p+i, we get p + 1 < 0(Ki,ii4pi4p) < 0(Ki,ii4p+ii4p+i). So we have 0(Ki,ii4pi4p) = 0 ( Ki, i ,4p+i ,4p+i) = p +1. When n = 4p + 3, from Theorem 3.2 , we have 0(K2j4p+2j4p+2) = p + 2. Since K2,4p+2,4p+2 c K1,1,4p+2,4p+2 c K1,1,4p+3,4p+3 c K1,1,4(p+1),4(p+1), and the ideas from the previous case establish, we have p + 2 < 0(Kiiij4p+2i4p+2) < 0(Kiiii4p+3j4p+3) < 0(Kijij4(p+i)j4(p+i)) = p + 2, which shows 0(Kiiij4p+2i4p+2 ) = 0(Kiiii4p+3j4p+3) = p + 2. Summarizing the above, the theorem follows. □ References [1] V. B. Alekseev and V. S. Goncakov, The thickness of an arbitrary complete graph, Mat. Sb. (N. S.) 101(143) (1976), 212-230, http://mi.mathnet.ru/eng/msb3 8 9 7. [2] L. W. Beineke and F. Harary, The thickness of the complete graph, Canad. J. Math. 17 (1965), 850-859, doi:10.4153/cjm-1965-084-2. [3] L. W. Beineke, F. Harary and J. W. Moon, On the thickness of the complete bipartite graph, Math. Proc. Cambridge Philos. Soc. 60 (1964), 1-5, doi:10.1017/s0305004100037385. [4] Y. Chen and X. Yin, The thickness of the Cartesian product of two graphs, Canad. Math. Bull. 59 (2016), 705-720, doi:10.4153/cmb-2016-020-1. [5] M. Kleinert, Die Dicke des n-dimensionalen Wurfel-Graphen, J. Comb. Theory 3 (1967), 1015, doi:10.1016/s0021-9800(67)80010-3. [6] T. Poranen, A simulated annealing algorithm for determining the thickness of a graph, Inform. Sci. 172 (2005), 155-172, doi:10.1016/j.ins.2004.02.029. [7] W. T. Tutte, The thickness of a graph, Indag. Math. (Proceedings) 66 (1963), 567-577, doi: 10.1016/s1385-7258(63)50055-9. [8] J. M. Vasak, The Thickness of the Complete Graph, Ph.D. thesis, University of Illinois at Urbana-Champaign, ProQuest Dissertations Publishing, 1976, https://search. proquest.com/docview/302820090. [9] Y. Yang, A note on the thickness of Kl>m,n, Ars Combin. 117 (2014), 349-351. [10] Y. Yang, Remarks on the thickness of Kn>n,n, Ars Math. Contemp. 12 (2017), 135-144, doi: 10.26493/1855-3974.823.068. /^creative ^commor ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 375-382 https://doi.org/10.26493/1855-3974.1228.d7d (Also available at http://amc-journal.eu) Touching perfect matchings and halving lines Micha A. Perles Institute of Mathematics, The Hebrew University, Jerusalem, Israel Horst Martini Fakultät fur Mathematik, Technische Universität Chemnitz, Germany Yaakov S. Kupitz Institute of Mathematics, The Hebrew University, Jerusalem, Israel Received 10 November 2016, accepted 4 March 2018, published online 10 July 2018 Abstract Let V be a set of 2m (1 < m < to) points in the plane. Two segments I, J with endpoints in V cross if relint I n relint J is a singleton. A (perfect) cross-matching M on V is a set of m segments with endpoints in V such that every two segments in M cross. A halving line of V is a line l spanned by two points of V such that each one of the two open half planes bounded by l contains fewer than m points of V. Pach and Solymosi proved that if V is in general position, then V admits a perfect cross-matching iff V has exactly m halving lines. The aim of this note is to extend this result to the general case (where V is unrestricted). Keywords: Bigraphs, cross-matching, halving lines, perfect matchings. Math. Subj. Class.: 05C62, 68R10, 52C35 1 Introduction, notions and main results Let V be a set of 2m distinct points in the plane R2 (1 < m < to). By a (perfect geometric) matching of V we mean a set M = [Ii,..., Im } of m non-degenerate closed line segments whose endpoints are (all) the points of V. The number of matchings of V is (2m - 1)!! = fi(2i - 1) = 2g . E-mail addresses: perles@math.huji.ac.il (Micha A. Perles), martini@mathematik.tu-chemnitz.de (Horst Martini), kupitz@math.huji.ac.il (Yaakov S. Kupitz) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ If V is in general position (no three points on a line), then two distinct segments I, J e M may be (a) disjoint (I n J = 0), (b) or they may cross, i.e., share a unique point that lies in the relative interior of both I and J. When V is unrestricted, two more possibilities arise. (c) The unique common point of I and J maybe an interior point of I and an endpoint of J (or vice versa). (d) If the four endpoints of I and J are collinear, then I and J may share a line segment. (This includes the possibility that I C relint J, or vice versa.) We shall say that two segments I, J touch if they have at least one point in common (In J = 0). We call M a simple matching (SM) if the segments of M are pairwise disjoint. It is well known and quite easy to show (see [2, Theorem 4.2]) that if V is in general position, then the number sm( V) of simple matchings on V is bounded from below by the m-th Catalan number Cm, i.e., Equality holds for m =1 or when V is the set of vertices of a convex 2m-gon. (It can be shown that if V is in general position but not in convex position, then sm(V) > Cm, with only one exception: when m = 3 and V consists of the vertices of a convex pentagon P plus a sixth point that lies in the interior of the pentagon formed by the diagonals of P.) Call M a cross-matching (CM) if each two distinct segments of M cross. Let us call M a touching matching (TM) if every two segments of M touch. 1.1 Halving lines Definition 1.1. A line L is a halving line of V if each of the two open half-planes L+, L-bounded by L contains fewer than m points of V. This clearly implies that |L n V | > 2, i.e., that the line L is spanned by V. When V is in general position, then necessarily |L n V| = 2, and |L- n V| = |L+ n V| = m — 1. When V is unrestricted we call L a halving line of order k if max(|L- n V |, |L+ n V |) = m — k (1 < k < m). In that case we may assume that, say, |L+ n V| = m — k, |L- n V| = m — k — e, and |L n V| = 2k + e, for some e, 0 < e < m — k. (See Figure 1.) 1.2 Halving lines and TMs If M is a TM on V, I is a segment of M, and L = aff I is the line spanned by I, then L is a halving line. Indeed, an open half-plane bounded by L contains no endpoint of I, and at most one endpoint of each other segment of M. The connection between the number h( V) of halving lines of V, and the existence of a cross-matching on V, in the case where V is in general position, was established by Pach and Solymosi in [3] as follows: They observed that each point of V lies on at least one halving line, hence h(V) > m. Then they found that either each point of V lies on just one (1.1) halving line, h( V) = m and V admits a unique CM, or at least one point of V lies on more than one halving line, h( V) > m, and V admits no CM at all. This result was generalized in [1] (see Theorem 1 and Corollary 3 there). In [4] we prove an extremal property of CMs, namely that if V admits a CM M, and M' is another (perfect) geometric matching on V, then the sum of the (Euclidean) lengths of the edges of M' is strictly less than the sum of the lengths of the edges of M. An analogous result holds for TMs. The geometric graph whose edges span (all) the halving lines of its vertex set V (with | V | even and V in general position) is said to be a bigraph. We refer to [5] regarding results on bigraphs. The aim of this note is to extend the result of [3] to arbitrary, unrestricted 2m-subsets V of R2. In the next section we define the notion of "a halving line at a point p e V", and show that a halving line of order k is a halving line at exactly 2k points. We also show that the number of halving lines at any point p e V is odd, hence > 1. The main results can be summarized as follows: Theorem 1.2. Suppose L1,... ,Lt (t = h(V)) are all the halving lines of V, with Li of order ki (1 < ki < m, i = 1,... ,t). If for each p e V there is just one halving line at p, then and V has no TM. In particular we have Corollary 1.3. The set V has a unique TM iff V has exactly m halving lines, each of order 1. The unique TM is a CM if each of the m halving lines contains just two points of V. i=1 and the number of TMs of V is precisely ncki!)- i= 1 If, for some p e V, there is more than one halving line at p, then i=i |V n L+1 = m - k ak ak-1 a1 b1 b2 be c1 c2 |V n L-1 = m - k - e Figure 1: A halving line of order k. 2 Proofs We start with the definition of "a halving line of V at p", where V is a set of 2m points in R2, and p G V. For a point p G V and a unit vector u = (u1,u2), denote by L(p, u) the directed line {p + Au : A G R}. (The direction is from small A to larger A.) Note that L(p, -u) is the same line, directed backwards. Define u+ = (-u2,u1), L(p,u)F = L(p,u) + : ^ > 0}, and L(p,u)B = L(p,u) + : ^ < 0}. F and B stand for "Front" and "Back", respectively. L(p, u)F and L(p, u)B are the two open half-planes bounded by L(p, u). Now move the unit vector u continuously on the unit circle in counterclockwise direction. Note that L(p, u) F and L(p, u)B switch when u is replaced by -u. As long as L(p, u) does not meet V \ {p}, we find that |V n L(p,u)F | + |V n L(p,u)B | = |V - {p}| = 2m - 1, and therefore one side of L(p, u) (the "major" side) contains at least m points of V, whereas the other side (the "minor" side) contains at most m - 1 points of V. As we change the direction u, the major side of L(p, u) will remain (Front or Back) as long as the rotating line L(p,u) does not meet V \ {p}. We call L(p,uo) a halving line of V at p if the major side of L(p,u) switches (from B to F or vice versa) as u passes through uo. Proposition 2.1. If L = L(p, uo) is a halving line of V at p, then L is a halving line of V. Proof. We must show that both open sides of L, L(p, uo)F and L(p, uo)B, contain fewer than m points of V each. If, say, | VnL(p, uo)F | > m, then VnL(p, u)F D VnL(p, uo)F, and therefore |V n L(p,u)F| > m, for all unit vectors u sufficiently close to uo, on both sides of uo, so the major side of L(p, u) does not switch at u = uo. □ Proposition 2.2. For each point p G V, the number of halving lines of V at p is odd (hence > 1). Proof. Choose an initial direction uo, such that V n L(p,uo) = {p}. Suppose the major side of L(p, uo) is, say, L(p, uo)F. Rotate the line throughp counterclockwise by 180°, i.e., move u along a semicircle, until we reach L(p, -uo). Now the major side is L(p, -uo)B (= L(p, uo)F). We conclude that on the way the major side switched (from F to B or vice versa) an odd number of times. □ Proposition 2.3. Suppose L is a halving line of V of order k (1 < k < m). Then L is a halving line of V at p for exactly 2k points of V. Proof. Assume, w.l.o.g., that |V n L-| = m - k - £, |V n L+| = m - k, and |V n L| = 2k + e, for some 0 < e < m - k. Label the points of V n L in order ak, ak-i, .. ., a1,b1, .. . ,b£,c1,. .., ck, as in Figure 1. Fix a point p G V n L, and consider a line that rotates counterclockwise through p. As the rotating line passes through the horizontal position (see Figure 1), the major side switches from Above to Below if p is one of the a»'s, and from Below to Above if p is one of the c's. But if p is one of the b»'s, then the major side remains Above (at least in a small neighborhood on both sides of the horizontal position). □ Next we show that if L is a halving line of V of order k, as in Figure 1, and M is a TM on V, then M matches the aj's with the c/s (and vice versa). Proposition 2.4. Suppose V = S U T is a partition of V into two sets of equal size (|S| = |T| = m), and conv S n conv T = 0. If M is a TM of V, then each segment I G M connects a point of S with a point of T. Proof. Assume, on the contrary, that some segment I G M has both endpoints in S. This leaves (at most) m - 2 points of S to be matched to points of T, and thus some other segment J G M has both endpoints in T. But then I n J C conv S n conv T = 0. □ Now look again at the halving line L in Figure 1. Define A = {a^ ..., ak}, B = jbi, ...,6J, C = {ci, ...,ck}, D_ = B U (V n L_) and D+ = V n L+ (|D_| = |D+| = m - k). Applying Proposition 2.4 twice, first with S = A U D_, T = C U D+, and then with S' = C U D_, T' = A U D+, we find: Proposition 2.5. If M is a TM of V, then each segment I G M with one endpoint in A has its other endpoint in C (and vice versa), and each segment J G M with one endpoint in D_ has its other endpoint in D+ (and vice versa). Note also that for any permutation 0 of {1,2,..., k}, the intersection of the k segments [a», c^)] (i = 1,..., k) is the segment [a1, c1], that connects the k'th point of V n L from the right with the k'th point of V n L from the left. We call this segment [a1, c1] the central segment of the halving line L. Suppose L1,..., Lt (t = h(V)) are all the halving lines of V, with L» of order k» for i = 1,... , t. For p G V, denote by h(p) the number of halving lines at p. In view of Propositions 2.1 - 2.3, we have 4 1 k» = 2 h(p) - m, i=1 pev with equality (= m) iff h(p) = 1 for all p G V. Proposition 2.6. If h(p) > 1 for some p G V, then there is no TM on V. Proof. Suppose, on the contrary, that V admits a TM M. Let I = [p, q] be a segment in M with one endpoint p. Let L, L' be two different halving lines of V at p (h(p) > 1). By Proposition 2.5 we have q G L n L'. But L n L' = {p}. □ Assume, from now on, that h(p) = 1 for all p G V. Thus J21=1 k» = m. In other words, on each line L» we can match two disjoint subsets of V n L», each of order k», A» (the kj "leftmost" points of V n L») and C» (the k» "rightmost" points of V n L»). L» is a halving line of Vat p iff p G A» U C». The sets A1, C1,..., At, Ct form a partition of V. As we have seen in Proposition 2.5, any TM of V will match the points of A» with those of C». There are k! ways to match A with Cj, and in each of these matchings, the intersection of the connecting segments is the "central segment" of the halving line L. To show that the individual TM's of A U Cj on L (i = 1,...,t) yield a TM of V, it suffices to show that the central segments of different halving lines L and Lj do meet (assuming, of course, that h(p) = 1 for all p G V). This will be done in the next proposition. Proposition 2.7. Suppose L is a halving line of V of order k, with V n L labelled ak,..., ai, bi,..., be, ci,..., ck as in Figure 1, A = {ak,..., ai}, C = {ci,..., ck}, and let L' be another halving line of V, of order k', with V n L' labelled similarly: a'k,,..., ai, bi, ..., b'E, ,ci,...,c'k/, A' = {a'k,, ...,ai}, C' = {ci,...,c'k,}. If the central segments [ai, ci] (of L) and [ai, ci] (of L') do not meet, then h(p) > 1 for some p G {ai, ci, ai, ci}. Proof. The two distinct lines L, L' cannot be parallel. If they are, and L' lies, say, above L, then the open side L+ of L includes the closed side cl L'+ of L', and therefore | V n L+1 > |V n cl L'+| > m, which is impossible. Let z be the crossing point of L and L', and suppose, w.l.o.g., that z misses the central segment [ai, ci] of L, and lies to the left of ai on L, see Figure 2. L' , L" L z / ai ci Figure 2: Proof of Proposition 2.7. Consider a directed line that rotates counter-clockwise through a1. As it passes through L (directed from left to right), the major side of V switches from Front to Back. As it reaches L'' (parallel to L'), or any direction sufficiently close to that of L', the major side of V is again Front, since the open half-plane to the left of L'' includes the closed half-plane to the left of L', which in turn contains at least m + k' points of V. Thus, there must have been another switch from Back to Front on the way, or, in other words, h(a1) > 1. □ 3 Algorithmic aspects The insights gained in the earlier sections of this note can be used to device an algorithm that decides whether a set P c R2 (|P| = 2m) admits a TM, and to find a TM (or all TMs) if one exists. The algorithm is conceptually simple, and seems to be also computationally quite effective, though not as efficient as the one proposed in [3] (m2 vs. m log m). Step 1: Find the point po = (x0, yo) G P that is the first in P with respect to the lexicographic order of points (x, y) G R2. p0 is a vertex of the convex hull [P] = conv P. Step 2: Calculate the slopes of the 2m - 1 segments [p0,p] (p G P \ {p0}), arrange them in non-decreasing order and find the median slope (this can be shared by several segments, of course). This slope determines the (unique) halving line L of P at p0. Find the number of points of P that lie below L, on L and above L, and order the points of P n L lexicographically. This enables us to determine the order k of the halving line L, and the sets A, C consisting of the first (resp. last) k points of P n L. These are the 2k points p € P n L such that L is a halving line at p. Erase these 2k points, and call the remaining set P' (|P= 2(m - k)). If P' = 0, stop. Otherwise, return to Step 1 with P replaced by P'. To see that this really works, we make the following observations: (A) If P admits a TM M, then M contains k segments (on L) that connect points of A with points of C. The rest of M is a TM of P' (= P \ (A U C)). Moreover, if L is any halving line of P other than L, of order k, then removal of A U C leaves L a halving line of P' of the same order k. This is clear when the central segments of L and of L meet at a point that is interior to the central segment [ai, ci] of L. In that case we lose k points on each side of L. The case when the common point of these two central segments is an endpoint, say a1, of [a1, c1], is shown in Figure 3. (The reason why C is included in L+ and not in Lis given below.) Figure 3: Two central segments whose common point is an endpoint in one of them. Since M matches P n L + with (P n L-) U B and a1 (ai G B and ai G A c P n L) with some point of C (Proposition 2.5), C c PnL+ (as in Figure 3). Thus, removing A U C will reduce |P n L+1 by k to (m - k) - P n B by 1 to e - 1 (> 0, since a1 G B?1), and |P n L-1 by k - 1 to (m - k) - k - (e - 1). (B) If M' is a TM of P', and N is a matching of A to C (on P n L), then M = M' U N is a TM of P iff the central segment [a1, c1] of L meets the central segment of each halving line of P'. Thus, if applying our algorithm to P' we find that P' has no TM, then the same holds for P. If P' does admit a TM, then P has a TM iff the central segment of L meets the central segment of each halving line of P'. To check this, we may need O(m2) operations. References [1] A. Andrzejak, B. Aronov, S. Har-Peled, R. Seidel and E. Welzl, Results on k-sets and j-facets via continuous motion, in: R. Janardan (ed.), Proceedings of the Fourteenth Annual Symposium on Computational Geometry, New York, NY, pp. 192-199, 1998, doi:10.1145/276884.276906, proceedings of the 14th ACM Symposium on Computational Geometry (SCG '98) held in Minneapolis, MN, June 07 - 10, 1998. [2] A. Garcia, M. Noy and J. Tejel, Lower bounds on the number of crossing-free subgraphs of KN, Comput. Geom. 16 (2000), 211-221, doi:10.1016/s0925-7721(00)00010-9. [3] J. Pach and J. Solymosi, Halving lines and perfect cross-matchings, in: B. Chazelle, J. E. Goodman and R. Pollack (eds.), Advances in Discrete and Computational Geometry, American Mathematical Society, Providence, RI, volume 223 of Contemporary Mathematics, pp. 245-249, 1999, doi:10.1090/conm/223/03141, proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Discrete and Computational Geometry: Ten Years Later held at Mount Holyoke College, South Hadley, MA, July 14- 18, 1996. [4] M. A. Perles, H. Martini, Y. S. Kupitz, H. Last and R. Pinchasi, Cross-matchings and circuits have maximal length, in preparation. [5] G. J. Simmons and R. C. Entringer, Some properties of components of bigraphs, Period. Math. Hungar. 3 (1973), 167-174, doi:10.1007/bf02018472. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 383-406 https://doi.org/10.26493/1855-3974.1282.378 (Also available at http://amc-journal.eu) Characterizing all graphs with 2-exceptional edges* Drago Bokalf Faculty of Natural Sciences and Mathematics, University ofMaribor, Slovenia Jesús Leaños * Academic Unit of Mathematics, Autonomous University of Zacatecas, Mexico Received 11 January 2017, accepted 20 June 2018, published online 6 Aúgúst 2018 Dirac and Shuster in 1954 exhibited a simple proof of Kuratowski theorem by showing that any 1-crossing-critical edge of G belongs to a Kuratowski subdivision of G. In 1983, Siran extended this result to any 2-crossing-critical edge e with endvertices b and c of a graph G with crossing number at least two, whenever no two blocks of G - b - c contain all its vertices. Calling an edge f of G k-exceptional whenever f is k-crossing-critical and it does not belong to any Kuratowski subgraph of G, he showed that simple 3-connected graphs with k-exceptional edges exist for any k > 6, and they exist even for arbitrarily large difference of cr(G) - cr(G - f). In 1991, Kochol constructed such examples for any k > 4, and commented that Siran's result holds for any simple graph. Examining the case when two blocks contain all the vertices of G - b - c, we show that graphs with k-exceptional edges exist for any k > 2, albeit not necessarily simple. We confirm that no such simple graphs with 2-exceptional edges exist by applying the techniques of the recent characterization of 2-crossing-critical graphs to explicitly describe the set of all graphs with 2-exceptional edges and noting they all contain parallel edges. In this context, the paper can be read as an accessible prelude to the characterization of 2-crossing-critical graphs. Keywords: Kuratowski subgraphs, crossing number, exceptional edges. Math. Subj. Class.: 05C10, 05C62 * Both authors would like to acknowledge the Crossing Number Workshop 2016inStrobl, Austria, where parts of this research took place. We deeply acknowledge the significant effort of the referee 1 for improving the clarity of some technical details of the arguments in our paper. tD. Bokal was partially supported by the Slovenian Research Agency projects L7-5459 and J1—8130. * Research started while on sabbatical leave at Maribor University. Partially supported by CONACyT Grant 179867 and by the grant Internationalisation of Slovene higher education within the framework of the Operational Programme for Human Resources Development 2007-2013. E-mail addresses: drago.bokal@um.si (Drago Bokal), jleanos@matematicas.reduaz.mx (Jesus Leafios) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ Abstract 1 Introduction The crossing number cr(G) of a graph G is the minimum number of pairwise crossings of edges in a drawing of G in the plane. An edge e of a graph G is said to be k-crossing-critical, if cr(G) > k > cr(G - e), and a graph is k-crossing-critical, if each its edge is k-crossing-critical. Therefore K3,3 and K5 are the only 3-connected 1-crossing-critical graphs. Any subdivision of K3,3 or K5 in G is called a Kuratowski subgraph of G and an edge e is a Kuratowski edge, if e belongs to a Kuratowski subgraph of G. Any edge of G which is not a Kuratowski edge, will be called a non-Kuratowski edge. Following [12], we call an edge e of G k-exceptional if e is k-crossing-critical and e is a non-Kuratowski edge. Note that the existence of a k-exceptional edge in G for k > 0 implies the existence of a Kuratowski subgraph, and hence that G is non-planar. Since loops are irrelevant for crossing number purposes, all graphs in this paper are loopless, but they may have multiple edges. In their simple proof of Kuratowski theorem from 1954, Dirac and Shuster established that any 1-crossing-critical edge e of a graph G belongs to a Kuratowski subdivision of G [6]. In 1983, Siran showed that the number of non-Kuratowski edges (and hence the number of exceptional edges) of a 3-connected simple non-planar graph of order at least 6 is at most 4 [13]. The following statement was exhibited in the same year. Statement 1.1 (Theorem 2 in [12]). Let e be a crossing-critical edge of a graph G, for which cr(G — e) < 1. Then e belongs to a Kuratowski subgraph of G. Figure 1: A minimal graph with two 2-exceptional edges. We have found a family of exceptions (see Figure 1) to Statement 1.1, i.e. a family of graphs with 2-exceptional edges. That such graphs exist was already exhibited by Kochol [8], who noted without proof that Sirffl's result may only be true for simple graphs. Closely investigating Siran's proof, it establishes [12] the following: Theorem 1.2 (Theorem 2 in [12]). Let e with endvertices b and c be a crossing-critical edge of a graph G for which cr(G — e) < 1. If no two blocks of G — b — c contain all its vertices, then e belongs to a Kuratowski subgraph of G. The correct statement indicates that the structure of graphs with 2-exceptional edges is limited, and the aim of the present paper is to characterize these graphs, i.e. to explicitly describe the family E of graphs with 2-exceptional edges. The rest of this paper is organized as follows. In the following section, we exhibit some known and new properties of Kuratowski edges in graphs, and offer their characterization in Theorem 2.6, as well as introduce our main result. In Section 3, we sketch our overall approach, which follows a simplified version of the recent characterizatizaton of 2-crossing-critical graphs [4]. The description of all 3-connected graphs with 2-exceptional edges is given in Section 4, along with the proof of the sufficiency direction of the characterization and some other properties of graphs with 2-exceptional edges. The remainder of the paper is devoted to proving necessity of the characterization of 3-connected graphs with 2-exceptional edges. A skeleton graph, the basic subgraph that is used to describe 3-connected graphs with exceptional 2-edges, is studied in Section 5. Bridges of the skeleton graph are studied in Section 6. Also the necessity of characterization is established there. We conclude with some corollaries bearing upon existence of k-exceptional edges and some open problems in Section 7. 2 Kuratowski edges First we introduce some notation, aligned with the notation of [4]. Any vertex of a graph G of degree at least 3 is called a node of G. A branch is a maximal path with no internal nodes connecting two nodes of G. Two distinct nodes u, v of a subdivision K of K33 are said to be independent if any u, v-path in K contains a node of K different from u and v, i.e. if there is no branch between them. Let A, B be either two subsets of V(G) or two subgraphs of G. Then, an A, B-path is a path with first end in A and last end in B that is internally disjoint from both A and B. When A = {s} and B = {t} are just vertices, we shorten the notation to just s, t-path. When the ends need to be emphasized, we write P = sPt = [sPt], the former emphasizing ends of P and the latter emphasizing that the complete path with ends is considered. When either end or both ends of P are removed from the path, we use P - t = [sPt), P - s = (sPt], and P - s - t = (sPt). We refer to these paths as the (semi) open paths P. Let G be a graph and H its subgraph. A path P is H-avoiding, if all the non-end vertices and all the edges of P are not in H. The ends of an H-avoiding path are allowed to be in H. In [13], J. Siran gave a characterization of Kuratowski edges in k-connected graphs with k > 3, cf. Lemmas 2.1 and 2.2. In this section, we extend his a characterization to any graph. Next two lemmas were proved in [13] and will be very useful for our purposes. Lemma 2.1 (Lemma 1 in [13]). Let K be a subdivision of K3,3 and let u, v be distinct vertices of K. Let K' := K + P be a graph obtained by joining u, v with a path P internally disjoint of K. Then any edge of P is Kuratowski edge of K' if and only if u, v are not independent nodes of K. Lemma 2.2 (Lemma 2 in [13]). Let G be a 3-connected non-planar graph. Let e = uv be an edge of G which belongs to no subdivision of K3,3 in G. Then u, v are independent vertices of any subdivision of K3,3 in G. Although [13] considered only simple graphs, it is easy to see that the two lemmas apply to multigraphs as well. Our first statement is an easy exercise. Lemma 2.3. Let G be a graph and let G' be a subdivision of G. Let e be an edge of G and let P be the path of G' obtained by subdividing e. The following are equivalent: (i) e is a Kuratowski edge of G, (ii) every edge of P is a Kuratowski edge of G', (iii) some edge of P is a Kuratowski edge of G'. The proof of our next result is essentially the same as that of Lemma 1 in [13]. Lemma 2.4. (i) Let K be a subdivision of K5 and let u,v be distinct vertices of K. Let K' := K + P be a graph obtained by joining u, v with a path P internally disjoint of K. Then any edge of P is a Kuratowski edge of K'. (ii) Let G be a 2-connected graph that contains a subdivision of K5 as a subgraph. Then every edge of G is a Kuratowski edge. Proof. For (i), one can easily check using the symmetry of K5 that there are exactly six homeomorphism classes of graphs to which K' can belong. In all of them, it is easy to find the required subdivision of K5 or K33 in K': if both u, v are on the same branch of K, a slight rerouting establishes the claim. In other cases we find a suitable K3,3 subdivision including a degree-three vertex in one part, and its nearest degree at least 3 vertices in the other part. For (ii), let e be an edge of G with ends u and v and let K be a subgraph of G, which is isomorphic to a subdivision of K5. If e G K, we are done. Thus, we may assume that e is not an edge of K. By Menger's theorem, G contains two disjoint paths P1 := uP1x1 and P2 := vP2x2 with xi, x2 G V(K) such that V(K) n Pi = {xi} for i = 1,2. Now by applying (i) to P := x1 P1uevP2x2 and K, we have that e is a Kuratowski edge. □ The following is immediate from the definition of exceptional edge, Kuratowski's theorem and Lemma 2.4. Lemma 2.5. If G is a 2-connected graph containing exceptional edges, then G contains at least one Kuratowski subgraph and every Kuratowski subgraph of G is a subdivision of K33. The above gives the following characterization of Kuratowski edges in general graphs: Theorem 2.6. Let G be a graph and let e be an edge of G. Then e is a Kuratowski edge of G if and only if G contains a Kuratowski subgraph K and a path P such that: (i) P contains e, (ii) P joins distinct vertices u, v of K, (iii) u, v are not independent nodes of K, and (iv) either P is contained in K or P is internally disjoint from K. Proof. The necessity part is immediate: if e is a Kuratowski edge of G, then e belongs to a Kuratowski subgraph K1 and K := K1, P := e satisfy conditions (i)-(iv). The sufficiency follows from previous lemmas: assume that G contains subgraphs K and P satisfying (i)-(iv), and apply Lemma 2.1 or 2.4 to the pair K,P depending on whether K is a subdivision of K33 or K5. □ This characterization is important due to the following corollary, which allows us to restrict ourselves to 2-connected graphs when characterizing all graphs with 2-exceptional edges: Corollary 2.7. Let G be a graph, let e be an edge of G and let B be the block of G containing e. Then e is a Kuratowski edge of G if and only if e is a Kuratowski edge of B. Following [12], for a (possibly empty) subset S of vertices of G, a pair (H, K) of subgraphs of G is an S-decomposition, if (i) each edge of G belongs to precisely one of H, K and (ii) H n K = S. For |S| = 0,1, let H+ = H, K+ = K, and for |S| = 2, let H+ (respectively K+) be obtained from H (respectively, K) by adding an edge between the two vertices of S. Lemma 2.8. Let G be a graph with a 2-exceptional edge and let (H, K) be an S-decomposition of G with |S| < 2. Then, precisely one of H+, K + is non-planar. Proof. If both are planar, so is G, a contradiction, so at least one is non-planar. Suppose that both are non-planar. As e is not on any Kuratowski graph of G, G - e has the same Kuratowski graphs as G. Let KH be a Kuratowski graph of H + and Kk a Kuratowski graph of K +. Since at most one branch of Kk and KH contains S, for each K g {Kk , Kh }, every drawing of K has a crossed edge that is not an edge of the graph in {Kk , Kh } \ {K}. This shows that any drawing of Kk U Kh has at least two crossings, implying that cr(G - e) > 2, a contradiction. □ For k = 0,1,2,3, let £k be the family of k-connected graphs that contain 2-exceptional edges. Our main theorem describes these sets, but recursive description of £2 \ £3 requires an additional lemma: Lemma 2.9 ([12]). Let (H, K) be an {u, v}-decomposition of G andsuppose that cr(H) = cr(H +). Then, cr(G) = cr(K + Auv) + cr(H), where A is the maximum number of edge-disjoint paths from u to v in H. Theorem 2.10. Let G be a graph that has a 2-exceptional edge e with endvertices b and c. Then 1. G g £0 \ £ 1 is disconnected, all but one of its components are planar, and the non-planar component belongs to £1. 2. G g £1 \ £2 is connected, but not 2-connected, all but one of its blocks are planar, and the non-planar block belongs to £2. 3. G g £2 \ £3 is obtained from a subdivision of G' g £3 by replacing its edge st of multiplicity p with a planar graph H containing vertices s and t, such that H + st is 2-connected and there are at least p edge-disjoint s, t-paths in H. 4. G g £3 is a cyclization of four tiles, as described in Theorem 4.2. Proof. Claims 1 and 2 follow from applying Lemma 2.8, with |S| = 0,1, respectively. As we defer the proof of Theorem 4.2 to the next sections, we only need to prove Claim 3. Suppose that (H, K) is a {u, v}-decomposition of G that has exceptional edges. By Lemma 2.8, we may assume H+ is planar. Then by Lemma 2.9, cr(G) = cr(K + Auv) and cr(G — e) = cr((K — e) + Auv). Therefore, e is exceptional in K + Auv. Applying this reduction to any (H, K) decomposition in which H has vertices not in K reduces G to a subdivision of a graph in E3. This reduction has a constructive counterpart: any edge f of a graph in E2 can be subdivided, yielding a graph of E2. If the original edge was exceptional, so are both new edges. Furthermore, if e = uv is not the only exceptional edge of G, then e can be replaced by any planar 2-connected H, for which H + uv is also planar. The resulting graph is again in E2. Moreover, if uv has multiplicity A, some of its edges can be replaced by H simultaneously, provided H has at least that many edge-disjoint u, v-paths. Thus, any graph of E2 can be obtained starting with a graph in E3, applying subdivisions and replacing edges by described planar graphs, proving Claim 3. □ 3 Tile decomposition method For clarity, we describe the structure of our characterization of graphs with 2-exceptional edges in this section. It will follow the ideas of recent characterization of 2-crossing-critical graphs [4]. The approach can be abstracted into the following steps, which allow to decompose an abstract graph with properties of interest into smaller pieces called tiles. The tiles are a tool often applied in the investigation of crossing critical graphs [2, 3, 4, 7, 9, 10, 11]. The method structures arguments as follows: 1. Limit connectivity of graphs of interest. In both instances, we focus on 3-connected graphs, showing how to obtain graphs of lower connectivity from these and identifying exceptional less connected instances. For us, this step is simple as all graphs fit the pattern; in characterization of 2-crossing-critical graphs, it involved analyzing the exceptional graphs. 2. Identify a skeleton graph K. In the case of 2-crossing-critical graphs, the skeleton graph K is V10. In our case, it is the graph K in which contracting the edge incident with two degree three vertices produces K5. 3. Study drawings or embeddings of the skeleton graph. In the case of 2-crossing-critical graphs, V10 has two nonhomeomorphic drawings in the plane, which turned to be better analyzed as two essentially different projective-planar embeddings. For a graph G with 2-exceptional edge e and the skeleton graph K, we show that in any optimal drawing of G - e, the subdrawing of K - e is determined up to homeomor-phism. 4. Find a skeleton graph H = K and its drawing/embedding that offers sufficient structure for finding tiles. Usually this amounts to a skeleton graph, for which in the selected embedding, all bridges lie in well-controlled faces. For 2-crossing-critical graphs, there were three steps (friendly embedding, pre-tidy V10, tidy V10). Showing existence of such embeddings and skeleton graphs turned to be an important step in both cases. After this step, a standard quadruple was introduced in both cases to carry the information about the investigated graph G, its selected drawing or embedding n, and the tidy skeleton graph K = H C G that were required for subsequent proofs. Once a special skeleton graph and its drawing are defined, introduce a standard labelling with respect to that skeleton and its drawing. 5. Restrict bridges of (parts of) the skeleton graph. In the case of 2-crossing-critical graphs, bridges of V10 are shown to be either edges or small stars, and their attachments are near in the V10 subdivision. In our case, we show that there exists a skeleton graph K, such that all its bridges that are not edges lie in the infinite face of some of its optimal drawings, and that after removing the K4 subgraph of the skeleton with its parallel edges, we (roughly) obtain a join of two 3-connected planar graphs. 6. Combine bridges into tiles. In the case of 2-crossing-critical graphs, this analysis relies on identifying types of edges in the V10 subdivision and then splitting pieces between two consecutive edges of a specific type. Our instance is simpler: we show that our bridges constitute a sequence of four tiles, whose cyclization yields a graph of interest. 7. Prove that every tiled structure yields a graph of interest. Once the structure is determined, this is usually an easy task, and to confirm intuition about the listed steps, it can even be done as soon as the tiles are conjectured. We conclude this section by introducing the needed notation related to tiles. As in our approach we do not need the gadgets limiting crossing numbers of tiles, only the very basics are needed. For most recent developments on the theory of tiles, see [3, 4]. A tile is a triple T = (G, A, p), where G is a graph and A, p are two disjoint sequences of distinct vertices of G, called the left and right wall of T, respectively. Two tiles T = (G, A, p) and T' = (G',A',p') are compatible, if |p| = |A'|. The join of two compatible tiles T and T' with p = (p1,..., pw) and A' = (A1...,AW) is defined as the tile T ( T' := (G'', A, p'), where G'' is the graph obtained from the disjoint union of G and G' by identifying pj with Ai, for i = 1,..., w. Specially, if pj = Aj is a vertex with precisely two neighbors (after the identification), we replace it with a single edge in G'' of multiplicity equal to the smaller of the multiplicities of the edges incident with pj = Ai. This technical detail is important when considering 3-connected graphs. Since the operation ( is associative, we can safely define the join of a compatible sequence of tiles T = (T0, T1,..., Tm) as (T = T0 ( T1 ( ... ( Tm. The cyclization of a self-compatible tile T = (G, A, p), denoted by o T, is the ordinary graph obtained from G by identifying Aj with pj for i = 1,..., w. The cyclization of a self-compatible sequence of tiles T = (T0, T\,..., Tm) is o T := o((T). Again, possible vertices with two neighbors are replaced with an edge maintaining smaller edge multiplicity, as above. We will also need the concept of a reversed tile of T, which is the tile with the two walls exchanged, T« = (G, p, A). 4 3-connected graphs with 2-exceptional edges In this section, we describe the class of graphs whose members are precisely all the 3-connected graphs containing at least one 2-exceptional edge. In particular, we define such a class and show that all its elements are 3-connected and have 2-exceptional edges. In the following sections, we show that any graph with 2-exceptional edges belongs to this class. For i > 1, let G1 = u^ be an edge of multiplicity i, and Oj = (G1, (uj), (vj)) a corresponding tile. Let O be the family of all tiles Oj, i > 1. For i, j > 1, let jG2 be the graph obtained by identifying the vertices vj and uj of Oj and Oj, respectively. Then lG\ has a vertex w of degree i + j and two vertices «j, vj of degree i, j, respectively, see Figure 2. By jQj = (jG^, («i), (vj, w)), we denote the tile constructed using jG^. Let si, s2, «3 and s4 be the vertices of K4. We use H = °H° ^ to denote the graph obtained from such a K4 by doubling the edges of the path s4s1s2s3 and adding to s3 and s4 a new edge leading to a new vertex w2 and w1, respectively (see Figure 2). The graph Hi = °Hj 1 is obtained from the disjoint union of H and G1 by identifying si of H with vi of G1, and letjHi = jHi,1 be the graph obtained from the disjoint union of G1 and Hi by identifying the vertex uj of G1 with s2 of Hi. Note that for i, j =0 the graph iHj is defined independently of G1, G"[, which only exist for i, j > 1. For k a positive integer, we denote by j Hlk the graph obtained from j Hi by increasing multiplicity of one of the edges s1s3 or s2s4 (but not both) to k. Finally, for l a positive integer, the graph j Hlk t is obtained from j Hlk by increasing the multiplicity of the edge S3S4 to l. For any integers i, j, k and l such that i, j > 0, and k, l > 1, we define a tile jRk ; = (jHk [, (ui, w2), (vj, w1)); for i, j = 0, we set u° = s1 and v° = s2, respectively. We use R to denote the family consisting of all the tiles j R®k t and all the tiles that can be obtained from these by arbitrarily increasing multiplicity of each edge on the path s4s1s2s3 (which must, however, remain at least two). Let P be the family that contains each tile T that can be obtained from any 3-connected planar G containing a degree three vertex x with neighbors u, v, w as T = (G - x, (u), (v, w)). In addition to these, let us assume that P also contain each tile iQj, with i, j > 1. A pre-exceptional sequence T of tiles has four tiles (T, T2, T3, T4), such that: (C1) T1 = Oil e O, (C2) T2 e P, (C3) T3 = i3 Rk3,13 eR, (C4) TT e P, and (C5) if T2 = i2 Qj2, then i3 > 1 (respectively, if T4T = i4 Qj4 then j3 > 1). Then there are exactly six types of pre-exceptional sequences: depending on whether T2 (respectively, T4) comes from a 3-connected planar graph, or T2 = i2 Qj2 (respectively, if T4t = i4Qj4), we have sixteen types of T's, which are reduced to six by considering (C5) and the symmetry. Such six types of T's are shown in Figure 3. The signature of a pre-exceptional sequence T = (I\, T2, T3, T4) is an ordered list of integers 2. The following technical observation is also needed in our proof. Lemma 4.1. Let G be a graph, and let {u, v, w} be a vertex cut of G such that {u, v, w} is not the neighborhood of a vertex in G. Let Gi be a non-trivial bridge of {u, v, w} in G, and obtain Gi by connecting a new vertex ti to each of u, v, w. Then, G is 3-connected, if and only if each of Gi is 3-connected. Proof. First we assume that G is 3-connected. Letp, q be any two vertices in Gi. If p = ti, choose as p any vertex of Gj for j = i. The three internally disjoint paths in G connecting p and q can be easily converted to three internally disjoint paths connecting ti and q. If both p and q are distinct from ti, there are three internally disjoint paths pP1q, pP2q, pP3q in G. At least two of these paths are in Gi, and if the third one is not, it uses two of the vertices u, v, w. We may assume it is pPuuP3vPv q. But then, pPuutivPv q is a path of Gi, internally disjoint from P1, P2, completing the necessity direction. For sufficiency direction, let p, q be two arbitrary vertices of G. If they are in the same Gi, there are three internally disjoint paths in Gi that connect them, which can easily be augmented to paths in G. If p e Gi, q e Gj, i = j, then let pPUputi, pPpvti, pPw wti be three internally disjoint paths between p and ti in Gi, and similarly qPuutj, qPqvtj, qPqwtj be three internally disjointpaths between q and tj in Gj. Then, pPpuPuq, pPpvPqq, pPp wPq q are three internally disjoint paths between p and q in G. □ Theorem 4.2. (I) A graph G is in E3, if and only if G can be described as a cyclization of an exceptional sequence T = (T1,T2,T3, T4). (II) G has two 2-exceptional edges if and only if k3 = 1 in T3 = 13 Rj l3. (III) If G € E0, then cr(G) = 2. Proof. We show Claims (II) and (III), and sufficiency of Claim (I). The necessity of Claim (I) is covered in the subsequent sections. Let G be a cyclization of T = (Ti, T2, T3, T4) as above. In this proof, we use the notation and drawings provided in Figure 3. Without loss of generality, we may assume that the edge of T3 with endvertices si and s3 has multiplicity k3. In order to show that G € E3, we need verify that it is 3-connected and that it contains 2-exceptional edges. From the construction of G and Lemma 4.1 it is not difficult to see that G is 3-connected. Thus it is enough to show that the edge e = s2s4 is a 2-exceptional edge of G. In other words, we need to show that: (11) cr(G - e) < 1, (12) e is not a Kuratowski edge of G, and (13) cr(G) > 2. For r € {2,4}, we assume that in any drawing of G under consideration, the restriction of such a drawing to Tr is a plane graph. Indeed, since there exists a drawing of Tr that has all the wall vertices on the same face, such a face can be made the infinite face by inversion and the resulting drawing or its mirror can be used to form the required drawing of G. Then, regardless the multiplicities of the edges in or T3, the drawings in Figure 3 imply that cr(G) < 2 and cr(G - e) < 1, which reduces (I1) to (I3). For (I2), seeking a contradiction, assume that e lies on some Kuratowski subgraph K of G. As G has exactly 4 vertices not in T2 U T4 (namely, si, s2, s3 and s4), then for some r € {2,4}, Tr contains at least one node of K. On the other hand, the planarity of Tr and the fact that Tr contains only three wall vertices imply that at least one node of K is not in Tr. Because Tr is joined to exactly three vertices of G - Tr, K is not homeomorphic to K5. Then K is homeomorphic to K3 3. Since any set of four edges with an end in Tr and the other in G - Tr contain at least one pair of parallel edges, the number of nodes of K in Tr must be exactly one. In particular, this implies that si, s2, s3, and s4 are nodes of K, and that each of T2 and T4 contains exactly one node of K. Since the node of K in T4 is joined to s2 and s4, these vertices belong to the same chromatic class in K, however, as e = s2s4 is an edge in K, s2 and s4 belong to distinct chromatic classes, a contradiction. Now we show that (I3) cr(G) > 2. We analyze separately two cases, depending on whether min p(T) =2 or min p(T) = 1. Case 1. min p(T) > 2. Let H be the graph that results by deleting from G all the vertices of T2 and T4 that are not in the face containing the wall vertices. Note that if T2 (respectively, T4) comes from a 3-connected planar graph, then T2 n H (respectively, T4 n H) is a cycle of length at least 3, and in the other case, T2 n H = T2 = i2 Qj2 (respectively, T4 n H = T4 = i4Qj4). Clearly, cr(G) > cr(H). Now we verify that cr(G) > 2 by showing that cr(H) > 2. Let D be an optimal drawing of H. As usual, we assume that parallel edges are placed very closely to each other in D, and hence, that they have the same number of crossings. Then if any edge of the path P := s4sis2s3 is crossed in D, we are done. Let h be an edge of H with endvertices si and s3. From Figure 3, we can see that H contains a subgraph J homeomorphic to K3,3 that avoids e and h (the thick edges). Indeed, the nodes of J are s1, s2, s3, s4 and p, q the endvertices of T1. If the restriction D[J] of D to J has at least two crossings, or at least one of e, h is crossed in D, we are done. Thus we assume that cr(D[J]) = 1 and that both e, h are clean in D. In particular, note that the restriction D' of D to the subgraph H' of H induced by s1, s2, s3 and s4 is a plane graph and that H' contains to K4 as subgraph. On the other hand, min p(T) > 2 implies that the number of parallel edges between p and q is at least 2, and hence both p and q are in the same face of D', or we are done. By using stereographic projection if necessary, we may assume that such a face is the infinite face of D'. Then exactly one vertex of H', say s', is in the triangular finite face formed by the other three vertices. Moreover, from the definition of J it follows that at least one of p or q is joined with s' by a path P', which is internally disjoint of H'. Since, for r = 1, 2, H contains at least two p, sr-paths edge disjoint and internally disjoint from H', then s' € {s1, s2}, or such p, sr-paths provide the required crossings. If s' € {s3, s4}, then P' crosses at least one edge of E(P) U {e, h}, which is impossible. Case 2. min p(T) = 1 and 13 > 2. Since T2 and T4 are connected and 13 > 2, then G contains a subgraph H which is homeomorphic to the graph shown in Figure 4. As before, we verify cr(G) > 2 by showing that cr(H) > 2. Let C be the double cycle of H whose vertices are s1, s2, s3, and s4, and let D be an optimal drawing of H. If any edge of C is crossed, we are done. Then we may assume that the restriction D[C] of D looks like in Figure 4. Then u and v must be in the same face of D[C]: otherwise, at least one edge of C is crossed by the edge with endvertices u and v and we are done. Without loss of generality, we assume that both are in the infinite face of D[C], as shown in Figure 4. Note that the paths s1us3 and s2vs4 cross each other because they have alternating ends in C. Similarly, if the edges s1s3 and s2s4 are in the same face of C, we have the required crossing. Then at least one of them is in the infinite face of C and such an edge must cross with some of s1us3 or s2vs4 providing the required crossing. This proves (I3) and hence sufficiency of (I). The inequality in (I3) was independently checked with the crossing number computing tool of Chimani et al. [5] SI s2 Figure 4: A drawing of H. Now we show (II) that G has two 2-exceptional edges if and only if k3 = 1 in T3 = 3 Rfca h. Note that, by symmetry, the argument used in (I2) also shows that any edge of G with endvetices s1 and s3 is a not a Kuratowski edge. Let us denote by K3 the set of edges of G with ends s1 and s3. On the other hand, from the definition ofi3 Rj t , we know that e is the only edge of G with ends s2 and s4. Then Lemmas 2.2 and 2.5 imply that s1 and s3 (respectively, s2 and v u s4) are in the same chromatic class of nodes of a subdivision K of K3,3 in G. We derive a contradiction from the assumption that G contains a non-Kuratowski edge e' G K3 U {e}. Then Lemma 2.2 implies that e' joins two nodes in the same chromatic class of nodes of K. Furthermore, since e' G K3 U {e}, then it must have an end in {si, s2, s3, s4} and the other in a node of K \ {si, s2, s3, s4}. The existence of such an e' implies that K U K3 U {e, e'} C G contains K5 as subdivision. This and Lemma 2.5 imply that all the edges of G are Kuratowski edges, a contradiction. Let us assume that k3 = 1 in T3 = i3 R^,3,1 . Then K3 consists of an edge h. From (I1) and (I3) we have cr(G) = 2. Now, if we draw e inside of the square s1s2s3s4 in Figure 3, we get, in all the cases, an optimal drawing of G in which h is crossed by e. This proves that h is 2-crossing-critical, and hence e and h are both 2-exceptional edges. On the other hand, since cr(G) = 2 for any k3 > 1, then if k3 > 2 we have that no edge in K3 is 2-crossing-critical and since K3 U {e} are the only non-Kuratowski edges of G, then k3 > 1 implies that e is the only 2-exceptional edge of G. This proves (II). Finally, we show (III) that if G G Eo, then cr(G) = 2. (1) If G g E3 we are done by (I1) and (I3). (2) If G G E2 \ E3 then, by Theorem 2.10(3), there exists G' G E3 such that cr(G) = cr(G'). Since cr(G') = 2, we are done. (3) If G G E1 \E2 then, by Theorem 2.10(2), all but one of blocks of G, say B, are planar and B g E2. Then cr(G) = cr(B). If B G E3 (respectively B G E2 \ E3) we are done by (1) (respectively (2)). (4) If G G E0 \ E1 then, by Theorem 2.10(1), all but one of the components of G, say C, are planar and C g E1. Then cr(G) = cr(C). Clearly, exactly one of the following is true: C g E3, C g E2 \ E3, or C g E1 \ E2. Note that these three cases have been studied, respectively, in (1), (2), and (3), and in all of them the conclusion is cr(C) = 2, as required. □ 5 The skeleton graph In this section, we present the skeleton graph, which is the essential structure of 3-connected graphs with 2-exceptional edges. First, we introduce some notation, aligned with the notation of [4]. Let H be a subdivision of a graph G and let e be an edge of G. If s and t are the ends of e, then we denote by sHt the s, t-path of H which results from subdividing e. We use vst to denote an arbitrary, but fixed, vertex of (sHt). Following this general notation, we turn our attention to the specific graph K'', which we show to constitute the skeleton of graphs in E3. It is depicted in Figure 5. We always use the labelling from the figure (and we call it standard labelling), so {{a, b, c}, {x, y, z}} constitute the bipartition of a subdivision K = K3 3, and bc, yz are the exceptional edges of K''. We will use K' for K + bc, and refer to it as apre-skeleton. A bypass of a non-Kuratowski edge e of K'' is the union of any two K3 3 branches that together with e form a cycle containing exactly 3 nodes of K''. A bypass is open, if the endvertices of e are removed from it and closed if they are contained in it. The common node t G {a, b, c, x, y, z} of the K'' branches used in the bypass is the peak of the bypass, and we denote the bypass by Kt. For instance, Kx = bK''xK''c and Kb = yK''bK''z. a b c Figure 5: The skeleton graph K''. We will be vague by using Kt both for open and closed bypass, but where distinction will be required, (Kt) is open and [Kt] is closed. Besides bypasses, claws at a and x will play a significant role. We define them by Da := aK'x U aK'y U aK'z and Dx := xK'a U xK'b U xK'c. A talon of a claw is its one degree vertex. A claw is open, if we remove its talons. Again, we will use [Da] and [Dx] for closed, and (Da), (Dx) for open claws. The graph K4' := K'' \ ((Da) U (Dx)) is a subdivision of K4. When H = K'' is a subdivision of K'', we extend the definition of bypasses and claws naturally to H. The next lemmas restrict the possible bridges of a skeleton graph in G. The first one shows that graphs in E3 do not contain a subdivision of a graph, obtained from K' by adding a path with ends in two distinct bypasses, except for three exceptions. Lemma 5.1. Let H := K' + P, where P is a path joining two distinct elements of {(Kx), (Ky), (Kz)} and internally disjoint from K'. Then every edge of H is a Kura-towski edge, or P joins distinct vertices of {x, y, z}. Proof. By Lemma 2.3, we may assume that H has no vertices of degree 2. In particular, P is an edge. Assume that P does not join distinct vertices of {x, y, z}. Let q and r be the endvertices of P. As (Kx), (Ky), and (Kz) are open, P does not join two vertices of {a, b, c}. By Lemma 2.1, we have that all the edges in H except bc are Kuratowski edges. It remains to show that in each case, bc belongs to a subdivision of K3 3. By the symmetry of K', we need only analyze the cases in which q G {y, vyb} and r G {vbz, vcz}. If q = y and r = vbz, then H \ {by, cx} is the required subdivision. If q = y and r = vcz, then H \ {bx, cy} is the required subdivision. If q = vyb and r = vbz, then H \ {cx, bvbz} is the required subdivision. If q = vyb and r = vcz, then H \ {x} is the required subdivision. □ The next lemma restricts paths adjacent to paths linking two nodes of K. Lemma 5.2. Let H := K' + P + Q, where P is a path joining two distinct elements of {x, y, z} and internally disjoint from K', and Q is a path joining an inner vertex p of P with a vertex q G V (K') \ V ((Da)) and internally disjoint from K' + P. Then every edge of H is a Kuratowski edge, or cr(H — bc) > 2, or q G P. Proof. Without loss of generality, we may assume P = yPz. By Lemma 2.3, we may assume that H has no vertices of degree 2. Lemma 2.1 implies that all the edges in H except bc and possibly edges of P U Q are Kuratowski edges. Since q G V((Da)) and H has no vertices of degree 2, then q is a node of H distinct from a. If q G {y, z}, we are done. If q = x, then (K' — bc) U P U Q is a subdivision of K3,4, and cr(H) > 2. So we assume that q G {b, c}. By symmetry, we may assume that q = b. In this case, H \ {cx, by, bz} is a subdivision of K3,3 that uses the edge bc and all edges of P U Q, concluding the proof. □ Lemma 5.1 implies the following useful structure of optimal drawings of G — e: Lemma 5.3. Let G G E3 and let e be its 2-exceptional edge with endvertices b and c, and let K3,3 = K C G. If D is an optimal drawing of G — e and DK is the induced subdrawing of K, then the ends of e lie on a face of DK that is not incident with its crossing. Proof. By Lemma 2.2, b, c are independent nodes of K, so they are on the boundary of some (possibly different) face(s) of DK. Up to homeomorphism, DK is drawn in Figure 6. The parts in the bipartition of K3,3 are {1,3,5} and {2,4,6}. Any pair of independent nodes of K lies on a common face of DK, and Ei, E2 are the only faces contradicting the conclusion of Lemma 5.3. 2 1 Figure 6: The unique drawing of K3,3, up to homeomorphisms. By symmetry, we may assume b, c lie in Ei, implying {b, c} = {2, 6}. As cr(G - e) < 1, the crossing of DK is the only crossing of D. As cr(G) > 2, there is an arc in D connecting the two segments of the boundary of Ei having b, c as ends. As this path avoids the only crossing of D, it is a path in G - e that connects two distinct open bc-bypasses, and at least one of its endvertices is not a node of K, contradicting Lemma 5.1. Therefore, b, c lie either in Oi or O2, and neither of these is incident with the crossing of D, as claimed. □ In the analysis, we use the following result from [12]. We also repeat some notation. Lemma 5.4 (Lemma 3 in [12]). Let G be a 3-connected non-planar graph, and let e be a non-Kuratowski edge of G with endvertices b and c. Then the graph G/e is 2-connected but not 3-connected, and the graph G - b - c is connected, but not 2-connected. Let H := G - b - c and let T(H) be the block-cutpoint tree of the graph H. Lemma 5.4 implies that T(H) is a non-trivial tree. According to Theorem 1.2, G - b - c has all vertices in two blocks for any graph G with 2-exceptional edge bc. By now, we are ready to establish that the pre-skeleton graph is a subdivision contained in any graph with a 2-exceptional edge. Theorem 5.5. Let G G E3 with e its 2-exceptional edge with endvertices b and c. There exist a pre-skeleton subgraph H with K' = H C G and an edge f of G, such that H + f is a subdivision of the skeleton graph K''. Proof. By Lemma 2.5, G has a subdivision K = K3,3. As e is not a Kuratowski edge, e is not in K. Let uPv be any maximal K-avoiding path containing e. As G is 3-connected, Theorem 2.6 implies that u, v are distinct nodes of K; we choose the standard labelling of K such that {u, v} = {b, c}. As G is 3-connected, P is either an edge, or there exists a K + P-avoiding path pQq connecting a vertex of p G (P) with a vertex q G G \ V (P). One of the paths bPQq and cPQq contains e, hence Theorem 2.6 applied to it implies q = a. Then, K U P U Q is a subdivision of K34 containing e, a contradiction to e not being a Kuratowski edge. Hence Q does not exist and bPc is just a single edge, showing that H := K + e is a pre-skeleton in G. Next we prove that there exists a K'-avoiding path Q, connecting two nodes from {x, y, z}. We may be forced to change K' for this. Claim 5.6. There exists a pre-skeleton subdivision K' in G, such that K' n K' contains the closed bypasses of e (in particular, b, c, x, y, z are nodes of K'), and there is an K'-avoiding path P of G connecting q, r G {x, y, z}. Proof. As there are two blocks of H = G - b - c containing all its vertices, at least two vertices p, q of {x, y, z} are in the same block B of H. We may assume without loss of generality that they are y and z. As B is 2-connected, there are two internally disjoint paths yPiz and yP2z in B. By Lemma 5.1, the intersection of Pi U P2 with Kx U Ky U Kz is contained in {x, y, z}. Suppose that x G P1 U P2 .If either P1 n (Da) or P2 n (Da) is empty, it is the required path and K' = K'. So we may assume they are both non-empty. Let a' be a vertex of P1 U P2, such that xDaa' has no vertex of P1 U P2. We may assume a' G P^ As (P1 U a'Dax) n K' is contained in [Da], K' := (K' - (Da)) U (P1 U a'DaaDax) is the required skeleton and P2 is the required K'-avoiding y, z-path. Now we may assume x G P1 U P2. Then x, y, z split C := P1 U P2 into three arcs Cxy := xCy, Cyz := yCz, and Czx := zCx, such that C = xCyyCzzCx. Let ax, ay, az be the a-closest vertices of P1 U P2 in xDaa, yDaa, zDaa, respectively; they may all be equal to a. If each segment of Cxy, Cyz, Czx contains a vertex of ax, ay, az, then let a'' be the one of ax, ay, az in Cyz and let a' be any other one. Then, C U (([axDaa] U [ayDaa] U [azDaa]) - (aDaa'') - (Cyz)) contains an x, y, z-claw T with center a' and is internally disjoint from Cyz. Hence, K' = (K' - (Da)) U T is the required pre-skeleton and Cyz the y, z-path internally disjoint from K'. If a segment Cxy, Cyz, Czx contains two vertices of ax, ay, az, and a segment C0 contains none, we relabel {x, y, z}, so that Cyz = C0. Then Da U P1 U P2 - (Cyz) contains a claw T with center a and talons x, y, z so that Cyz is internally disjoint from it; again, K' = (K' - (Da)) U T is the required pre-skeleton and Cyz the y, z-path internally disjoint from ^'. If a segment C3 of Cxy, Cyz, Czx contains all three vertices of ax, ay, az, then in C U Da, there is a C-avoiding path pRq from (C3) to C - [C3]. We relabel {x, y, z}, so that p G Cxy and q G Cxz. Then, (C UpRq) - (Cyz) contains an x, y, z-claw T, with center in p, so that Cyz is internally disjoint from it; again, K' = (K' - (Da)) U T is the required pre-skeleton and Cyz the y, z-path internally disjoint from K'. □ Without loss of generality, we label the nodes of K so that P = yPz. Claim 5.7. The path yPz from Claim 5.6 is an edge. Proof. Seeking a contradiction, assume that P has an internal vertex v. Consider an optimal drawing D of G - e. Since K c G - e and cr(G - e) < 1, we have cr(G - e) = 1. Thus the drawing D restricted to K is homeomorphic to the drawing DK in Figure 6. Because G is 3-connected, G - y - z contains a path Q from v to K - y - z, which is internally disjoint from K U P .If q is the endvertex of Q in K, Lemma 5.2 implies q G V (Da). Since the crossing d of DK is the only crossing of D, no edge of P U Q is crossed in D. Hence P is drawn in a face of DK incident with two independent nodes. By the symmetry of Dk, we may assume that v G Ei or v G Oi. See Figure 6. If v G Oi, then {y, z} = {4, 6} or {y, z} = {1,5}. By the symmetry of DK, we may assume {y, z} = {4,6}, and hence x = 2. This implies that a =1 or a = 5. If a = 5, then {b, c} = {1,3}. Since cr(G - e) < cr(G), then there must be a simple arc a of D contained in O2, with endpoints on its boundary and separating b from c (1 from 3 in Figure 6). Since d is the only crossing of D, a corresponds to a path R of G which joins two vertices of V(K') \ V(Da). Lemma 5.1 implies R joins x and z. Now it is easy to see that K' U P U R contains a subdivision of K5, contradicting Lemma 2.4. For the final case, v G Ei. Then, without loss of generality, P connects y = 2 and z = 6, implying x = 4. As q is on some path in the boundary of Ei, a can be any of the vertices 1, 3, or 5. Suppose a =1. This implies bc = 35, contradicting Lemma 5.3. Suppose next a = 3, implying bc = 15. As cr(G) > 2, there is an arc in D separating 1 from 5 in Oi. By Lemma 5.1 and as there is only one crossing in D, this arc is a path R from 4 to 6. As K' U P U R has a subdivision of K5, it contradicts Lemma 2.4. The subcase a = 5 is similar, with bc = 13 and 2R4. □ Thus f is an edge connecting y and z, and K'Uf is a subdivision of K'', as claimed. □ Proposition 5.8. Let G G £3 and let K'' = H C G be its skeleton subgraph with standard labelling. Then G does not contain a path P internally disjoint from K'' with endvertices in any of the pairs {a, b}, {a, c}, {x, y}, {x, z}. Proof. Let u, v be the endvertices of P. If {u, v} G {{a, b}, {a, c}, {x, y}, {x, z}} then the subgraph (K'' + P) c G contains K5 as subdivision. This and Lemma 2.4 imply that G has no exceptional edges, a contradiction. □ Corollary 5.9. Let G G £3. Any non-Kuratowski edge g of G is parallel to e or f. Proof. Let u, v be the endvertices of g. By Lemma 2.2, we know that u, v are independent vertices of K and by Proposition 5.8, we have that {u, v} = {b, c} or {u, v} = {y, z}. □ Let e and f be the exceptional edges of G g£3. The graph obtained from G by adding a parallel edge f' to f is also an exception to Statement 1.1, but such a graph contains only e as an exceptional edge, because both edges f and f' are non-critical. These observations yield the following corollary to Theorem 4.2: Corollary 5.10. Let G G £3. The number of 2-exceptional edges of G is at most two. Proof. By Corollary 5.9, it is enough to show that if f and f' are parallel edges of G, then they are not critical. Suppose an arbitrary of them is and let it be f. Then cr(G - f) < 1. As K C G - f, there exists an optimal drawing D of G - f in which f' G K has no crossings. By drawing f very close to f' in D, we get a drawing of G with exactly one crossing, a contradiction implying that f is not critical and hence not exceptional. □ 6 Bridges of the skeleton graph Let H be a subgraph of a graph G. An H-bridge is either an edge not in H together with its two incident vertices that are in H or is obtained from a component J of G - V(H) by adding all edges incident with a vertex of J together with their incident vertices in H. This concept will be helpful for the remainder of this section. A bridge is trivial, if it is just an edge, and non-trivial otherwise. For a graph H and its bridge B, any vertex of att(B) := V(H) n V(B) is an attachment of B. First we exhibit the structure of an optimal drawing of G - e. Lemma 6.1. Let G G E3, let K'' be its skeleton graph, let e be its 2-exceptional edge with endvertices b and c, and let f be a non-Kuratowski edge not parallel to e. If D is an optimal drawing of G - e, then the drawing D restricted to K + f is homeomorphic to the drawing in Figure 7 (right) and b, c are the ends of e. Proof. Let D be an optimal drawing of G - e and K the K3 3 subdivision in K''. As e is a 2-exceptional edge, D has a unique crossing and D restricted to K is homeomorphic to the drawing DK in Figure 7 (left). Using symmetry, stereographic projection, and Lemma 5.3, we may assume that the ends of e are b and c. Hence, f G {xy, yz, xz}. If f G {xy, xz}, cr(G) > 2 implies there is a path P of G - e that is by Lemma 5.1 drawn from y to z in E, yielding a K5 subdivision in K'' U P, contradicting Lemma 2.4. Thus y and z are the ends of f and the drawing D restricted to K + f is homeomorphic to the drawing in Figure 7 (right), as required. □ b b x x Figure 7: If D is an optimal drawing of G - e, then the drawing D restricted to K + f is homeomorphic to the right drawing. In what follows, we call (G, H, e, D) a standard quadruple, abbreviated sq, if G G £3, K" = H C G, such that H has standard labelling, e is a 2-exceptional edge of G, and D is an optimal drawing of G - e, with the induced subdrawing of H - e drawn as in Figure 7. Lemma 6.2. Let G G £3. Then, there exists a standard quadruple (G, H, e, D) containing G. Proof. As G € £3, there exists a 2-exceptional edge e of G. Theorem 5.5 guarantees existence of the skeleton graph H in G, that is a subdivision of K''. Finally, Lemma 6.1 yields existence of the desired drawing D of G - e. □ Lemma 6.3. Let G € £3, let K'' = H C G with the standard labelling. If B is a bridge of H, v € att(B), and v is not a node of H, then att(B) C [Da] U [Dx]. Proof. Let v be as in the statement. Seeking a contradiction, suppose that there exists u € att(B) \ ([Da] U [Dx]). Then u € (Ky) U (Kz) (or equivalently, u € (K6) U (Kc)). Since B is connected, it contains an u, v-path, say P. From Lemma 5.1 and the hypothesis that v is not a node of H, we have that v € (aHx). By the symmetry of K4, we need only analyze the case in which u = vby. But in such a case, (H U P) \ {ay, bx, cz} is a subdivision of K3 3 containing both bc and yz as edges, which contradicts that bc is a 2-exceptional edge. □ Let G € £3. In what follows, we will denote with H4 as the subgraph of G induced by the four vertices that are ends of the non-Kuratowski edges of G. By Corollary 5.9, H4 is well-defined for any G, i.e. it is independent of the choice of H. Lemma 6.4. Let (G, H, e, D) be a standard quadruple of a graph G € £3. The subgraph H4 of G is isomorphic to K4 with some multiple edges, and it has only one bridge that contains both a and x and the only crossing of D. Proof. As G is 3-connected and H4 is induced in G, H4 has no trivial bridges. As there exists an H4-avoiding path aHx, a and x are in the same H4-bridge B, and that bridge is crossed in D. If B is the only bridge, we are done, otherwise let B' be any other bridge. As G is 3-connected, each of B, B' has at least three attachments. As e € B' and the only crossing of D is in B, B' is drawn planarly in D. Then D implies the attachments of B' are either b, y, z or c, y, z, both contradicting Lemma 5.2. Let {u, v} € {{c, z}, {b, z}, {b, y}, {c, y}}. Now consider the branch of H connecting u to v. If such branch is not an edge, then it has one internal vertex, say w. Using the 3-connectivity, the drawing D, and the fact that B is the only bridge in H4, we know that there is a path, that is internally disjoint from H, connecting w to a vertex in (Da) U (Dx). However, no such path exists by Lemma 6.3. □ For a graph G € £3, we will denote the only bridge of its graph H4 by B4. Lemma 6.5. Let K'' = H C G and let P be a path from u € (Ka) to v € [Kx] with {u, v} = {a, x}, and internally disjoint from H. Then every edge of H + P is a Kuratowski edge. The claim also holds with the role of a and x exchanged. Proof. By Lemma 2.3, we may assume that H has no vertices of degree 2. It is enough to show that in each case H has a subdivision of K3 3 or K5 containing either bc or yz. By the symmetry of H, we need only analyze the following cases; the same arguments also show the claim with the role of a and x interchanged: If u = vaz and v = vcx, (H U P) \ {by, cz, xHv} is a subdivision of K3,3. If u = vaz and v = c, (H U P) \ {xc, cz, by} is a subdivision of K3,3. If u = a and v = vcx, (H U P) \ {xv} is a subdivision of K5. If u = a and v = c, (H U P) — xc is a subdivision of K5. □ Lemma 6.6. Let (G, H, e, D) be a standard quadruple of a graph G G £3. If P is a b, y-path of length at least 2 contained in B4, then P intersects [aHx]. Proof. Seeking a contradiction suppose that P n [aHx] = 0. By Lemma 6.5 and our hypothesis, at least one of P n (Ka) or P n (Kx) is empty. By symmetry, we may assume that P n (Ka) = 0. If P n (Kx) is also empty, then P is internally disjoint from H. Then, (H — (by)) U P is a different choice of H for G whose structure contradicts Lemma 6.4, as the H4 produced by this H has a subdivided edge by. Hence, P n (Kx) is nonempty. Lemma 6.1 and disjointness of P from (Da) + x imply that there is a H-bridge B with attachments in y and (xHb]. By the previous paragraph, at least one attachment is in (xHb). However, any path in B from y to (xHb) contradicts Lemma 6.5, concluding the proof. □ Lemma 6.7. Let (G, H, e, D) be a standard quadruple of a graph G G £3. Then, there exist vertices vc and vz in B4, such that vcc and vzz are the only attaching edges of B4 at c and z. Moreover, these edges are crossed in D. Proof. We show the claim for vcc, the claim for vzz is analogous. Let x be the crossing of D. By Lemma 6.5, there is no H-avoiding path in B4 from (xHc) to [Da] avoiding x. Let F be any face of D incident with the segment (x Hc). As V(dF) C V([Da]) U V((xHc)), existence of a vertex in (xHc) would contradict 3-connectivity of G. Hence, (xHc) lies on some edge cvc of B4, and cvc is crossed in D. Analogously, we can conclude that (x Hz) lies on some edge vzz of B4, and hence cvc and vzz are the only two crossing edges of D. By Lemma 6.5 we know that G does not have a path internally disjoint from H, with an end in c and the other end belonging to (Ka). Thus, the existence of any other edge of B4 attaching at c, together with the location of c in D imply that at least one of vz z or zy is crossed by some edge in B4 — cvc, contradicting that cr(D) = 1. □ Lemma 6.8. Let (G, H, e, D) be a standard quadruple of a graph G G £3. There exists K" ^ h' C G and an optimal drawing D' of G — e, such that a'z and cx' are edges of H', and any face incident with the crossing of D' has no bridges of H' drawn in it. Proof. By Lemma 6.7, there exist vertices vc and vz, such that cvc and zvz are edges of G. Moreover, cvc crosses zvz in D and such a crossing x is the only crossing of D. Let D'' be the subdrawing of D, induced by G — c — z. Since x is the only crossing of D, then b, vc, vz and y lie in the same face F of D''. Note that F contains (in the interior) vertices c and z. By Lemma 6.1, the boundary walk dF can be decomposed into bPivcP2vz P3yb. Note that if some P G {P1, P2, P3 } is not a path, then P must have a cut vertex, say w. By Lemma 6.7 we know that vc, y and vz, b are the only vertices of B4 adjacent to c and z, respectively. From this and the supposition that w is a cut vertex of P it follows that w is a cut vertex of G — e, which contradicts the connectivity of G — e. Thus we can assume that P1, P2 and P3 are paths. Define H' = H — (Da) — (Dx) U [bP1vcP2vzP3y] U cvc U zvz. We relabel a' := vz and x' := vc. Observing how H' is drawn in D, the claim follows with D = D'. □ A standard quadruple (G, H', e, D') from Lemma 6.8 is called a tidy standard quadruple, abbreviated tsq. Lemma 6.9. Let (G, H, e, D) be a tsq of a graph G G E3. Then, B = B4 — c — z has two cut vertices u, v in [aHx], and uHv is an edge of G, and any u, v-path in G avoiding H is an edge. Proof. Let F be the face of the subdrawing of D of the cycle C = bHxHaHyb not containing the crossing of D. As C is clean in D, dF = C. Tidiness implies that D[B] is contained in F and that D [B + by] is planar. Now, let F' be the face of the subdrawing D[B] containing the edge by. Note that D[B] is a drawing of B4 — c — z and hence contains no edges of H4. We decompose the boundary of F' into two paths, bHxHaHy and P as follows: As H is tidy, bHxHaHy is on the boundary of F', and let P be the remaining part of the boundary, i.e. dF' = yPbHxHaHy. As P is a b, y-path in B, P intersects [aHx] in a vertex v by Lemma 6.6. As v appears twice in the boundary of F', it is a cut-vertex of B. Let P = yP1vP2b and assume that P1 — v, P2 — v do not intersect [aHx]. Then, H4 + P1 + P2 + vHxc + vHaz is a subdivision of K5, a contradiction to Lemma 2.4. Thus there is a vertex u g P n [aHx], u = v, and u is another cut-vertex of B. Now consider any {u, v}-bridge B' in B with attachments in both u and v that has a vertex w distinct from u, v. As G is 3-connected, either B' contains (i) b or (ii) y, or (iii) an attachment of B on H + cx + az. The latter option (iii) is dismissed by tidiness: az, cx are the only edges from c, z to B in a tsq, and the vertices b, y can be interpreted as (i) or (ii). The former two options (i) and (ii) both contradict the claim that both u, v are cut vertices of B. As any H-avoiding u, v-path is either an edge uv or contained in a B-bridge with attachments u, v, the lemma follows. □ Lemma 6.10. Let (G, H, e, D) be a tsq. If there are no two internally disjoint b, x- (respectively a, y-) paths in B4, then there is a cut vertex v G bHx (respectively, v' G aHy), such that bv (yv') is an edge and any H-bridge B' C B4 attaching at b (y) is an edge bv (yv'). Proof. We prove the claim for bHx, the proof for aHy is analogous. Suppose there are no two disjoint b, x-paths, implying there is a cut-vertex v G bHx. Let u be any vertex of bHv. As G is 3-connected, there is a path from u to G — (bHv) in G — b — v. This path is in B, and by Lemma 5.1, it does not attach to H4. Therefore it attaches in (Da) U (Dx) — bHv, a contradiction to v being a cut. Therefore if there is a cut vertex v, then bv is an edge. The same argument implies that any H-bridge B' C B attaching at b has attachments only at b and v, and is therefore a trivial bridge. □ Lemma 6.11. Let (G, H, e, D) be a tsq. The edges by, bz, cy have multiplicity at least two. Proof. Suppose on the contrary that at least one of the mentioned edges has multiplicity one. First we handle bz and cy with a slightly modified drawing with the existing crossing of xc, az replaced by a crossing of an edge with assumed multiplicity one, and for by we also twist the modified drawing: Augment the sub-drawing D[B4] by contracting the edges xc, az slightly with c, z following their edge's drawings past the crossing, so that xc, az no longer cross, and call this new drawing D' (cf. Figure 8 left). As B4 contains all the vertices of G and all edges not in B4 are connecting nodes of H4, it is a routine exercise to extend the drawing D' to a drawing of G with just one crossing, in which either bz is crossing cx or cy is crossing az, contradicting criticality of bc whenever either bz or cy have multiplicity one. Figure 8: A twist in a drawing demonstrating multiplicity of certain edges. For by, we need to twist D': As u is a cut-vertex of the bridge B by Lemma 6.9, the outer face of D' has the following boundary: bHxcxHuvHazaHyP1vuP2b. Let D'' be obtained by twisting D' at u, so that D'' is a drawing of B with the outer face bHxcxHuP^HazaHuP2 b (cf. Figure 8, right). Then D'' can be augmented to a drawing of G in which the only crossing is between by and az, a contradiction. □ Lemma 6.12. Let (G, H, e, D) be a tsq. If there do not exist two edge disjoint p, q-paths in B4 for any of {p, q} G {{a, x}, {a, y}, {b, x}}, then cz has multiplicity at least two. Proof. In the proof, we use the drawings D' and D'' defined in the proof of Lemma 6.11. For {p, q} = {b, x}, suppose there are no two edge disjoint b, x-paths. Therefore, there are no internally disjoint b, x-paths. By Lemma 6.10, there is a cut-vertex v in bHx and bv is an edge, and any other H-bridge attaching at b is a trivial edge bv. If the edge bv has multiplicity two, then there are two edge-disjoint b, x paths in B4: vP2uHx and vHx (they are edge disjoint, as them sharing an edge would imply G is not 3-connected). Hence the edge b, v has multiplicity one. If cz is a single edge, then the drawing D'' can be modified to a drawing of G in which the only crossing is of cz with bv, a contradiction establishing cz is a double edge. Symmetric arguments apply to the case {p, q} = {y, a}. In the final case of {p, q} = {a, x}, Lemma 6.9 implies there is an edge uv in G, such that u, v are cut-vertices, uv is a single edge and D'' is obtained from D[B] by twisting at the vertex u. If cz is a single edge, D'' can be augmented to a drawing of G by cz crossing uv as its only crossing, the final contradiction establishing the claim. □ Now we have all the ingredients to establish necessity in Theorem 4.2. Proof of necessity in Theorem 4.2, (i). Let G G £3, and let (G, H, e, D) be its tidy standard quadruple, whose existence is guaranteed by Lemma 6.8. By Lemma 6.9, there exist two vertices p, q in [aHx], such that G contains an edge h with endvertices p and q and any H-avoiding p, q-path is an edge. Let O be the union of all these edges; then T1 := (O, (p), (q)) G O. Without loss of generality, we may assume that aHx = aHqhpHx. By definition of the tidy standard quadruple, az and cx are edges of G. Furthermore, Lemma 6.10 asserts that either there is an edge g with endvertices b and v such that any b H-bridge B' C B4 attaching at b is an edge parallel to g, or there are two internally disjoint b, x-paths (in this case, we let b = v) in the bridge B4. Symmetrically by the same lemma, there is an edge g' with endvertices u, y with H-bridges within B4 restricted to edges parallel to g', or there are two internally-disjoint a, y-paths in B4 (in this case, we let y = u). Let H4 be the subgraph of G, induced by the vertices {b, c, y, z}. We let R to be obtained from H4 by adding the two edges az and cx, and, if b = v, all the edges which are parallel to g, and, if y = u, all the parallel edges to g'. Note that when T2 (respectively, T4) is a Q-tile, we have v = x (respectively u = a) in G due to suppression of vertices with two neighbors when joining tiles, but in R, we always have u = a and v = x. Lemma 6.11 implies that the edges by, bz, and cy have multiplicity at least 2. As bc is a single edge, we have that T3 = (R, (a, u), (x, v)) is a tile in R. Now consider the vertices p, v, and x. As cx is an edge, v is a vertex-cut in B4, which disconnects b from x, and x, v are two attachments of an R-bridge B'. As p is a vertex-cut in B4 disconnecting a from x, then {p, x, v} form a cut in G, or they are all equal. If they are a cut, then they are all three distinct as G is 3-connected. We first analyze the case when they are all distinct. Let P be a bridge of {p, x, v} disjoint from R, and let P' be the graph obtained from P by adding a vertex t adjacent to precisely its three attachments. As G is 3-connected, Lemma 4.1 implies that P' is 3-connected, so the tile T2 = (P, (p), (x, v)) is a tile in P. Suppose now that p = x = v. Then v = b. Let i be the multiplicity of the edge pq, and let j be the multiplicity of the edge vb. We set T2 = (iQj, (q), (p, b)), which is a tile in P. Symmetric arguments applied to y, a, q, and u imply that there is a tile T4, such that T4t e P. As vertices with just two neighbors are suppressed when joining tiles, we have that G is a join oT of a pre-exceptional sequence T = (T, T2, T3, T4). To see that T is exceptional, assume that min p(T) = 1. Thus, after joining the tiles, one of the edges pq, bv, or yu is a single edge. This implies that in G, there are no two internally disjoint w, s-paths for one of {w, s} e {{a, x}, {a, y}, {b, x}}, and Lemma 6.12 implies cz has multiplicity at least two. In terms of 2 as desired. □ 7 Conclusions We conclude with some comments regarding the existence of k-exceptional edges. Theorem 4.2 immediately gives the following corollary, claimed by Siran [12] and Kochol [8]: Corollary 7.1. Let G be a simple graph and e its crossing-critical edge with cr(G — e) < 1. Then, e is a Kuratowski edge of G. It is also easy to obtain the following: Corollary 7.2. For any integer k > 2, there exist infinitely many 3-connected graphs with k-exceptional edges. Proof. For k = 2, the claim follows from Theorem 4.2. For higher k, we only sketch the proof by induction; an attentative reader will be able to provide the technical details. Let Fk be the family of 3-connected graphs with k-exceptional edges containing a tidy skeleton subdivision H, such that G — az — bc — cz is planar. By induction and Theorem 4.2, we assume that Fk-1 is infinite for k > 3. Let G e Fk-1 be arbitrary. Assuming the standard labelling of H, we produce a graph G' e Fk as follows: i) For Gi, we make any edge of G - az - bc - cx have multiplicity at least k. Note that Gi is still planar. ii) For G2, we add to Gi single edges az, cx; this graph has crossing number 1 and any optimal drawing of G2 has az crossing cx. iii) For G3, we add the edge bc with multiplicity k - 1. In any optimal drawing, bc edges cross the edge az, implying the crossing number of the graph G to be at most k. Should any other edge be crossed, that would add at least k crossings, implying crossing number > k. As the edges of G need to cross at least twice (this is the technical detail we omit, but it is true due to the construction of graphs in F2), cr(G3) = k and cr(G3 - e) < k for any edge parallel to bc, hence G' = G3 has k-exceptional edges and the claim follows. □ Note that the graphs of Corollary 7.2 cannot be made simple by subdividing edges and connecting the new vertices in a cycle (the operation is called n-subdivision in [1]), as was done in [8]: then the new vertices introduced in n-paths would violate Lemma 2.5 and a K5 would be introduced in the new graph. Hence, the following remain open: Problem 7.3 ([8]). What is the smallest k, for which simple 3-connected graphs with k-exceptional edges exist? Clearly, 3 < k < 4. Hence, the simple graphs obtained by Kochol do not follow our tile structure, but as all the K3 3's of G need to share the endvertices of the exceptional edges, there may still exist an explicit description of graphs with k-exceptional edges. We therefore conclude with the following: Problem 7.4. Is there a descriptive characterization (i.e. a tile description) of 3-connected graphs with k-exceptional edges? References [1] D. Bokal, On the crossing numbers of Cartesian products with paths, J. Comb. Theory Ser. B 97 (2007), 381-384, doi:10.1016/j.jctb.2006.06.003. [2] D. Bokal, Infinite families of crossing-critical graphs with prescribed average degree and crossing number, J. Graph Theory 65 (2010), 139-162, doi:10.1002/jgt.20470. [3] D. Bokal, M. Bracic, M. Dernar and P. Hlineny, On degree properties of crossing-critical families of graphs, in: E. Di Giacomo and A. Lubiw (eds.), Graph Drawing and Network Visualization, Springer, Cham, volume 9411 of Lecture Notes in Computer Science, pp. 75-86, 2015, doi:10.1007/978-3-319-27261-0_7, revised selected papers from the 23rd International Symposium (GD 2015) held in Los Angeles, CA, September 24 - 26, 2015. [4] D. Bokal, B. Oporowski, R. B. Richter and G. Salazar, Characterizing 2-crossing-critical graphs, Adv. Appl. Math. 74 (2016), 23-208, doi:10.1016/j.aam.2015.10.003. [5] M. Chimani and T. Wiedera, An ILP-based proof system for the crossing number problem, in: P. Sankowski and C. Zaroliagis (eds.), 24th Annual European Symposium on Algorithms, Schloss Dagstuhl - Leibniz-Zentrum fur Informatik, Wadern, volume 57 of Leibniz International Proceedings in Informatics, pp. 29:1-29:13, 2016, papers from the symposium (ESA 2016) held in Aarhus, August 22 - 24, 2016. [6] G. A. Dirac and S. Schuster, A theorem of Kuratowski, Indag. Math. (Proceedings) 57 (1954), 343-348, doi:10.1016/s1385-7258(54)50043-0. [7] M. Kochol, Construction of crossing-critical graphs, Discrete Math. 66 (1987), 311-313, doi: 10.1016/0012-365x(87)90108-7. [8] M. Kochol, Linear jump of crossing number for non-Kuratowski edge of a graph, Rad. Mat. 7 (1991), 177-184. [9] B. Pinontoan and R. B. Richter, Crossing numbers of sequences of graphs II: Planar tiles, J. Graph Theory 42 (2003), 332-341, doi:10.1002/jgt.10097. [10] B. Pinontoan and R. B. Richter, Crossing numbers of sequences of graphs I: General tiles, Aus-tralas. J. Combin. 30 (2004), 197-206, https://ajc.maths.uq.edu.au/pdf/30/ ajc_v30_p197.pdf. [11] R. B. Richter and C. Thomassen, Minimal graphs with crossing number at least k, J. Comb. Theory Ser. B 58 (1993), 217-224, doi:10.1006/jctb.1993.1038. [12] J. Siran, Crossing-critical edges and Kuratowski subgraphs of a graph, J. Comb. Theory Ser. B 35 (1983), 83-92, doi:10.1016/0095-8956(83)90064-3. [13] J. Siran, Edges and Kuratowski subgraphs of nonplanar graphs, Math. Nachr. 113 (1983), 187190, doi:10.1002/mana.19831130118. /^creative ^commor ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 407-424 https://doi.org/10.26493/1855-3974.1345.ae6 (Also available at http://amc-journal.eu) Isomorphisms of generalized Cayley graphs* Xu Yang School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, China, and School ofMathematics and Statistics, Central South University, Changsha, Hunan, 410083, China Weijun Liu, Lihua Feng t School ofMathematics and Statistics, Central South University, Changsha, Hunan, 410083, China Received 5 March 2017, accepted 2 December 2017, published online 12 August 2018 Abstract In this paper, we investigate the isomorphism problems of the generalized Cayley graphs, which are generalizations of the traditional Cayley graphs. We find that there are two types of natural isomorphisms for the generalized Cayley graphs. We also study the GCI-groups among the generalized Cayley graphs, and the Cayley regressions of some groups. We mainly showed that, for an odd prime power n, Z2n (resp. D2n) is a restricted GCI-group if D2n (resp. Z2n) is a CI-group. We also obtain that the cyclic group of order 2n is a 4-quasi-Cayley regression if and only if n = 3. Keywords: Generalized Cayley graph, natural isomorphism, GCI-group, Cayley regression. Math. Subj. Class.: 05C25, 20D20 1 Introduction Let G be a finite group, S C G be a subset and a e Aut(G). If G, S, a satisfy the following three conditions: *The authors would like to express their sincere thanks the referees for their valuable comments, corrections and suggestions which lead to a great improvement of this paper. L. Feng, as the corresponding author, would like to thank SDIBT for their hospitality. This work was supported by NSFC (Nos. 11671402, 11271208, 11371207), Hunan Provincial Natural Science Foundation (2016JJ2138, 2018JJ2479), Mathematics and Interdisciplinary Sciences Project of CSU. t Corresponding author. E-mail address: xcubicy@163.com (Xu Yang), wjliu6210@126.com (Weijun Liu), fenglh@163.com (Lihua Feng) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ (i) a2 = 1; (ii) if g € G, then a(g-1)g € S; (iii) if g, h € G and a(g-1)h € S, then a(h-1)g € S, then the structure r = GC(G, S, a) is called a generalized Cayley graph with V(r) = G, E(r) = {{g, h} | a(g-1)h € S}. The neighborhood of a vertex g € G is the set of vertices adjacent to g, denoted by N(g). Then N(g) = {a(g)s | s € S}. According to condition (i), a is either the identity of Aut(G) or an involution. When a is the identity, then the definition of GC(G, S, a) is just the same as that of Cayley graphs, and thus GC(G, S, a) = Cay(G, S). In this case, S is symmetrical, i.e., S = S-1 = {s-1 | s € S} and for a € Aut(G), we have that a acts on V(r) naturally as V(r) = G. Also, if T = Sff, then there is a bijection from r to r = Cay(G, T) induced by a, defined as a: V(r) ^ V(rff), g ^ ga. It follows r = r. This kind of isomorphism between Cayley graphs induced by the automorphisms of G is called the Cayley isomorphism. It should be mentioned that not all isomorphisms between Cayley graphs are Cayley isomorphisms. In fact, there are pairs of isomorphic Cayley graphs with no Cayley isomorphism between them. This encourages us to investigate the so-called Cl-graphs and Cl-groups defined below. Definition 1.1. A Cayley graph Cay(G, S) is called a Cl-graph of G, if for any Cayley graph Cay(G, T), Cay(G, S) = Cay(G, T) implies S= T for some a € Aut(G). In this case, S is called a Cl-subset. Furthermore, G is called a Cl-group if any symmetrical subset not containing the identity is a CI-subset. For those graphs having particular transitive properties, such as Cayley graphs and bi-Cayley graphs, their isomorphism problems are well studied in the literature (recall that a bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits on the vertices). The isomorphism problem for Cayley graphs was proposed decades ago and has been investigated deeply up to now. It was initiated by Adam in 1967 who conjectured that any cyclic group is a DCI-group, where a DCI-group satisfies that any subset not containing the identity and not necessarily symmetrical is a CI-subset. Although this conjecture was soon denied by Elspas and Turner [4], it stimulated the study of CI- and DCI-groups. Alspach, Parsons [1] and Babai [3] presented a criteria for CI-graphs. Muzy-chuk [18, 19] obtained a complete classification of the CI-groups in finite cyclic groups. Li [14] showed that all finite CI-groups are solvable. The isomorphism problem and the automorphism groups for bi-Cayley graphs have also been studied flourishingly; one may refer to [10, 11, 28]. Other related results could be found in [15, 16, 23, 24, 26, 27]. The concept of generalized Cayley graphs was introduced by Marusic et al. [17] when they dealt with the double covering of graphs. Answering a question in [17], the authors in [8] found some vertex-transitive generalized Cayley graphs which are not Cayley graphs. Further, the authors in [25] studied the isomorphism problems of generalized Cayley graphs and found that the alternating group An is a restricted GCI-group if and only if n = 4. The present paper can be regarded as the continuance of the above work, and also provides support to the question at the end of [8], where the authors asked for the classification of all generalized Cayley graphs arising from cyclic groups. The structure of this paper is as follows. In Section 2, we give several properties of the generalized Cayely graphs and some lemmas which will be used later. In Section 3, we introduce two types of natural isomorphisms for any generalized Cayley graph. In Section 5, we study the GCI-groups in cyclic groups. We show that when G is a dihedral group of order 2n with n an odd prime power, if G is a Cl-group, then Z2n is a restricted GCI-group. In Section 6, we study the GCI-groups in dihedral groups. We show that when G is a cyclic group of order 2n with n an odd prime power, if G is a CI-group, then D2n is a restricted GCI-group. In Section 7, we study the Cayley regressions, a concept relating to both Cayley graphs and generalized Cayley graphs. We show that the cyclic group Z2n is a 4-quasi-Cayley regression if and only if n = 3. Finally, we propose some questions for future research. 2 Preliminaries All graphs considered in the paper are simple, finite and undirected. All the automorphisms in the paper that induce generalized Cayley graphs are assumed to be some involutions. Let G be a finite group that admits an automorphism a of order two. For g =1, we have a(h-1) G S whenever h G S, by condition (iii), implying a(S) = S-1. Let wa: G ^ G be the mapping defined by wa(g) = a(g-1)g for any g G G. Note that wa is not necessarily a bijection. Let wa(G) = {wa(g) | g G G}. We use the same notation and terminology as in [8]. Suppose s G S, then a(s) G a(S), and thus a(s) G S-1. Therefore s G S if and only if a(s-1) G S. Let Qa be the set containing all elements satisfying a(g) = g-1 in G \ wa(G), and be the set containing all elements in G satisfying a(g) = g-1. Let Ka = {g G G | a(g)g = 1}. Then we have Proposition 2.1 ([25]). Let GC(G, S, a) be a generalized Cayley graph of G. Then (1) S n wa(G) = 0. Conversely, if S n wa(G) = 0, a is an involution in Aut(G) and a(S) = S-1, then G, S, a can induce a generalized Cayley graph. (2) G = Ka U and Ka = wa(G) U Qa. Furthermore, wa(G), are all symmetrical. (3) S = S1 U S2, where S1 C and S2 C Proposition 2.2. Let G be a finite group admitting two automorphisms a, ft of order two. If a, ft are conjugate in Aut(G), then Cay(G, wa(G) \ {1}) = Cay(G, w^ (G) \ {1}). Proof. By Proposition 2.1, we have wa(G) = wa(G)-1 and (G) = (G)-1. Since a, ft are conjugate, there exists some 7 G Aut(G) such that ft = YaY-1 = aY. Therefore 7(wa(G)) = {Y(a(g-1)g) | g G G} = {YaY-1Y(g-1)Y(g) 1 g G G} = {ft(Y(g)-1)Y(g) I Y(g) G G} = w^ (G). It follows that Y(wa(G) \ {1}) = w^(G) \ {1}. Hence the result follows. □ Theorem 2.3. Let G be a finite group admitting an automorphism a of order two, S C G such that S n wa(G) = 0. Let $(g) = a(g)Sg-1. If S is symmetrical and $(g) = S for any g G G, then GC(G, S, a) = Cay(G, S). Proof. Let r = GC(G, S, a) and r2 = Cay(G, S). Let ^: V(ri) ^ V(r2), x ^ x-1 be a bijection between these two graphs. For any {g, h} e E(r1), there exists some s e S such that h = a(g)s. {g, h}^ = {g-1, h-1}. Note that gh-1 = gs-1a(g)-1 = a(a(g))s-1a(g)-1. Since S is symmetrical and $(g) = S for any g e G, we have a(a(g))s-1a(g)-1 e S. This implies {g, h}^ e E(^), and thus GC(G,S,a) = Cay(G,S). □ Theorem 2.3 can be regarded as a criteria to judge whether some generalized Cayley graphs are Cayley graphs or not. Theorem 2.4. Let G be any finite group admitting an automorphism a of order 2. Then we have GC(G, S, a) = Cay(G, S), where S = Ua, Qa or G \ wa(G). Proof. By proposition 2.1, G = wa(G) U Qa U Ua. For any g e G, G = a(g)Gg-1. For any x e wa(G), there exists some h e G such that x = a(h-1)h. So a(g)xg-1 = a(g)a(h-1)hg-1 = a((hg-1)-1)hg-1, and hence wa(G) = a(g)wa(G)g-1. Asaresult, U = a(g)^ag-1 U a(g)^ag-1. For any s e assume that a(g)sg-1 e then a(a(g)sg-1)-1 e and a(g)sg-1 = a(a(g)sg-1)-1. Since a(a(g)sg-1)-1 = a(g)a(s-1)g-1, we have that s = a(s-1), which is a contradiction as s e This means a(g)sg-1 e Thus, Qa = a(g)^ag-1 and = a(g)^ag-1. By Theorem 2.3, we get the result. □ Let Fix(a) = {g e G | a(g) = g}. So Fix(a) < G and we have the following lemma. Lemma 2.5 ([8]). |wa(G)| = jFiaYi • Note that some references also use CG(a) to denote Fix(a). Those papers mainly investigate the properties of the finite groups which admit involutory automorphisms; one can refer to [2, 13, 21, 22]. Although those problems are not considered in this paper, we borrow the following well-known result. Lemma 2.6 ([7]). Let G be a finite group of odd order admitting an automorphism ^ of order two. Then the following statements hold. (1) G = FK = KF, F n K =1, and |K| = |G : F|, where F = CG(^) and K = K0; (2) Two elements of K conjugate in G are conjugate by an element of F; (3) If H is a subgroup of F, then NG(H) = CG(H)Nf(K). By Lemmas 2.5 and 2.6, we get Proposition 2.7. Let G be a group of odd order admitting an automorphism a of order two. Then Qa = 0. Proof. By Lemmas 2.5 and 2.6, |Ka| = |wa(G)| = | F|Ga)|. As Ka = wa(G) U we obtain tta = 0. □ Remark 2.8. By Proposition 2.7, for any generalized Cayley graph GC(G, S, a), if |G| is odd, S C Ua. We present an alternative proof avoiding Lemmas 2.5 and 2.6. If Qa = 0, assume that = 0. Then G is an abelian group of odd order by Proposition 2.1. Thus a is a fixed-point-free automorphism of G. Then Ka = wa(G) = G according to [7, Lemma 10.1.1], which is a contradiction. This implies that Ua = 0. Since the S in GC(G, S, a) are choosen from Qa and Ua. Therefore |S| must be odd, which is a contradiction as, there are no regular graphs of odd order with odd valency. This implies Qa = 0. It is well known that a finite group G of odd order is solvable by Feit-Thompson Theorem [5]. From above, we can see that the classification of GC(G, S, a) of finite group G of odd order seems to be more clear as the elements of S can only be chosen from since Qa = 0. In [8], Hujdurovic et al. defined the following set Aut(G, S, a) = jp G Aut(G) | p(S) = S, ap = pa}. Moreover, one sees that Aut(G, S, a) = Aut(G, S) n CAut(G)(a), where Aut(G, S) = Aut(G, S, 1). Proposition 2.9. Let S be the set as in (3) of Proposition 2.1. Then Aut(G, S, a) = Aut(G,Si,a) n Aut(G,S2,a) = Aut(G,Si) n Aut(G, S2) n CAut(G)(a). Furthermore, the couples of the form like js, a(s-1)} are imprimitive blocks of Aut(G, S, a). Proof. For any s G S1 and s' G S2, if there exists some p G Aut(G, S, a) such that s = p(s'), then ap(s') = a(s). Since ap = pa and s = a(s-1), pa(s'-1) = s. This implies a(s') = s'-1, which is a contradiction as s' G S2. Hence p(S1) = S1 and p(S2) = S2 for any p G Aut(G, S, a). Let A = js, a(s-1 )} be a couple in S2. For any p G Aut(G, S, a), Av Ç S2. If A n Av = 0, then s = p(s) or s = pa(s-1). If s = p(s), then a(s-1) = pa(s-1). If s = pa(s-1), then a(s-1) = p(s). This implies that A = Av. Thus A is an imprimitive block. □ Let GC(G, S, a) be a generalized Cayley graph of G. Under the condition of Proposition 2.9, S n S-1 = (S1 u S2) n (S1 u S2)-1 = (S1 n S-1) u (S1 n S-1) u (S2 n S-1) U (S2 n S-1). Note that S1 n S-1 = S2 n S-1 = 0, it follows that S n S-1 = (S1 n S-1) U (S2 n S-1). Since S1 Ç and Qa is symmetrical, so S1 n S-1 Ç Similarly, S2 n S2-1 Ç Ua. Let T = S n S-1. It follows that GC(G,T,a) is still a generalized Cayley graph of G. We call GC(G, T, a) the induced generalized Cayley graph of GC(G, S, a). Note that T-1 = T, this encourages us to consider the Cayley graph Cay(G, T), called the induced Cayley graph of GC(G, S, a). Next we consider Aut(G, S, a), Aut(G, T, a) and Aut(G, T). Proposition 2.10. Aut(G, S, a) < Aut(G,T, a) < Aut(G,T). Furthermore, Aut(G, S, a) < Aut(G, T, a) if S is not symmetrical; Aut(G, T, a) = Aut(G, T) if a G Z(Aut(G)). Proof. For any p G Aut(G, S, a), we have p(S) = S and p(S-1) = S-1, thus p(T) = T, p G Aut(G, T, a). If S is not symmetrical, we have a G Aut(G, S, a) as a(S) = S-1 = S, but a G Aut(G, T, a) as a(T) = T. Aut(G, T, a) < Aut(G,T) is obvious by the definition. Since Aut(G, T, a) = Aut(G, T) n CAut(G)(a), we get the result. □ Finally, we introduce a lemma about the connectivity of the generalized Cayley graph. Lemma 2.11. Let G be a group, A C G and a G Aut(G) of order 2. The generalized Cayley graph X = GC(G, A, a) is connected if and only if A is a left generating set for (G, *), where f * g = a(f )g for all f, g G G. 3 Two basic types of isomorphisms In this section, we will introduce two types of natural isomorphisms of generalized Cayley graphs for any finite group. First, we introduce the first type of natural isomorphism found by A. Hujdurovic et al. Theorem 3.1 ([9]). GC(G, S, a) = GC(G, Sp, ap) for any fi G Aut(G), where ap = fiafi-1. Remark 3.2. From Theorem 3.1, one can see that if a, 7 are conjugate, then there is a generalized Cayley graph GC(G, S, a) if and only if there is a generalized Cayley graph GC(G, Sp, y) with y = ap such that these two graphs are isomorphic. Hence, if we intend to study all the generalized Cayley graphs of some group G, we only need to study the generalized Cayley graphs related to the representatives of the conjugacy classes of elements in Aut(G). Corollary 3.3. GC(G, S, a) = GC(G, S-1, a). Proof. Let fi = a. Then GC(G, S, a) = GC(G, a(S), aa) by Theorem 3.1. Note that a(S) = S-1, this completes the proof. □ Next, we introduce the second type of natural isomorphism. Theorem 3.4. Let GC(G, S, a) be a generalized Cayley graph. Then GC(G, a(g)Sg-1, a) is also a generalized Cayley graph of G for any g G G. Furthermore, GC(G, S, a) = GC(G, a(g)Sg-1, a). Proof. For any x G G, if a(x-1)x G a(g)Sg-1, a(g-1)a(x-1 )xg G S, that is, a((xg)-1)xg G S, which conflicts with condition (ii). If a(x-1)y G a(g)Sg-1, then we have a((xg)-1)yg G S. Thus a((yg)-1)xg G S by condition (iii). It follows that a(y-1)x G a(g)Sg-1. Therefore, GC(G, a(g)Sg-1, a) is also a generalized Cayley graph of G for any g G G. Let r = GC(G, S, a) and rg = GC(G, a(g)Sg-1, a). Let 0: V(r) ^ V(rg), a ^ ag-1. So 0 is abijection. For any {a, b} G E(r), a(a-1)6 G S. Since a((ag-1)-1)(bg-1) = a(g)(a(a-1)b)g-1 G a(g)Sg-1, we have {ag-1 ,bg-1} G E(rg). Therefore {a, b} G E(r) if and only if {a, 6}e G E (ra). Thus they are isomorphic. □ According to Theorem 3.1, r = rp for any fi G Aut(G), we call the mapping x ^ xp the the first basic type of isomorphism of r. By Theorem 3.4, r = rg for any g G G, we call the mapping x ^ xg-1 the second basic type of isomorphism of r. For any g G G, R(g): x ^ xg is a permutation of G. Set R(H) = {R(h) | S = a(h)Sh-1}. Theorem 3.5. Let r = GC(G, S, a) be a generalized Cayley graph. Then R(H) < Aut(r). Proof. For any {a, b} G E(T), it suffices to show that {a, b}R(h) G E(r) for any R(h) G ). Since {a,b} G E(r), a(a-1)b G S = a(h)Sh-1. It follows that a((ah)-1)bh G S, which implies that {ah,bh} G E(r). Thus R(h) G Aut(r). For any R(h),R(h') G R(H), S = a(h)Sh-1 and S = a(h')Sh'-1. Therefore S = a(h/-1h)S(h/-1h)-1, thus R(h'-1h) G R(H). This implies that R(H) < Aut(r). □ 4 GCI, restricted GCI and strongly GCI groups Similarly to the Cl-groups in Cayley graphs and BCI-groups in bi-Cayley graphs, we propose the following definitions relating to generalized Cayley graphs. Definition 4.1. Let G be a finite group. Let M be the set of all Cayley graphs and N be the set of all generalized Cayley graphs constructed by automorphisms of order two. Then 1. G is called a GCI-group if both of the following are satisfied: (i) for any two nontrivial generalized Cayley graphs GC(G, S, 1) and GC(G, T, 1) in M, whenever GC(G, S, 1) = GC(G, T, 1), there exists 5 G Aut(G) such that S5 = T. (ii) for any two nontrivial generalized Cayley graphs GC(G, S, a) and GC(G, T, P) in N, whenever GC(G, S, a) = GC(G, T, P), there exists 5 G Aut(G) such that P = a5 = 5a5-1 and T = a5(g)S5 g-1. 2. G is called a restricted GCI-group if (ii) is satisfied. 3. G is called a strongly GCI-group if for any nontrivial GC(G, S, a), whenever GC(G, S, a) = GC(G, T, P), there exists 5 G Aut(G) such that P = a5 = 5a5-1 and T = a5 (g)S5 g-1. Remark 4.2. 1. The definition is based on Theorems 3.1 and 3.4 and Definition 1.1. The two basic types of isomorphisms and their compositions are called the natural isomorphisms of generalized Cayley graphs. For instance, GC(G, S, a) = GC(G, SY, aY) by Theorem 3.1, GC(G, SY, aY) = GC(G, aY(g)SYg-1,aY) by Theorem 3.4, then we have GC(G, S, a) = GC(G, aY(g)SYg-1, aY). 2. The word 'nontrivial' in the definition means that the null graph is not considered. In fact, if it is included, for a finite group G which has an automorphism a of order 2, GC(G, 0,1) and GC(G, 0, a) are both isomorphic to the null graph. By the definition, G cannot be a strongly GCI-group, otherwise it will make the definition meaningless, thus the null graph is not considered in the definition. 3. If a finite group G has no automorphisms of order two, then we still consider that (ii) is satisfied for G. 4. By definition, strongly GCI-group implies GCI-group, GCI-group implies CI-group and restricted GCI-group. However, restricted GCI does not imply GCI and does not imply CI either. If G is not a restricted GCI-group or a CI-group, then it is not a GCI-group either. Next we will give some examples of finite groups satisfying special conditions: Example 4.3. Let G = Z4. Then G is a GCI group by Theorem 5.2. However, let a: x ^ —x be an involution. Thus GC(G, {1}, a) is a generalized Cayley graph of G. Also, GC(G, {2}, 1) is a generalized Cayley graph of G. Although GC(G, {1}, a) = GC(G, {2}, 1) but, a is not conjugate to 1, that means G is not a strongly GCI group. Therefore Z4 is a GCI but not strongly GCI group. Let G = Z8. Then G is a CI group [19]. However, Z2n is a GCI group if and only if it is Z2 or Z4 by Theorem 5.2. It follows that G is not a GCI group. Thus Z8 is a CI but not GCI group. Though we find example of CI but not restricted GCI groups, like Z8, we have not found out the example of restricted GCI but not CI groups up to now. Thus we propose the following question: Question 4.4. Is every restricted GCI group a CI group? The next theorem is useful to determine whether a group is a restricted GCI-group or not. Theorem 4.5. Let G be a finite group admitting two automorphisms a, ft of order two. If a, ft satisfy the following three conditions: (1) a and ft are not conjugate; (2) K(G)| = |Ka|; (3) (G)| = K |, then G is not a restricted GCI-group. Proof. Assume |G| = n. If these three conditions are satisfied, then n is even by Proposition 2.7. Furthermore, there must exist two generalized Cayley graphs, say GC(G, {s}, a) and GC(G, {s'}, a), which are both isomorphic to nK2. But there is no natural automorphism as a and ft are not conjugate. Hence G is not a restricted GCI-group. □ To conclude, we give the characterization of strongly GCI-groups. Theorem 4.6. A finite group G is a strongly GCI-groups if and only if G is a Cl-group and one of the following is true for G: (1) G has no involutory automorphisms; (2) all involutory automorphisms are fixed-point-free. Proof. First we show the necessity. If G is a strongly GCI-groups, then G must be a CI-group. If not all involutory automorphisms of G are fixed-point-free automorphisms or, as we will show that G has no automorphisms of order two. If there exists some involutory automorphism which is not fixed-point-free, say a, this means | Fix(a) | = 1. By Lemma 2.5, we get wa(G) = G. Since G = wa(G) U Qa U by Proposition 2.1, it follows that Qa U = 0. Thus at least one of Qa and Ua, say is not an empty set. According to Theorem 2.4, GC(G, a) = GC(G, 1) which is not a null graph. This is a contradiction to the fact that G is a strongly GCI-group. Therefore G has no automorphisms of order two since otherwise all automorphisms of order two of G are fixed-point-free automorphisms. If G has no automorphisms of order two, then G must be a CI-group as G is a strongly GCI-group. Next we show the sufficiency. Suppose that all automorphisms of order two of G are fixed-point-free. Let a G Aut(G) be such an involution. Then G = wa(G) by Lemma 2.5, so any generalized Cayley graph induced by involutory automorphism is a null graph. □ 5 The cyclic GCI groups Theorem 5.1. The cyclic group of order pn with p an odd prime is a GCI-group if and only if it is a CI-group. Proof. Let G = . Then G has only one automorphism of order two, that is a: x ^ —x. Note that Wa(G) = {a(g-1)g | g G G} = {2g | g G G}, it follows that S = 0 as any non-identity of G is a square since |G| is odd. Thus the only generalized Cayley graph of G induced by automorphisms of order two is GC(G, 0, a) = pnK1. □ Babai [3] classified the Cl-groups of cyclic groups of order 2p with p a prime. God-sil [6] classified the Cl-groups of cyclic groups of order 4p. Next we will classify the GCI-groups of cyclic groups of even order. We will deal with the problem step by step in this section. Theorem 5.2. Let G be a finite cyclic group of order 2n. Then G is a GCI-group if and only if n = 1, 2. Proof. Let G = Z2n = {0,1,..., 2n — 1}. When n =1, Aut(G) = 1, there are no automorphisms of order two in Aut(G). Therefore G is a GCI-group by Definition 4.1. When n = 2, then Aut(G) = Z2, there is a unique element of order two in Aut(G) since Aut(G) is cyclic, say a: x ^ —x. If g G G, then a(g-1)g = 2g G S. Hence S C {1,3}. Therefore there are only three generalized Cayley graphs of G, with S being {1}, {3} and {1, 3}, respectively. Let r = GC(G, {1}, a), r2 = GC(G, {3}, a). Note that —1 = 3 (mod 4), and so r1 = r2 by Corollary 3.3. When n > 3, then Aut(G) = Z2 x Z2n-2, and there are only three automorphisms of order two in Aut(G), say, a: x ^ —x, ft: x ^ (2n-1 — 1)x, y: x ^ (2n-1 + 1)x. Let S = {1,2n-1 + 1}. Since 1 ^ 2n-1 + 1 (mod 2n) and they are both odd, we have S n wa(G) = 0 as wa(g) = a(g-1)g = 2g is even for any g G G. Further, S n wg (G) = 0 as wg (g) = ft(g-1)g = 2n-1g + 2g (mod 2n) is also even for any g G G. Recall that ft( —1) = 2n-1 + 1, a( —1) = 1 and a(—(2n-1 + 1)) = 2n-1 + 1, hence a(S) = S-1 and ft(S) = S-1. Therefore both GC(G, S, a) and GC(G, S, ft) are generalized Cayley graphs of G. Let r = GC(G, S, a). Since |S| = 2, the valency of r is two. For any x G V(r0, N(x) = {a(x) + y | y G S} = {—x + 1, —x + 2n-1 + 1}. Consider the vertex 2n-1 + x (mod 2n), it follows that x ^ 2n-1+x (mod 2n). N(2n-1+x) = {a(2n-1 +x)+y | y G S} = {2n-1 — x +1, —x + 1}. Thus 'x ^ (—x +1) ^ (2n-1 + x) ^ (2n-1 — x + 1) ^ x' is a 4-cycle in r1. Therefore r1 = 2n-2C4. Let r2 = GC(G, S, ft). Since |S| =2, the valency of r2 is two. For any x G V(^), N(x) = {ft(x) + y | y G S} = {(2n-1 — 1)x + 1, (2n-1 — 1)(x — 1)}. We consider the vertex 2n-1 + x (mod 2n). Then N(2n-1 + x) = {ft(2n-1 + x) + y | y G S} = {(2n-1 — 1)x — 2n-1 + 1, (2n-1 — 1)x +1}. Thus 'x ^ (2n-1 — 1)x +1 ^ (2n-1 + x) ^ (2n-1 — 1)(x — 1) ^ x' is a 4-cycle in r1. Therefore r2 = 2n-2C4. From above, GC(G, S, a) = GC(G, S, P) = 2n 2C4, but a and P are not conjugate in Aut(G) as Aut(G) is abelian, hence G is not a restricted GCI-group by Definition 4.1. □ Theorem 5.3. Let G be a finite cyclic group of order 2apb with p an odd prime and a, b > 0. If G is a restricted GCI-group, then a = 1. Proof. Since G is a finite cyclic group of order 2apb, let G = Gi x G2, where Gi = Z2a and G2 = Zpb. We claim that a < 2. Now we suppose a > 3. By Theorem 5.2, a: (g1,g2) ^ (-g1,g2) and P: (g1,g2) ^ ((2n-1 - 1)g1,g2) are two different automorphisms of G with order two when a > 3. Let S = {(1,0), (2n-1,0)}. Then GC(G,S,a) and GC(G, S, P) are two generalized Cayley graphs of G. According to Theorem 5.2, we have GC(G, S, a) = GC(G, S, P) = 2n-2pbC4. Note that a and P are not conjugate in Aut(G), it follows that a < 2. Assume a = 2. Note that a: (#1,02) ^ (-01,02),P: (01,02) ^ (01, -02) are two automorphisms of G. Furthermore, wa(G) = {(0,0), (2,0)}, Ka = {(g1,0) | g1 G G1}. Therefore Qa = {(1,0), (3,0)}. wg(G) = {(0,02) | 02 G G2}, Kg = {(0,02), (2,02) | 02 G G2}. Thus Qg = {(2,02) | 02 G G2}. Let S1 = {(1,0)} and S2 = {(2,0)}. We can see that GC(G, S1, a) = GC(G, S2, P) = 2pbK2. However a and P are not conjugate as Aut(G) is abelian. It follows from above discussion that a =1. □ Theorem 5.4. Let G be a finite cyclic group of order n, where n is even with at least two different odd prime divisors. Then G is not a restricted GCI-group. Proof. Suppose that n = p00 • p11 • • • p6kk, where po = 2, pi,pj are different odd primes for any i, j G {1,..., k} and st > 1 is an integer for any t G {0,1,..., k}, k > 2. It follows that G can be decomposed into the direct product of some cyclic groups, say G = G0 x • • • x Gk = Z2=o x Zpsi x • • • x Zpsfc, where Gi = Z , i = 0,1,..., k. p1 Let and (xo ,x1,...,xk) ^ (xo, -x1,...,xk) P: (xo, X1, X2,... ,xk) ^ (xo,X1, -X2,..., xk). Since k > 2, then such a, P can not appear in Aut(G). Obviously wa(G) = {(0, x1,0,..., 0) | x1 G G1} and wg (G) = {(0,0,x2,..., 0) | x2 G G2}. Let 0i G Gi, i G {0,1,...,k} and 0o the element of order two. Then (0o, 01,0,..., 0) G Qa and (0o, 0,02,0,..., 0) G Qg. Therefore GC(G, {(0o,01,0,..., 0)},a) and GC(G, {(0o, 0,02,0,..., 0)},P) are both generalized Cayley graphs of G. In fact, they are both isomorphic to f K2, but a and P are not conjugate in Aut(G). Thus G is not a restricted GCI-group by Theorem 4.5. □ Theorem 5.5. Let G = Z2n, where n is an odd prime power. Then G is not a strongly GCI-group. Proof. Let G = (a, b | an = b2 = 1, ab = ba). It can be checked that the mapping a: a ^ a-1,b ^ b is the only automorphism of G of order two. Also Qa = {aib | i G {1,..., n}} and = 0 by direct computation. Let GC(G, S, a) be any generalized Cayley graph of G. Then S C Qa. Let H = (a', b' | a'n = b'2 = 1, b'a'b' = a'-1) and ^: as ^ a's, atb ^ a'-tb. It follows that ^ is a bijection from G to H. Furthermore, a GC(G,S,a) = Cay(H,y(S)). Let S = {ab,a2b}. Then y>(S) = {a-1b,a-2b}, this implies Cay(H, = H. Let T = {ab,a-1b}. Then (T> = G, therefore Cay(G,T) = C2n. Thus Gc(G,S,a) ^ GC(G,T, 1) = C2n, which means that G is not a strongly GCI-group from Definition 4.1. □ Theorem 5.6. Let G = Z2n, H = D2n, where n is an odd prime power. Then G is a restricted GCI-group if H is a CI-group. Proof. Let G = (a, b | an = b2 = 1, ab = ba) and H = (a', b' | a'n = b'2 = 1,b'a'b' = a'-1). It is easy to see that a: a ^ a-1, b ^ b is the only automorphism of G of order two. Then we have = {ajb | i e {1,..., n}} and = 0 by direct computation. Let GC(G, S, a) be any generalized Cayley graph of G. Then S C Let ^: as ^ a's,a4b ^ a'-tb. Obviously ^ is a bijection from G to H. Furthermore, GC(G, S, a) = Cay(H, y>(S)). Assume that GC(G, S1, a) ^ GC(G, S2, a), then Cay(H, ^(S1)) = Cay(H, y>(S2)). Since H is a CI-group, there exists some 7 e Aut(H) such that y(^(S1)) = y(S2). Without loss of generality, assume that there exist k, l satisfying (k, n) = 1 and 1 < l < n such that 7 is the mapping a' ^ a'k, b' ^ a'1 b'. Let ^: a ^ ak, b ^ b. Then ^ e Aut(G). Since n is some odd prime power, there must exist some 1 < m < n such that a1 = a2m. Therefore there exist ^ e Aut(G) and g = a-m such that S2 = a(g)Sj>g-1. Hence G is a restricted GCI-group. □ Theorem 5.7. Let G be a finite cyclic group of odd order n, where n has at least two different prime divisors. Then G is not a strongly GCI-group. Proof. Let G = G1 x G2 x • • • x Gs where Gj = Z^, p is some odd prime. Let a: (g1, g2,..., gs) ^ ( g1, g2,..., gs). Then wa(G) = G1, and thus G \ wa(G) = 0. By Theorem 2.4, GC(G, S, a) = GC(G, S, 1), where S = G \ wa(G). It follows that G is not a GCI-group. □ 6 The GCI-groups in dihedral groups Theorem 6.1. Let G = D2n (n > 3) be a dihedral group. If G is a restricted GCI-group, then n is some odd prime power. Proof. Let G = D2n = (a, b | a" = b2 = 1,bab-1 = a-1} be a GCI-group. Assume first that n is even. Let a: a ^ a-1, b ^ b. Then a G Aut(G) is of order two. ^a(G) = {a(g-1)g | g G G} = ja® G G | i is even}. {a(g)g = 1 | g G G} = ja®,b, anb | a® G G}. It follows that Oa(G) = ja®,b, anb | i is odd}. This implies GC(G, {a}, a) and GC(G, jb}, a) are always generalized Cayley graphs of G. Note that they are both isomorphic to nK2. Furthermore, a(g)aYg-1 = a-2®+j if g = a® and aY = aj, a(g)aYg-1 = a-2®-j if g = a®b and aY = aj. It follows that a(g)aYg-1 G (a}. Since G is a GCI-group, a(g)aYg-1 = b for some g G G and 7 G Aut(G), but this is impossible. Thus n is not even. Assume n is odd and has at least two different prime factors, say n = p^1 • • • is the prime decomposition and t > 2. By [20, Lemma 3.4], Aut(G) = Aut(G1) x • • • x Aut(Gt), where G® = (a®, b} and (a} = (a1} x • • • x (at}. It can be checked that there must exist two automorphisms a: a1 ^ a-1, a® ^ a®, b ^ b and ft: a2 ^ a-1, a¿ ^ a¿, b ^ b in Aut(G). Notice that each is of order two, and they are not conjugate in Aut(G) as they belong to Aut(G1) and Aut(G2) respectively, and Aut(G) is the direct product of these Aut(Gj). Furthermore, b G Oa(G) n Op(G). Thus GC(G, jb},a) and GC(G, {b},ft) are two generalized Cayley graphs of G which are isomorphic to nK2. However a and ft are not conjugate. Thus n is some odd prime power. □ Theorem 6.2. Let G = D2n, H = Z2n, with n odd prime power. Then G is a restricted GCI-group if H is a Cl-group. Proof. Let G = (a, b | an = b2 = 1, bab = a-1) and H = Z2n = {0,1,..., 2n - 1}. We will show first that any two automorphisms of G of order two are conjugate. Let a: a ^ a®, b ^ ajb and ft: a ^ ak, b ^ alb be two automorphisms of order two. Then i, k = -1. Let y: a ^ as, b ^ a4b with (n, s) = 1. Then 7 G Aut(G). We can see that y-1 : a ^ ar, b ^ a-rtb with rs = 1 (mod n). It follows that 7a7-1: a ^ a-1, b ^ a2t+sjb. For any j, l, the equation 2t + sj = l (mod n) has a solution. It follows that a, ft are conjugate. According to the Remark 3.2, it suffices to consider the isomorphisms of the generalized Cayley graphs induced by the same automorphisms. Without loss of generality, we consider a: a ^ a-1, b ^ b. Let s = n—1 and I = {1,..., s}. Then (G) = {a(g-1)g | g G G} = (a). Ka = {b} U (a). Thus tta = {b} and Ua = Uie/{a®b, a-ib}. Let GC(G, S, a) and GC(G, T, a) be any two isomorphic generalized Cayley graphs. We divide the proof into two cases. Case 1: Qa C S. If C S, then Qa C T. Suppose S = Ui£/lc/{a®b,a-ib} U Qa and T = Uie/2c/{a®b, a-ib} U Let p: G ^ H, as ^ 2s, a4b ^ n — 2t. Then p is a bi-jection from G to H. Furthermore, GC(G, S, a) = Cay(H, p(S)) and GC(G, T, a) = Cay(H, p(T)), where p(S) = Uie/lC/{n — 2i, n + 2i} U {n} and p(T) = Uie/2c/{n — 2i, n + 2i} U {n}. Since H is a CI-group, there exists some automorphism ^ G Aut(H) such that p(T) = ^(p(S)). Since n is the unique involution in H, ^(n) = n and ^(n — 2i) = ^(n) — 2^>(i) = n — 2^>(i) for any i G 11 C I. This implies that ^ can induce an automorphism ^ of G with rules a® ^ , b ^ b. As ^a = a^, there exist ^ and g =1 G G such that the isomorphism between GC(G, S, a) and GC(G, T, a) is a natural isomorphism. Case 2: £ S. If £ S, then Qa £ T. The rest of the proof is similar to that of Case 1. □ The next result is about the graph structure. Recall that a graph r is Hamiltonian if it contains a cycle passing through all vertices of r. Theorem 6.3. Let G = D2n with n odd prime power. Then any connected generalized Cayley graph of G is Hamiltonian. Proof. Let H = Z2n and p: G ^ H, as ^ 2s, a4b ^ n —2t be the bijection from G to H. Then any generalized Cayley graph GC(G, S, a) of G is isomorphic to the Cayley graph Cay(H, p(S)) of H. Therefore GC(G, S, a) is connected if and only if Cay(H, £(£)) is connected. It is well known that Cay(H, p(S)) is connected if and only if (p(S)) = H. (p(S)) = H if and only if there exist some a®b, a-® b G S satisfying (i, n) = 1 as p(a®b) = n — 2i. Then there always exists a Hamilton cycle GC(G, {a®b, a-ib}, S) in a connected generalized Cayley graph of G. This completes the proof. □ 7 Cayley regression First of all, we give the following related definitions. Definition 7.1. Let G be a finite group. (1) G is called a Cayley regression if any generalized Cayley graph of G is isomorphic to some Cayley graph of G. (2) G is called an a-Cayley regression if any generalized Cayley graph of G induced by a e Aut(G) is isomorphic to some Cayley graph of G. (3) G is called a quasi-Cayley regression if any generalized Cayley graph not induced by the automorphism x ^ x-1 is isomorphic to some Cayley graph of G. (4) G is called an m-Cayley regression if any generalized Cayley graph of G with valency at most m is isomorphic to some Cayley graph of G. (5) G is called an m-quasi-Cayley regression if any generalized Cayley graph not induced by the automorphism x ^ x-1 of G with valency at most m is isomorphic to some Cayley graph of G. (6) G is called a skew Cayley regression if any generalized Cayley graph of G is isomorphic to some generalized Cayley graph of another finite group. It is well known that every Cayley graph is also a generalized Cayley graph. But many examples, see [8] for instance, reveal that the converse is not true. Therefore a natural question arises. Question 7.2. Characterize Cayley regressions. Remark 7.3. If a: x ^ x-1 is an automorphism of G, then G is abelian. This case is very special as Ka = G and = 0. In fact, Hujdurovic et al. in [9] had already noticed this situation. They studied the relationship between the generalized Cayley graphs induced by involutory automorphism and Cayley graphs. They obtained two families of generalized Cayley graphs induced by involutory automorphisms on Z2m x Z2n and Z2 x Z2 x Z2k+1 respectively (where m > 1, n > 2, k > 1) are not vertex-transitive. Therefore we propose the definition of 'quasi-Cayley regression' and 'm-quasi-Cayley regression'. Also, we propose the following problem: Are there finite groups which are quasi-Cayley regressions but not Cayley regressions? When G is an abelian simple group, then G is a cyclic group of prime order and Aut(G) is not necessarily a Cayley regression. Example 7.4. For the prime p = 61, obviously, Aut(Zp) = Zp-1 = Z6o. Let G = Aut(Zp), S = {±6, ±12,5, 25} and a(x) = 31x. By [17, Theorem 4.4], we have GC(G, S, a) is not a Cayley graph. Thus G is not a Cayley regression. Theorem 7.5. Let G be a finite cyclic group of odd order n. Then G is a Cayley regression if and only if n is some odd prime power. Proof. The sufficiency is obvious by Theorem 5.1, it suffices to show the necessity. Assume on the contrary that n has at least two different odd prime divisors, say n = q^2m, where q1 and q2 are different prime powers and (qi, = 1, then we have G = G1 x G2 x G3, where |G11 = q1, |G2| = q2 and |G31 = m. Let a: G ^ G, (g1,g2,g3) ^ (-g1,g2,g3). It is easy to see that the order of a is two, so a can induce some generalized Cayley graphs of G. Note that wa(G) = {(g1,0,0) | g1 e G1}. Let S = {(0,1,0), (0, q2 - 1,0)}. Then r = GC(G, S, a) is a generalized Cayley graph of G. Consider the vertex of the form (0, g2,g3) in r for any g2 e G2, g3 e G3. For any fixed g3, there are q2 vertices of the form {(0,g2,g3) | g2 e G2} which induce a cycle of length q2. For any other vertex of the form (g1, g2, g3) with g1 = 0, there are 2q2 vertices {(g1, g2, g3), (—g1, g2, g3) | g2 e G} which induce a cycle of length 2q2. Therefore r1 = mCq2 U (qi-1)m C2q2, which is not vertex-transitive. Thus GC(G, S, a) is not a Cayley graph, and hence G is not a Cayley regression. □ Theorem 7.6. Let G = Zn x ■ ■ ■ x Zn with n odd, s > 2. Then G is not a Cayley regression. Proof. Let a: (i1, i2, i3 ..., is) ^ (i2, i1, i3,..., is) for all it e Zn. So a e Aut(G) and o(a) =2. Therefore a can induce generalized Cayley graphs of G. Let S = {(1, 0,..., 0), (0, n - 1,0,..., 0)}. It follows that GC(G, S, a) is a generalized Cayley graph of G. Consider vertex (0,..., 0), then vertices like (i, i, 0,..., 0) and (i, i - 1,0,..., 0) are in the same cycle with (0,..., 0). Thus (0,..., 0) is in a cycle of length 2n. Consider vertex (0, n—1, 0,..., 0), then vertices like (i, n—1 + i, 0,..., 0) and (^ + i, i,..., 0) are in the same cycle with (0, , 0,..., 0). It follows that (0, , 0,..., 0) is in a cycle of length n. Therefore GC(G, S, a) is not vertex-transitive, and thus GC(G, S, a) is not a Cayley graph. That completes the proof. □ From Theorem 7.5, we see that the cyclic group of odd non prime power order is not an m-Cayley regression for any m > 0. So we only consider the cyclic groups of even order in the rest of this section. Corollary 7.7 ([9]). Let G = Z2n. Then GC(G, S, a) is isomorphic to a Cayley graph on D2n, where a: x ^ -x. According to Corollary 7.7, we can see that Z2pn (with p an odd prime) is a skew-Cayley regression since a: x ^ -x is the only automorphism of G of order two. Theorem 7.8. Let G be a finite cyclic group of order 2n with n > 3. Then (1) G is a 3-quasi-Cayley regression; (2) G is a 4-quasi-Cayley regression if and only if n = 3. Proof. Assume that G = {0,1,..., 2n - 1}. By Theorem 5.2, we have that a: x ^ (2n—1 - 1)x and P: x ^ (2n-1 + 1)x are the only two automorphisms of G of order two except the automorphism x ^ -x. Also, the valency of the generalized Cayley of G induced by a or P are even as x + a(x) = 0 and x + P(x) =0 for any x e G. (1) Consider the generalized Cayley graphs induced by a, ^ respectively. For any g € G, wa(G) = {a(-g)g | g € G} = {(2"-1 - 1)(-g)+ g | g € G} = {2"-1g + 2g (mod 2") | g € G} = {g | g = 0 (mod 2)} = Ka. Wf(G) = {£(-g)g | g € G} = {(2"-1 + 1)(-g)+ g (mod 2") | g € G} = {-2"-1g (mod 2") | g € G} = {0, 2"-1} = Kf. It follows that for any generalized Cayley graph GC(G, S, a), S contains no even integers and, for any generalized Cayley graph GC(G, S, ,0), 0 and 2"-1 are not contained in S. For any g € G with g odd, a(-g) = (2"-1 - 1)(-g) (mod 2") = g - 2"-1g (mod 2") = 2"g + g - 2"-1g (mod 2") = 2"-1g + g (mod 2") = 2"-1 + g (mod 2"). This implies that there are 2"-2 couples can be included in S, that is, S1 — {1,2" 1 + 1}, S3 = {3, 2"-1 + 3},..., S2n-i-1 = {2"-1 - 1, 2" - 1}. For any g € G \ Wf (G), £(-g) = (2"-1 + 1)(-g) (mod 2") = -g - 2"-1g (mod 2n) = 2ng - g - 2n-1g (mod 2n) = 2n-1g - g (mod 2n). Then we have (2--. if g is»dd; 12" - g, if g is even. This implies that there are (2n-1 - 1) couples which could be included in S, they are: S1 = {1, 2"-1 - 1}, S3 = {3, 2"-1 - 3}, S2n-2-1 = {2n-2 - 1, 2"-2 + 1}, S2n-i+1 = {2n-1 + 1, 2" - 1}, S2n-1+2n-2-1 = {2"-1 + 2"-2 - 1, 2"-1 + 2"-2 + 1}, T2 = {2, 2" - 2}, T2n-i-2 = {2"-1 - 2, 2"-1 + 2}. Let r = GC(G, S, a), where S = S,, then r = 2"-2C4 by Theorem 5.2. Let r = GC(G,S,£). If S = Si, we have GC(G,S,^) = as S, is the left generating set for (G, *) by Lemma 2.11. If S = T,, then GC(G, S, £) is isomorphic to 2"-fcC2fc if 2ki = 0 (mod 2n); and isomorphic to C2n otherwise. In conclude, all of the 2-valent generalized Cayley graphs of G induced by a or ft are Cayley graphs and, this implies that G is a 3-quasi-Cayley regression. (2) When n = 3, it is easy to check that those 4-valent generalized Cayley graphs induced by a and respectively, are all Cayley graphs. Next we construct a family of generalized Cayley graphs which is not vertex-transitive to show the necessity. Let S = Si U Tj, where i € {1,..., 2n-2 - 1} U {2n-1 + 1,..., 2n-1 + 2n-2 - 1} is odd and j € {2,..., 2n-1 - 2} is even. If x is odd, then N(x) = {2n-1 + x + i, x -i, 2n-1 + x + j, 2n-1 + x - j }. If x is even, then N (x) = {x + i, 2n-1 + x - i, x+j, x - j }. Suppose X is the bicirculant such that the vertex set V(X) can be partitioned into to subsets U = {uk | k € Z2n-i} and V = {vk | k € Z2n-i}, and there is an automorphism of X such that p(uk) = wk+1 and p(vk) = vk+1, k € Z2n-i. The edge set E(X) can be partitioned into three subsets: L = Ukez2„-i {uk, uk+i | l € L}, M = Ukez2„-i {uk, vk+m | m € M}, R = Ukez2„-i {vk, vk+r | r € R}, so we have L = {±(2n-2 + |)}, M = {2n-2 + i+1, -^}, R = {±§}. Then X = BC2n-i [L, M, R]. Let 7 be the mapping as follows: {x ^ u x-i, if x is odd; x ^ v x, if x is even. It follows that r = X. Note that BC2n-i [L, M, R] = BC2,-i [aL, aM + b, aR] with a, 6 € Z2n-i and a invertible [12]. Then r = BC2n-i [L, M', R] with M' = M + ^ = {0, 2n-2 + i}. In particular, r is connected since (L, M', R) = Z2n-i. When j = 2i, there are no triangles with three vertices of the form {uk, Vk+2n-2+ ¿+i, vk- i-i }, but there is a triangle with three vertices as {vk/, uk,+ i-i, uk,_2„-2_i+i } since for n > 3, k + 2n-2 + 1+1 ± j ^ k - i-1 (mod 2n-1) 2 2 ^ 2 v 7 k' + - (V-2 + 0 = k' - 2n-2 - (mod 2n-1). This implies that there is no automorphism of X which permutates uk and vk/. So X is not vertex-transitive when n > 3. This completes the proof. □ At last, we propose the following questions for further research. Question 7.9. Classify finite GCI-groups, such as Zm where m is odd with at least two different prime divisors, abelian groups, dihedral groups and some classes of finite simple groups. Question 7.10. Characterize the structure of the automorphism group of any generalized Cayley graph. Question 7.11. Classify Cayley regressions for certain types of group, such as the cyclic groups and the dihedral groups. References [1] B. Alspach and T. D. Parsons, Isomorphism of circulant graphs and digraphs, Discrete Math. 25 (1979), 97-108, doi:10.1016/0012-365x(79)90011-6. [2] A. O. Asar, Involutory automorphisms of groups of odd order, Arch. Math. (Basel) 36 (1981), 97-103, doi:10.1007/bf01223675. [3] L. Babai, Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329-336, doi:10.1007/bf01895854. [4] B. Elspas and J. Turner, Graphs with circulant adjacency matrices, J. Comb. Theory 9 (1970), 297-307, doi:10.1016/s0021-9800(70)80068-0. [5] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775-1029, http://projecteuclid.org/euclid.pjm/1103053 94 3. [6] C. D. Godsil, On Cayley graph isomorphisms, Ars Combin. 15 (1983), 231-246. [7] D. Gorenstein, Finite Groups, Harper's Series in Modern Mathematics, Harper & Row, New York, 1967. [8] A. Hujdurovic, K. Kutnar and D. Marusic, Vertex-transitive generalized Cayley graphs which are not Cayley graphs, European J. Combin. 46 (2015), 45-50, doi:10.1016/j.ejc.2014.11.007. [9] A. Hujdurovic, K. Kutnar, P. Petecki and A. Tanana, On automorphisms and structural properties of generalized Cayley graphs, Filomat 31 (2017), 4033-4040, doi:10.2298/fil1713033h. [10] W. Jin and W. Liu, A classification of nonabelian simple 3-BCI-groups, European J. Combin. 31 (2010), 1257-1264, doi:10.1016/j.ejc.2009.11.003. [11] W. Jin and W. Liu, On Sylow subgroups of BCI groups, Util. Math. 86 (2011), 313-320. [12] I. Kovacs, B. Kuzman, A. Malnic and S. Wilson, Characterization of edge-transitive 4-valent bicirculants, J. Graph Theory 69 (2012), 441-463, doi:10.1002/jgt.20594. [13] L. G. Kovacs and G. E. Wall, Involutory automorphisms of groups of odd order and their fixed point groups, NagoyaMath. J. 27 (1966), 113-120, http://projecteuclid.org/ euclid.nmj/1118801619. [14] C. H. Li, Finite CI-groups are soluble, Bull. London Math. Soc. 31 (1999), 419-423, doi: 10.1112/s0024609399005901. [15] C. H. Li, On isomorphisms of finite Cayley graphs—a survey, Discrete Math. 256 (2002), 301-334, doi:10.1016/s0012-365x(01)00438-1. [16] A. Malnic, D. Marusic and P. Sparl, On strongly regular bicirculants, European J. Combin. 28 (2007), 891-900, doi:10.1016/j.ejc.2005.10.010. [17] D. Marusic, R. Scapellato and N. Zagaglia Salvi, Generalized Cayley graphs, Discrete Math. 102 (1992), 279-285, doi:10.1016/0012-365x(92)90121-u. [18] M. Muzychuk, Adam's conjecture is true in the square-free case, J. Comb. Theory Ser. A 72 (1995), 118-134, doi:10.1016/0097-3165(95)90031-4. [19] M. Muzychuk, On Adam's conjecture for circulant graphs, Discrete Math. 167/168 (1997), 497-510, doi:10.1016/s0012-365x(96)00251-8. [20] J. Pan, X. Yu, H. Zhang and Z. Huang, Finite edge-transitive dihedrant graphs, Discrete Math. 312 (2012), 1006-1012, doi:10.1016/j.disc.2011.11.001. [21] P. Shumyatsky, Involutory automorphisms of groups of odd order, Monatsh. Math. 146 (2005), 77-82, doi:10.1007/s00605-004-0289-5. [22] J. N. Ward, Involutory automorphisms of groups of odd order, J. Austral. Math. Soc. 6 (1966), 480-494, doi:10.1017/s1446788700004961. [23] M.-Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998), 309-319, doi:10.1016/s0012-365x(97)00152-0. [24] X. Yang, L. Feng and W. Liu, Some properties of graphs constructed from 2-designs, Appl. Math. Comput. 284 (2016), 1-11, doi:10.1016/j.amc.2016.02.045. [25] X. Yang, W. Liu, J. Chen and L. Feng, GCI-groups in the alternating groups, Appl. Math. Comput. 303 (2017), 42-47, doi:10.1016/j.amc.2017.01.022. [26] X. Yang, W. Liu, H. Liu and L. Feng, Incidence graphs constructed from t-designs, Appl. Anal. Discrete Math. 10 (2016), 457-478, doi:10.2298/aadm160914021y. [27] J.-X. Zhou and Y.-Q. Feng, Cubic bi-Cayley graphs over abelian groups, European J. Combin. 36 (2014), 679-693, doi:10.1016/j.ejc.2013.10.005. [28] J.-X. Zhou and Y.-Q. Feng, The automorphisms of bi-Cayley graphs, J. Comb. Theory Ser. B 116 (2016), 504-532, doi:10.1016/j.jctb.2015.10.004. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 425-440 https://doi.org/10.26493/1855-3974.1537.97c (Also available at http://amc-journal.eu) On constructing expander families of G-graphs* Mohamad Badaoui Normandie Univ-Caen, GREYC CNRS UMR-6072, Campus II, Bd MarechalJuin BP 5186, 14032 Caen cedex, France, and Lebanese University, Laboratoire de Mathématique, EDST, Rafic Hariri University Campus, Hadath P.O. Box 5, Beirut, Lebanon Alain Bretto Normandie Univ-Caen, GREYC CNRS UMR-6072, Campus II, Bd Marechal Juin BP 5186, 14032 Caen cedex, France David Ellison RMIT University, 124 Little La Trobe St, Melbourne VIC 3000, Australia Bassam Mourad Department of Mathematics, Faculty of Science, Lebanese University, Beirut, Lebanon Received 1 December 2017, accepted 11 July 2018, published online 14 August 2018 Like Cayley graphs, G-graphs are graphs that are constructed from groups. A method for constructing expander families of G-graphs is presented and is used to construct new expander families of irregular graphs. This technique depends on a relation between some known expander families of Cayley graphs and certain expander families of G-graphs. Several other properties of expander families of G-graphs are presented. Keywords: Cayley graph, diameter of a graph, abelian group, G-graph, expander family. Math. Subj. Class.: 05C40, 05C42, 05C69 *The authors sincerely thank the referee for many valuable suggestions and useful comments. The first and the fourth authors are supported by the Lebanese University research grants program. E-mail addresses: mohamad.badaoui1@gmail.com (Mohamad Badaoui), alain.bretto@unicaen.fr (Alain Bretto), davidellison@polytechnique.edu (David Ellison), bmourad@ul.edu.lb (Bassam Mourad) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ 1 Introduction Expander graphs are sparse graphs that have strong connectivity properties. Expander constructions have found extensive applications in computer science [13, 16], in constructing of algorithms, error correcting codes [12], random walks [23], and sorting networks [1]. If one chooses at random a family of d-regular graphs, it is almost certain to be an expander graph [10]. Nevertheless, constructing expander families is not an easy task. Most constructions use deep algebraic and combinatorial techniques; mainly through Cayley graphs and the Zig-Zag product (see for example [15, 19]). Like Cayley graphs, G-graphs are defined from groups, but they correspond to an alternative construction. These graphs, introduced in [6], have highly regular properties. In particular, because the algorithm for constructing G-graphs is simple, it appears to be a useful tool to construct new symmetric and semi-symmetric graphs [7]. Several extensively studied problems in graph theory such as the hamiltonicity of Cayley graphs (see e.g. [3,18] for the latest development on this problem) may as well be approached using these objects. For instance, G-graphs are used to characterize new classes of Hamiltonian Cayley graphs [4], and to improve some upper bounds in the cage graphs problem [6]. Recently in [9], the authors studied some robustness properties of G-graphs such as edge/vertex-connectivity and vertex/edge-transitivity. It turns out, that several families of G-graphs are optimally connected where an optimally connected graph can be thought of as a graph whose vertex-connectivity is equal to its minimum degree. Because of their nice properties, it is natural to consider the problem of constructing an expander family of G-graphs. One of the chief tools for constructing a family of expander graphs is the concept of Cayley graphs. The main advantage for using such graphs is that at first it enables us when fixing the size of the generating set, to construct a large family of sparse graphs in an effective and concise way. Additionally, the underlying properties of a group G and its generating set S can give us an insightful gaze on the expansion properties of its corresponding Cayley graph Cay(G, S). Generally speaking, it is hard to prove that a certain family of Cayley graphs is an expander family. Concerning this, a huge amount of research in the last few decades has been devoted to dealing with the following question: which sequence of groups corresponds to an expander family of Cayley graphs? Using some algebraic techniques that depend mainly on Kazhdan constant, many partial results were obtained. In fact, most of these results gave negative answers to this question for certain groups (see [14] and [17], see also Example 3.2 below). The purpose of this article is to present a technique for constructing such families. Our construction is based on a relation between some known expander families of Cayley graphs and certain expander families of G-graphs. More precisely, for a group G and a subset S of G with S* = |Js£S (s) \ {e} (i.e. if S = {si,..., sk}, then S* = {si,..., s0(si)-i,..., sk,..., sk(sk)-1}, where o(sj) denotes the order of s4), we prove the following main result (see below for terminology). Theorem 1.1. If {Cay(Gn,Sn*), n G N+} is an expander family, then {i>(G„,S„), n G N+ } is also an expander family. The rest of the paper is organized as follows. In Section 2, we give a review of some basic facts concerning groups, multigraphs, G-graphs and expander graphs that are needed for our purposes. In Section 3, we shall prove the preceding theorem. In addition, just like in the case of Cayley graphs, we prove that abelian groups can not yield an expander family of G-graphs. In Sections 4 and 5, we first identify a new method for generating an infinite regular family of Cayley graphs from another one by switching specific edges. This leads to a new infinite expander family of Cayley graphs on the projective special linear group PSL(2, Z/pZ). Consequently, we construct several new infinite families of expander G-graphs on the special linear group SL(2, Z/pZ) and projective special linear group PSL(2, Z/pZ). These families are formed of irregular graphs, in particular semi-regular, which are of the very few ones. 2 Preliminaries This section has been designed to give a general introduction to some of the basic facts needed from the theory of groups, multigraphs, expanders and G-graphs. This introduction is given here to provide a convenient repository for all readers. We discuss briefly the material we shall require from these theories and for more details on any of these subjects, see for example [2, 11, 14, 17, 20]. 2.1 Groups Throughout this paper, all groups are assumed to be finite. Let (G,., e) be a group, where e denotes the identity element of G and "." denotes the group operation (multiplicative notation). For every g in G we define the order of g, denoted by o(g), as the smallest integer l such that g1 = e. Let S = {si,...,sfc } be anon-empty subset of G, and let Omax(S) and Omin(S) be respectively the maximum and the minimum o(sj) for all i e {1,..., k}. A subset S of G is said to be symmetric if every element in S has its inverse in S. We define (S} to be the smallest subgroup of G which contains S. If (S) = G, then set S is said to be a generating set of G, or G is generated by S. If H is a subgroup of G then the set Hx is called right coset of H in G, and we denote by G/H to be the set of all right cosets of H in G. A subset TH of G is said to be a right transversal for H if TH contains exactly one element from each right coset of H in G. Let A and B be subsets of a set U, then we denote B \ A = {x e B and x e A} and A = U \ A. The special linear group SL(2, Z/qZ) is defined as follows: The projective special linear group PSL(2, Z/qZ) = SL(2, Z/qZ)/{± 12}, where 12 is the 2 x 2 identity matrix. 2.2 Multigraphs All multigraphs considered in this paper are undirected and finite. Generally, we define an undirected multigraph r as the triple (V (T), E(T), £r), where V (T) is the set of vertices, E(r) is the set of edges, and £r is an incidence function that associates with each edge of r an unordered pair of vertices of r. In addition, we denote by {u, v} the multi-edge that links vertices u and v. The multiplicity of the multi-edge {u, v} is the cardinality of the set of edges that links u and v. A multi-edge with identical end-points is called a loop. A multigraph is a simple graph if it has neither loops nor multi-edges with multiplicity greater than or equal to 2. The neighborhood of vertex u denoted by N(u) is the set of all vertices that are adjacent to u. The degree of a vertex v in a multigraph r, denoted by d(v) is the number of edges of r incident to v where each loop counts as two edges. The maximum and minimum degree of a multigraph r are denoted by A(T) and ¿(r) respectively. A multigraph r is d-regular if d(u) = d for all u G V(r). A family of d-regular multigraphs is formed of regular multigraphs where each has degree d, while a family of regular multigraphs is formed of regular multigraphs with possibly varying degrees. The distance d(u, v) between two vertices u and v is the number of edges in a shortest path that connects u and v. The diameter diam(r) of a multigraph r is defined by: diam(r) = max{d(u, v); u, v G V(r)}. Let r = (Vi, Ei, ^i) and r2 = (V2, E2, £2) be two multigraphs, a homomorphism from r1 to r2 is a couple (f, f #) where f: V ^ V2 and f # : E1 ^ E2 such that if ^1(a) = {u,v} then £2(f#(a)) = {f(u),f(v)}. A graph isomorphism is the couple (f, f #) where f and f # are bijective. We say that r1 is isomorphic to r2 if there exists an isomorphism between r1 and r2. In such a case, we write r ~ r'. A multigraph r = (V, E, £r) is k-partite if there is a partition of V into k parts such that each part is a stable set. We will write r = ([JieI Vi; E) where I = {1,..., k}. A multigraph is minimum k-partite (k > 1) if it is k-partite and not (k - 1)-partite. It is easy to verify that for any multigraph r, there exists k such that r is minimum k-partite. If a multigraph r is k-partite, then we will say that (Vi)ie{1j2 ...jfc} is a k-representation of r. Cayley graphs offer a combinatorial depiction of groups and their generators. More precisely, the Cayley graph Cay(G, S) is the multigraph with vertex set G and two elements x and y of G are adjacent if and only if y = s.x for some s G S. It is well-known that Cay(G, S) is connected if and only if G = (S) (see for example [14]). 2.3 G-graphs Definition 2.1. Let (G,., e) be a finite group. Let S be a nonempty subset of G. For s G S, consider the cycles (s)x = (x, sx,..., so(s)-1 x) of permutation gs: x ^ sx. Note that (s)x is the set {x, sx,..., so(s)-1x}. We define the G-graph $(G, S) in the following way: 1. The vertex set of $(G, S) is V = |Js£S Vs where Vs = {(s)x, x G T(s)} where T(s) is a right transversal for the subgroup (s). 2. For each (s)x, (t)y G V, there exists edge between (s)x and (t)y labeled g for each g G (s)x n (t)y, such an edge will be denoted by ({(s)x, (t)y}, g). If card((s)x n (t)y) = p, p > 1 then there exists p labeled edges between (s)x and (t)y, or {(s)x, (t)y} is a multi-edge with multiplicity p. Since the cosets of (s) form a partition of G, (Vs)seS is a |S|-representation of $(G, S). Every vertex (s)x has o(s) loops. We denote by $(G, S) the multigraph $(G, S) without loops. The multigraph (G, S) = ([JseS Vs; E) be a G-graph with \G\ = n and \S\ = k. Then the following holds. d((s)x) = o(s)(k — 1), for all (s)x G Vs, d((s)x) = n(k — 1), for all s G S, (s)xevs \E(*(G,S))\ = . 2.3.1 New results on G-graphs Proposition 2.6. Let $(G, S) be any G-graph such that \S\ = {si,..., sk}. Then the following are equivalent: 1 This definition is due to [4]. i. $(G, S) is d-regular, ii. o(si) = kzi, for all i € {1, ...,k}, iii. |Vsi | = | VSj |, for all i,j € {1,..., k}. Proof. Let (s)x € Vs, where s € S. From Proposition 2.5, we have d((s)x) = o(s)(k - 1) or o(s) = ^ifM, k - 1 and then = JG[ = |G|(k - 1) 1 s| o(s) d((s)x) . Therefore o(si) = o(sj) if and only if |VSi | = |VSj |, for all i, j € {1,..., k}. □ Remark 2.7. When $(G, S) is a regular multigraph, we use the notation O instead of o(s) for any s € S. The following lemma can be found in [22]. Lemma 2.8 ([22]). Let $(G, S) be a G-graph with S = {si,..., sk} a generating set of G, then all the multi-edges between levels Vsi and Vsj have the same multiplicity l(si)n(sj )|. As a result, we have the following corollary. Corollary 2.9. Let $(G, S) be a G-graph with S = {si,..., sk}. Then $(G, S) is a simple graph if and only if (si) n (sj) = {e} for all i, j € {1,..., k} with i = j. 2.4 Expanders Before we define expander graphs, we need to define some expansion parameters. Let r = (V, E, £r) be a non-oriented multigraph with |V| > 2 and V' be a subset of V. The edge boundary of V' in r denoted by dV'(r) (or simply dV' when no ambiguity occurs) is defined as follows: 5V'(r) = {a € E; £r(a) € V' x V'}. In other words, this is the set of edges emanating from the set V' to its complement. The rate of expansion of r is then defined as follows: h(r)= min . ( ) o) < r for all i G N+. That is jr, i G N+} is a sequence of bounded degree multigraphs, iii. There exists e G R+ such that r is an e-expander for all i G N+, then this family is called an expander family and an element of this family is an expander graph. If r is a d-regular multigraph, then in [14] it is proved that logd |V(r)| < diam(r). The next proposition is a simple generalization of this result. Proposition 2.12. Let r be a connected multigraph such that A(r) < r G N+. Then logr |V(r)| < diam(r). Proof. Consider v G V(r) and define B^v) = ju G V(r); d(v,u) < l}. We show by induction that |B;(v)| < rl. The result is trivial for l = 0. Suppose it is true up to l - 1 and let's prove it for l. Since every vertex in Bl_i(v) has at most r - 1 neighbors in Bl_i(v), then |Bl(v)| < (r- 1)|Bl_1 (v)| + |Bl_1(v)| = r|Bl_1(v)| < rrl-1 = rl. If l = diam(r), then Bl (v) = V(r) and therefore | V(r)| < rdiam(r_. □ 3 Cay-expanders and G-expanders In this section, we are mainly concerned with proving Theorem 1.1. First, we need to introduce more auxiliary materials. We start with the following definition which is virtually an interpretation of Definition 2.11 for the G-graph and Cayley graph cases. Definition 3.1. Let {Gj, i G N+} be a family of finite groups. We say that {Gj, i G N+} is a G-expander family, if for every i G N+ there exists a generating subset Sj of Gj such that j$(Gj,Sj), i G N+} is an expander family. More precisely, {Gj, i G N+} is a G-expander family if the following 3 conditions are satisfied: i. |V($(Gj, Sj))| = |Gj| EseSi oi) — ~ as i — to. ii. There exists a positive integer r such that A(<&(Gj, Sj)) < r for all i G N+ which by Proposition 2.5 means that for every (s)x G Vs we have d((s)x) = (|Sj| - 1)o(s) < A(<&(Gj,Sj)) < r G N+ for all i G N+. This in turn means that there exists r1,r2 G N+ such that 2 < |Sj| < r1 and o(s) < r2 for all s G Sj and for all i G N+. In addition, since A((Gj, Sj)) < r for all i G N+, then clearly Condition i. is equivalent to saying that |Gj | —> ^o as i — to. iii. There exists an e G R+ such that e < h($(Gj, Sj)) for all i G N+. Note that 2 < |Sj| since otherwise <&(Gj, Sj) will be a disconnected multigraph so that h( |Y|. In Cay(Gn,S;), we have |dW | > e|W |.Let f: dW ^ dH, {x,y} ^ ({(s, )x, (sj )y},y), where x € W, y € W, i and j are chosen so that xy-1 € (s,) and y € U(s)xeH- (s)x (note here that there may be several possible choices for i and j). Now observe that if f (x, y) = f (x',y'), then xx'-1 € (s,) and y = y'. So for all a € dH, |f-1(a)| < Omax(Sn). Hence, |dH| > IdWI > e|W| > emax, |Xj| > e|X| Omax(Sn) °max(Sn) °max(Sn) °max(Sn) |Sn1 Using |dH| > |Y| and |H| = |X| + |Y|, we obtain |dH|>1mi" {omaxcSn)!^ •1}|H|> 2min {• 1}|H|. This completes the proof. □ The following results are obvious consequences of Theorem 1.1. Corollary 3.5. If {Gn, n € N+} is a Cay-expanderfamily, then it is also a G-expander family. Corollary 3.6. If {Cay(Gj,S*), i € N+} is an expander family, then {$(Gj,Sj), i € N+} is also an expander family. Proof. By Theorem 1.1, {$(Gj, S,), i € N+} is an expander family. By Definition 3.1, there exists r € N+ such that o(sj) < r, for every sj € S,. Then | (sj1) n (sj2) | < r for all sj1, sj2 € Sj. Thus h($((;i'Si)) < h(2 i|o(si)=2 1 10 and S = {±1, ±2}. Then Cay(Z/nZ, S) is 4-regular multigraph on n vertices. Let H be a subgraph of Cay(Z/nZ, S) such that V(H) = {1, 2,3,7}. Let s1 = +1 and s2 = +2. Then Nsi (H) = {4,8}, Ns-i (H) = {o, 6}, NS2 (H) = {4, 5,9}, and Ns-i (H) = {0, 5,n - 1}. Thus 1 |dH(Cay(Z/nZ, S))| = 2(|N,i (H)| + |Ns2 (H)|) = 10. Next, we shall show that it is possible to construct an expander family of Cayley graphs from another one by switching some of its edges. Corollary 5.4. Let {Cay(G,, { s±1,s±1}); i G N+} be an expander family. If o(s1), o(s2), and o(s1s2) > 2, then {Cay(Gj, {s^1, s1s2, s-1s-1}); i G N+} is also an expander family. Proof. Let V(H) = {x1,..., xt} G G. Define d'H, d"H to be the sets of emanating edges from V(H) in the multigraphs Cay(Gj, { s±\s^1}) and Cay(Gj, {s^1, S1S2, s-1s-1}) respectively. By Lemma 5.2, we have: |d'H| = 2|Nsi(H)| + 2|Ns2(H)|, and (1) |d''H | = 2|Nsi (H )| + 2|Nsis2 (H )|. (2) Let y G Ns2 (H), y = s2x for some x G H. i. If s1y G H, then s1s2x G H and s1s2x G Nsis2 (H). ii. And if s1y G H, then s1s2x G H. Let H1 and H2 be the set of vertices of H defined as follows: H1 = {x G H/s2x G H and s1s2x G H}, H2 = {x G H/s2x G H and s1s2x G H}. From equalities (1) and (2), we have 2|Nsi (H )| + 2|Ns2 (H )(H1)| + 2^ (H )(H2 )| = |d'H |, 2|Nsi(H)| +2|Nsis2(H)(H1)| + 2^(H)(H2)| < |d''H|. From the definition of H2, we have |NSlS2 (H)(H2)| = 0, then 2|NS1 (H)| + 2|nSiS2(H)(Hi)| < |d''H|. Therefore, it holds that 2|Nsi(H)| + 4|Nsis2(H)(Hi)| - 2^(H)(Hi)| - 2^(H)(H2)| < 2|d''H| - |d'H|. From the definition of H1, we have |NSlS2(H)(H1)| = |NS2(H)(H1)| and similarly from the definition of H2, we have |NS2(H)(Hf)| = |Ns-i (H) nNS2 (h)|. Thus, 2|Nsi(H)| + 2|Ns2(H)(Hi)|- 2|Ns-i(H) n (H)| < 2|d''H| - |d'H|. Noticing that |Ns-i (H) n Ns2 (H)| < |Ns-i (H)| = |Nsi (H)|, then 2|Ns2(H)(Hi)| < 2|d''H| - |d'H|. Finally, we obtain 0 « (PSL(2, Z/pZ), {S-1, S2, S-1S-1}); p e P} are all isomorphic to {$(PSL( 2, Z/pZ), {S2,S|, S2S3}); p e P}. Table 2: Comparison of some graph invariants between Cay(G, S*) and S) for S = L and S = W. Cay(G, L*) $(G, L) Order |G| dtcS) = 12|G| Degree 5-regular multigraph d(u) =4 for all u G VS2 d(v) = 3 for all v G Vs2s3 Size 2 |G| |G| Cay(G, W*) a)+d(a>v) — ^ w(u)w(a)xd(u'a) — uev(Gi) uev (Gi) vev (g2) - J2 w(a)w(v)xd(a,v) + w(a) vev (G2) = H(Gux)+ H(G2,x) + . Ha(Gl,x) Ha(G2,x) urn \ u in \ i f \2 +-----T^--Ha(Gi,x) - Ha(G2, x) + w(a) = w(a) w(a) = H(Gi, x) + ii(G2, x) + w(^2 (Ha(Gi, x) - w(a)2) (iia(G2, x) - w(a)2) . We used the fact that, because G1 and G2 share the vertex a, E = E + E . ueV(G 1) ueV(G 1) u=v=a veV (G2) v£V (G2) u=v The second equation holds because HHa (G,x)= ^ w(a)w(v)xd(a'v) = veV (G) = E w(a)w(v)xd(a'v) + E w(a)w(v)xd(a'v) - w(a)2 = v£V (Gi) veV (G2) = Ha(Gi,x)+ Ha(G2,x) - w(a)2. □ Remark 4.5. The statement of Lemma 4.4 is a generalization of Theorem 2.1 from [4] for the case when two double weighted connected graphs Gi and G2 are point-attached to obtain G. However, it is easy to see that Lemma 4.4 can be generalized to the general case with any finite number of graphs. As the proof is short, we write and prove the following theorem for completeness of presentation. Theorem 4.6. Let Gi be graphs with one common vertex a and let Gi - a be disjoint. If g=u n = 1 Gi, the Hosoya polynomial of the graph G equals to n H(G, x) = E HH(Gi, x) + i=i and n Ha(G, x) = E Ha(Gi,x) - (n - l)w(a)2. i=i Proof. For n = 2 the result follows from Lemma 4.4, and for arbitrary n the result follows by induction. Suppose the equation (4.1) is valid for n - l, let Go = Un-11 Gi and G = G0 U Gn. Then, using Lemma 4.4, + w(a)2 (Ha(Go,x) - w(a)2j (Ha(Gn,x) - w(a)2^j H(G, x) = H(G0, x) + H(Gn, x) + 1 w(a) 2 n-1 = Y H(Gi,x)+H(Gn,x) + i=i n-2 n-1 i + YY W{a)^(^a(Gi,x) - w(a)2^ (Ha(Gj,x) - w(a)2^ + i=1 j=i+1 n-1 w(a) 2 W v i=1 + Wla^i ( Y ^a(Gi, x)) - (n - 2)w(a)2 - w(a)2) (Ha(Gn, x) - w(a)2) = n-2 n-1 1 Y H(Gi,x) + Y Y wUrfftaG ,x) - w(a)2^( Ha(Gj ,x) - w(a)2) + / \2 \ ^a( w( a) 2 w(a)2 = 1 i=1 j=i+1 v ' n-1 i + Y w(a)2 i^a(Gi,x) - w(a)2) (jia(Gn,x) - w(a)2) = 1 n 1 n i=1 i=1 j=i+1 and Y H(Gi,x) + Y Y wia]2(^a(Gi,x) - w(a)2^(Ha (Gj ,x) - w(a)2^j Ha(G,x) = Ha(Go,x) + Ha(Gn,x) - w(a)2 = n-1 Y Ha(Gi,x) - (n - 2)w(a)2 + Ha(Gn,x) - w(a) i=1 n 2 Y Ha(Gi,x) - (n - 1)w(a)2 . □ 1 5 Cycle-like and unicyclic graphs We now consider the case when the specific vertices a1,a2,... ,an in G are vertices of a cycle. Lemma 5.1. Let Gai, Ga2,..., Gan be disjoint rooted graphs and denote by GC the union of Gai, Ga2,..., Gan, joined by the edges a1a2, a2a3,..., an-1an and ana1, see Figure 4. In this case the Hosoya polynomial of GC equals to H(GC, x) = Y H(Ga lC,x) = y n(Gai,x) 1 n 1 n + < W A (Gai ,x) ■ Haj (Gas ,x) ■ xd(ai>a > (5.1) i=1 j=i+1 w(ai)w(aj) j j and Hai (GC,x) = Ha (Gai ,x) + V ^Hj (Gj ,x) ■ xd(aia j j=1 w(«j )" j=i for every i = 1, 2,... ,n. Figure 4: A cycle-like graph GC. Remark 5.2. In Lemma 5.1, the graph GC can be any connected graph with a cycle such that all vertices a1,a2,... ,an of the cycle C are cut-vertices, and the Gai can be any subgraphs. Clearly, this includes as a special case the unicyclic graphs. Proof of Lemma 5.1. Following the idea of the proof of Lemma 4.1, we write H(GC ,x)= ^ w(u)w(v)xd(u'v) = {u,v}C V (G) n— 1 n J2 w(n)w(v)xd(u'v) + ^ J2 J2 w(u)w(v)x d(u,v) _ = 1 {u,v}CV (Gai) =1 j=i+1 uev (Gai) vev (g„.) n 1 n Y,H(Gai + Hai (Gai , x) ■ xd(ai>a) ■ Ha, (Ga, , x) i=1 i=1 j=i+1 w(ai)" w(aj) and Hai (GC, x) = w(ai)w(v)x vev (G) d(ai,v) _ ^^ w(ai)w(v)xd(ai'v) + ^^ ^^ w(ai)w(v)x d(ai )+d(a, ,v) vev (Gai) j=1vev (g) j=i j = (Gai,x) + V Ha, (Ga, ,x) • xd(ai'a>>, z—' w(aj) J J j=i as claimed in Lemma 4.1. □ Example 5.3. Let C be a cycle on three vertices a, b and c, with edges ab, bc and ca (see Figure 5). From Lemma 5.1 it follows H(C, x) = H(Ga, x) + H(Gb, x) + H(Gc, x) + + , * Ha(Ga,x)Hb(Gb,x)xX(ab) + w(a) w( b) + . . Hb(Gb,x)Hc(Gc,x)xx(bc) + . .1 . . Hc(Gc,x)Ha(Ga,x)xx(ca) = w(b) w(c) w(c) w( a) = w(a)w(b)xx(ab) + w(b)w(c)xx(bc) + w(c)w(a)xx(ca). The result is the same as expected, from the definition (3.3). A similar reasoning applies to larger cycles. Figure 5: A cycle graph C. Recall that our original motivation was to design a linear algorithm for calculating the Hosoya polynomial of a cactus graph. Observe that from equation (5.1) it appears that a double sum needs to be calculated which yields quadratic complexity. Therefore, we are going to consider this case more carefully and provide an alternative expression that will later be used to show the existence of a linear algorithm. First, we will consider path-like graphs, and introduce, for technical reasons, polynomials of two variables that will in turn allow a natural generalization to handle cycle-like graphs. Let GP be a path-like graph, i.e. the union of disjoint graphs Gai ,Ga2,... ,Gan, rooted at a1,a2,... ,an respectively, and joined by the edges aia2, a2a3,. .., an-1 an. The Hosoya polynomial H(GP, x) and the polynomial Han (GP, x) can be calculated recur- sively using Lemma 4.1 (n - 1) times. For such a path-like graph GP, we use the notation Hi — Ga i i j Hj — Gai U jaia2, a2a3,. .., a,j-ia,j} , 1 where GP = Hn. By Lemma 4.1, the Hosoya polynomial and the corresponding polynomials Haj are, for j = 1: H(Hux) — H(Gai ,x), Hai (Hi,x) — Hai (Gai ,x), and, for j > 1: H(Hj, x) — H(Ga, ,x) + H(Hj—i,x) + + Wjj) Haj (Gaj ,x)Haj-i j ^^ ' Ha, (Hj ,x) — Haj (Gaj ,x) + Ha;-i (Hj — i, x)xX(a>-ia> > . w(aj—i) The recursion above implies that, given polynomials H(Gai,x) and Hai(Gai,x), i = 1,2,..., n, we need 3(n - 1) additions and 2(n - 1) multiplications (of polynomials) to obtain all H(Hi, x) and Ha. (Hi, x). From the definition of the graphs Hi and the recursions written above, we also have Lemma 5.4. For the graphs Hj, j = 2,... ,n, the following is true 3 H(Hj ,x)=£ H(Gai ,x) + = 1 j-l j + } ^ } ^ / \ / \ Hai (Gai ,x)Hae (Gaf, x)x^~jk i ( + ), h Ai w(ai)w(a^) j—i ( ) Haj (Hj ,x) — Ha, (Ga, ,x)+£ Hai (Gai, x)^ k-i X(a"ak+i ) . Proof. Lemma follows directly by the induction on j, using Lemma 4.1 and the recursive formulae above. □ Before generalizing from path-like to cycle-like graphs, we introduce auxiliary polynomials of two variables. For technical reasons, to distinguish the exponents based on the distance on the path and off the path, i.e. the exponents based on the distance within the graphs Ga1, we introduce a second variable y. For example, assume that a shortest path from u G V(Gai) to v G V(Gaj) has distance d(u, v) — d(u, ai) + d(ai, aj) + d(aj, v). Then the contribution to the auxiliary polynomial is w(u)w(v)xd(u'ai)yd(ai'aj)xd(aj'v). More formally, j HiHj ,x,y) :=£ H{Gai , x) + (5.2) 3 yj- / , " v a. i=i 3'-i 3 I + E E ( ) ( )HHai(Gai,x)Ha,(G„,,x)y^=^fc+O, 3-1 ( ) Ha3 (Hj, x, y) := Haj (Gaj, x) + ^ —jHai ^, x)yji *(akak+!) . .=1 w(ai) After the introduction of the new variable y, the recursion formulae become H(Hi,x, y) = ii(Ga! ,x), (5.3) HHa! (Hi,x,y)= HHai (Ga! ,x) (5.4) for j = 1 and at every step of the recursion we have F(Hj, x, y) = Hf(Ga,. ,x) + tf(Hj_i,x)+ (5.5) + -r Haj (Gaj , x)Haj_ i (Hj-i, x)y*(aj-iaj), —(aj)—(a3--i) j j j HHaj (H3, x, y) = Haj (Gaj ,x) + ——3HHaj-i (Hj-i, x)yA(aj-iaj) . (5.6) ^^ (aj_i) It is obvious that HXHj, x, x) = Hi(Hj, x) and Haj (Hj, x, x) = Haj (Hj, x). Let GC be a cycle-like graph, i.e. the union of disjoint graphs Gai, Ga2,..., Gan, rooted at ai,a2,...,a„ respectively, and joined by the edges &i&2, a2&3, .. ., an-ian and a„ai. Denote by L the girth of the cycle C on vertices ai,o2,...,a„, specifically L = A(a„ai) + ^ A(ajai+i) . ^-1 " (a. i=i Define new modified polynomials of two variables of the path-like graph GP as follows H m(GP, x, y) := £ Hi (Gai ,x) + i E 1 min^ A(afcafc + i),L- £ A(akak+i)} 7 r ^ -Hi (Gai ,x)Hf(G a,, x)y k=i k=i , Hamn (GP, x, y) := Han (Ga„ ,x) + n — i n-i n-i E—(an) H min{ E A(afcafc+i),L- E ^(afcafc+i)} , \ Hai (Gai ,x)y fc=i fc=i . —(a.) .=1 According to Lemma 5.1, the next statement is obvious. Proposition 5.5. HHm(GP,x,x) = HH(GC,x) and HH^(GP,x,x) = Han(GC,x). Example 5.6. Let GC be a cycle-like graph which is the union of disjoint graphs Gai, Ga2, Ga3 and Ga4 and edges and We assume that the polynomials HHa. (Ga., x) and H"(Gai,x) are given and that A(a1a2) = 2, A(a2a3) = 5, A(a3a4) = 3 and A(a4a1) = 1 as we see in Figure 6. In this case L = 11. The computations below a4 a1 a4 ai 3 2 la3 a2 Figure 6: A cycle-like graph with given lengths of cycle's edges. are following the recursion for the path-like graph and the idea of the separation of the exponents, and, in addition, we observe that the distances on the path change when the path is closed to a cycle with the edge a4a1. The base of the recursion is HHai (Hi,X,y) = HHai (Gai ,x), #(#i,x,y) = Hi (Gai ,x). Other steps are clearly HHa 2 (H2,x,y) = HHa 2 (Ga2, x) + HHai (Hi,x,y)y2 = w(ai) = Ha2 (Ga2 , x) + ( )HHai (Gai , x)y^ w(ai) Hf(H2, x, y) = ii(Ga2, x) + Hf(Hi, x, y) + +--;—^—7HHa2 (Ga2, x)HHai (Hi, x, y)y2 1 w(a2)w(ai) F(Ga2 ,x) + Hf(Gai , x) + w(a2)w(ai) Ha2 (Ga2 ,x)Hai (Gai , x)y , HHa3 (H3, x, y) = HHa3 (Ga3, x) + H^ (h-2,x, y)y5 = w(«2) = HHa3 (Ga3 , x) + w(°3) HHa2 (Ga2 , x)y5 + w(°3) HHai (Gai , x)y7, w(«2) w(ai) H(H3,x,y) = H(Ga3 ,x) + H(H2,x,y) + + —,-^-7Ha3 (Ga3 ,x)Ha2 (H2,x,y)y5 = w(a3)w(a2) = H(Ga3, x) + H(Ga2, x) + H(Gai, x) + +--7—^—rHal (Gal, x)Ha2 (Ga2, x)y2 + w(a2)w(ai) + 7 \ 7 Ha3 (Ga3 ,x)Ha2 (Ga2 : x)y + w(a3)w(a2) + 7 \ 7 THa3 (Ga3 : x)Hai (Gai i x)y ■ w(a3)w(a\) Similarly, w (0,4) Ha4 (H4, x, y) = Ha4 (Ga4, x) +---—-Ha3 (H3,x,y)y = w(a3) = -Ha4 (Ga4 , x) + Ha3 (Ga3 , x)y3 + w(a3) + ^ Ha2 (Ga2 , x)y8 + ^ Hai (Gai , x)y10 w(a2) w(ai) and H(H4 ,x,y)= H(Ga4 ,x) + H(H,x,y) + + , \ , . Ha 4 (Ga4 ,x)Ha3 (H3 ,x,y)y3 = w(a4 )w(a3) = H(Ga4, x) + H(Ga3, x) + H(Ga2, x) + H(Gai , x) + + 7 n 7 rHai (Gai 1 x)Ha2 (Ga2 1 x)y'2 + w(a2)w(ai) + 7 7 ^Ha3 (Ga3 ,x)Ha2 (Ga2 ,x)y + w(03)W(02) +--7-^-rHa3 (Ga3 , x)Hai (Gai , x)y7 + w(a3)w(ai) + , 1 , , Ha4 (Ga4 , x)Ha3 (Ga3 , x)y3 + w(a4)w(a3) 1 + 7 \ 7 rHa4 (Ga4 1 x)Ha2 (Ga2 1 x)yS + w(04)W(02) + 7-n-7-r Ha4 (G a4 1 x)Hjai (Gai 1 x)yW ■ w(a4)w(ai) Than the modified polynomials of two variables are HZ (H41 x1 y) = Hia 4 (Ga4 1 x) + ^Pl Hi a 3 (Ga3 , x)y3 + 4 w(a3) , w(a4) jj tr< S3, w(a4) jj (n , +--7-rHa2 (Ga2 1x)y +---THai (Gai 1 x)y w(a2) 2 2 w(a1) i i = HH04 (G04, x) + Httl (Gai, x)y + w(ai) + ( ^ Ha3 (Ga3 ,x) + ^ Ha2 (G02 ,x))y3 and Fm(F4, x, y) = F(G04, x) + ii(G03, x) + F(G02, x) + i/(G„i, x) + + 7 Ñ 7 rHHa1 (Gai, x)HHa2 (Ga2 , x)y2 + w(a2)w(ai) + 7 \ 7 Ha3 (Ga3 , x)HHa2 (Ga2 , x)y + w(a3)w(a2) +--7—^—rHHa3 (G03, x)HH0i (Gai, x)y4 + w(a3)w(ai) +--7-^-rHHa4 (Ga4 ,x)HHai (Gai, x)y1 + w(«4)w(ai) + (—/-Ñ-/-ÑHHa4 (Ga4 , x)HHa3 (Ga3 , x) + yw(a4)w(a3) + , w ^ HHa4 (Ga4 , x)HHa2 (Ga2 , x) ) y3 . w(a4)w(a2 ) / Hence, the Hosoya polynomial and the polynomial HHa4 (GC, x) are F(GC,x)= î"m(H4,x,x) and HHa4(GC,x) = HH^(H4,x,x). Observe that the time complexity of the transformation of a polynomial of two variables x and y to the polynomial of one variable x (where y ^ x) is comparable to time complexity of multiplication of polynomials. Thus, we can conclude: Theorem 5.7. The Hosoya polynomial of a cycle-like graph can be computed using the recursion (5.3), (5.4), (5.5), (5.6) in linear time, in the model where addition and multiplication of polynomials are atomic operations. 6 The Hosoya polynomial in terms of edge contributions It is well-known that the Wiener number can be expressed as a sum of edge contributions, see for example [19]. Recall, for example, the version for weighted graphs. Lemma 6.1 ([22]). For a weighted graph G, -y n*( n*(a, b) W(G)= £ A(e) • £ ^^T w(«)w(b), where P*6 3 e denotes a shortest path between a and b, n*(a, 6) is the number of shortest paths with endpoints a and b and n*(a, b, e) is the number of all shortest paths with endpoints a and b traversing edge e. Hence, the quotient ""i""^ represents the proportion of all shortest paths between a in b including e, among all shortest paths between a and b. On a tree, there is a unique shortest path between any pair of vertices, thus n* (a, b, e) = n* (a, b) = 1 for all a, b. If G is a cactus graph, ""i""^ can only have value 1 or 1. Clearly, ""i""^ = 1 exactly when a and b are opposite vertices of a cycle and edge e is on this cycle. (More precisely, with opposite vertices of a cycle we mean that d(a, b) = L/2 where L is the girth of the cycle.) It may be interesting to note that this formulation has an interesting meaning when considering the weighted graphs as communication networks [6, 18, 24]. In this case the Wiener number is interpreted as the total communication traffic in the network, where naturally the communication between nodes u and v contributes d(u, v)w(u)w(v) (distance times population sizes of the two nodes). Assuming that all the communication is routed on the shortest paths and that it is evenly distributed among shortest paths if there are many of them, the edge contribution corresponds to the communication load on the edge. Example 6.2. Let G be a communication network represented in Figure 7, where all edges have lengths 1. There are exactly three shortest paths between vertices u and v. Ratios indicating the part of the communication load are attached to the edges on the shortest paths. As shown in [26], the Hosoya polynomial can be represented as a sum of the contributions of all shortest paths: Lemma 6.3 ([26]). For a weighted graph G, 1 ^ T-r A(e) .ff(G,x)= y, En*(a b) w(a)w(b) nX {a,b}CV(G) P:,b ( , ) eepi,t where P"b denotes a shortest path between a and b and n*(a, b) is the number of all shortest paths with endpoints a and b. Representing the Hosoya polynomial in terms of edge contributions is hence somewhat more complicated: For each path crossing the edge, one needs to know the amount of traffic (the intensity of the traffic corresponds to "„(" b) w(a)w(b)) but also the length of the paths. In case when G is a weighted tree, the Hosoya polynomial was expressed as a sum of edge contributions and a recursive formula for computing the Hosoya polynomial was given in [26]. In this section we show that the Hosoya polynomial can be similarly expressed as a sum of edge contributions on general graphs, and then provide a somewhat more elaborated expression that holds for cactus graphs. Lemma 6.4. The modified Hosoya polynomial, defined by (3.5), is a sum of edge contributions M(G,x) = £ X(e) ■ Me(G,x), (6.1) eeE(G) where Me(G, x) is given by Me(G,x)= V n*(u,v,e) w(u)w(v)xd(u'v). (6.2) —^ n* (u v) PU ,, 3e (u,v) Here P*v 3 e denotes a shortest path between u and v including e, n* (u, v) is the number of all shortest paths with endpoints u and v and n*(u, v, e) is the number of all shortest paths with endpoints a and b including edge e. Proof. To see this, it is enough to sum up the contribution of each edge to M(G,x) in two different ways. Each pair of vertices u, v contributes d(u, v)w(u)w(v)xd(u'v) to the modified Hosoya polynomial. This can be regarded as a contribution of the pair u, v or it can be divided into n* (u, v, e)/n* (u, v) path contributions, which can be further regarded as a sum of edge contributions along the path. An edge contributes as many times as it appears on various shortest paths. Hence, one can sum up the lengths of all shortest paths, or, equivalently, sum up the contributions of all edges. □ Let G be a cactus graph. Recall that each edge e of a cactus graph is on at most one cycle, in other words, either e is not on a cycle or there is a unique cycle C with e G C .On the other hand, a vertex in a cactus graph can lie on more than one cycle. In case when the edge e = ab does not lie on a cycle, we can write our graph G as disjoint union of two graphs, denote them Ga and Gb, connected with edge e = ab (defined in Lemma 4.1), see Figure 1. On the other side, when edge e with endpoints a and b lies on a cycle C, we can use notations from Lemma 5.1, see Figure 4: e is one of the edges named aiai+i with ai = a and ai+\ = b, Gai = Ga, Gai+1 = Gb for some i G {1, 2,... ,n}. We can also say that Ga is the connected component of G - E(C) with a G G - E(C), where G - E(C) denotes the graph G without edges of the cycle C. According to Lemma 4.1 and Lemma 5.1, we can derive the Hosoya polynomials H(G, x) and M(G, x) for a cactus graph G as sums of edge contributions. Theorem 6.5. The modified Hosoya polynomial M(G, x) on a weighted cactus graph G is a sum of edge contributions x M(G.x)= £ (/^ ,\ lf ^ + e=abEE(G) K ' K ' Yo / \o e not on a cycle let s g( ( / ^ / ^ ^ e on a cycle In case when e = ab is on a cycle C with girth L and vertices b = bo, bi, b2, ..., bM, aK, a^-i, .. ., ai, ao = a, M S1, d(ai, bj) = L/2 we define Nij = < 10, otherwise. Proof. As every edge e of a cactus graph G does not lie on a cycle or there is unique cycle including e, we can discuss separately the two cases. Case 1: The edge e with endpoints a and b does not lie on a cycle. Then G = Ga U {e} U Gb n*(u,v,e) n* (u,v) (see Figure 1) and, obviously, " *Unv) = 1 for allu e Ga and all v e Gb. Using the definition (3.6) Ha(Ga,x) • Hb(Gb,x) = w(a)w(b) w(u)w(v)x d(u,a)+d(b,v) uev (Ga) vev (Gb) and Ha(Ga,x) • Hb(Gb,x) • xx(e) ^ d(u v) - = y, w(u)w(v)xd(u'v), j(a)w(b) w w uev (Ga) vev (Gb) since d(u, a) + A(e) + d(b, v) = d(u, v). So, the contribution (6.2) of the edge e in Case 1 is equal to „ , Ha(Ga,x) • Hb(Gb,x) • xx(e) Me(G,x) = --^ w(a) w( b) \0 (a)w(b) T dt ———dt xx(e) I fMa(Ga,t) u Mb(Gb,t) Case 2: The edge e with endpoints a and b lies on a cycle C with girth L. Let A = {a = a0,al,a2,... ,aK} be the set of vertices of C that are closer to a than to b, i.e. d(a, ai) < d(b, ai). B = {b = b0,b1:b2,... ,bM} the set of vertices of C that are closer to b than to a, i.e. d(b, bi) < d(a, bi). Clearly, for a pair of vertices ai e A, bj e B the edge e lies on the unique shortest path between them exactly when d(a, b) = d(a, ai) + X(e) + d(b, bj) < L/2. Furthermore, e is on one of the two shortest paths exactly when d(a, b) = d(a, ai) + A(e) + d(b, bj) = Lj2 and is not on a shortest path between ai and bj when d(a, b) < d(a, ai) + A(e) + d(b, bj). Denote Hai (Gai,x)= Y!, w(u)w(ai)xd(u'ai) = w(ai) ^ w(u)x ueG*i ueG*i i = 0,1,... ,K d(u, ai) i Hbj (Gbj, x) = ^ w (bj)w(v)xd(b''v) = w(bj) J2 w(v)xd(bj'v) veQb- veQb j = 0,1,..., M. Since Hai (Gai,x) • Hbj (Gbj ,x) • xd(°i'bj) ^^ w(u)w(v)x d(u,v) w(ai)w(bj) v ^ v j uev(Qai) vev (obj) the contribution (6.2) of the edge e in Case 2 is equal to KM ( 1 \Ni3 Ha-(Ga-,x) • lib-(Gb. ,x) • xd(ai'bj) Me(G,x) = VV - a,v a" ' , j j '-= e( , ) ¿0j=0V y w(ai)w(bj) =11 (2 r(/ ™ -v(/- ¿=0 j=0 ^ ^ where N |1, d(ai, bj) = L/2 ¿j 0, otherwise. As we mentioned earlier, the case "„f^fv'f = 2 appears only when u G Ga., v G Gbj and ai and bj are opposite vertices of a cycle C, such that d(ai, bj) = L/2. In all other cases ""^e) =1. n 7 Linear algorithm In this section we give some details of the algorithm for computing Hosoya polynomial of a weighted cactus graph that is based on results provided in previous sections. Before writing the algorithm outline we recall the skeleton structure of a cactus graph and the depth first search algorithm. The algorithm and analysis of its time complexity are given in Subsection 7.2. The section is concluded with an example. 7.1 The structure of cactus graph and DFS algorithm In the skeleton structure (elaborated for example in [2]) that corresponds to every cactus graph G = (V(G), E(G)), the vertices are of three types: • C-vertex is a vertex on a cycle of degree 2, • G-vertex is a vertex not included in any cycle, • H-vertex or a hinge is a vertex which is included in at least one cycle and is of degree > 3. The depth first search (DFS) algorithm is a well known method for exploring graphs. It can be used for recognizing cactus graphs providing the data structure (see [17,21,22,23]). Let Gr be a rooted cactus graph with a root r. After running the DFS algorithm, the vertices of Gr are DFS ordered. The order is given by the order in which DFS visits the vertices. (Note that the DFS order of a graph is not unique as we can use any vertex as the starting vertex (the root) and can visit the neighbors of a vertex in any order. However, here we can assume that the DFS order is given and is fixed.) For any vertex v e V(G) we denote by DFN(v) the position of v in the DFS order and we set DFN(r) = 0. DFN is called the depth first number. Following [22], it is useful to store the information recorded during the DFS run in four arrays, called the DFS (cactus) data structure: • FATHER(v) is the unique predecessor (father) of the vertex v in the rooted tree, constructed with the DFS. • ROOT(v) is the root vertex of the cycle containing v i.e. the first vertex of the cycle (containing v) in the DFS order. If v does not lie on a cycle, then ROOT(v) = v by definition. We set ROOT(r) = r. (In any DFS order, if DFN(w) < DFN(v) and w is the root of the cycle containing v and v is the root of another cycle (it is a hinge), then ROOT(v) = w.) • For vertices on a cycle (i.e. ROOT(v) = v), orientation of the cycle is given by ORIEN(v) = z, where z is the son of ROOT(v) that is visited on the cycle first. If ROOT(v) = v, then ORIEN(v) = v. • IND(v) := |{u | FATHER(u) = v}| is the number of sons of v in the DFS tree. We omit detailed description of DFS algorithm here, as it is well known, see for example [21]. The pseudocode of the DFS algorithm is also written in [17]. Some properties of the DFS ordered vertices of cactus and the relationship between the definitions of C, G, H-vertices in a rooted cactus Gr and arrays in the DFS table is described in [17]. In the rest of the paper the following notations are used. For a given cactus graph G and a vertex v e V(G) let Gv be the rooted induced subgraph of G with the root v. The set of vertices of Gv is the set V(Gv) = {w e V(G) | DFN(w) > DFN(v)}. Let u = FATHER(v) and let the edge uv not be an edge of a cycle of G (i.e. ROOT(v) = v). The graph Gu is the induced rooted subgraph of G with the root u. The set of vertices of Gu is the set V(Gu) = {w e V(G) | DFN(u) < DFN(w) < DFN(v)}. The sketch of rooted graphs Gv, Gu and Gu is shown in Figure 8. Figure 8: A rooted graph G, 7.2 The algorithm The linear algorithm consists of three steps. First, the representation of a given weighted cactus is found, then, in Step 2 the initialization for the recursive algorithm is done and in the third step, the Hosoya polynomials of certain rooted subgraphs are computed recursively that finally give the Hosoya polynomial of the whole graph. More precisely, in Step 3 we traverse the DFS tree of the cactus in the reversed DFS order and for each vertex v compute Hv (Gv ,x) and H(Gv ,x). The algorithm continues until the last vertex in the back DFS order is considered, which is the root r of the cactus. The result follows from the fact that H(G, x) = H(Gr, x). We now give more details of each step. Step 1: Cactus recognition Using a DFS algorithm on the rooted cactus G (any vertex chosen for a root) the data structure of cactus graph can be derived, including arrays DFN(v), FATHER(v), ROOT(v), ORIEN(v) and IND(v). Step 2: Initialization For every vertex v we set Hv (Gv ,x) = Hv ({v},x) = w(v)2 and H(Gv ,x) = H({v},x) = 0. Step 3: Computation of polynomials H Start with v, the last vertex in the DFS order. Let u = FATHER(v). While v = u (i.e. v = u is not the root of G) do (3a) or (3b): (3a) If the edge e = uv is not an edge of a cycle of G (i.e. ROOT(v) = v): • If DFN(u) = DFN(v) - 1 (i.e. DFN(u) < DFN(v) - 1), there exists rooted subgraph Gu (see Figure 8). The algorithm calls itself recursively for the subgraph Gu, the rooted subcactus with root u and vertices in DFS table With DFN(u),..., DFN(v) - 1. We obtain H(Gu,x) = H(Gu,x) and Hu(Gu,x) = Hu(Gu,x). • After the recursion or when u and v are the sequential vertices in the DFS order, polynomials Hu (Gu, x) and H(Gu, x) are calculated according to Lemma 4.1: H(Gu,x) = H(Gu,x) + H(GV,x) + w(u}w(v) Hu(Gu,x)Hv(Gv,x)xx(uv) Hu(Gu, x) = Hu(Gu, x) + WuHv (Gv,x)xx(uv). • v = u and u = FATHER(v). (3b) If the edge e = uv lies on a cycle C (i.e. r = ROOT(v) = v): • We have to read and remember all cycle's vertices. Denote them by a\,a2,..., an where ai = v, an-1 = ORIEN(v) and an = r = ROOT(v). • If DFN(aj) < DFN(aj-i) - 1, denote by Kaj the rooted subcacti on vertices with DFN: DFN(aj) < DFN < DFN(aj-lL) for j = 2,3,...,n - 1. Re- cursively calculate polynomials HT(Kaj, x) and iHaj (Ka,, x) and repair polynomials HT(Gaj, x) and HTaj (Ga,, x) following Lemma 4.4: HT(Gaj , x) = i/(Gaj , x) + i/(Kaj , x) + + W1F (HHa,- (Gaj ,x) - w(Oj)2)(iHa3 (Ka,, x) - w(a,j)2) Ha, (Ga, , x) = Ha, (Ga, , x) + Ha, (Ka, , x) - w(a,j )2 . • According to the discussion in Section 5 we calculate HTr (Gr, x) and HT(Gr, x) using details Hr(Ga,, x) and HHa, (Ga,, x), j = 1, 2,..., n. • u is the vertex with DFN(u) = DFN(ORIEN(v)) - 1. • v = u and u = FATHER(v). We conclude the subsection summarizing the time complexity. Step 1: It is well-known that traversing the graph with DFS algorithm and computing arrays DFN(v), FATHER(v), ROOT(v), ORIEN(v) and IND(v) can be done within O(m) time. Obviously, Step 2 can be computed in O(m) time. In Step 3, existence of implementation that uses O(m) additions and multiplications of polynomials follows from Lemmata 4.1,4.4, 5.1, and Theorem 5.7. Hence we can conclude that the algorithm runs in linear time. Theorem 7.1. The algorithm for the Hosoya polynomial on a weighted cactus graph (given in Subsection 7.2) correctly calculates the polynomial, in linear time in the model where the addition and multiplication ofpolynomials are atomic operations. 7.3 Example Example 7.2. Let G be a cactus graph in Figure 9 (high) with representing DFS tree (one of possibilities) in Figure 9 (low) and its DFS structure in Table 1. Table 1: The DFS structure of graph G. v DFN(v) FATHER(v) ROOT(v) ORIEN(v) IND(v) Vl 0 Vl Vl V2 2 V2 1 Vl Vl V2 3 V3 2 V2 V3 V3 0 V4 3 V2 V4 V4 0 V5 4 V2 Vl V2 2 V6 5 V5 V6 V6 1 V7 6 V6 V7 V7 0 V8 7 V5 Vl V2 1 Vg 8 V8 Vg Vg 0 Starting from the initialization (Step 2) and following the algorithm (Step 3), we obtain • v = vg, u = vg: Gv9 = {vg}, Gv8 = {vg, vg} U vgvg, - step 2: iiV9 (Gv9, x) = 1, Hf(GV9, x) = 0, - step (3a): HHV8 (Gv8, x) = x +1, HH(Gv8, x) = x. Figure 9: A weighted cactus graph G (high) and its DFS tree (low). • v = vs, u = V5: V5V8 is on cycle with ai = vs, = V5, = = vi, - step (3b): Kv& = {v5,v6, vr}U V5V6 U V6V7, Kv2 = jv2,v3,v4}U v2v3 U v2v4, - recursively: H (Kv5, x) = 2x3 + 2x2 + x, HV6 (Kv5, x) = 2x3 + 2x2 + 4, HH (KV2, x) = 2x3 + 4x2 + 2x, HHV2 (KV2, x) = 4x2 + 2x + 4, - from Lemma 4.4: {v5, v6, vr, vs, vg} U v5v6 U v6vr U v5vg U vgvg, {v2, v3, v4, v5, v6, v7, vs, vg} U v2v3 U v2v4 U v2v5 U v5v6 U U v6v7 U v5vs U vsvg, x6 + 2x5 + x4 + 4x3 + 4x2 + 2x, 4x3 + 4x2 + 4, 5x6 + 8x5 + 7x4 + 14x3 + 14x2 + 8x, 4x4 + 4x3 + 4x2 + 4x + 4, C — Gv6 = Gv2 = k(gv6 ,x) = H^V6 (gv6 , x) H(Gv2 , x) = V2 (GV2 , x) - according to discussion in Section 5 we calculate (Gvi = G) H(Gvi, x) = 6x6 + 9x5 + 9x4 + 17x3 + 13x2 + 9x, Hvi (Gvi ,x) = x6 + x5 + 2x4 + 3x3 + 3x2 + x +1, • v = v1,u = vi: end of the algorithm. Finally, the Hosoya polynomial of graph G equals H(G, x) = H(Gvi, x) = 6x6 + 9x5 + 9x4 + 17x3 + 13x2 + 9x. References [1] F. M. Briickler, T. Doslic, A. Graovac and I. Gutman, On a class of distance-based molecular structure descriptors, Chem. Phys. Lett. 503 (2011), 336-338, doi:10.1016/j.cplett.2011.01. 033. [2] R. E. Burkard and J. Krarup, A linear algorithm for the pos/neg-weighted 1-median problem on a cactus, Computing 60 (1998), 193-215, doi:10.1007/bf02684332. [3] G. G. Cash, Relationship between the Hosoya polynomial and the hyper-Wiener index, Appl. Math. Lett. 15 (2002), 893-895, doi:10.1016/s0893-9659(02)00059-9. [4] E. Deutsch and S. Klavzar, Computing the Hosoya polynomial of graphs from primary subgraphs, MATCH Commun. Math. Comput. Chem. 70 (2013), 627-644, http://match.pmf.kg.ac.rs/electronic_versions/Match7 0/n2/ match70n2_627-644.pdf. [5] T. Doslic, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1 (2008), 66-80, doi:10.26493/1855-3974.15.895. [6] B. Elenbogen and J. F. Fink, Distance distributions for graphs modeling computer networks, Discrete Appl. Math 155 (2007), 2612-2624, doi:10.1016/j.dam.2007.07.020. [7] M. Eliasi and A. Iranmanesh, Hosoya polynomial of hierarchical product of graphs, MATCH Commun. Math. Comput. Chem. 69 (2013), 111-119, http://match.pmf.kg.ac.rs/ electronic_versions/Match6 9/n1/match69n1_111-119.pdf. [8] J. Galtier, I. Pesek, K. Prnaver and J. Zerovnik, Oriented networks design problem, J. Inf. Sci. Eng. 26 (2010), 1231-1242, http://www.iis.sinica.edu.tw/page/jise/2010/ 201007_05.html. [9] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34 (1994), 1087-1089, doi:10.1021/ci00021a009. [10] I. Gutman and B. Furtula (eds.), Distance in Molecular Graphs - Theory, volume 12 of Mathematical Chemistry Monographs, University of Kragujevac, Kragujevac, 2012, http: //match.pmf.kg.ac.rs/mcm12.html. [11] I. Gutman, S. KlavZar, M. Petkovsek and P. Zigert, On Hosoya polynomials of benzenoid graphs, MATCH Commun. Math. Comput. Chem. 43 (2001), 49-66, http://match.pmf. kg.ac.rs/electronic_versions/Match43/match4 3_4 9-66.pdf. [12] H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988), 239257, doi:10.1016/0166-218x(88)90017-0. [13] S. Klavzsar and I. Gutman, Wiener number of vertex-weighted graphs and a chemical application, Discrete Appl. Math. 80 (1997), 73-81, doi:10.1016/s0166-218x(97)00070-x. [14] S. Klavzar and M. Mollard, Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes, MATCH Commun. Math. Comput. Chem. 68 (2012), 311-324, http://match.pmf. kg.ac.rs/electronic_versions/Match68/n1/match68n1_311-32 4.pdf. [15] M. Knor, R. Skrekovski and A. Tepeh, Mathematical aspects of Wiener index, Ars Math. Con-temp. 11 (2016), 327-352, doi:10.26493/1855-3974.795.ebf. [16] X. Lin, S.-J. Xu and Y.-N. Yeh, Hosoya polynomials of circumcoronene series, MATCH Commun. Math. Comput. Chem. 69 (2013), 755-763, http://match.pmf.kg.ac.rs/ electronic_versions/Match6 9/n3/match69n3_755-7 63.pdf. [17] T. Novak and J. Zerovnik, Weighted domination number of cactus graphs, Int. J. Appl. Math. 29 (2016), 401-423, doi:10.12732/ijam.v29i4.1. [18] I. Pesek, M. Rotovnik, D. Vukicevic and J. Zerovnik, Wiener number of directed graphs and its relation to the oriented network design problem, MATCH Commun. Math. Comput. Chem. 64 (2010), 727-742, http://match.pmf.kg.ac.rs/electronic_ versions/Match64/n3/match64n3_727-7 42.pdf. [19] T. Pisanski and J. Zerovnik, Edge-contributions of some topological indices and arboreality of molecular graphs, Ars Math. Contemp. 2 (2009), 49-58, doi:10.26493/1855-3974.68.51b. [20] B. E. Sagan, Y.-N. Yeh and P. Zhang, The Wiener polynomial of a graph, Int. J. Quantum Chem. 60 (1996), 959-969, doi:10.1002/(sici)1097-461x(1996)60:5(959::aid-qua2)3.0.co;2-w. [21] K. Thulasiraman and M. N. S. Swamy, Graphs: Theory and Algorithms, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1992, doi:10.1002/9781118033104. [22] B. Zmazek and J. Zerovnik, Computing the weighted Wiener and Szeged number on weighted cactus graphs in linear time, Croat. Chem. Acta 76 (2003), 137-143, https://hrcak. srce.hr/103089. [23] B. Zmazek and J. Zerovnik, The obnoxious center problem on weighted cactus graphs, Discrete Appl. Math. 136 (2004), 377-386, doi:10.1016/s0166-218x(03)00452-9. [24] B. Zmazek and J. Zerovnik, Estimating the traffic on weighted cactus networks in linear time, in: 9th International Conference on Information Visualisation, IEEE Computer Society, Los Alamitos, 2005 pp. 536-541, doi:10.1109/iv.2005.48, held in London, UK, July 6-8, 2005. [25] B. Zmazek and J. Zerovnik, On generalization of the Hosoya-Wiener polynomial, MATCH Commun. Math. Comput. Chem. 55 (2006), 359-362, http://match.pmf.kg.ac.rs/ electronic_versions/Match55/n2/match55n2_35 9-3 62.pdf. [26] B. Zmazek and J. Zerovnik, The Hosoya-Wiener polynomial of weighted trees, Croat. Chem. Acta 80 (2007), 75-80, https://hrcak.srce.hr/12 82 7. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 467-485 https://doi.org/10.26493/1855-3974.1355.576 (Also available at http://amc-journal.eu) On Jacobian group and complexity of the I-graph I(n, k, I) through Chebyshev polynomials Ilya A. Mednykh * Sobolev Institute of Mathematics, Koptyuga 4, Novosibirsk, 630090, Russia Novosibirsk State University, Pirogova 1, Novosibirsk, 630090, Russia 4= Received 21 March 2017, accepted 24 May 2018, published online 26 August 2018 Abstract We consider a family of I-graphs I(n, k, l), which is a generalization of the class of generalized Petersen graphs. In the present paper, we provide a new method for counting Jacobian group of the I-graph I(n, k,l). We show that the minimum number of generators of Jac(I(n, k, l)) is at least two and at most 2k + 21 - 1. Also, we obtain a closed formula for the number of spanning trees of I(n, k, l) in terms of Chebyshev polynomials. We investigate some arithmetical properties of this number and its asymptotic behaviour. Keywords: Spanning tree, Jacobian group, I -graph, Petersen graph, Chebyshev polynomial. Math. Subj. Class.: 05C30, 39A10 1 Introduction The notion of the Jacobian group of a graph, which is also known as the Picard group, the critical group, and the dollar or sandpile group, was independently introduced by many authors ([1, 2,4, 9]). This notion arises as a discrete version of the Jacobian in the classical theory of Riemann surfaces. It also admits a natural interpretation in various areas of physics, coding theory, and financial mathematics. The Jacobian group is an important algebraic invariant of a finite graph. In particular, its order coincides with the number of spanning trees of the graph, which is known for some simplest graphs, such as the wheel, *The author is grateful to professor D. Lorenzini for helpful comments on the preliminary results of the paper and professor Young Soo Kwon, whose remarks and suggestions assisted greatly in completion of the text. Also, author is thankful to an anonymous referee for valuable recommendations. The results of this work were partially supported by the Russian Foundation for Basic Research (grants 16-3100138, 18-01-00420 and 18-501-51021) and by the program of fundamental researches of the SB RAS no.I.1.2., project 0314-2016-0007 and the Slovenian-Russian grant (2016-2017). E-mail address: ilyamednykh@mail.ru (Ilya A. Mednykh) ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ fan, prism, ladder, and Mobius ladder [6], grids [23], lattices [25], prism and anti-prism [26]. At the same time, the structure of the Jacobian is known only in particular cases [4, 7, 9, 17, 20, 21] and [22]. We mention that the number of spanning trees for circulant graphs is expressed is terms of the Chebyshev polynomials; it was found in [8, 27], and [28]. We show that similar results are also true for the I-graph I(n, k, l). The generalized Petersen graph GP(n, k) has vertex set and edge set given by V (GP (n,k)) = {ui,vi | i = 1, 2,...,n} E (GP (n,k)) = {uiui+i, UiVi, ViVi+k | i = 1, 2,...,n}, where the subscripts are expressed as integers modulo n. The classical Petersen graph is GP(5,2). The family of generalized Petersen graphs is a subset of so-called I-graphs ([3, 14]). The I-graph I(n, k, l) is a graph of the following structure V (I (n,k,l)) = {ui,vi | i = 1, 2,...,n} E(I(n,k,l)) = {uiui+i, UiVi, ViVi+k | i = 1, 2, .. . ,n}. where all subscripts are given modulo n. Since I(n, k, l) = I(n, l, k) we will usually assume that k < l. In this paper we will deal with 3-valent graphs only. This means that in the case of even n and l = n/2 the graph under consideration has multiple edges. The graph I(n, l, k) is connected if and only if gcd(n, k,l) = 1. If gcd(n, k,l) = m > 1, then I(n, k, l) is a union of m copies of the graph I(n/m, k/m,l/m). If m = 1 and gcd(k, l) = d, then the graphs I(n, k, l) and I(n,k/d,l/d) are isomorphic [5, 16, 24]. In the case of l = 1 it easy to see that the graph I(n, k, 1) coincides with the generalized Petersen graph GP(n, k). The number of spanning trees and the structure of Jacobian group for the generalized Petersen graph were investigated in [19]. The spectrum of the I-graph was found in [11]. Even though the number of spanning trees of a given graph can be computed through eigenvalues of its Laplacian matrix, it is not easy to find the number of spanning trees for I(n, k, l) using them. In this paper, we obtained a closed formula for the number of spanning trees for I(n, k, l), investigate some arithmetical properties of this number and provide its asymptotic behavior. Also, we suggest an effective way for calculating Jacobian of I(n, k, l) and find sharp upper and lower bounds for the rank of Jac(I(n, k,l)). 2 Basic definitions and preliminary facts Consider a connected finite graph G, allowed to have multiple edges but without loops. We endow each edge of G with the two possible directions. Since G has no loops, this operation is well defined. Let O = O(G) be the set of directed edges of G. Given e e O(G), we denote its initial and terminal vertices by s(e) and t(e), respectively. Recall that a closed directed path in G is a sequence of directed edges e-i e O(G), i = 1,... ,n such that t(ei) = s(ei+1) for i = 1,... ,n - 1 and t(en) = s(e^). Following [1] and [2], the Jacobian group, or simply Jacobian Jac(G) of a graph G is defined as the (maximal) Abelian group generated by flows u(e), e e O(G), obeying the following two Kirchhoff laws: Ki. the flow through each vertex of G vanishes, that is J2eeo t(e)=x ^(e) = 0 for all x e V(G); ' K2: the flow along each closed directed path W in G vanishes, that is J2eew w(e) = 0. Equivalent definitions of the group Jac(G) can be found in papers [1, 2, 4, 9, 12, 18, 20]. We denote the vertex and edge set of G by V(G) and E(G), respectively. Given u, v e V(G), we set auv to be equal to the number of edges between vertices u and v. The matrix A = A(G) = {auv }u,veV(G), called the adjacency matrix of the graph G. The degree d(v) of a vertex v e V(G) is defined by d(v) = J2u auv. Let D = D(G) be the diagonal matrix indexed by the elements of V(G) with dvv = d(v). Matrix L = L(G) = D(G) - A(G) is called the Laplacian matrix, or simply Laplacian, of the graph G. Recall [20] the following useful relation between the structure of the Laplacian matrix and the Jacobian of a graph G. Consider the Laplacian L(G) as a homomorphism Z|V 1 ^ Z|V|, where |V| = |V(G)| is the number of vertices in G. The cokernel coker(L(G)) = Z|V 1 / im(L(G)) — is an Abelian group. Let coker(L(G)) = Zdl 0 Zd2 © • • • © Zd|V| be its Smith normal form satisfying the conditions dj |di+1, (1 < i < | V|). If the graph is connected, then the groups Zdl, Zd2 — are finite, and \ IV | Z. In this case, Jac(G) = Zt1 © Zt2 © • • • © Zd IV |-1 is the Jacobian of the graph G. In other words, Jac(G) is isomorphic to the torsion subgroup of the cokernel coker(L(G)). Let M be an integer n x n matrix, then we can interpret M as a homomorphism from Zn to Zn. In this interpretation M has a kernel ker M, an image im M, and a cokernel coker M = Zn/ im M. We emphasize that coker M of the matrix M is completely determined by its Smith normal form. In what follows, by In we denote the identity matrix of order n. We call an n x n matrix circulant, and denote it by circ(a0, a1,..., an-1) if it is of the form / circ(ao, ai, .. ., a„_i) «0 ai a2 «n-1 «o «1 ai a2 a3 an-1 a^ 2 ao J Recall [10] that the eigenvalues of matrix C = circ(a0, a^ ..., an-1) are given by the following simple formulas Xj = p(en), j = 0,1,..., n -1 where p(x) = a0 + a1x + • • • + an-1xn-1 and e„ is the order n primitive root of the unity. Moreover, the circulant matrix C = p(T), where T = circ(0,1,0,..., 0) is the matrix representation of the shift operator T: (xo, X1,... ,x„_2, xn_1 ) ^ (x1,x2,..., x„_1 ,xo). By [15, Lemma 2.1] the 2n x 2n adjacency matrix of the I-graph I(n, k, l) has the following block form A(1 (n,k,1)) In cn where CÎ is the n x n circulant matrix of the form Ck = circ(0,..., 0,1, 0,..., 0,1,0,..., 0). k times i-2fc-1 k — 1 times V1 k n n Denote by L = L(I(n, k, l)) the Laplacian of I(n, k, l). Since the graph I(n, k, l) is three-valent, we have L = 3I2n - A(I(n, k, l)) = (3I"_- ^^ 3 3 Cokernels of linear operators Let P(z) be a bimonic integer Laurent polynomial. That is P(z) = zp + aizp+1 + • • • + as-1zp+s-1 + zp+s for some integers p, a1, a2,..., as-1 and some positive integer s. Introduce the following companion matrix A for the polynomial P(z): 0 I Is_ 1 ' .4 = -1, — «1,. .., —a. s — 1 where Is—1 is the identity (s - 1) x (s - 1) matrix. We will use the following properties of A. Note that det A = (-1)s. Hence A is invertible and inverse matrix A—1 is also integer matrix. The characteristic polynomial of A coincides with z—pP(z). Let A = (ay, j G Z) be a free Abelian group freely generated by elements ay, j G Z. Each element of A is a linear combination J2 j cj aj with integer coefficients cy. Define the shift operator T: A ^ A as a Z-linear operator acting on generators of A by the rule T: ay ^ aj+1, j G Z. Then T is an endomorphism of A. Let P(z) be an arbitrary Laurent polynomial with integer coefficients, then A = P (T) is also an endomorphism of A. Since A is a linear combination of powers of T, the action of A on generators ay can be given by the infinite set of linear transformations A: ay ^ J2i ai,ja, j € Z. Here all sums under consideration are finite. We set ftj = J2i ai,jai. Then im A is a subgroup of A generated by ftj, j G Z. Hence, coker A = A/ im A is an abstract Abelian group (xi, i G Z | J2i ai,jxi = 0, j G Z) generated by xi, i G Z with the set of defining relations J2i ai,jxi =0, j G Z. Here Xj are images of a.j under the canonical homomorphism A ^ A/ im A. Since T and A = P(T) commute, subgroup im A is invariant under the action of T. Hence, the actions of T and A are well defined on the factor group A/ im A and are given by T: Xj ^ xj+1 and A: Xj i ai,jxi respectively. This allows to present the group A/ im A as follows (xi, i G Z | P(T)xj = 0, j G Z). In a similar way, given a set P1(z), P2(z),..., Ps(z) of Laurent polynomials with integer coefficients, one can define the group (xi, i G Z | P^T)xj = 0, P2(T)xj = 0,...,Ps(T)xj =0, j G Z). We will use the following lemma. Lemma 3.1. Let T: A ^ A be the shift operator. Consider endomorphisms A and B of the group A given by the formulas A = P(T), B = Q(T), where P(z) and Q(z) are Laurent polynomials with integer coefficients. Then B: A ^ A induces an endomorphism B coker a of the group coker A — A/ im A defined by B|coker A(a + im A) — B(a)+im A, a G A. Furthermore (xi, i G Z | A(T)xj = 0, B(T)xj = 0, j G Z) = coker A/ im(B|coker a) = coker(B|TOker a). Proof. The images im A and im B are subgroups in A. Denote by (im A, im B) the subgroup generated by elements of im A and im B. Since P(z) and Q(z) are Laurent polynomials, the operators A = P(T) and B = Q(T) do commute. Hence, subgroup im A is invariant under endomorphism B. Indeed for any y = Ax G im A, we have By = B(Ax) = A(Bx) G im A. This means that B: A ^ A induces an endomorphism of the group coker A = A/ im A. We denote this endomorphism by B|coker A. We note that the Abelian group (xj, i G Z | A(T)xj = 0, B(T)xj = 0, j G Z) is naturally isomorphic to A/ (im A, im B). So we have A/ (im A, im B) = (A/ im A)/im(B|coker a) = coker A/ im(B|TOker a) = coker(B|coker a). The lemma is proved. □ 4 Jacobian group for the /-graph I(n, k, l) In this section we prove one of the main results of the paper. We start in the following theorem. Theorem 4.1. Let L = L(I(n, k, l)) be the Laplacian of a connected I-graph I(n, k, l). Then coker L = coker(An — I), where A is 2(k + l) x 2(k + l) companion matrix for the Laurent polynomial (3 — zk — z—k )(3 — zl — z—l) — 1. Proof. Let L be the Laplacian matrix of the graph I(n, k, l). Then, as it was mentioned above, L is a 2n x 2n matrix of the form L _ Z' 3In Cn ^n V —^n 3In — Ch where Ck = circ(0,..., 0,1,0,..., 0,1,0,..., 0). k times k — 1 times Consider L as a Z—linear operator L: Z2n ^ Z2n. In this case, coker(L) is an abstract Abelian group generated by elements x1, x2,..., xn, y1, y2,..., yn satisfying the system of linear equations 3xj — xj—k — xj+k — yj = 0, 3yj — yj—l — yj+1 — xj =0 for any j = 1,..., n. Here the indices are considered modulo n. By the property mentioned in Section 2, the Jacobian of the graph I(n, k, l) is isomorphic to the finite part of cokernel of the operator L. To study the structure of coker(L) we extend the list of generators to the two bi-infinite sequences of elements (xj jeZ and (yj )jeZ setting xj+mn = xj and yj+mn = yj for any m G Z. Then we have the following representation for cokernel of L: coker(L) = (xj, yj, i G Z | 3xj — xj+k — xj—k — yj = 0, 3yj — yj+i — yj—i — xj = 0,xj+n = xj ,yj+n = yj, j g Z). Let T be the shift operator defined by the rule T: xj ^ xj+1, yj ^ yj+1, j G Z. Consider the operator P(T) defined by P(T) = (3 — Tk — T—k)(3 — T1 — T—l) — 1. We use the operator notation from Section 3 to represent the cokernel of L. Then we have coker(L) = (x,, y,, i e Z | (3 - Tk - T-k)xcj = y^-, (3 - T1 - T-1)yj = x^-, Tnxj = xj, Tnyj = yj ,j e Z> = (x,, i e Z | (3 - T1 - T-1 )(3 - Tk - T-k^ = x^-,Tnxj = x^-, j e Z> = (x,, i e Z | ((3 - Tk - T-k)(3 - T1 - T-1) - 1)xj = 0, (Tn - 1)xj = 0, j e Z> = (x,, i e Z | P(T)xj = 0, (Tn - 1)xj = 0, j e Z>. To finish the proof, we apply Lemma 3.1 to the operators A = P(T) and B = Q(T) = Tn - 1. Since the Laurent polynomial P(z) = (3 - zk - z-k)(3 - z1 - z-1) - 1 is bimonic, it can be represented in the form P(z) = z-k-1 + aiz-k-1+1 + • • • + a2k+21-1zk+1-1 + zk+1, where a1, a2,..., a2k+21-1 are integers. Then the corresponding companion matrix A is (_0 | l2k+21-1_A y -1, -«1,..., -a2k+2i-1 J It is easy to see that det A =1 and its inverse A-1 is also integer matrix. For convenience we set s = 2k + 21 to be the size of matrix A. Note that for any j e Z the relations P(T)xj = 0 can be rewritten as xj+s = - xj - a1xj-+1 - • • • - as-1xj+s-1. Let xj = (xj+1, xj+2,..., xj+s)4 be s-tuple of generators xj+1, xj+2,..., xj+s. Then the relation P(T)xj = 0 is equivalent to xj = Axj-1. Hence, we have x1 = Ax0 and x-1 = A-1 x0, where x0 = (x1, x2,..., xs)'. So, xj = Aj x0 for any j e Z. Conversely, the latter implies xj = A xj-1 and, as a consequence, P(T)xj = 0 for all j e Z. Consider coker A = A/ im A as an abstract Abelian group with the following representation (x,, i e Z | P(T)xj = 0, j e Z>. Our present aim is to show that coker A = Zs. We have coker A = (x,, i e Z | P(T^ = 0, j e Z> = (xj, j e Z | x£ + 01x^+1 +-----+ as-1x£+s-1 + x£+s = 0, i e Z> = (xj, j e Z | (x^+1, x£+2,..., x^+s)4 = A(x£,x£+1,..., x^+s—1)4, i e Z> = (xj, j e Z | (x^+1, x^+2,..., x^+s)4 = A£(x1, x2,..., xs)4,i e Z> = (x1,x2,...,xs | 0> = Zs. Now we describe the action of the endomorphism B|coker A on the coker A. Since the operators A = P(T) and T commute, the action T|coker A: xj ^ xj+1, j e Z on the coker A is well defined. First of all, we describe the action of T |coker A on the set of generators x1, x2,..., xs. For any i = 1,..., s - 1, we have T|coker(x,) = x,+1 and T|coker a(xs) = xs+1 = -x1 - 01x2 - • • • - 0s-2xs-1 - as-A. Hence, the action of T |coker A on the coker A is given by the matrix A. Considering A as an endomorphism of the coker A, we can write T | coker a = A. Finally, B| coker a = Q(T | coker a ) = Q(A). Applying Lemma 3.1, we finish the proof of the theorem. □ Corollary 4.2. The Jacobian group Jac(I(n, k, 1)) of a connected I-graph I(n, k, 1) is isomorphic to the torsion subgroup of coker(An - I), where A is the companion matrix for the Laurent polynomial (3 - zk - z-k )(3 - z1 - z-1) - 1. The Corollary 4.2 gives a simple way to find Jacobian group Jac(I (n, k, l)) for small values of k, l and sufficiently large numbers n. The numerical results are given in the Tables 2 and 3. 5 Counting the number of spanning trees for the I-graph I(n, k, l) In what follows, we always assume that the numbers k and l are relatively prime. To get the result for an arbitrary connected I-graph I(n, k, l) with gcd(n, k, l) = 1 and gcd(k, l) = d > 1 we observe that I(n, k, l) is isomorphic to I(n, k', l'), where the numbers k' = k/d and l' = l/d are relatively prime. Theorem 5.1. The number of spanning trees of the I-graph I(n, k, l) is given by the formula k+l-1 , k + l 1 rp ( \ -i Tk,i(n) = ( —1)(n-1)(k+l)n n T"(Ws) - 1, s = 1 W - 1 where ws, s = 1,2,..., k + l — 1 are roots of the order k + l — 1 algebraic equation (3 - 2Tk(w))(3 - 2Ti(w)) - 1 =Q w — 1 and Tj (w) is the Chebyshev polynomial of the first kind. Proof. By the celebrated Kirchhoff theorem, the number of spanning trees rk}i (n) is equal to the product of nonzero eigenvalues of the Laplacian of a graph I(n, k, l) divided by the number of its vertices 2n. To investigate the spectrum of Laplacian matrix we note that matrix C% = Tk + T-k, where T — circ(0,1,..., 0) is the n x n shift operator. The latter equality easily follows from the identity Tn = In. Hence, T = /3/n — Tk — T-k —In L = ' In ^ n The eigenvalues of circulant matrix T are en, where en = e ~. Since all eigenvalues of T are distinct, the matrix T is conjugate to the diagonal matrix T = diag(1, en,..., e;-1), where diagonal entries of diag(1, en,..., e;-1) are 1, en,..., e;-1. To find spectrum of L, without loss of generality, one can assume that T = T. Then the blocks of L are diagonal matrices. This essentially simplifies the problem of finding eigenvalues of L. Indeed, let A be an eigenvalue of L and (x, y) = (x1,..., xn, y1,..., yn) be the corresponding eigenvector. Then we have the following system of equations ( (3In - Tk - T-k)x - y = Ax \-x + (3In - Tl - T-l)y = Ay . From here we conclude that y = (3In - Tk - T-k)x - Ax = ((3 - A)In - Tk - T-k)x. Substituting y in the second equation, we have (((3 - A)I„ - Tl - T-l)((3 - A)I„ - Tk - T-k) - 1)x = 0. Recall the matrices under consideration are diagonal and the (j +1, j + 1)-th entry of T is equal to en. Therefore, we have ((3 - A - enk - e-jk )(3 - A - e;1 - e-jl) - 1)xj-+1 = 0 and yj+1 = (3 - A - e;1 - e-jl So, for any j = 0,..., n - 1 the matrix L has two eigenvalues, say Aij and A2 j satisfying the quadratic equation (3 - A - enk - e—jk)(3 - A - enl - e—jl) - 1 = 0. The corresponding eigenvectors are (x, y), where x = ej+1 = (0,..., ,..., 0) and (j+1)-th y =(3 - A - Tk - T—k)ej+i. In particular, if j =0 for A1j0, A2,0 we have (1 - A)(1 - A) - 1 = A(A - 2) = 0. That is, A1j0 = 0 and A2,0 = 2. Since A1 j and A2j are roots of the same quadratic equation, we obtain Aid A-j = P (en), where P (z) = (3 - zk - z—k )(3 - zl - z—l) - 1. Now we have n—1 n—1 n—1 2nA2,o n A1,j A2,j=n n A1,j A2,j=n np (en: j=1 j=1 j=1 To continue we need the following lemma. Lemma 5.2. The following identity holds (3 - zk - z—k)(3 - zl - z—l) - 1 = (3 - 2Tk(w))(3 - 2Tl(w)) - 1, where Tk (w) is the Chebyshev polynomial of the first kind and w = -1- (z + z—1). Moreover, if k and l are relatively prime then all roots of the Laurent polynomial (3 - zk - z—k)(3 - zl - z—l) - 1 counted with multiplicities are 1, 1, z1, 1/z1,..., zk+l—1, 1/zk+l—1, where we have |zs | = 1, s = 1, 2,..., k + l - 1. So, the right-hand polynomial has the roots 1, w1, ..., wk+l—1, where ws = 1 for all s = 1, 2,..., k + l - 1. Proof. Let us substitute z = e®v. It is easy to see that w = - (z + z—1) = cos y>, so we have Tk (w) = cos(k arccos w) = cos(k^). Then the first statement of the lemma is equivalent to the following trigonometric identity (3 - 2cos(k^))(3 - 2cos(ly)) - 1 = (3 - 2Tk(w))(3 - 2Tl(w)) - 1. To prove the second statement of the lemma we suppose that the Laurent polynomial P(z) = (3 - zk - z—k)(3 - zl - z—l) - 1 has a root zo such that |zo| = 1. Then zo = eiV0, ^o e R. Now we have (3 - 2cos(kyo))(3 - 2cos(lyo)) - 1 = 0. Since 3 - 2 cos(k^0) > 1 and 3 - 2 cos(l^0) > 1 the equations holds if and only if cos(k^0) = 1 and cos(l^0) = 1. So k^0 = 2ns0 and l^0 = 2nt0 for some integer s0 and t0. As k and l are relatively prime, so there exist two integers p and q such that kp + ql = 1. Hence ^>0 = ^0(kp + lq) = 2n(ps0 + qt0) e 2nZ. As a result z0 = e®V0 = 1. Now we have to show that the multiplicity of the root z0 = 1 is 2. Indeed, P(1) = P'(1) = 0 and P"(1) = -2(k2 + l2) = 0. □ Let us set H(z) = f]s=1(z - zs)(z - zs 1), where m = k + l - 1 and zs are roots of P(z) different from 1. Then by Lemma 5.2, we have P(z) = (zk+) H(z). Lemma 5.3. Let H(z) = f]m=i(z - zs)(z - zs and H(1) = 0. Then n-1 m rr , n , n h (,i)=n , j=1 s=1 s where ws = ^ (zs + z-1), s = 1,..., m and Tn (x) is the Chebyshev polynomial of the first kind. Proof. It is easy to check that "-i(z - en) = z—r if z = 1. Also we note that 1 (zn + z-n) = Tn( 1 (z + z-1)). By the substitution z = e®the latter follows from the evident identity cos(ny) = Tn(cos y). Then we have n—1 n—1 m n h (en) = nn (en - zs)(4 - z—1) j=1 j=1 s = 1 m n- 1 = nn (zs - en)(zs-1 - en) s = 1 j=1 = n zn - 1 z—n - 1 = m Tn(ws) - 1 J=1 zs - 1 z—1 - 1 J=1 Ws - 1 . Note that n™=i (1 - pn) = lim, FIJ=i1(z - 4) = lim Z-T = « and n^! pn (- 1)n- 1. As a result, taking into account Lemma 5.2 and Lemma 5.3, we obtain 1 n-1 1 n-1 (pj - 1)2 Tk,i(n) = n n P (p n) = n n j+rH (p n) j=1 j=1 2 n-1 n ^ ' n ^ (en)fc+1 (_1)(n-1)(fc+0 n ^ ^—- n H (pn) j=1 k + 1 — 1 /"Tl / \ -1 (-i)(n-1)(fc+i)n rr Tn(Ws) - 1. □ S = 1 W - 1 Ik+r Un-1 xMW 2 where ws, s = 1, 2,..., k are Corollary 5.4. rfc,;(n) = n []S=1 Un the same as in Theorem 5.1 and Un-1(w) is the Chebyshev polynomial of the second kind. Proof. Follows from the identity — = (y'^+p ) . □ The following theorem appeared after fruitful discussion with professor D. Lorenzini. Theorem 5.5. Let t (n) = rk,; (n) be the number of spanning trees of the graph I (n, k, l). Then there exist an integer sequence a(n) = ak,; (n), n G N such that 1° t(n) = n a2(n) when n is odd, 2° t(n) = 6n a2(n) when n is even and k + l is even, 3° t(n) = n a2(n) when n is even and k + l is odd. Proof. Recall that all nonzero eigenvalues are given by the list |A2,o, Ai,j, A2,j, j = 1,..., n — 1}. By the Kirchhoff theorem we have 2nr(n) = A2,0 n^-i A1,jA2,j. Since A2 0 = 2, we have nr(n) = f]A1,jA2,j. We note that A1,jA2,j = P(en) = P(en-j) = A1,n-jA2,n-j. So, we get nr(n) = (nj=-1)/2 A1,jA2,j)2 if n is odd and nr(n) = A1,n A2,n (nn=1-1 A1,jA2,j)2, if n is even. The value A1,nA2,n = P( —1) = (3 — 2( —1)k)(3 — 2( —1)1) — 1 is equal to 4 if k and l are of different parity and 24 if both k and l are odd. The case when both k and l are even is impossible, since k and l are relatively prime. The graph I(n, k, l) admits a cyclic group of automorphisms isomorphic to Zn which acts freely on the set of spanning trees. Therefore, the value t(n) is a multiple of n. So is an integer. Hence / \ t of"-1)/2 A A x 2 I 0 t(n) = [ 11, = 1 A1,jA2,j n when n is odd, 20 T(n) = 6 ( 2 n Al,j A2,j ) when n is even and k + l is even, n/ 2 -1 2 30 ^ = ( nTn/2-^^ \ when n is even and k + l is odd. nn Each algebraic number A^ comes into both products n(=-1)/2 A1,j A2 j and j-1 A1,j A2,j with all its Galois conjugate elements. Therefore, both products are integer numbers. From here we conclude that in equalities 10, 20 and 30 the value that is squared is a rational number. Because T(n) is integer and 6 is a squarefree, all these rational numbers are integer. Setting a(n) = —n—~—~ if n is odd and a(n) = if n is even, we finish the proof of the theorem. □ 2HI ?=1-1 A From now on, we aim to estimate the minimum number of generators for the Jacobian of I-graph I(n, k, l). Lemma 5.6. For any given I-graph I(n, k, l) the number of spanning trees t (n) satisfies the inequality t(n) > n3. Proof. Recall that for any j = 0,..., n — 1, the Laplacian matrix L of I(n, k, l) has two eigenvalues, say A1 j and A2 j, which are roots of the quadratic equation Q (A) = (3 — A — enk —e-jk)(3—A—en1 —e-j1) —1 = 0. So, AuA2,j = (3—enk —e-jk)(3— e^ — e-j1) — 1 = P (en). Note that A1,0 = 0 and A2,0 = 2. Furthermore {A1,j, A2,j | j = 0,..., n — 1} is the set of all eigenvalues of L. The Kirchhoff theorem states the following n- 1 n- 1 ,0 2nTfc,;(n) = 2nT(n) = A2,o JJ A1,jA2,j = 2 JJ A1,jA2,j. j=1 j=1 Hence nT (n) = ¿1 P (en), where P (en) = (3 — 2cos( ))(3 — 2cos( j-)) — 1. It is easy to prove the following trigonometric identity 3 - 2cos ( j^)) (3 - 2cos (j^)) - 1 = 4sin2 f—) + 4sin2 (—) + 16 sin2 (—) sin2 (—) . V n J V n J V n J V n J Connectedness of I-graph implies gcd(n, k,l) = 1. It may happen that gcd(n, k) m = 1 and gcd(n, l) = m' =1. We will use the notation n = m q = m'q', k = p m, l p'm'. We introduce three sets, J, Jk and J; in the following way J = {1, 2,..., n - 1}, Jk = {j | j = dq, d = 1,..., m - 1} and J; = {j | j = d' q', d' = 1,..., m' - 1}. If j G Jfc then sin( ^ ) = 0 and if j G J| then sin( ) =0. We note that Jk and J do not intersect. Otherwise, for j G Jk n J; we have A1 j A2 j = P (en) = 0. Then at least one of the eigenvalues A1 j and A2 j is equal to zero. This leads to contradiction, as we have the unique zero eigenvalue A1j0 = 0. Now we are going to find a low bound for r(n). As nr (n) = f]"—j1 P (en) we evaluate the product n-1 ij=i J vcn n-1 np(en)=n 4si sin2 ( j—j +4 sin 2 jn) +i6sin2 jn) sin2 ( ^) V n / V n / Vn/ > n 4sin2 ( j^) n 4sin2 ( ^) n ^ (^) s^ (jin) jeJk jeJi je J\(JfcUJi) sin - sin . . nn n 4s,n= ^) n 4^ (ç). jeJ\Jk jeJ \Ji n Now we analyze individual component of the product. We make use of the following simple identity cos(2jpn) = cos(2(j+q)pn). n 4 sin2 (jf ) = n(2 - 2 jeJ\Jk jeJ\Jk cos (^)) = n (2 - 2cos ^ jeJ\Jk n (2 - 2« jeJ \Jk n ( 2jpn ) v q q-1 ) =11 2 - 2 j=1 /2j'mpf\ - 2 cos (- mq / 2jpn q The Chebyshev polynomial Tq(x) = cos(q arccos(x)) has the following property. The roots of the equation Tq(x) - 1=0 are cos( j), j = 0,1,..., q - 1. Since the leading coefficient of Tq(x) is 2q-1, for x =1 we have the identity q-1 j=1 n(2x - 2cos(j t,(x) - 1 x1 As p and q are relatively prime we obtain q-1 n (2 - 2 cos j=i 2jpn q-1 n 2 2 cos 2>X q ^ Hence ^Q 4 sin je J In a similar way we obtain lim 2 (?) T(x) - 1 x — 1 / n \ 2m \mJ (q2) ,2\ m m 2m n 4 sin 2 ( ^ ) = ( n -LJ- V n / Vm' jeJ \Ji 2m' To get the final result we use the following trivial inequality. For any integers a > 2 and b > 2 we have ab > ab. Since q = n/m > 2 and q' = n/m' > 2, we conclude a-1 2m n 2 j=1 » = 11P(j) > (n)( m > n2 n2 = n4. □ Using Lemma 5.6, one can show the following theorem. Theorem 5.7. For any given I-graph I(n, k, /) the minimum number of generators for Jacobian Jac(I(n, k, /)) is at least 2 and at most 2k + 2/ — 1. Proof. The upper bound for the number of generators follows from Theorem 4.1. Indeed, by this theorem the group coker(L(I(n, k, /)) = Jac(I(n, k, /)) © Z is generated by 2k + 2/ elements. One of these generators is needed to generate the infinite cyclic group Z. Hence Jac(I(n, k, /)) is generated by 2k + 2/ — 1 elements. To get the lower bound we use Lemma 5.6. Let us suppose that Jac(I(n, k, /)) is generated by one element. Then it is the cyclic group of order t (n). Denote by D be a product of all distinct nonzero eigenvalues of I(n, k, /). By Proposition 2.6 from [20] the order of each element of Jac(I(n, k, /)) is divisor of D. Hence, t(n) is divisor of D and we have inequality D > t (n). By the Kirchhoff theorem we have 2nT (n) = A2,0 rj—i A1 A2j. We note that all algebraic numbers Ai}j comes into product together with its Galois conjugate, so 2nT (n) is a multiple of D. In particular 2nT (n) > D. _ fn("-l)/2 From the proof of Theorem 5.5 we have nT(n) = j=i A2,j)2 if n is odd and nr(n) = A1,n A2,n (rij=i-1 A^jA2,j)2 if n is even. Moreover, the value A1,nA2,n is equal to 4 if k and l are of different parity and 24 if both k and l are odd. The case when both k and l are even is impossible as k and l are relatively prime. Now, we have 4nr(n) = (^nj=-1)/2 A1,jA2j-)2 if n is odd. Again, all algebraic numbers Aj,j comes into the product p = 2\\j=-1)/2 A1,j A2j together with its Galois conjugate. Therefore, the product p is an integer number and contains all distinct nonzero eigenvalues. Hence p is a multiple of D. So we obtain 4nr(n) = p2 > D2 > t(n)2. m m q m m Also we get 4nAi,n A2,n t(n) = (2Ai,n A2,n n"=1-1 A2,j)2 if n is even. By a similar argument, taking into account the inequality 24 > A1; n A2, n we obtain 96nT(n) > 4nA1,nA2,nt(n) > D2 > t(n)2. As result, by Lemma 5.6 we have 4n > t(n) > n3 if n is odd and 96n > t(n) > n3 if n is even. For n > 10 this is impossible. So, the rank of Jac(1 (n, k, 1)) is at least two for all n > 10. For n less than 10 this statement can be proved by direct calculation. □ For graphs I(4,2, 3) and I(6, 3,4), the Jacobian group Jac(1 (n, k, 1)) is generated by 2 elements. The upper bound 2k + 21 — 1 for the minimum number of generators of Jac(1 (n, k, 1)) is attained for graph I(34, 2,3) and I(170,3,4). See Tables 2 and 3 in Section 7. So the lower bound 2 and the upper bound 2k + 21 — 1 for the minimum number of generators of Jac(1 (n, k, 1)) are sharp. 6 Asymptotic for the number of spanning trees The asymptotic for the number of spanning trees of the graph I(n, k, 1) is given in the following theorem. Theorem 6.1. Let P(z) = (3 — zk — z-k)(3 — z1 — z-1) — 1. Suppose that k and 1 are relatively prime and set = n P O)=o |z|>1 | z |. Then the number Tk,;(n) of spanning trees of the graph I(n, k, 1) has the asymptotic TM(n) ~ AM, n Proof. By Theorem 5.1 we have k + 1 rp / \ -i Tfc,i(n) = ( — 1)("-1)(k+0n n T"(Ws) — 1, J=1 Ws — 1 where ws, s = 1, 2,...,k + 1 — 1 are roots of the polynomial Q( ) (3 — 2Tk(w))(3 — 2Ti(w)) — 1 Q(w) =-;-. w — 1 So we obtain fc+i-1 fc+i-1 ,fc+i-1 Tfc,i(n) = n T"wWWs_) 1 1 = n |T„(ws) — 1| / |ws — 1|. s=1 s s = 1 ' s=1 By Lemma 5.2 we have T"(ws) = 2(z" + z-"), where the zs and 1/zs are roots of the polynomial P(z) with the property |zs | = 1, s = 1, 2,..., k + 1 — 1. Replacing zs by 1/zs, if it is necessary, we can assume that all |zs| > 1 for all s = 1, 2,..., k +1 — 1. Then T"(ws) — 1 z" as n tends to to. So |T"(ws) — 1| — 2>|zs|" as n ^ to. Hence fc+i-1 1 fc+i-1 1 1 1 1 1 1 11 1 1 4" n |T"(ws)—1| - ^ro n |zs|"=^¡^11 |z|"=^fc+i-ra s=1 s=1 P(z) = 0, |z|>1 Now we directly evaluate the quantity f] k=1 1 |ws - 11. We note that Q(w) = aowk+i-1 + aiwk+l-2 +-----+ ak+i-2w + afc+i-1 is an integer polynomial with the leading coefficient a0 = 2k+i. From here we obtain k+i-1 k+i-1 n iws - ii = n ii -wsi aL Q(i) a0 2(k2 + l2) k2 + l2 2 k+l 2k+1-1 ' s=1 s=1 Indeed, Q = lim (3 - 2Tk(w))(3 - 2T1(w)) - 1 w — 1 = -2Tk(1)(3 - 2Ti(1)) - 2T/(1)(3 - 2Tk(1)) = -2kUk-1(1)(3 - 2Ti(1)) - 21U-1(1)(3 - 2Tk(1)) = -2(k2 + l2) and a0 = 2k+l. In order to get the statement of the theorem we combine the above mentioned results. Then Af,i / k2 + l2 n f \ Ak,i Ik2 +12 n Tk,i(n) ~ n 2+1-1/ = klTp Ak,i as n ^ □ Remark 6.2. It was noted by professor A. Yu. Vesnin that constant Ak l coincides with the Mahler measure of Laurent polynomial P (z) = (3 - zk - z-k )(3 - zl - z-1) - 1. It gives a simple way to evaluate Ak l using the following formula Ak,i =exp(^1 log IP(e2nii)|d^. See, for example, [13, p. 6] for the proof. The numerical values for Ak i, where k and 1 are relatively prime numbers 1 < k < 1 < 9 will be given in Table 1 in the Section 7. 7 Examples and tables 7.1 Examples 1° The Prism graph I(n, 1,1). We have the following asymptotic T1,1(n) = n(T„(2) - 1) — |(2 + V3)n, n ^ to. 2° The generalized Petersen graph GP(n, 2) = I(n, 1,2). The the number of spanning e T1,2(n) - f A?,2, n ^ c 7+ V5+ ^38 + 14V5 trees (see [19]) behaves like r1,2(n) — f Af,2, n ^ to, where A1,2 = —^-—-!-— = 4.39026. 3° The smallest proper /-graph I(n, 2,3) has the following asymptotic for the number of spanning trees T2,3(n) ^ 13^^ n ^ ^ Here A2,3 = 4.84199 is a suitable root of the algebraic equation 1 - 7x + 13x2 - 35x3 + 161x4 - 287x5 + 241x6 - 371x7 + 577x8 - 371x9 + 241x10 - 287X11 + 161x12 - 35x13 + 13x14 - 7x15 + x16 = 0. Here is the table for asymptotic constants Ak,i for relatively prime numbers 1 < k < l< 9. Table 1: Asymptotic constants Ak,, k\1 1 2 3 4 5 6 7 8 9 1 3.7320 4.3902 4.7201 4.8954 4.9953 5.0559 5.0945 5.1203 5.1382 2 - 4.8419 - 5.0249 - 5.1033 - 5.1414 3 - 5.0054 5.0541 - 5.1137 5.1320 - 4 - 5.0802 - 5.1244 - 5.1504 5 - 5.1201 5.1346 5.1461 5.1554 6 - 5.1438 - - 7 - 5.1589 5.1649 8 - 5.1691 7.2 The tables of Jacobians of I-graphs Theorem 4.1 is the first step to understand the structure of the Jacobian for I(n, k, l). Also, it gives a simple way for numerical calculations of Jac(I(n, k, l)) for small values of k and l. See Tables 2 and 3. The first example of Jacobian Jac(I(n, 3,4)) with the maximum rank 13: n = 170, Jac(I(170, 3, 4)) = Z2 © Z8 © Z6108 ® Z30540 © Z22.3-5-103-509-1699-11593-p-q ® Z22-3-5 -17-103-509 -1699 -11593 p q, and T3,4(170) = 225 • 34 • 53 • 17 • 1032 • 5094 • 16992 • 115932 • p2 • q2 where p = 16901365279286026289 and q = 34652587005966540929. Table 2: Graph I(n, 2, 3). n Jac(I(n, 2, 3)) T2,3(n) = I Jac(I(n, 2, 3))| 4 Z7 e Z28 196 B Z19 e Z95 1S0B 6 Z19 e Z114 2166 T Z83 e Z581 4S223 S Zl6l e Z1288 20T36S 9 Z289 e Z2601 TB16S9 10 Zl558 e Z3895 606S410 11 Zl693 e Zl8623 31B2ST39 12 Z5 e Z5 e Z665 e Z7980 13266TB00 13 Z25 e Z325 e Z325 e Z325 SBS20312B 14 Z17513 e Z245182 4293ST2366 1B Z37069 e Z556035 2061166141B 16 Z84847 e Z1357552 11B1S4214B44 1T Z2 e Z23186 e Z394162 BS4S9SB6S44S 1S Z400843 e Z7215174 2S921B19916S2 19 Z898243 e Z17066617 1B3299692B3931 20 Z49 e Z5453 e Zl09060 TTB02443441TS0 21 Z4301807 e Z90337947 3SS616412TT0229 22 Z9536669 e Z209806718 2000SBT223B42342 23 Z20949827 e Z481846021 10094B90TS0BSS36T 24 Z5 e Z5 e Z9192295 e Z220615080 B0B9S9T242021B000 2B Zl01468531 e Z2536713275 2BT396B69BS244902B 26 Z25 e Z325 e Z8923525 e Zl7847050 12939T60994164062B0 2T Z490309597 e Zl3238359119 6490S94B24BTS16B043 2S Z49 e Z154342069 e Z4321577932 326S30626S9111444092 29 Z2376466133 e Z68917517857 163TS014T1BTBS32369S1 30 Z19 e Z19 e Z275089049 e Z8252671470 S19B492B624T41B262S30 31 Zl1507960491 e Z356746775221 410B42TT94B3492BT93B11 32 Z25318259953 e Z810184318496 20B124BT1SBB2BST39906SS 33 Z55700389051 e Z1838112838683 1023S36002342S11024B9S33 34 Z2 e Z4 e Zl9l5580948 e Z32564876116 B11022336096BS23B2633SB6 3B Z269747901677 e Z9441176558695 2B46T3TB660T00B60T9431B1B Table 3: Graph I(n, 3,4). n Jac(I(n, 3,4)) T3,4(n) = | Jac(I(n, 3, 4))| 5 Z2 © Z10 © Z10 © Z10 2000 6 Z19 © Z114 2166 7 Z71 © Z497 35287 8 Z73 © Z584 42632 9 Z289 © Z2601 751689 10 Z2 © Z12 © Z60 © Z60 © Z60 5184000 11 Z1541 © Z16951 26121491 12 Zn © Zn © Z209 © Z2508 63424812 13 Z5 © Z5 © Z1555 © Z20215 785858125 14 Z16969 © Z237566 4031257454 15 Z2 © Z10 © Z17410 © Z52230 18186486000 16 Z71321 © Z1141136 81386960656 17 Z2 © Z23186 © Z394162 584898568448 18 Z400843 © Z7215174 2892151991682 19 Z37 © Z37 © Z23939 © Z454841 14906272578931 20 Z8 © Z12 © Z120 © Z79080 © Z79080 72042006528000 21 Z4487981 © Z94247601 422981442583581 22 Z10002631 © Z220057882 2201157792287542 23 Z22138559 © Z509186857 11272663275719063 24 Z187 © Z187 © Z259369 © Z6224856 56458663080288216 25 Z2114 © Z52850 © Z52850 © Z52850 312061332000250000 References [1] R. Bacher, P. de la Harpe and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France 125 (1997), 167-198, http://www. numdam.org/item?id=BSMF_19 97_125_2_16 7_0. [2] M. Baker and S. Norine, Harmonic morphisms and hyperelliptic graphs, Int. Math. Res. Notices 15 (2009), 2914-2955, doi:10.1093/imrn/rnp037. [3] N. Biggs, Three remarkable graphs, Canad. J. Math. 25 (1973), 397-411, doi:10.4153/ cjm-1973-040-1. [4] N. L. Biggs, Chip-flring and the critical group of a graph, J. Algebraic Combin. 9 (1999), 25-45, doi:10.1023/a:1018611014097. [5] M. Boben, T. Pisanski and A. Zitnik, I-graphs and the corresponding configurations, J. Combin. Des. 13 (2005), 406-424, doi:10.1002/jcd.20054. [6] F. T. Boesch and H. Prodinger, Spanning tree formulas and Chebyshev polynomials, Graphs Combin. 2 (1986), 191-200, doi:10.1007/bf01788093. [7] P. Chen, Y. Hou and C. Woo, On the critical group of the Mobius ladder graph, Australas. J. Combin. 36 (2006), 133-142, https://ajc.maths.uq.edu.au/pdf/3 6/ajc_v36_ p133.pdf. [8] X. Chen, Q. Lin and F. Zhang, The number of spanning trees in odd valent circulant graphs, Discrete Math 282 (2004), 69-79, doi:10.1016/j.disc.2003.12.006. [9] R. Cori and D. Rossin, On the sandpile group of dual graphs, European J. Combin. 21 (2000), 447-459, doi:10.1006/eujc.1999.0366. [10] P. J. Davis, Circulant Matrices, AMS Chelsea Publishing, 2nd edition, 1994. [11] A. S. S. de Oliveira and C. T. M. Vinagre, The spectrum of an i-graph, 2015, arXiv:1511.03513 [math.CO]. [12] D. Dhar, P. Ruelle, S. Sen and D.-N. Verma, Algebraic aspects of abelian sandpile models, J. Phys.A 28 (1995), 805-831, http://stacks.iop.org/0305-44 70/2 8/8 05. [13] G. Everestand T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer Science & Business Media, 2013, doi:10.1007/978-1-4471-3898-3. [14] R. M. Foster, The Foster Census, Charles Babbage Research Centre, Winnipeg, MB, 1988, with an introduction by I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star. [15] R. Gera and P. Stanica, The spectrum of generalized Petersen graphs, Australas. J. Combin. 49 (2011), 39-45, https://ajc.maths.uq.edu.au/pdf/4 9/ajc_v4 9_p039.pdf. [16] B. Horvat, T. Pisanski and A. Zitnik, Isomorphism checking of I-graphs, Graphs Combin. 28 (2012), 823-830, doi:10.1007/s00373-011-1086-2. [17] Y. Hou, C. Woo and P. Chen, On the sandpile group of the square cycle C^, Linear Algebra Appl. 418 (2006), 457-467, doi:10.1016/j.laa.2006.02.022. [18] M. Kotani and T. Sunada, Jacobian tori associated with a finite graph and its abelian covering graphs, Adv. Appl. Math. 24 (2000), 89-110, doi:10.1006/aama.1999.0672. [19] Y. S. Kwon, A. D. Mednykh and I. A. Mednykh, On Jacobian group and complexity of the generalized Petersen graph GP(n, k) through Chebyshev polynomials, Linear Algebra Appl. 529 (2017), 355-373, doi:10.1016/j.laa.2017.04.032. [20] D. Lorenzini, Smith normal form and Laplacians, J. Comb. Theory Ser. B 98 (2008), 12711300, doi:10.1016/j.jctb.2008.02.002. [21] A. D. Mednykh and I. A. Mednykh, On the structure of the Jacobian group of circulant graphs, Dokl. Math. 94 (2016), 445-449, doi:10.1134/s106456241604027x. [22] I. A. Mednykh and M. A. Zindinova, On the structure of Picard group for Moebius ladder, Sib. Elektron. Mat. Izv. 8 (2011), 54-61, http://semr.math.nsc.ru/v8/p54-61.pdf. [23] S. D. Nikolopoulos and C. Papadopoulos, The number of spanning trees in Kn-complements of quasi-threshold graphs, Graphs Combin. 20 (2004), 383-397, doi:10.1007/ s00373-004-0568-x. [24] M. Petkovsek and H. Zakrajsek, Enumeration of I-graphs: Burnside does it again, Ars Math. Contemp. 2 (2009), 241-262, doi:10.26493/1855-3974.113.3dc. [25] R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, J. Phys. A 33 (2000), 3881-3902, doi:10.1088/0305-4470/33/21/303. [26] W. Sun, S. Wang and J. Zhang, Counting spanning trees in prism and anti-prism graphs, J. Appl. Anal. Comput. 6 (2016), 65-75, doi:10.11948/2016006. [27] Y. Zhang, X. Yong and M. J. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000), 337-350, doi:10.1016/s0012-365x(99)00414-8. [28] Y. Zhang, X. Yong and M. J. Golin, Chebyshev polynomials and spanning tree formulas for circulant and related graphs, Discrete Math. 298 (2005), 334-364, doi:10.1016/j.disc.2004.10. 025. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 487-497 https://doi.org/10.26493/1855-3974.1359.b33 (Also available at http://amc-journal.eu) The isolated-pentagon rule and nice substructures in fullerenes* * Hao Li Laboratoire de Recherche en Informatique, UMR 8623, C.N.R.S.-Université Paris-sud, Received 22 March 2017, accepted 6 September 2017, published online 5 September 2018 After fullerenes were discovered, Kroto in 1987 proposed first the isolated-pentagon rule (IPR): the most stable fullerenes are those in which no two pentagons share an edge, that is, each pentagon is completely surrounded by hexagons. To now the structures of the synthesized and isolated (neutral) fullerenes meet this rule. The IPR can be justified from local strain in geometry and n-electronic resonance energy of fullerenes. If two pentagons abut in a fullerene, a 8-circuit along the perimeter of the pentalene (a pair of abutting pentagons) occurs. This paper confirms that such a 8-circuit is always a conjugated cycle of the fullerene in a graph-theoretical approach. Since conjugated circuits of length 8 destabilize the molecule in conjugated circuit theory, this result gives a basis for the IPR in n-electronic resonance. We also prove that each 6-circuit (hexagon) and each 10-circuit along the perimeter of a pair of abutting hexagons are conjugated. Two such types of conjugated circuit satisfy the (4n + 2)-rule, and thus stabilise the molecule. Keywords: Fullerene, patch, stability, isolated pentagon rule, Kekule structure, conjugated cycle, cyclic edge-cut. Math. Subj. Class.: 05C70, 05C10, 92E10 »This work was supported by NSFC (Grant Nos. 11371180, 11871256). t Corresponding author. E-mail addresses: Hao.Li@lri.fr (Hao Li), zhanghp@lzu.edu.cn (Heping Zhang) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ F-91405, Orsay, France Heping Zhang t School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China Abstract 1 Introduction The fullerenes are closed carbon-cage molecules such that every carbon atom has bonds to three other atoms, and the length of each carbon ring is either 5 or 6. Ever since the first fullerene, Buckministerfullerene C6o, was discovered by Kroto et al. in 1985 [15], the stabilities of fullerenes have attracted many theorist's attentions. The simple Huckel molecular orbital model that predicts reliably the relative stabilities of planar aromatic hydrocarbons is not generally found to work so well for fullerenes. Kroto [14] in 1987 proposed first the isolated-pentagon rule (IPR): the most stable fullerenes are those in which no two pentagons share an edge, that is, each pentagon is completely surrounded by hexagons. Schmalz et al. [23] gave a more theoretical discussion of the rule in support of the fullerene hypothesis. Indeed the structures of the synthesized and isolated fullerenes meet this rule. The IPR can be justified from local strain and n-electronic resonance of fullerenes; for details, also see a book due to Fowler and Manolopoulos [7]. Pentagon adjacency leads to higher local curvature of the molecule surface and increases the strain energy. On the other hand, according to Huckel (4n+2)-rule, conjugated circuits of length 6,10,14,... stabilize the molecule, whereas conjugated circuits of length 4, 8,12,... destabilize the molecule. Here a conjugated circuit is a cycle of alternating single and double bonds within a Kekule structure. If two pentagons abut in a fullerene, the conjugated or resonant 8-circuit along the perimeter of the pentalene may occur, and this leads to resonance destabilization [22]. This is an interpretation of IPR in n-electronic resonation stabilization. However, a problem occurs: In a fullerene, is every 8-length circuit conjugated? To now we have not seen any definite answer to this problem in mathematics. In this article we investigate nice patches of a fullerene by applying some small cyclic edge-cuts of graphs and present a positive answer to the above problem (a patch of a fullerene is nice if its Kekule structure can be extended to a Kekule structure of the entire fullerene). As immediate consequences of our main theorems, we have that every 8-length circuit of a fullerene surrounds a pentalene (a pair of abutting pentagons) and is conjugated or alternating with respect to a Kekule structure (see Corollary 3.4). This confirms the destabilization of any pentalene as a nice substructure to the entire fullerene and thus gives a mathematical support for the IPR of fullerenes. Furthermore we also show that in a fullerene every hexagon is a conjugated 6-circuit (see Corollary 3.3) and the boundary along a naphthalene (i.e. a pair of abutting hexagons) is a conjugated 10-circuit (see Corollary 4.2). The former has already been proved (see [26]). In conjugated circuit theory [10, 19, 20], conjugated 6-circuits and 10-circuits contribute stabilizations of fullerenes and the small conjugated circuits have the greatest effects (positive and negative) on stability. For recent discussions on the IPR of fullerenes about steric strain factor and n-electronic resonance factor, see [1, 2, 8, 13, 21]. For mathematical aspects of fullerenes, see a recent survey [3]. 2 Preliminary To obtain the above end we now start our arguments in a graph-theoretical approach. As a molecular graph of a fullerene, a fullerene graph is a 3-connected planar cubic graph with only pentagonal and hexagonal faces. It is well known that a fullerene graph on n vertices exists for every even n > 20 except n = 22 [9]. By Euler's polyhedron formula, every fullerene graph with n vertices has exactly 12 pentagonal faces and (n/2 - 10) hexagonal faces. Let G be a graph with vertex-set V(G) and edge-set E(G). An edge set M of a graph G is called a matching if no two edges in M have a common endvertex. A matching M of G is perfect if every vertex of G is incident with one edge in M. In organic molecular graphs, perfect matchings correspond to Kekule structures, playing an important role in analysis of the resonance energy and stability of polycyclic aromatic hydrocarbons. The following classical theorem is Tutte's 1-factor theorem on the existence of perfect matching of a graph [24]. For detailed monograph on matching theory, see Lovasz and Plummer [17]. Theorem 2.1. A graph G has a perfect matching if and only if odd(G — S) < |S | for each S C V (G), where odd(G — S) denotes the number of odd components in subgraph G — S. Subgraph G' of a graph G is called nice if G — V(G') has a perfect matching. In particular, an even cycle C of a graph G is nice if G has a perfect matching M such that C is an M-alternating cycle, i.e. the edges of C alternate in M and E(G) \ M .A nice even cycle is also called resonant or conjugated cycle (or circuit) in chemical literature. For convenience, a cycle of length k is said to be a k-cycle or k-circuit. For nonempty subsets X, Y of V(G), let [X, Y] denote the set of edges of G that each has one end-vertex in X and the other in Y .If X = V (G) \ X = 0, then V(X) := [X, X] is called an edge-cut of G, and k-edge-cut whenever | [X, X] | = k. The edges incident with a single vertex form a trivial edge-cut. For a subgraph H of G, let H := G — V(H). We simply write V(H) for V(V(H)). Lemma 2.2 ([25]). Every 3-edge-cut ofafullerene graph is trivial. Lemma 2.3 ([25]). Every 4-edge-cut ofafullerene graph isolates an edge. An edge-cut S = V(X) of G is cyclic if at least two components of G—S each contains a cycle. The minimum size of cyclic edge-cuts of G is called cyclic edge-connectivity of G, denoted by cA(G). Theorem 2.4 ([6, 12, 18]). Let F be anyfullerene graph. Then cA(F) = 5. From the definition with the above properties we know that each fullerene graph has the girth 5 (the minimum length of cycles), and each of its 5-cycles and 6-cycles bounds a face. A cyclic k-edge-cut of a graph isolating just a k-cycle will be called trivial. Theorem 2.5 ([12, 16]). A fullerene graph with a non-trivial cyclic 5-edge-cut is a nan-otube with two disjoint pentacaps (see Figure 1), and each non-trivial cyclic 5-edge-cut must be an edge set between two consecutive concentric cycles of length 10. A fullerene patch is a 2-connected plane graph with all faces pentagonal or hexagonal except one external face, all internal vertices (not incident with the external face) of degree 3 and those incident with the external face having degree 2 or 3. The cycle bounding the external face is the boundary of the patch. We can count the pentagons of a fullerene patch as internal faces as follows. Lemma 2.6 ([4]). For fullerene patch G, let p5 denote the number of pentagonal faces other than the external face. Then P5 = 6 + k3 — k2 =6 + 2k3 — l, (2.1) where k2 and k3 denote the number of vertices of degree 2 and 3 on the boundary of G, respectively, and l is the boundary length. For T C V(G), the induced subgraph of G by T consists of T and all edges whose endvertices are contained in T, denoted by G[T]. In the next two sections we will investigate nice patches of fullerene graphs in cyclic 6-edge-cut and 8-edge-cut cases, respectively. 3 Cyclic 6-edge-cut We first consider a more general case than fullerene patches. Theorem 3.1. Let F0 be a connected induced subgraph of a fullerene graph F such that interior faces of F0 exist and each one is a pentagon or hexagon. If F has exactly six edges from F0 to the outside F0 = F — V(F0), then F0 has a perfect matching. Proof. Let n0 and e0 denote the numbers of vertices and edges of F0 respectively. Then 3n0 = 2e0 + 6, which implies that n0 is even, i.e. F0 has an even number of vertices. We will prove that Fo has a perfect matching by Tutte's theorem. To the contrary suppose that F0 has no perfect matchings. By Theorem 2.1, there exists a subset X0 c V(Fo) such that odd(Fo — Xo) > |Xo|. (3.1) For the sake of convenience, let a := odd(F0 — X0). Since a and |X0| have the same parity, we have a > |Xo| + 2. (3.2) Let Gi,..., Ga and Ga+i,..., Ga+^ denote respectively the odd components and the even components of F0 — X0, where ft denotes the number of even components of F0 — X0. For i = 1, 2,..., a + ft, let mj denote the number of edges of F0 which are sent to X0 from Gj, and 7j (resp. 70) the number of edges of F from Gj (resp. X0) to F0. Since V(F0) is a 6-edge-cut of F, we have a+P |V(F0)| = ^ 7i = 6. (3.3) i=0 Since F is 3-connected, for i = 1,..., a,..., a + ft we have |V(Gj)| = mj + 7j > 3. (3.4) Figure 2: Illustration for the proof of Theorem 3.1. By taking the number of edges of F from the components Gi to Fo and Xo into account and by using Equation (3.3) and Inequalities (3.2) and (3.4) we have a+p 3(a + £) < ^ (mi + Yj) i=1 a+p < 3|Xo|- Yo + ^ Yi (3.5) i=l = 3|Xo| +6 - 2yo < 3a — 2yo, which implies that ft = 0, yo =0 and equalities always hold. Hence J2a=1 Yi = 6, and a = |Xo| + 2. Further, the second equality in (3.5) implies that Xo is an independent set of Fo. The first equality in (3.5) implies that mi + Yi = 3 for each 1 < i < a, that is, V(Gi) is a 3-edge-cut of F. So by Lemma 2.2 it is a trivial edge-cut and each Gi is a singleton. Let Yo denote the set of all singletons Gi. Then Fo is a bipartite graph with partite sets Xo and Yo. If Fo has no vertices of degree one, then Fo is 2-connected. Otherwise, Fo has a bridge, the deletion of which results in two components each containing a cycle. So the bridge together with at most three edges in V(Fo) form a cyclic edge-cut, contradicting that cA(F) = 5 (Theorem 2.4). Hence Fo is a fullerene patch. Since k2 = |V(Fo)| = 6, by Lemma 2.6 we have that the number p5 of pentagons contained in Fo is equal to the number k3 of vertices of degree three lying on the boundary of Fo. Since Fo is bipartite, k3 = p5 = 0, which implies that Fo is just a hexagon, contradicting that a = | Yo | = |Xo | +2. If Fo has a vertex x of degree one, let xy be the edge of Fo, and xy1 and xy2 be the other two edges in F incident with x. Then V(Fo — x) = (V(Fo) \ {xy1, xy2}) U {xy} forms a cyclic 5-edge-cut of F since Fo — x contains all cycles of Fo and Fo — x can be obtained from F — Fo by adding a 2-length path y1xy2 and contains at least seven pentagons. Since Fo — x is bipartite, cyclic 5-edge-cut V(Fo — x) is not trivial, and Fo — x is always 2-connected from Theorem 2.5. By Lemma 2.6 we have p5 = k3 + 1 for the fullerene patch F0 — x, which implies that F0 has at least one pentagon, contradicting that F0 is bipartite. □ Corollary 3.2. For each cyclic 6-edge cut E0 of a fullerene graph F, both components of F — E0 have a perfect matching. Proof. It follows that F — E0 has exactly two components from Lemma 2.2 and 3-edge-connectedness of F. Such two components fulfil the conditions of Theorem 3.1 and thus each has a perfect matching. □ 0 <£> Figure 3: Some nice substructures of fullerene graphs. Figure 4: Some nice patches of fullerene graphs with six 2-degree vertices. From Corollary 3.2 we can find many nice substructures of fullerene graphs, examples of which are shown in Figures 3 and 4. It should be mentioned that the third nice substructure fulvene in Figure 3 has been discovered by Doslic applying 2-extendability of fullerenes [5, 27], and the first one has been proved in investigating k-resonance [26, 11]; see the following. Corollary 3.3 ([26]). Each hexagon of a fullerene graph is resonant. Corollary 3.4. Each 8-length cycle (if exists) of a fullerene graph bounds a pentalene (a pair of abutting pentagons) and is thus resonant. Proof. Let C be a 8-length cycle of a fullerene graph F .If F has an edge e whose endver-tices both lie in C but e ^ E(C), then e is called a chord of C. If C has no chords, then the eight edges issuing from C can be classified into two edge-cuts of size from 3 to 5, which lie in the interior and the exterior of C respectively. If one is a 3-edge-cut, then Lemma 2.2 implies that it is trivial, and thus a triangle or quadrilateral appear, a contradiction. If both are 4-edge-cuts, then Lemma 2.3 implies that F has only 12 vertices, also a contradiction. So C must have a chord. Further, this chord and C form a pair of 5-length cycles sharing this chord, which must bound pentagonal faces of F by Theorem 2.4. That is, C bounds a pentalene and is resonant from Corollary 3.2. □ 4 Cyclic 8-edge-cut Theorem 4.1. If E0 is a cyclic 8-edge-cut of a fullerene graph F and E0 is a matching, then F — E0 has a perfect matching. Proof. There exists a nonempty and proper subset X of vertex set V(F) such that E0 = V(X) = [X,X]. Let Fo := F[X] and Fo := F[X]. We claim that both Fo and Fo are connected and E0 is a minimal edge-cut. If not, then one of F0 and F0, say F0, is disconnected. Then F0 has exactly two components since F is 3-connected. Since E0 is a matching, F0 and each component of F0 have the minimum degree 2 and contain a cycle. So a cyclic edge-cut of at most four edges occurs in F, a contradiction. So the claim is verified. Hence each of F0 and F0 has exactly one face of size more than six, which has exactly 8 two-degree vertices on its boundary. We only show that F0 has a perfect matching (the same for F0). If F0 has a bridge, then it follows that F0 can be obtained from two pentagons by adding one edge between them by Theorems 2.4 and 2.5. In this case F0 has a perfect matching. So in the following we always suppose that F0 is a patch of F. We adopt similar arguments and notations as in the proof of Theorem 3.1 (see Figure 2). It is known that F0 has an even number of vertices. Suppose to the contrary that F0 has no perfect matchings. By Tutte's theorem we can choose a minimal subset X0 c V(F0) satisfying a := odd(F0 — X0) > |X0| + 2. Let Gi,..., Ga and Ga+i,..., Ga+p denote respectively the odd components and the even components of F0 — X0. For i = 1, 2,..., a + ,0, let mi denote the number of edges of F0 which are sent to X0 from Gj, and 7i (resp. 70) the number of edges of F from Gi (resp. X0) to the patch F0. By |V(F0) | = Y,0=cf Yi =8 and Inequality (3.4), we have a+p 3(a + £) < Y (mi + Yi) i=i a+p < 3|X0|— 70 + Y Yi (4.1) i=i = 3|X0| + 8 — 270 < 3a + 2 — 2y0, which implies that ft = 0, 0 < 70 < 1, and |X0| + 2 = a. So the forth equality in Inequality (4.1) holds. If 70 = 1, then |[X — X0,X]| = ^jf Yi = 7 and all equalities in Inequality (4.1) hold. Like the proof of Theorem 3.1 we have that X0 is an independent set, mi + Yi = 3 for each 1 < i < a and each Gi is a singleton. Hence F0 is a bipartite graph. By Lemma 2.6 we have that F0 has two three-degree vertices on the boundary of F0. That implies that F0 is just the graph obtained by gluing two hexagons along an edge. So F0 has the same cardinalities of two partite sets, which contradicts that |X0| + 2 = a. From now on we suppose that y0 = 0. That is, each vertex of X0 has degree 3 in F0. We claim that second equality in Inequality (4.1) must hold. Otherwise, F0 [X0] has exactly one edge, say uv, and the first equality holds, so each Gj is a singleton. Without loss of generality, suppose that y1 and y2 are two neighbors of u other than v, and V(Gi) = jyi} and V(G2) = {y2}. Let X0 := X0 \ {u}, and Xi := {u,yi,ys}. Then Gi := F0[Xi] is a 3-vertex path obtained by combining Gi and G2 with vertex u. Hence F0 - X0 has the odd components Gi, G3,..., Ga, and odd(F0 - X0) = a - 1 = |X01 + 2, contradicting the minimality of X0. Hence X0 is an independent set of F0, and the first inequality is strict. Since for each 1 < i < a, mj + 7j is always odd, there exists an i0 such that mio + Yio =5 and mj + 7j = 3 for all i = i0. For convenience, we may suppose that i0 = 1. So Gi is an odd component with at least three vertices and G2,..., Ga are all singletons. Let Y0 denote the set of all singletons Gj (2 < i < a). Then H := (X0, Y0) is a bipartite graph as the induced subgraph of fullerene graph F. If Gi is a tree, then it is a 2-length path, say xyz, since V(Gi) has exactly five edges. For F0, by Lemma 2.6 we have p5 = k3 - 2. Since E0 is a matching, x and z both have neighbors in X0, so y1 < 3. The latter implies J2a=2 > 5. That is, the boundary of F0 contains at least 5 two-degree vertices belonging to Y0. We assert that p5 < 2. Since H is bipartite, any pentagon P of F0 must intersect G1. If P only intersects a vertex of G1, say z, then P - z is a path of length 3 in H which connects two vertices of X0, contradicting that any path between two vertices in the same partite set of a bipartite graph has an even length. Similarly we have that P cannot contain both edges of G1. If F0 has two distinct pentagons sharing the same edge of G1, then one pentagon must have two edges G1, a contradiction. So the assertion holds. By the assertion and p5 = k3 - 2 we have k3 < 4. This implies that the boundary of F0 has at most 4 vertices in X0. Let C be the boundary of F0. Then C - V(C) n X0 has at most |V(C) n X0| components. On the other hand, C - V(C) n X0 has all singletons in V(C) n Y0 as components. But |V(C) n Y0| > 5, contradicting |V(C) n X0| < 4. From now on suppose that G1 contains a cycle. Then V(G1) is a cyclic 5-edge-cut of F. By Theorem 2.5 V(G1) is a matching and G1 is also a patch (precisely, G1 is a pentagon or contains a pentacap according as the cyclic 5-edge-cut V(G1) is trivial or not), so each vertex of H has degree at least two, and each component of H contains a cycle. If H is disconnected, then H has exactly two components H1 and H2 since F0 is 2-edge-connected and V(G1) has exactly five edges. Further, between G1 and each Hj has at least two edges. So V(G1) has two consecutive edges along the boundary of G1 separately from G1 to H1 and H2. These two edges must be contained in a cycle of length at least 8 bounding a face of F, a contradiction. Hence H is connected. Since G1 and F0 are two connected subgraphs of F with exactly one face of size more than six, there are two possible cases to be considered. Case 1. G1 and F0 lie in different faces of H. Suppose that G1 lies in a bounded face f of H and F0 does in the exterior face of H. Then the boundary df of f is a 10-length cycle since 5 neighbors of G1 in H belong to X0 and are separated by 5 vertices in Y0. Hence F is a nanotube with two pentacaps and F0 has exactly 6 pentagons. By Lemma 2.6 the boundary of F0 has exactly 8 vertices of degree 3 in F0. Hence the boundary of F0 is an alternating cycle of three-degree and two-degree vertices. But in this nanotube there is only H. Li and H. Zhang: The isolated-pentagon rule and nice substructures infullerenes 495 10-length cycle as such boundary of a patch, a contradiction. Figure 5: Illustration for Case 2 in the proof of Theorem 4.1 (the vertices in X0 are colored white and other vertices black). Case 2. G1 and F0 lie in the exterior face of H. Then the boundary of F0 is formed by a path P of H and a path P1 of Gi and two edges between them. So 0 < y1 < 3, and there are 8 - y1 two-degree vertices lying on P, which belong to Y0 and are thus non-adjacent mutually. So there are at least 7 - y1 three-degree vertices in X0 on P that can separate them. Since the four end-vertices of P and P1 are all of degree three in F0, there are at least 11 - y1 vertices of degree three of F0 on the boundary. That is, for F0, k3 > 11 - y1. On the other hand, if G1 is a pentagon, then F0 has at most 5 - y1 pentagons, so k3 < 7 - y1 by Lemma 2.6, a contradiction. Otherwise, V(G1) is a non-trivial cyclic 5-edge-cut and F0 has exactly 6 pentagons. Hence, by Lemma 2.6 we have that for F0, k3 = 8. So y1 = 3. Take two consecutive edges e and f of V(G1) along the boundary of G1 separately from G1 to F0 and H. Since V(G1) is a non-trivial cyclic 5-edge-cut, by Theorem 2.5 we have that e and f have non-adjacent end-vertices in G1. So these two edges belong to a cycle of length at least 7 bounding a face of F (see Figure 5). But this is impossible. □ From Theorem 4.1 we further find many nice substructures of fullerene graphs, which are listed in Figure 6. In particular, the first one is the naphthalene (a pair of abutting hexagons), whose boundary is a resonant cycle of length 10. Corollary 4.2. Any adjacent hexagons of a fullerene graph form a nice substructure, and the boundary (10-length cycle) is thus resonant. However, not all 10-length cycles of fullerene graphs are resonant. For example, see Figure 1. The following corollary gives a criterion for a 10-length cycle of a fullerene graph to be resonant. Corollary 4.3. A 10-length cycle C of a fullerene graph F is resonant if and only if it bounds either the naphthalene or the second patch in Figure 4. Proof. The sufficiency is immediate from Corollaries 3.2 and 4.2. So we only consider the necessity. Suppose that 10-length cycle C of a fullerene graph F is resonant. Let F0 be Figure 6: Some nice patches of fullerene graphs with eight 2-degree vertices. the patch of F bounded by 10-length cycle C with p5 < 6. So F0 has an even number of vertices, and we can have that k3 and k2 both are even. By Lemma 2.6 we have p5 = 2k3 -4 and 2 < k3 < 5. The possible values of k3 are 2 and 4. If k3 = 2, then C bounds a pair of adjacent hexagons. If k3 = 4, then F0 has exactly two vertices in the interior of C which are adjacent by Lemma 2.3. In fact, F0 is the second patch in Figure 4. □ References [1] J. Aihara, Bond resonance energy and verification of the isolated pentagon rule, J. Am. Chem. Soc. 117 (1995), 4130-4136, doi:10.1021/ja00119a029. [2] J. Aihara, Graph theory of aromatic stabilization, Bull. Chem. Soc. Japan 89 (2016), 14251454, doi:10.1246/bcsj.20160237. [3] V. Andova, F. Kardos and R. Skrekovski, Mathematical aspects of fullerenes, Ars Math. Con-temp. 11 (2016), 353-379, doi:10.26493/1855-3974.834.b02. [4] J. Bornhoft, G. Brinkmann and J. Greinus, Pentagon-hexagon-patches with short boundaries, European J. Combin. 24 (2003), 517-529, doi:10.1016/s0195-6698(03)00034-9. [5] T. Doslic, On some structural properties of fullerene graphs, J. Math. Chem. 31 (2002), 187195, doi:10.1023/a:1016274815398. [6] T. Doslic, Cyclical edge-connectivity of fullerene graphs and (k, 6)-cages, J. Math. Chem. 33 (2003), 103-112, doi:10.1023/a:1023299815308. [7] P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995. [8] P. W. Fowler, S. Nikolic, R. De Los Reyes and W. Myrvold, Distributed curvature and stability of fullerenes, Phys. Chem. Chem. Phys. 17 (2015), 23257-23264, doi:10.1039/c5cp03643g. [9] B. Grunbaum and T. S. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Canad. J. Math. 15 (1963), 744-751, doi:10.4153/cjm-1963-071-3. [10] W. C. Herndon, Resonance energies of aromatic hydrocarbons: Quantitative test of resonance theory, J. Am. Chem. Soc. 95 (1973), 2404-2406, doi:10.1021/ja00788a073. [11] T. Kaiser, M. Stehlik and R. Skrekovski, On the 2-resonance of fullerenes, SIAM J. Discrete Math. 25 (2011), 1737-1745, doi:10.1137/10078699x. [12] F. Kardos and R. Skrekovski, Cyclic edge-cuts in fullerene graphs, J. Math. Chem. 44 (2008), 121-132, doi:10.1007/s10910-007-9296-9. [13] A. R. Khamatgalimov and V. I. Kovalenko, Molecular structures of unstable isolated-pentagon-rule fullerenes C72-C86, Russ. Chem. Rev. 85 (2016), 836-853, doi:10.1070/rcr4571. [14] H. W. Kroto, The stability of the fullerenes Cn, with n = 24, 28, 32, 36, 50, 60 and 70, Nature 329 (1987), 529-531, doi:10.1038/329529a0. [15] H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl and R. E. Smalley, C60: Buckminster-fullerene, Nature 318 (1985), 162-163, doi:10.1038/318162a0. [16] K. Kutnar and D. Marusic, On cyclic edge-connectivity of fullerenes, Discrete Appl. Math. 156 (2008), 1661-1669, doi:10.1016/j.dam.2007.08.046. [17] L. Lovasz and M. D. Plummer, Matching Theory, AMS Chelsea Publishing, Providence, Rhode Island, 2009, doi:10.1090/chel/367, corrected reprint of the 1986 original. [18] Z. Qi and H. Zhang, A note on the cyclical edge-connectivity of fullerene graphs, J. Math. Chem. 43 (2008), 134-140, doi:10.1007/s10910-006-9185-7. [19] M. Randic, Conjugated circuits and resonance energies of benzenoid hydrocarbons, Chem. Phys. Lett. 38 (1976), 68-70, doi:10.1016/0009-2614(76)80257-6. [20] M. Randic, Aromaticity and conjugation, J. Am. Chem. Soc. 99 (1977), 444-450, doi:10.1021/ ja00444a022. [21] A. Sanz Matías, R. W. A. Havenith, M. Alcami and A. Ceulemans, Is C50 a superaromat? Evidence from electronic structure and ring current calculations, Phys. Chem. Chem. Phys. 18 (2016), 11653-11660, doi:10.1039/c5cp04970a. [22] T. G. Schmalz and D. J. Klein, Fullerene structures, in: W. E. Billups and M. A. Ciufolini (eds.), Buckminsterfullerenes, VCH Publishers, New York, chapter 4, pp. 83-101, 1993. [23] T. G. Schmalz, W. A. Seitz, D. J. Klein and G. E. Hite, Elemental carbon cages, J. Am. Chem. Soc. 110 (1988), 1113-1127, doi:10.1021/ja00212a020. [24] W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947), 107-111, doi:10.1112/jlms/s1-22.2.107. [25] Q. Yang, H. Zhang and Y. Lin, On the anti-forcing number of fullerene graphs, MATCH Commun. Math. Comput. Chem. 74 (2015), 673-692, http://match.pmf.kg.ac.rs/ electronic_versions/Match7 4/n3/match7 4n3_67 3-6 92.pdf. [26] D. Ye, Z. Qi and H. Zhang, On k-resonant fullerene graphs, SIAM J. Discrete Math. 23 (2009), 1023-1044, doi:10.1137/080712763. [27] H. Zhang and F. Zhang, New lower bound on the number of perfect matchings in fullerene graphs, J. Math Chem. 30 (2001), 343-347, doi:10.1023/a:1015131912706. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 499-522 https://doi.org/10.26493/1855-3974.1409.e54 (Also available at http://amc-journal.eu) Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime* Yan-Quan Feng t, Da-Wei Yang , Jin-Xin Zhou Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R. China Received 23 May 2017, accepted 10 May 2018, published online 9 September 2018 A regular cover of a connected graph is called cyclic or dihedral if its transformation group is cyclic or dihedral respectively, and arc-transitive (or symmetric) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentava-lent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined. Keywords: Symmetric graph, Cayley graph, bi-Cayley graph, regular cover. Math. Subj. Class.: 05C25, 20B25 1 Introduction All groups and graphs considered in this paper are finite, and all graphs are simple, connected and undirected, unless otherwise stated. Let G be a permutation group on a set Q and let a e Q. Denote by Ga the stabilizer of a in G, that is, the subgroup of G fixing the point a. We say that G is semiregular on Q if Ga = 1 for every a e Q, and regular if G is transitive and semiregular. Denote by Zn, Z*n, Dn, An and Sn the cyclic group of order n, the multiplicative group of units of Zn, the dihedral group of order 2n, the alternating and symmetric group of degree n, respectively. For two groups M and N, we use MN, M.N, M x N and M x N to denote the product of M and N, an extension of M by N, a split extension of M by N and the direct product of M and N, respectively. For a subgroup H *This work was supported by the National Natural Science Foundation of China (11731002, 11571035, 11711540291, 11671030) and by the 111 Project of China (B16002). t Corresponding author. E-mail addresses: yqfeng@bjtu.edu.cn (Yan-Quan Feng), dwyang@bjtu.edu.cn (Da-Wei Yang), jxzhou@bjtu.edu.cn (Jin-Xin Zhou) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ of a group G, CG(H) means the centralizer of H in G and NG(H) means the normalizer of H in G. For a graph r, we denote its vertex set, edge set and full automorphism group by V(r), E(r) and Aut(r), respectively. An s-arc in r is an ordered (s + 1)-tuple (v0, vi,..., vs) of vertices of r such that {vi-1, v¿} G E(r) for 1 < i < s, and vi-1 = vi+1 for 1 < i < s. A 1-arc is just an arc. A graph r is (G, s)-arc-transitive for a subgroup G of Aut(r) if G acts transitively on the set of s-arcs of r, and (G, s)-transitive if r is (G, s)-arc-transitive but not (G, s + 1)-arc-transitive. A graph r is said to be s-arc-transitive or s-transitive if it is (Aut(r), s)-arc-transitive or (Aut(r), s)-transitive, respectively. In particular, 0-arc-transitive means vertex-transitive, and 1-arc-transitive means arc-transitive or symmetric. A graph r is edge-transitive if Aut(r) is transitive on the edge set E(r). Let r be a graph and N < Aut(r). The quotient graph rN of r relative to the orbits of N is defined as the graph with vertices the orbits of N on V(r) and with two orbits adjacent if there is an edge in r between those two orbits. In particular, for a normal subgroup N of Aut(r), if r and rN have the same valency, then rN is a normal quotient of r, and if r has no proper normal quotient, then r is basic. To study a symmetric graph r, there is an extensive used strategy consisting of two steps: the first one is to investigate normal quotient graph rN for some normal subgroup N of Aut(r) and the second one is to reconstruct the original graph r from the normal quotient rN by using covering techniques. This strategy was first laid out by Praeger (see [31]), and it is usually done by taking the normal subgroup N as large as possible and then the graph r is reduced a basic graph. In the literature, there are many works about basic graphs (see [1, 14, 16] for example), while the works about the second step, that is, covers of graphs, are fewer. An epimorphism n: f ^ r of graphs is called a regular covering projection if Aut(f) has a semiregular subgroup K whose orbits in V (f) coincide with the vertex fibres n-1 (v), v G V (r), and whose arc and edge orbits coincide with the arc fibres n-1((u, v)) and the edge fibres n-1({u,v}), {u, v} G E(r), respectively. In particular, we call the graph f a regular cover or a K-cover of the graph r, and the group K the covering transformation group. If K is dihedral, cyclic or elementary abelian, then r is called a dihedral, cyclic or elementary abelian cover of r, respectively. An automorphism of r is said to be fibre-preserving if it maps a vertex fibre to a vertex fibre, and all such fibre-preserving automorphisms form a group called the fibre-preserving group, denoted by F. It is easy to see that F = NAut(p) (K). If r is F-arc-transitive, we say that r is an arc-transitive cover or a symmetric cover of r. For an extensive treatment of regular cover, one can see [3, 26, 27]. Covering techniques have long been known as a powerful tool in algebraic and topo-logical graph theory. Application of these techniques has resulted in many constructions and classifications of certain families of graphs with particular symmetry properties. For example, by using covering techniques, Djokovic [10] constructed the first infinite family of 5-arc-transitive cubic graphs as covers of Tutte's 8-cage, and Biggs [4] constructed some 5-arc-transitive cubic graphs as covers of cubic graphs that are 4-arc-transitive but not 5-arc-transitive. Gross and Tucker [18] proved that every regular cover of a base graph can be reconstructed as a voltage graph on the base graph. Later, Malnic et al. [26] and Du et al. [12] developed these ideas further in a systematic study of regular covering projections of a graph along which a group of automorphisms lifts. Based on the approaches studied in [12, 26], many arc-transitive covers of symmetric graphs of small orders and small valencies have been classified. For example, Pan et al. [29] studied arc-transitive cyclic covers of some complete graphs of small orders. One may see [2, 27] for other works. Moreover, a new approach was proposed by Conder and Ma [7, 8] by considering a presentation (quotient group) of a universal group, which can be obtained from Reidemeister-Schreier theory, and representation theory and other methods are applied when determining suitable quotients. As an application, arc-transitive abelian covers of the complete graph K4, the complete bipartite graph K3 3, the 3-dimensional hypercube Q3, the Petersen graph and the Heawood graph, were classified. Later, arc-transitive dihedral covers of these graphs were determined by Ma [25]. For arc-transitive covers of infinite families of graphs, Du et al. studied 2-arc-transitive elementary abelian and cyclic covers of complete graphs Kn in [11, 13] and Kn n - nK2 in [32, 33]. Recently, Pan et al. [28] determined arc-transitive cyclic covers of the complete bipartite graph Kp,p of order 2p for a prime p. Compared with symmetric covers of graphs of small orders and valencies, there are only a few contributions on symmetric covers of infinite families of graphs. Arc-transitive covers of non-simple graphs were also considered in literature. For example, regular covers of the dipole Dipk (a graph with two vertices and k parallel edges) were extensively studied in [2, 16, 26, 27, 34]. Such covers are called Haar graphs, and in particular, cyclic regular covers of dipoles are called cyclic Haar graphs, which can be regarded as a generalization of bipartite circulants and were studied in [21] (also see [15]). Construction of Haar graphs have aroused wide concern. Marusic et al. [26] studied elementary abelian covers of the dipole Dipp for a prime p. In particular, symmetric elementary abelian covers and Zp x Zp-covers for a prime p of the dipole Dip5 were classified completely in [16] and [34], respectively. Let p be a prime. Pentavalent symmetric graphs of order 2p were classified by Cheng and Oxley in [6], which are the complete graph K6 of order 6 and a family of Cayley graphs CDp with p = 5 or 5 | (p - 1) on dihedral groups (see Proposition 3.4). It has been shown that many pentavalent symmetric graphs are regular covers of them, see [16, 34]. In this paper, we consider arc-transitive cyclic and dihedral covers of these graphs. For K6, the cyclic covers have been classified in [29], which should be the complete bipartite graph K6,6 and the Icosahedron graph I12 (note that Ii2 is missed in [29]). For CDp, the cyclic covers consist of six infinite families of graphs, which are Cayley graphs on generalized dihedral groups. In particular, one family of graphs are cyclic Haar graphs and the other five families are non-cyclic Haar graphs. What is more, the full automorphism groups of them are determined. Arc-transitive dihedral covers of K6 and CDp are also classified, and there are only four sporadic graphs of order 24, 48, 60 and 120, respectively. A similar work about cubic graphs was done by Zhou and Feng [37]. Different from regular covers of graphs mentioned above, the method to classify arc-transitive cyclic covers used in this paper is related to the so called bi-Cayley graph. A graph r is a bi-Cayley graph over some group H if Aut(r) has a semiregular subgroup isomorphic to H having exactly two orbits on V(r). Clearly, a Haar graph is a bipartite bi-Cayley graph. Recently, Zhou and Feng [38] gave a depiction of the automorphisms of bi-Cayley graphs (see Section 4), and based on this work, we classify the cyclic covers. In particular, all these covers are bi-Cayley graphs over some abelian groups. Note that vertex-transitive bi-Cayley graphs of valency 3 over abelian groups were determined in [36], while the case for valency 5 is still elusive. Indeed, even for arc-transitive pentavalent bi-Cayley graphs over abelian groups, it seems to be very difficult to give a classification, and one may see [2, 16, 34] for partial works. The paper is organized as follows. After this introductory section, in Section 2 we give some notation and preliminary results. In Section 3, several infinite families of connected pentavalent symmetric graphs are constructed as Cayley graphs on generalized dihedral groups Dih(Zmpe x Zp), where e, m are two positive integers and p is a prime such that (m, p) = 1. In Section 4, it is proved that these Cayley graphs include all arc-transitive normal bipartite bi-Cayley graphs over Zmpe x Zp, and using this result, arc-transitive cyclic and dihedral covers of connected pentavalent symmetric graphs of order 2p are classified in Sections 5 and 6, respectively. In Section 7, the full automorphism groups of these covers are determined. 2 Preliminaries In this section, we describe some preliminary results which will be used later. The following result is important to investigate symmetric pentavalent graphs. Proposition 2.1 ([19, Theorem 1.1]). Let r be a connected pentavalent (G, s)-transitive graph for some G < Aut(r) and s > 1, and let v G V(r). Then one of the following holds: (1) s = 1 and Gv = Z5, D5 or D10; (2) s = 2 and Gv = F20, F20 x Z2, A5 or S5, where F20 is the Frobenius group of order 20; (3) s = 3 and Gv = F20 x Z4, A4 x A5, S4 x S5 or (A4 x A5) x Z2 with A4 x Z2 = S4 and A5 xZ2 = S5; (4) s = 4 and Gv = ASL(2,4), AGL(2,4), ASL(2,4) or ArL(2,4); (5) s = 5 and Gv = Z2 x rL(2,4). From [24, Theorem 9], we have the following proposition. Proposition 2.2. Let r be a connected G-arc-transitive graph of prime valency, and let N be a normal subgroup of G. If N has at least three orbits, then it is semiregular on V(r) and the kernel of G on the quotient graph rN. Furthermore, rN is G/N-arc-transitive, and r is a regular cover of rN with N as the covering transformation group. Let G and E be two groups. We call an extension E of G by N a central extension of G if E has a central subgroup N such that E/N = G, and if further E is perfect, that is, if it equals its derived group E', we call E a covering group of G. Schur proved that for every non-abelian simple group G there is a unique maximal covering group M such that every covering group of G is a factor group of M (see [22, V, § 23]). This group M is called the full covering group of G, and the center of M is the Schur multiplier of G, denoted by Mult(G). Lemma 2.3. Let G be a group, and let N be an abelian normal subgroup of G such that G/N is a non-abelian simple group. If N is a proper subgroup of CG(N), then G = G'N and G' n N < Mult(G/N). Proof. Since N is aproper subgroup of CG(N), we have 1 = CG(N)/N < G/N, forcing CG (N)/N = G/N because G/N is simple. Thus G = CG (N) and it is a central extension of G/N by N. Since G/N = (G/N)' = G'N/N = G'/(G' n N), we have G = G'N, and since G' = (G'N)' = (G')', G is a covering group of G/N. Hence G n N < Mult(G/N). □ Denote by soc(G) the socle of G, that is, the product of all minimal normal subgroups of G. A list of all proper primitive permutation groups of degree less than 1000 was given by Dixon and Mortimer [9, Appendix B], and based on the list, we have: Lemma 2.4. Let G be a primitive permutation group on a set Q and let a G Q, where |Q | G {2, 4, 6, 8,12,16, 24, 72,144, 288, 576}. If Ga is solvable, then either G < AGL(n, 2) and |Q| = 2n with 1 < n < 4, or soc(G) = PSL(2,p), PSL(3, 3) or PSL(2,q) x PSL(2, q) with |Q| = p + 1, 144 or (q + 1)2 respectively, wherep G {5,7,11,23,71} and q G {11, 23}. Proof. If |Q| = 2 or 4, then G < S2 ^ AGL(1, 2) or G < S4 = AGL(2,2), respectively. Let |Q| > 6 and write N := soc(G). Then N < G and Na < Ga. Since Ga is solvable, Na is solvable. By [9, Appendix B, Tables B.2 and B.3], G is an affine group, N = A|fi|, or G is isomorphic to one group listed in [9, Tables B.2 and B.3]. If G is affine, then |Q| is a prime power and thus |Q| = 2n with n =3 or 4. By [9, Theorem 4.1A (a)], we have G < AGL(n, 2). If N = A|q| then Na = A|q|-i, which is insolvable because |Q| — 1 > 5, a contradiction. In what follows we assume that G is isomorphic to one group listed in [9, Tables B.2 and B.3]. Note that all groups in the tables are collected into cohorts and all groups in a cohort have the same socle. Assume that |Q| = 144. By [9, Table B.4, pp. 324], there are one cohort of type C, two cohorts of type H and four cohorts of type I (see [9, Table B.1, pp. 306] for types of cohorts of primitive groups) of primitive groups of degree 144. For the cohort of type C, by [9, Table B.2, pp. 314], N = PSL(3, 3) and Na = Z13 x Z3. For the two cohorts of type H, by [9, Table B.2, pp. 321], they have the same socle N = M12 and Na = PSL(2,11). For the four cohorts of type I, by [9, Table B.3, pp. 323], N = A12 x A12, PSL(2,11) x PSL(2,11), M11 x M11 or M12 x M12 and Na = A11 x A11, (Z11 x Z5) x (Z11 x Z5), M10 x M10 or M11 x M11, respectively. Since Na is solvable, we have N = PSL(3, 3) or PSL(2,11) x PSL(2,11). For |Q| G {6, 8,12,16,24,72,288, 576}, by [9, Tables B.2, B.3 and B.4], a similar argument to the above paragraph implies that either N = PSL(2,23) x PSL(2,23) with degree 232 = 576 and Na = (Z23 x Zn) x (Z23 x Zn), or N = PSL(2,p) with degree p + 1 and Na = Zp x Zp-l wherep G {5,7,11,23,71}. □ 3 Graph constructions as Cayley graphs Let G be a finite group and S a subset of G with 1 G S and S-1 = S. The Cayley graph r = Cay(G, S) on G with respect to S is defined to have vertex set V(r) = G and edge set E(r) = {{g, sg} | g g G, s g S}. It is well-known that Aut(r) contains the right regular representation R(G) of G, the acting group of G by right multiplication, and r is connected if and only if G = (S), that is, S generates G. By Godsil [17], NAut(r)(R(G)) = R(G) x Aut(G, S), where Aut(G, S) = {a G Aut(G) | Sa = S}. A Cayley graph r = Cay(G, S) is said to be normal if R(G) is normal in Aut(r), and in this case, Aut(r) = R(G) x Aut(G, S). For an abelian group H, the generalized dihedral group Dih(H) is the semidirect product H x Z2, where the unique involution in Z2 maps each element of H to its inverse. In particular, if H is cyclic, then Dih(H) is a dihedral group. In this section, we introduce several infinite families of connected pentavalent symmetric graphs which are constructed as Cayley graphs on generalized dihedral groups. Example 3.1. Let Dih(Zf) = {a, b,c,h | a5 = b5 = c5 = h2 = [a, b] = [a, c] = [b, c] = 1, ah = a-1, bh = b-1,ch = c-1}, and define CgV53 = Cay(Dih(Zf), {h, ah, bh, ch, a-1b-1c-1h}). By [34, Theorem 1.1], Aut(C£D53) ^ Dih(Zf) x S5 and CGD53 is the unique connected pentavalent symmetric graph of order 250 up to isomorphism. Let m be a positive integer. Consider the following equation in Zm x4 + x3 + x2 + x +1 = 0. (3.1) In view of [15, Lemma 3.3], we have the following proposition. Proposition 3.2. Equation (3.1) has a solution r in Zm if and only if (r, m) e {(0,1), (1,5)} or m = 5tpe11 p22 ■ ■ ■ pess and r is an element in Z*m of order 5, where t < 1, s > 1, ei > 1 and pi's are distinct primes such that 5 | (p^ — 1). The following infinite family of Cayley graphs was first constructed in [23]. Example 3.3. Let m > 1 be an integer such that Equation (3.1) has a solution r in Zm. Then m = 5, 11 or m > 31. Let CDm = Cay(Dm, {b, ab, ar+1b, ar'2+r+1b, ar'3+r'2 +r+1b}) be a Cayley graph on the dihedral group Dm = {a,b | an = b2 = 1, ab = a-1}. For m = 5 or 11, by [6], Aut(CDm) = (S5 x S5) x Z2 or PGL(2,11), respectively. In particular, CV5 = K55. For m > 31, by [23, Theorem B and Proposition 4.1], Aut(CDm) = Dm x Z5, and obviously, if m has a prime divisor p with p < m, then Aut(CDm) has a normal subgroup Zm/p, and by Proposition 2.2, CDm is a symmetric Zm/p-cover of a connected pentavalent symmetric graph of order 2p. By [6], we have the following proposition. Proposition 3.4. Let r be a connected pentavalent symmetric graph of order 2p for a prime p. Then r = K6 or CDp with p = 5 or 5 | (p — 1). In the remaining part of this section, we construct five infinite families of Cayley graphs on some generalized dihedral groups, and for convenience, we always assume that G = Dih(Zm x Zpe x Zp) = {a, b,c,h | am = bp° = cp = h2 = [a, b] = [a, c] = [b, c] = 1, ah = a-1 ,bh = b-1,ch = c-1} and r is a solution of Equation (3.1) in Zm, that is, r4 + r3 + r2 + r + 1 = 0 (mod m). By Proposition 3.2, m is odd and 52 { m. Example 3.5. Assume that e > 2 and p is a prime such that (m,p) = 1 and 5 | (p — 1). Let A be an element of order 5 in Zpe. Then A is a solution of Equation (3.1) in Zpe. Set T1 (r, A) = {h, hab, har+1bx+1c, har2+r+1bx2+x+1cx4+x+1, ha^+^+^b^+^+^c}, T2 (r, A) = {h, hab, har+1bx+1c, har'2+r+1bx2+x+1cx3+x+1, har'3+r'2+r+1bx3+x2+x+1cx}, T3 (r, A) = {h, hab, har+1bx+1c, har2+r+1bx2+x+1cx2+x+1, har3+r"+r+1bx3+x2+x+1 cx"}. It is easy to see that each of T\ (r, A), T2 (r, A) and T3 (r, A) generates G. Define CGVi„ipexp = Cay(G, Ti(r, A)), i = 1, 2, 3. The maps X X 4 a ^ ar, b ^ b c, c ^ c , h ^ hab; a ^ ar, b ^ bXc, c ^ cX , h ^ hab; a ^ ar ,b ^ b c, c ^ c ,h ^ hab induce three automorphisms of order 5 of G, denoted by a1, a2 and a3 respectively, and a fixes the set Tj (r, A) and permutes its five elements cyclicly. It follows that for each i = 1,2, 3, a G Aut(G, Tj(r, A)) and (R(G), Oj) = G x Z5, which is arc-transitive on CQVmpe xp" The graphs CQVlmpexp, CQV2mpexp and CQV3mpexp for (m, e) = (1,2) have been introduced in [34, Example 4.4], and they are not isomorphic to each other by [34, Lemma 4.5]. Indeed, we can also prove that the graphs CQVlmpe xp, CQV2mpe xp and CQV3mpe xp for each integers m > 1 and e > 2 are not isomorphic to each other. Since the proof is similar to [34, Lemma 4.5], we omit it, and one can see [35] for a detailed proof. Example 3.6. Let p be a prime such that p = 5 or 5 | (p ± 1). Assume that e =1 and (m,p) = 1. Then G = Dih(Zm x Zp x Zp). For p = 5, let A = 0, and for 5 | (p ± 1), let A G Zp satisfying the equation A2 =5 in Zp. Set S(r, A) = {h, hab, har+1c, ha^+V2"^ X)c2"(1+X), ha^+^+V2-1^X)c}. It is easy to see that S(r, A) generates G. Define CGD^pxp = Cay(G,S(r,A)). The map a ^ ar, b ^ b-1c, c ^ b-2 1(3+X)c2 1(1+X) and h ^ hab induces an automorphism of the group G, denoted by a4, which permutes the elements in S(r, A) cyclicly. Then a4 G Aut(G, S(r, A)) and (R(G), a4) = G x Z5 acts arc-transitive on CQVAmpxp. Moreover, for m = 1 or 5, we have r = 0 or 1 respectively, and the map a ^ a-1, b ^ b-2 (1+X)c, c ^ b-2 (1+X)c2 (1+X), h ^ h induces an automorphism p of G. It is easy to check that p G Aut(G, S(r, A)) and (a4,P) = D5. In particular, by [16, Theorem 6.1], CQT>5x5 is the unique connected pentavalent symmetric graph of order 50 up to isomorphism. Example 3.7. Assume that e = 1 andp is a prime such that (m,p) = 1 and 5 | (p - 1). By [34, Case 2, page 14], x4 + 10x2 + 5 = 0 has a root A in Zp. Set S(r, A) = {h, hab, har+1c, har2 +r+1b8-1( X3-X2+7X+V~1( X+1), har3+r2+r + 1b-8-1( X3 + X 2+7 X-1)c8-1( X 3 + X2 + 11 X+3)} It is easy to check that S(r, A) generates G. Define CGD^pxp = Cay(G, S(r, A)). The map a ^ ar, b ^ b-1c, c ^ b8 1(*3-*2+7*-7)c2 1(a+i) and h ^ hab induces an automorphism of the group G, denoted by a5, which permutes the elements in S(r, A) cyclicly. Then a5 G Aut(G, S(r, A)) and (R(G),a5) = G x Z5 acts arc-transitive on Let r = CGVlnpexp withp = 5 or 5 | (p - 1), where 1 < i < 5. By Examples 3.53.7, Aut(r,) contains an arc-transitive subgroup R(G) x (oj) for each 1 < i <5. Let N be a subgroup of R(G) x (oj) as listed in Table 1. In particular, for r4 with 5 | (p - 1), since A2 = 5 in Zp, the equation x4 + 10x2 +5 = 0 has a root t such that t2 = 2A — 5 (see Example 3.7). It is easy to compute that Nj = Zmpe and Nj < R(G) x (aj) for each 1 < i < 5 (see [35] for a detailed computation). By Proposition 2.2, we have the following lemma. Table 1: Subgroups of Aut(CGD mpe xp ) for 1 < i < 5. Tj P Nj CGDmpexp 5 (p — 1) (R(a),R(b5 c3^2*2-^1)) CGDmpexp 5 (p — 1) (R(a),R(b-5c2A3+4A2+A+3)) CGDmpexp 5 (p — 1) (R(a),R(b-5c4A3+3A2+2A+1)) CGDmpxp p = 5 (R(a),R(b2c4)) 5 (p — 1) (R(a), R(bi+1cA-3)) (t2 = 2A — 5) CGDmpxp 5 | (p — 1) (R(a),R(b2(A2+5)-1(A3+10A+5)-(A+3)c4)) Lemma 3.8. Let p be a prime such that p = 5 or 5 | (p — 1). Then for each 1 < i < 5, CGDmpexp is a connected symmetric cyclic cover of a connected pentavalent symmetric graph of order 2p. 4 Pentavalent symmetric bi-Cayley graphs over abelian groups Given a group H, let R, L and S be three subsets of H such that R-1 = R, L-1 = L, and 1 ^ R U L. The bi-Cayley graph over H relative to the triple (R, L, S), denoted by BiCay(H, R, L, S), is the graph having vertex set {h0 | h G H} U {h1 | h G H} and edge set {{ho,go} | gh-1 G R} U {{hi,gi} | gh-1 G L} U {{ho,gi} | gh-1 G S}. For a bi-Cayley graph r = BiCay(H, R, L, S), it is easy to see that R(H) can be regarded as a semiregular subgroup of Aut(r) with two orbits, which acts on V(r) by the rule hR(g) = (hg)j, i = 0,1, h, g G H. If R(H) is normal in Aut(r), then r is a normal bi-Cayley graph over H. Let r = BiCay(H, 0,0, S) be a connected bi-Cayley graph over an abelian group H. Then r is bipartite. By [38, Lemma 3.1], we may always assume that 1 G S. Moreover, r = BiCay(H, 0, 0, Sa) for a G Aut(H), and H = (S). Since H is abelian, there is an automorphism of H of order 2, denoted by 7, induced by g ^ g-1, Vg G H. For a G Aut(H) and x G H, define ¿7,1,1 : h0 ^ (h-1)i, hi ^ (h-1)0, Vh G H; &a,x : h0 ^ (ha)0, h1 ^ (xha)1, Vh G H. Set F = jaa,x | a G Aut(H), Sa = x-1S}. Then ¿Y,1,1 G Aut(r) and F < Aut(r)1o (see [38, Lemma 3.3]). Since r is connected, F acts on Nr(10) faithfully. By [38, Theorem 1.1 and Lemma 3.2], we have the following proposition. Proposition 4.1. Let r = BiCay(H, 0, 0, S) be a connected bi-Cayley graph over an abelian group H, and let A = Aut(r). Then NA(R(H)) = R(H)(F, ¿7,1,1) with vertex stabilizer (NA(R(H)))1o = F, andr is isomorphic to the Cayley graph Cay(Dih(H), 7S), where Dih(H) = H x (7). The following lemma is from [2, Theorem 1.1]. Lemma 4.2. Let n be a positive integer and p a prime such that p > 5. Let r be a connected pentavalent symmetric bi-Cayley graph over Znp. Then r = CVnp, as defined in Example 3.3. Let H = (x) x (y) x (z) = Zm x Zpe x Zp, where m and e are two positive integers and p is a prime such that p > 5 and (m,p) = 1. In the remaining of this section, we always let r = BiCay(H, 0,0, S) be a connected pentavalent bi-Cayley graph over H such that NAut(r)(R(H)) is arc-transitive on r. Assume that S = {1, a, b, c, d}. Then H = (a, b, c, d). By Proposition 4.1, there exists a oag G F of order 5 permuting the neighborhood {I1, a1, b1, C1, d1} of I0 in r cyclicly. One may assume that 1 1 = a1 , which implies that g = a because 1aia'a = g1, and that b1 = a[a'a, c1 = b°ia'a, d1 = c[a'a and 11 = ¿1a'a. It follows that aa = ba-1, ba = ca-1, ca = da-1, da = a-1. (4.1) For h G H, denote by o(h) the order of h in H. Since aa = ba-1 by Equation (4.1), o(ba-1) = o(aa) = o(a), forcing that o(b) | o(a). Similarly, since da = a-1 and ca = da-1, we have o(d) = o(a) and o(c) | o(a). Since H = (a, b, c, d), we have o(x) | o(a) for any x G H, and since H = Zm x Zpe x Zp, we have o(a) = mpe and |H : (a)| = p. Suppose that b G (a), say b = a® for some integer i. By Equation (4.1), aa = ba 1 = a®-1 G (a) and ca-1 = ba = (a®)a = ai(i-1) G (a), implying that c G (a). Similarly, d G (a) because d = a • ca. Since H = (a, b, c, d), we have H = (a) = Zmpe, a contradiction. Hence b G (a), and since |H : (a)| = p, we have H = (a, b) and p | o(b). Let A = Aut(r). Since r is Na(R(H))-arc-transitive, F = Na(R(H))10 acts transitively on Nr(10). Let a^,g G F for some ft G Aut(H) and g G H such that 1^'9 = 11. Then 11 = (^g)1 = g1, forcing that g =1. Hence Fu = {03,1 | ft G Aut(H), S^ = S}, that is, F1l = Aut(H, S). By Proposition 4.1, |Na(R(H))| = 2|H||F| = 2|H| • |NC1q)||F1i| = 10|H|| Aut(H,S)|. Observation 4.3. o(a) = mpe, p | o(6), H = (a, 6} and |na(r(h))| = 10|H|| Aut(H,S)|. In the following two lemmas we consider the two cases: e > 2 and e =1, respectively. Lemma 4.4. If e > 2, then 5 | (p - 1), r = C£Dm„pexp for some 1 < i < 3 and |na(r(h))| = 10|H |. Proof. By Observation 4.3, o(a) = mpe, p | o(6) and H = (a, 6} = (x, y, z} = Zm x Zpe x Zp, where (m,p) = 1. Then H has an automorphism mapping xy to a, and thus we may assume a = xy, which implies that 6 = xr+1yA+1zl for some r +1 G Zm, A +1 G Zpe and 0 = i G Zp because H = (a, 6}. Furthermore, H has an automorphism fixing x, y and mapping z to zl, and so we may assume 6 = xr+1yA+1z. Let c and d = xfcy£z4, where i, k G Zm, j, ^ G Zpe and s, t G Zp. = x®yjzs 1, et > 1 and pt's are distinct primes such that 5 | (pt - 1) for 1 < i < f. Note that e > 2. By Equation (4.3), j = A2 + A+1 (mod pe-1) and by Equations (4.6) and (4.7), I = A3 + A2 + A +1 (mod pe-1) and A4 + A3 + A2 + A +1 = 0 (mod pe-1), implying A5 = 1 (mod pe-1). It follows from Proposition 3.2 that either (A,pe-1) = (1,5), or 5 | (p - 1) and A is an element in Zpe_i of order 5, forcing that A = 0 and i-1 = (-A4)-1 = -A. Furthermore, one may assume that j = A2 + A + 1 + sip6-1 (mod pe), i = A3 + A2 + A + 1 + s2pe-i (mod pe) and A4 + A3 + A2 + A +1 = ipe-1 (mod pe) for some s1, s2, i G Zp. In what follows all equations are considered in Zp, unless otherwise stated. As p | pe-1, the following equations are also true in Zp: j = A2 + A + 1, i = A3 + A2 + A +1, A4 + A3 + A2 + A + 1 = 0, i-1 = -A. By s x (4.9) - t x (4.8), s = i-1(jt - t2) = -A(jt - t2), and by Equation (4.9), we have At3 - (A3 + A2 + A)t2 - (A + 1)t + (A3 + A2 + A +1) =0. Combined with A4 + A3 + A2 + A +1 = 0 and A = 0, we have (t - 1)(t - A)(t - A2) = 0, which implies that t =1, A or A2. Recall that j = A2 + A + 1 and s = -A(jt - t2). Thus (t, s) = (1, A4 + A + 1), (A, A3 + A + 1) or (A2, A2 + A + 1). Since j = A2 + A +1 + s1pe-1 (mod pe) and i = A3 + A2 + A +1 + s2pe-1 (mod pe), by Equations (4.6) and (4.7) we have: I (A + s)sipe 1 = s2pe 1 (mod pe) I tsipe-1 + As2pe-1 = -(A4 + A3 + A2 + A +1) (mod pe) (4.10) Recall that either (A,pe-1) = (1,5) or 5 | (p - 1). Suppose that pe-1 = 5. Then p = 5, e = 2 and (A, s,t) = (1,3,1). By Equation (4.10), we have 5s2 = 20s1 and 52s1 + 5 = 0 in Z52, a contradiction. Hence 5 | (p - 1). Again by Equation (4.10), we have -(t + A2 + As)s1pe-1 = ipe-1 (mod pe), where ipe-1 = A4 + A3 + A2 + A + 1. Furthermore, Ut + a2 + As)s1 = -1 (411) \(t + A2 + As)s2 = -i(A + s) Since (t, s) = (1, A4 + A + 1), (A, A3 + A +1) or (A2, A2 + A +1), we have t + A2 + As = 2A2 + A + 2, A4 + 2A2 + 2A or A3 + 3A2 + A, respectively, and since (2A2 + A + 2)(A4 + 2A2 + 2A) = 6(A4 + A3 + A2 + A) + 1 = -5 and (A3 + 3A2 + A)(A4 - 2A3 + A2) = A4 + A3 + A2 + A - 4 = -5, we have (t + A2 + As)-1 = -5-1 (A4 + 2A2 + 2A), -5-1(2A2 + A + 2) or -5-1(A4 - 2A3 + A2), respectively. By Equation (4.11), (sb s2) = (5-1i(A4+2A2 +2A), 5-1 i(-3A4+A3+2A2)), (5-1i(2A2 +A+2), 5-1i(-3A4+2A3+A)) or (5-1i(A4 - 2A3 + A2), 5-1i(-2A4 + A2 + a)). Note that a = xy, b = xr+1yA+1z, and (c,d) = (xr2+r+1yA2+A+1+SlPe-1 zA4+A+1,xr3+r2+r+1 z), (xr2+r+1yA2 + A+1+SlPe-1 ZA3+A+1 xr3+r2+r+1 yA3 + A2+A+1+S2Pe-1 zA) or (xr2+r+y2 + A+1+SiPe-1 ZA2+A+1 xr3+r2+r+1 yA3 + A2+A+1+S2Pe-1 zA2 ) we have S = Si, S2 or S3, where 51 = {1,xy,xr+y+1z,xr2+r+y2+A+1+5-1(A4+2A2+2A)lPe-1 zA4+A+1, xr3 +r2 +r+1yA3 + A2 +A+1+5-1 (-3A4+A3 + 2A2 )ipe-1 z} 52 = {1,xy,xr+1yA+1z,xr2+r+1yA2+A+1+5-1(2A2+A+2)lPe-1 za3+a+1, xr3+r2+r+1yA3+A2 + A+1+5-1(-3A4 + 2A3 + A)ipe-1 zA } 53 = {1,xy,xr+y+1z,xr2+r+y2+A+1+5-1(A4-2A3+A2)lPe-1 zA2+A+1, xr3+r2+r+1yA3 + A2 + A+1+5-1(-2A4+A2 + A)ipe-1 zA2 Since x5 = 1 (mod pe) implies that x5 = 1 (mod pe 1), there exists f G Zp such that A1 = A + fpe-1 is an element of order 5 in Z*e. Then A = A1 - fpe-1, A® = 1 and A| + A? + A2 + A1 + 1 =0 in Zpe. Hence ipe-1 = A4 + A3 + A2 + A +1 = (A1 - fpe-1)4 + (A1 - fpe-1)3 + (A1 - fpe-1)2 + (A1 - fpe-1) + 1 = -(4A3 + 3A1 + 2A1 + 1)fpe-1 Upe. 1UC11 A = A1 - fp , A1 Uati/^a i/ne 1 — -•p 2 , n 1\ i 1 _ /n3 , in Zpe, and thus S1 = {1,xy,xr+1yA+1z,xr2+r+1yA2+A+1+5-1(A4+2A2+2A)lPe-1 zA4+A+1, xr3+r2+r+1yA3+A2 + A+1+5-1(-3A4 + A3+2A2)ipe-1 z} = {1, xy, xr+1yA1+1y-fPe-1 z,xr2+r+1 y^+^y-M+^+W"1 zA!+A+1, xr3+r2+r+1 yA3 + A?+A1 + 1y-/pe-1 z}. Let f be the automorphism of H induced by x ^ x, y ^ y and z ^ yfpe 1 z. Then (S1)^ = {1, xy, xr+1 yA1 + 1z, xr2+r+1yA2+A1 + 1zA4+A1+1, xr3+r2+r+1yA3+A1+A1+1z}. Since BiCay(H, 0,0, S1) = BiCay(H, 0, 0, Sf), we may assume that A = A1 is an element of order 5 in Zpe, and S1 = {1, xy, xr+1yA+1z, x^+y^+V^1, xr3+r2 +r+y3+A2 +A+1z}. Similarly, we can also assume that 52 = {1, xy, xr+1yA+1z, x^+y^+y^1, x^+^+V^+^V}, 53 = {1, xy, xr+1yA+1z, x^+y^+V^1, x^+^+y^^+V2}. By Proposition 4.1 and Example 3.5, r = BiCay(H, 0,0, S) = CGD^^ with 1 < i < 3. Note that |NA(R(H))| = 10|H|| Aut(H,S)| (see Observation 4.3). For S1, let ft G Aut(H,S1). Then Sf = S1. Since (y, z) is characteristic in H = (x) x (y) x (z) = Zm x Zpe x Zp, we have {y, yA+1 z, yA2+A+1 zA4+A+1, yA3+A2+A+1z}f = {y, yA+1z, yA2+A+1 za4+a+1, y^+^z}. It follows that yf = ysz4 with (s, t) = (1,0), (A + 1,1), (A2 + A + 1, A4 + A + 1), or (A3 + A2 + A + 1,1). Furthermore, we have (y • yA+1z • yA2+A+1zA4+A+1 • yA3+A2+A+1z)f = y • yA+1 z • yA2+A+1zA4+A+1 • yA3+A2+A+1z and (y8 y-1)A3+2A2+3A+4 = (z-a4-a-3)8 za4+a+3 In particular, (y8y-1)(A +2A +3A+4)p = 1. Note that A4 + A3 + A2 + A +1 = 0 in Zpe implies that A4 + A3 + A2 + A +1 =0 in Zp. If A3 + 2A2 + 3A + 4 = 0 in Zp, then A3 = -2A2 -3A - 4, A4 = A • A3 = A2 +2A + 8, and thus 0 = A4 + A3 + A2 + A +1 = 5, contrary to 5 | (p - 1). Hence A3 + 2A2 + 3A + 4 = 0 in Zp and (y8y-1)p = 1. Suppose that (s,t) = (1,0). Then y8y-1 = ys-1z4 with s - 1 = A, A2 + A or A3+A2+A. Since A4+A3+A2+A+1 =0 in Zp, we have A = 0,-1 and thus (s-1,p) = 1. This implies that y8y-1 = ys-1z4 has orderpe, and since e > 2, we have (y8y-1)p = 1, a contradiction. Hence (s, t) = (1,0), that is, y8 = y. It follows that (yA+1z)8 = yA+1z8 G {yA+1z,yA2 +A+1zA4+A+1,yA3+A2+A+1z}, and thus z8 G {z,yA2zA4+A+1,yA3+A2z}. If z8 = yA2 zA4+A+1 or yA3 +a2 z, then (yA2 )p = 1 or (yA3+A2 )p = 1. It forces that A2 = 0 or A3 + A2 =0 in Zpe-i, and A = 0, -1, a contradiction. Hence z8 = z. Noting that (x) is characteristic in H, we have (xy)8 = x8y G S8 = S1. Then it is easy to check that (xy)8 = xy and thus x8 = x. It implies that ft is the identity automorphism. Hence | Aut(H, S1)| = 1 and |NA(R(H))| = 10|H|. By a similar argument as above, for S2 and S3, we also have | Aut(H,S2)| = | Aut(H,S3)| = 1 and |Na(R(H))| = 10|H|. □ Lemma 4.5. If e =1, that is, H = Zm x Zp x Zp, then one of the following holds: (1) p = 5 or 5 1 (p ± 1) and r = CGD^pxp as defined in Example 3.6. Furthermore, (i) |Na(R(H))| = 10|HI if m =1, 5; (ii) |Na(R(H))| = 20|H| if m = 5; (iii) |Na(R(H))| = 20|H| if m = 1 andp = 5; and (iv) |Na(R(H))| = 40|H| if m = 1 andp = 5; (2) 5 | (p - 1), r = CGD;mpxp as defined in Example 3.7 and |NA(R(H))| = 10|H|. Proof. Note that (m,p) = 1. By Observation 4.3, we have o(a) = mp, p | o(b) and H = (a, b) = (x,y, z) = Zm x Zp x Zp. Then H has an automorphism mapping xy to a, and we may assume a = xy, implying that b = xr+1yAzl for some r +1 G Zm, i, A G Zp and i = 0 because H = (a, b). The group H also has an automorphism fixing x, y and mapping z to yAzl, and we may further assume b = xr+1z. Let c = x®yjzs and d = xky£z4, where i, k G Zm, j, s, t G Zp. By Equation (4.1), aa = ba-1, that is, (xy)a = xry-1z. Since both (x) and (y, z) are characteristic in H, we have xa = xr and ya = y-1z. Again by Equation (4.1), since (xr+1z)a = ba = ca-1 = xi-1yj-1zs, we have za = (x-r-1)a • ba = x-r2-r-1+i • yj-1zs, implying that za = yj-1zs and -r2 - r - 1 + i = 0 (mod m). (4.12) Moreover, we have cfc-y-V = da-1 = ca = (x4 yj zs)a = (xr )4(y-1z)j (yj-1zs)s = xri y-j+s(j-1)zj+s and x-1 y-1 = a-1 = da = (xk yV)a = (xr )k (y-1z) V'-^)4 = xrk y-i+j-1)izsi+£. Considering the powers of x, y and z, we have Equations (4.13) - (4.18). As shown in these equations, in what follows all equations are considered in Zp, unless otherwise stated: k — 1 = ri (mod m); (4.13) i — 1 = —j + s(j — 1); (4.14) t = j + s2; (4.15) — 1 = rk (mod m); (4.16) —1 = —i + (j — 1)t; (4.17) 0 = st + i. (4.18) By Equation (4.12), we have i = r2 + r +1 (mod m) and by Equations (4.13) and (4.16), k = r3 + r2 + r +1 (mod m) and r4 + r3 + r2 + r +1 = 0 (mod m). It follows from Proposition 3.2 that either (r , m) G {(0 , 1), (1,5)} or r is an element of order 5 in Z^ and the prime decomposition of m is • • • pf with t < 1, f > 1, et > 1 and 5 | (pt — 1) for 1 < i < f. By Equation (4.15), t = j + s2, and by Equations (4.14), (4.17) and (4.18), t = 1 — j + s(j — 1), t = 1 + (j — 1)t = 1 + (j — 1)(j + s2) and t = —st = —sj — s3. It follows j2 + (s2 — s)j — (s2 — s) = 0; (4.19) (2s — 1)j + s3 — s + 1=0. (4.20) By Equation (4.19), (2s — 1)2j2 + (2s — 1)2(s2 — s)j — (2s — 1)2(s2 — s) = 0, and since (2s — 1)j = —(s3 — s + 1), we have s6 — 3s5 + 5s4 — 5s3 + 2s — 1=0, that is, (s2 — s — 1)(s4 — 2s3 + 4s2 — 3s + 1) =0. Hence, either s2 — s — 1 =0 or s4 — 2s3 +4s2 — 3s +1 = 0. Case 1: s2 — s — 1=0. Let A = 2s — 1. Then s = 2-1(1 + A) and A2 = 5, and thus (A,p) = (0,5) or 5 | (p ± 1) by [34, Example 4.6]. By Equations (4.19) and (4.20), j2 + j — 1 = 0 and (2s — 1)j + (s + 2) = 0. For (A,p) = (0,5), j2 + j — 1 = 0 implies that j = 2 = —2-1(1 + A). For 5 | (p ± 1), we have A = 0, and since 2s — 1 = A and (2s — 1)j + (s + 2) = 0, we have j = — (2s — 1)-1(s + 2) = —A-1 • 2-1(A + 5) = —2-1(1 + A) (note that 5 = A2). Itfollows from Equations (4.15) and (4.18) that t = j + s2 = 1 and i = —st = —2-1(1 + A). Recall that i = r2 + r + 1 (mod m) and k = «3 + «2 + i + 1 (mod m). Hence c = xr2+r+1y-2-1(1+A)z2-1(1+A) and d = xr3+r2+r+1y-2-1(1+A)z. Now, S = {1, xy, xr+1z, xr2+r+1y-2-1(1+A)z2-1(1+A), xr3+r2+r+1y-2-1(1+A)z}. By Proposition 4.1 and Example 3.6, r = BiCay(H, 0,0, S) = CGD^pxp. For (m,p) = (1,5), we have A = 0 and S = {1, y, z, y-3z3,y-3z}. By Magma [5], |Na(R(H))| = 40|H|. Assume that (m,p) = (1, 5), and let ^ G Aut(H, S). Then S^ = S, and since both (x) and (y, z) are characteristic subgroups of H, we have {x xr+1 xr2+r+1 xr3+r2+r+1}^ = {x xr + 1 xr2+r+1 xr3+r2+r + 1} {y,z,y-2-1(1+A)z2-1(1+A),y-2-1(1+A)z}^ = {y, z, y-2"(1+A)z2-1(1+A), y-2"(1+A)z}. Similarly to Lemma 4.4, the two equations imply that for m = 1,5 (r = 0, ±1), ft is the identity automorphism of H, and for m = 1 or 5 (r = 0 or 1), ft has order 2 that are inducedby x ^ xr +r +r+1, y ^ y-2 (1+A)z,and z ^ y-2 (A+1)z2 (A+1) (onemay also see [35] for a detailed computation). It implies that | Aut(H, S)| = 1 for m = 1,5 and | Aut(H,S)| = 2 for m = 1 or 5. By Observation 4.3, we have |NA(R(H))| = 10|H| or 201H |, respectively. Case 2: s 4 - 2s3 + 4s2 - 3s + 1 = 0. By Case 1, we may assume that s2 - s - 1=0. If p = 5, then s4 - 2s3 + 4s2 - 3s + 1=0 implies that s = 3 and thus s2 - s - 1 = 0, a contradiction. Hence p = 5. By [34, Lemma 5.4, Case 2], we have 5 | (p - 1) and s = 2-1(1 + A), where A4 + 10A2 +5 = 0 and A = 0, ±1. Since s4 - 2s3 + 4s2 - 3s +1 = 0, we have (2s - 1)(8s3 - 12s2 + 26s - 11) = -5, and since p = 5, we have (2s - 1)-1 = -5-1(8s3 - 12s2 + 26s - 11). Noting that s4 = 2s3 - 4s2 + 3s - 1, we have s5 = -5s2 + 5s - 2 and s6 = -5s3 + 5s2 - 2s. By Equation (4.20), j = -(2s - 1)-1(s3 - s + 1) = 5-1(8s3 - 12s2 + 26s - 11)(s3 - s + 1) = s3 - 2s2 + 3s - 1 = 8-1(A3 - A2 + 7A + 1) and by Equations (4.15) and (4.18), t = j + s2 = s3-s2 + 3s-1 = 8-1(A3 + A2 + 11A+3) and I = -st = -s3 + s2-2s + 1 = -8-1(A3 + A2 + 7A - 1). It follows that S = {1, xy, xr+1z,xr2+r+1y8-1(A3-A2+7A+1)z2-1(1+A), y xr3+r2+r+1y-8-1(A3 + A2+7A-1)z8-1(A3 + A2 + 11A+3)} By Proposition 4.1 and Example 3.7, r = BiCay(H, 0,0, S) = CGD^pxp. Let ft G Aut(H, S). Then S^ = S. Since (x) and (y, z) are characteristic in H, we have 8-1(A3-A2+7A+1) z2-1(1+A) y-8-1(A3+A2 + 7A-1) z8-1(A3+A2 {y, Z,yS ^+7A+1)(1+A),y-8 (A" +A" + 7A-i)z8 = {y Z y8-1 (A3-A2 + 7A+1)Z2-1(1+A) y-8-1(A3 + A2+7A-l)Z8-1(A3 + A2 + llA+3)} and since A = 0, ±1, we have y^ = y and z^ = z (also see [35] for a detailed computation). Since (xy)^ = x^y G S, it is easy to check that (xy)^ = xy and thus x/ = x. Hence ft is the identity automorphism of H and | Aut(H, S)| = 1. By Observation 4.3, |Na(R(H))| = 10|H |. □ 5 Cyclic covers In this section, we classify connected symmetric cyclic covers of connected pentavalent symmetric graphs of order twice a prime. Denote by K6,6 - 6K2 the complete bipartite graph of order 12 minus a one-factor and by I12 the Icosahedron graph. Edge-transitive cyclic covers of K6 were classified in [29, Theorem 1.1], and by [29, Line 20, pp. 40], such graphs have order 12 and thus isomorphic to K6,6 - 6K2 or I12 by [20, Proposition 2.7] (note that the graph I12 is missed in [29, Theorem 1.1]). Theorem 5.1. Let r be a connected pentavalent symmetric graph of order 2p for a prime p, and let r be a connected symmetric Zn-cover of r with n > 2. Then r = K6,6 — 6K2, I12, CDnp, or CGDmpexp for 1 < i < 5 with n = mpe, (m,p) = 1, 5 | (p — 1) and e > 1, which are defined in Examples 3.3, 3.5, 3.6 and 3.7. Proof. By Proposition 3.4, r = K6 for p = 3, K5,5 for p = 5, or CDp for 5 | (p — 1). If r = K6 then r = K6,6 — 6K2 or I12 by [29, Theorem 1.1] (also see the proof in [2, Theorem 3.6]). In the following, we assume that p > 5. Let A = Aut(r). Let K = Zn and F = NA (K). Since r is a symmetric K-cover of r, F is arc-transitive on f and F/K is arc-transitive on = r. Let B/K be a minimal arc-transitive subgroup of F/K. By Proposition 3.4, B/K = Dp x Z5 for p > 11; by Magma [5], B/K ^ D11 x Z5 for p =11, and B/K = Z2 x Z2, Z2 x Z4 or Z2 x Z8 for p = 5. Each minimal normal subgroup of B/K is isomorphic to Zp or Z5 with p = 5 and B/K = Z5 x Z8. Clearly, B is arc-transitive on f and B/K is non-abelian. Set C = Cb(K). Since K is abelian, K < Z(C) < C, where Z(C) is the center of C. Suppose K = C. Then B/K = B/C < Aut(K) = Zn, which forces that B/K is abelian, a contradiction. Hence K < C and 1 = C/K < B/K. It follows that C/K contains a minimal normal subgroup of B/K, say L/K. Then L < B and L < C < B. Furthermore, L/K = Zp, or L/K = Z5 withp = 5 and B/K = Z2 x Z8. Clearly, L and L/K have two orbits on V(r) and V(rK), and r and rK are bipartite graphs with the two orbits of L and L/K as their bipartite sets, respectively. Since K < Z(C) and L < C, K < Z(L). First, assume L/K = Zp. Since K < Z(L), L is abelian, and so L = Znp or Zn x Zp with p | n. For the latter, L = Zm x Zpe x Zp with n = mpe, (m,p) = 1 and e > 1. Since L/K is semiregular on V (fK ), L is semiregular on V (f) and thus r is a bi-Cayley graph over L. Noting that L < B, we have that NA(L) is arc-transitive on T, forcing that f = BiCay(L, 0,0, S) for some subset S Ç L. Recall thatp > 5. By Lemmas 4.2-4.5, f = CDnp or C£Dmpexp (1 < « < 5), as required. Now, assume L/K = Z;2. Then p = 5 and B/K = Z5 x Z8. Since K < Z(L) and K = Zn, L = P x H, where P and H are the Sylow 5-subgroup and the Hall 5'-subgroup of L, respectively. Note that H < K is abelian, but P may not. Since (L/K)vk = Z5, we have Lv = Pv = Z5, where v g V(r and vK is an orbit of K on V(r containing v. Note that P < B^s P is characteristic in L and L < B. By Proposition 2.2, P has at most two orbits on V(r) because Pv = 1, and since L has exactly two orbits on V(r), P and L have the same orbits. It follows that L = PLv = PPv = P, forcing that H =1 and K is a 5-group. Suppose |K| = 54 with t > 2. Since K is cyclic, K has a characteristic subgroup N such that |K/N| = 25, and since K < B, N < B. By Proposition 2.2, fN is a connected pentavalent B/N-arc-transitive graph of order 10|K|/|N| = 250, and by Example 3.1, rw = CGD53. Since B/K = Z5 x Z8 and |K/N| = 52, all Sylow 2-subgroups of B/N are isomorphic to Z8 and |B/N| =8 • 54. However, by MAGMA [5], Aut(C£D53) has no arc-transitive subgroup of order 8 • 54 that has a Sylow 2-subgroup isomorphic to Z8, a contradiction. Since K = 1, we have |K| = 5 and |V(f)| = 10|K| = 50. By Example 3.6, f = x5, as required. □ 6 Dihedral covers In this section, we aim to classify symmetric dihedral covers of connected pentavalent symmetric graphs of order twice a prime. First, we introduce four graphs which are from [30]. Example 6.1. Let I(122) = Cay(D12, {b, ba, ba2, ba4, ba9}), G48 = Cay(D24, {b, ba, ba3, ba11 ,ba20}). be two Cayley graphs on the dihedral groups D12 = (a,b | a12 = b2 = 1,ab = a-1} and D24 = (a,b | a24 = b2 = 1,ab = a-1}, respectively. By Magma [5], Aut(l12)) = A5 x D4 and Aut(G48) = SL(2, 5) x D4, and their vertex stabilizers are isomorphic to F20. Example 6.2. Let Geo = Cay(A5, {(1 4)(2 5), (1 3)(2 5), (1 3)(2 4), (2 4)(3 5), (1 4)(3 5)}) be a Cayley graph on A5. By Magma [5], it is a connected pentavalent symmetric graph of order 60 and Aut(Ge0) = A5 x D5 with vertex stabilizer isomorphic to D5. Example 6.3. Let G be a subgroup of S7 generated by the elements a = (1 4)(2 5)(6 7), b = (1 3)(2 5)(6 7), c = (1 3)(2 4)(6 7), d = (2 4)(3 5)(6 7) and e = (1 4)(3 5)(6 7), and define G120 = Cay(G, {a, b, c, d, e}). By Magma [5], G = A5 x Z2 and G120 is a connected pentavalent symmetric graph of order 120. Moreover, Aut(G120) = A5 x D10 with vertex stabilizer isomorphic to D5. A list of all pentavalent G-arc-transitive graphs on up to 500 vertices with the vertex stabilizer Gv = Z5, D5 or F20 was given in Magma code by Potocnik [30]. Based on this list, we have the following lemma. Lemma 6.4. Let r be a G-arc-transitive graph of order 24, 48, 60, 120 or 240 with vertex stabilizer Gv = Z5, D5 or F20 for some G < Aut(r) and v G V(r). Then r is a connected symmetric dihedral cover of Ke if and only if r = I^, G48, Ge0 or G120. Proof. To show the necessity, let r be a connected symmetric dihedral cover of Ke. Then Aut(r) has an arc-transitive subgroup having a normal dihedral subgroup of order |V(r)|/6. Since r is G-arc-transitive with Gv = Z5, D5 or F20, by [30] r is isomorphic to one of the seven graphs: three graphs of order 24, 48 and 60 respectively, two graphs of order 120 and two graphs of order 240. For the orders 24, 48 and 60, r = I^, G48 or Ge0 by Examples 6.1 and 6.2. For the order 120, by Magma [5] one graph is isomorphic to G120 and the other has no arc-transitive group of automorphisms having a normal dihedral subgroup of order 20; in this case r = G120. For the order 240, again by Magma [5] none of the two graphs has an arc-transitive group of automorphisms having a normal dihedral subgroup of order 40. Now, we show the sufficiency. By Magma [5], Aut(l12)) has a normal subgroup N = D2. Clearly, N has more than two orbits on V(I^), and by Proposition 2.2, the quotient graph (I^ )N is a connected pentavalent symmetric graph of order 6, that is, the (2) complete graph Ke. Thus I12 is a D2-cover of Ke. Similarly, one may show that G48, Ge0 or G120 is a symmetric D3-, D5- or D10-cover of Ke, respectively. □ Now, we are ready to classify symmetric dihedral covers of connected pentavalent symmetric graphs of order 2p for any prime p. Clearly, we have p > 3. Theorem 6.5. Let r be a connected pentavalent symmetric graph of order 2p with p a prime, and le Ö60 or Gl20- (2) prime, and let r be a connected symmetric Dn-cover of r with n > 2. Then r = , G48 Proof. Let K = Dn and let F be jhe fibre-preserving group. Since r is a symmetric K-cover of r, F is arc-transitive on r and F/K is arc-transitive on rK = r. Assume n = 2. Then |V(f)| = 2n • |V(r)| = 8p. Recall that p > 3. By [20, Proposition 2.9], f = I^ or a graph G248 of order 248 with Aut(G248) = PSL(2, 31). Since PSL(2, 31) has no proper subgroup of order divisible by 248 by Magma [5], Aut(r) is the unique arc-transitive group of automorphisms of T, that is, F = PSL(2, 31). It implies that T C G248 because F has no normal subgroup isomorphic to Dn. Hence f C I(2) r = I12 . Assume n > 2. Let Zn be the cyclic subgroup of K = Dn of order n. Then Zn is characteristic in K and so Zn < F as K < F. By Proposition 2.2, rZn is a connected pentavalent F/Zn-arc-transitive graph of order 4p, and by [20, Proposition 2.7], fZn = I12 or K6,6 - 6K2. Thus r is a symmetric Zn-cover of K6,6 - 6K2 or I12. Note that |V (f)| = 12n. Let rZn = K6,6 - 6K2. Since each minimal arc-transitive subgroup of Aut(K6,6 -6K2) is isomorphic to A5 x Z2 or S5 by MAGMA [5], F/Zn has an arc-transitive subgroup B/Z„ = A5 x Z2 or S5. It follows that |Bv | = 10 forv G V (f), and form Proposition 2.1 that Bv = D5. In particular, B is arc-transitive on f and B/Zn has a normal subgroup M/Zn = A5, which is edge-transitive on f Zn and has exactly two orbits on V (rZn ). Thus M < B is edge-transitive and has two orbits on V(r). Since |B : M| = 2, we have Mv = D5. Clearly, Z„ < Cm(Z„). If Z„ = Cm(Z„), then A5 = M/Z„ = M/Cm(Z„) < Aut(Zn) = Zn, which is impossible. Hence Zn is a proper subgroup of CM(Zn), and since Mult(A5 ) = Z2, Lemma 2.3 implies that either M = M' x Zn = A5 x Zn or M = M'Zn = SL(2, 5)Zn with M' n Zn C Z2. In particular, M/M' is cyclic. Since M' is characteristic in M and M < B, we have M' < B. If M' has at least three orbits on V(f ), by Proposition 2.2, M' is semiregular on V (f ) and f M/ is a connected pentavalent B/M'-arc-transitive graph. The stabilizer of a G V(rM/) in M/M' is isomorphic to Mv = D5, but this is impossible because M/M' is cyclic. Thus M' has at most two orbits on V (f ) and so |V(f)| | 2|M'|, that is, 6n | |M'|. If M = A5 x Zn, then M' = A5 and 6n | |M'| implies that n = 5 or 10 as n > 2. It follows that |V(f )| = 60 or 120. Since Bv = D5, we have r C G60 or G120 by Lemma 6.4. If M = SL(2,5)Z„ with M' = SL(2,5) and SL(2,5) n Zn = Z2, then n is even and 6n | |M'| implies that n = 4, 10 or 20. It follows that | V(f ) | = 48, 120 or 240, and from Lemma 6.4 that r C G48 or G120. Let fZn = I12. By MAGMA [5], under conjugation Aut(I12) has only one minimal arc-transitive subgroup isomorphic to A5, and so F/Zn has an arc-transitive subgroup B/Zn = A5. By a similar argument as the previous paragraph, B = B'Zn and B'nZn < Mult(A5) by Lemma 2.3, forcing that either B = B' x Zn = A5 x Zn or B = B'Zn = SL(2, 5)Zn with SL(2, 5) n Zn = Z2. Furthermore, B is arc-transitive on T with Bv = Z5 for v G V(r), and B/B' is cyclic. If B' has more than two orbits on V(T), then rB is a connected pentavalent B/B'-arc-transitive graph by Proposition 2.2, which is impossible because B/B' is abelian. Thus B' has at most two orbits on V(f) and so 12n | 2|B'|. If B = A5 x Zn, then B' = A5, and 12n | 2|B'| implies that n = 5 or 10. It follows that |V(f)| = 60 or 120. Since Bv = Z5, we have f = Geo or G120 by Lemma 6.4. If B = SL(2, 5)Zn with SL(2, 5) n Zn = Z2, then B' = SL(2, 5) and n is even. Since 12n | 2|B'|, we have n = 4, 10 or 20, and so |V(f)| = 48, 120 or 240. It follows from Lemma 6.4 thatf = G48 or G120. □ 7 Full automorphism groups of covers Let r be a symmetric Dn- or Zn-cover of a connected symmetric pentavalent graph of order 2p, where n > 2 is an integer and p is a prime. In this section, we aim to determine (2) the full automorphism group of r. For Dn, by Theorem 6.5, r = I12, G48, Geo or Gi2o and by Examples 6.1 -6.3, Aut(T) is known. For Zn, by Theorem 5.1, r = K6,6 — 6K2, Ii2, CDnp (see Example 3.3), or CGVlmpexp with 1 < i < 5 (see Examples 3.5, 3.6 and 3.7). In particular, for the graph CGD„pexp, we have mpe = n and m is given by m = 5tp11 • • • pe/ s.t. t < 1, s > 0, ej > 1, 5 | (p^ — 1) for 0 < j < s, (7.1) where m,p, e satisfy the conditions as listed in the second column in Table 2. Note that m is odd by Equation (7.1). By Magma [5], Aut(K6,6 — 6K2) = S6 x Z2 and Aut(I12) = A5 x Z2, and by Example 3.3, Aut(CDnp) = Dnp xZ5. Hence we only need to determine the full automorphism groups of CGD'lmpexp for 1 < i < 5. All theses graphs are connected symmetric cyclic covers of some pentavalent symmetric graph of order 2p except CGD>mpxp with 5 | (p + 1), which are connected symmetric bi-Cayley graphs over Zmp x Zp. Theorem 7.1. Aut(CGD„pe xp) for 1 < i < 5 is isomorphic to one group listed in Table 2. Table 2: Full automorphism groups of CGD mpe xp for 1 < i< 5. r Conditions: (m,p) = 1, m: Eq. (7.1) Aut(r) CGD^pexp,» =1, 2 5 | (p - 1) and e > 2 Dih(Zmpe x Zp) x Z5 m =1, 5, and p = 5 or 5 | (p ± 1) Dih(Zmp x Zp) x Z5 CGDmpxp m =1 or 5, and 5 | (p ± 1) Dih(Zmp x Zp) x D5 m =1 and p = 5 (Dih(Z5) x F2q).Z4 CGDmpxp 5 | (p - 1) Dih(Zmp x Zp) x Z5 Proof. Let r = CGDmpexp for 1 < i < 5 and A = Aut(r). For (m,p) = (1, 5), we have r = CGD5x5 and by [16, Theorem 4.3 (1)], Aut(r) = (Dih(Z2) x ^2o).Z4. In what follows we assume that (m,p) = (1, 5). By Examples 3.5, 3.6 and 3.7, A has an arc-transitive subgroup F isomorphic to Dih(Zmpe x Zp) x Z5 for CGDmpe xp (i = 1, 2, 3), Dih(Zmp x Zp) x Z5 for CQV^p^p with m =1, 5 andp = 5 or 5 | (p ± 1), Dih(Zmp x Zp) x D5 for CGD^pxp with m =1 or 5 and 5 | (p ± 1), and Dih(Zmp x Zp) x Z5 for CgV^pxp with 5 | (p - 1). Note that Fv = Z5 or D5 for v G V(r). Furthermore, F has a normal semiregular subgroup K = Zmpe x Zp having two orbits on V(r), and hence r is an F-arc-transitive bi-Cayley graph over K. By Lemmas 4.4 and 4.5, |NA(K)| = |F|, implying that NA(K) = F. Note that |F| = 10|K| or 20|K|, that is, |F| = 10mpe+1 or 20mpe+1 with p = 5 or 5 | (p ± 1), and by Equation (7.1), both m and |K| are odd. In particular, |V(r)| = 2|K| = 2mpe+1 is twice an odd integer. Clearly, K = Zmpe x Zp has a characteristic Hall 5'-subgroup, say H. Then H < F as K < F .If H = K, then 5 | mpe+1 and H has at least three orbits. For p = 5, we have 5 | m, and since 52 { m by Equation (7.1), we have |K : H| = 5. For p = 5, by Table 2, r = CGDmpxp with (m, 5) = 1 and K = Zm x Zp x Zp, implying that |K : H| = 52. By Proposition 2.2, rH is a connected pentavalent F/H-arc-transitive graph of order 2 • 5 or 2 • 52. By Proposition 3.4 and Example 3.6, rH = K5,5 or = CGl4x5. Since |F| = 10|K| or 20|K| and |K| is odd, H is the characteristic Hall {2, 5}'-subgroup of F. Thus we have the following claim. Claim 7.2. H is the characteristic Hall {2, 5}'-subgroup of F, and we have H = K, or '5x5. |K : H| = 5 and Th = K55, or |K : H| = 25 and Th = To finish the proof, we only need to show that A = F. Suppose to the contrary that A = F. Then A has a subgroup M such that F is a maximal subgroup of M. Since F is arc-transitive on r, M is arc-transitive, and since NA(K) = F, we have K ^ M. By the definitions of the graphs CGDmpexp (1 < i < 5) in Examples 3.5, 3.6 and 3.7, r has the 6-cycle (1, h, a-r-1b-A-1c-1, ha-rb-Ac-1, a-rb-Ac-1, hab, 1) for 1 < i < 3, and the 6-cycle (1, h, a-r-1 c-1, ha-r bc-1, a-r bc-1, hab, 1) for 4 < i < 5. Suppose that r is (M, 4)-arc-transitive. Then each 4-arc lies in a 6-cycle in r and so r has diameter at most three. It follows that |V(r)| = 2mpe+1 < 1 + 5 + 5 • 4 + 5 • 4 • 4 = 106, that is, mpe+1 < 53. Since p = 5 or 5 | (p ± 1) and e +1 > 2 (see the second column of Table 2), we have p = 5 and m < 2. Since m is odd, (m,p) = (1,5), contrary to assumption. Thus r is at most 3-arc-transitive, and by Proposition 2.1, we have |Mv | G {5,10, 20, 40, 60, 80,120, 720,1440, 2880}. Note that |M : F| = |Mv : Fv | G {2,4, 6,8,12,16,24,72,144, 288, 576} because M = F and |F„ | = 5 or 10. Let [M : F] be the set of right cosets of F in M. Consider the action of M on [M : F] by right multiplication, and let FM be the kernel of this action, that is, the largest normal subgroup of M contained in F. Then M/FM is a primitive permutation group on [M : F] because F/FM is maximal in M/FM, and (M/FM)f = F/FM, the stabilizer of F g [M : F] in m/Fm. It follows that |m/Fm | = |M : F||F/Fm | and so |F/Fm| = |M/FM|/|M : F|. Since |M : F| G {2,4, 6,8,12,16,24,72,144, 288, 576}, by Lemma 2.4 we have M/FM < AGL(t, 2) with |M : F| = 24 and 1 < t < 4, or soc(M/FM) = PSL(2, q), PSL(3, 3) or PSL(2,r) x PSL(2,r) with |M : F| = q + 1, 144 or (r + 1)2 respectively, where q G {5, 7,11, 23, 71} and r G {11,23}. Suppose M/FM < AGL(2,2) and |M : F| = 4. Since a 2-group cannot be primitive on [M : F], we have 3 | |m/Fm| and so 3 | |MM/Fm|/|M : F| = |^F/Fm|. Since |F| = 10mpe+1 or 20mpe+1 with p = 5 or 5 | (p ± 1), we have 3 | m, which is impossible by Equation (7.1). Thus M/FM < AGL(2, 2). Similarly, since 7 { m, we have M/Fm < AGL(3, 2), and if M/Fm < AGL(4,2), then M/Fm is a {2,5}-group. Furthermore, soc(M/FM) ^ PSL(2, q), PSL(3, 3) or PSL(2, r) x PSL(2, r) for q G {7, 23, 71} and r = 23 because otherwise one of 7, 23,13, 23 is a divisor of m. It follows that M/Fm = Z2 with |M : F| = 2, M/Fm < AGL(4, 2) with |M : F| = 24 and M/Fm a {2, 5}-group, soc(M/FM) = PSL(2, q) with |M : F| = q +1 and q G {5,11}, or soc(M/FM) = PSL(2,11) x PSL(2,11) with |M : F| = 144. First assume that M/Fm = Z2 with |M : F| =2. Then F < M and H < M as H is characteristic in F by Claim 7.2. Let C = CM (H). Since K is abelian, H < K < C. Let P be a Sylow 5-subgroup of C containing the unique Sylow 5-subgroup of K. Since H is the Hall 5'-group of K, K < HP = H x P. Clearly, HP/H is a Sylow 5-subgroup of C/H. Recall that |F/K| | 20 and |K/H| | 25 (see Claim 7.2). Since |M| = 2|F|, we have |M/H| | 23 • 53, and by Sylow theorem, M/H has a normal Sylow 5-subgroup. In particular, C/H has a normal Sylow 5-subgroup, that is, HP/H < C/H. This implies H x P < C, and since C < M and P is characteristic in C, we have P < M. Since (m,p) = (1,5) and |V(r)| = 2mpe+1, P has at least three orbits on V(r). By Proposition 2.2, P is semiregular on V(r). Thus |P| | |V(r)| and |P| | |K|. It follows that |HP| = |H||P| | |K|, and since K < HP, we have K = HP < M, a contradiction. Assume that M/Fm < AGL(4,2) with |M : F| = 24 and M/Fm a {2,5}-group. Then M/Fm has a regular normal subgroup of order 24, say L/FM, and hence L < M, 24 | |L| and 5 | |M : L|. If L is semiregular then 24 | |V(r)| = 2mpe+1, which is impossible. Thus L is not semiregular, and so 5 | |Lv |. By Proposition 2.2, L has one or two orbits, yielding that |L| = |V(r)||L„| or |L| = |V(r)||L„|/2. Since |M| = |V(r)||M„|, we have |M : L| = |Mv : Lv | or 2|Mv : Lv |, and since 52 f |Mv|, we have 5 f |M : L|, a contradiction. Assume that soc(M/FM) = PSL(2, 5) with |M : F| = 6. Then M/Fm = PSL(2, 5) or PGL(2,5), and |F/Fm| = |M/Fm|/|M : F| = 10 or 20. Since H is the unique normal Hall {2, 5}'-subgroup of F, we have H < FM and so H is characteristic in FM. This implies H < M because Fm < M. Since M/Fm = (M/H)/(Fm/H), M/H is insolvable, and since K ^ M, we have H = K .By Claim 7.2, Th = K5,5 or cgd5x5. If rH = cgd4x5 then Aut(Fn) = (Dih(Z|) x F2o).Z4 is solvable and so M/H is solvable, a contradiction. If rH = K5 5 then as Aut(K5,5) — (S5 x S5) x Z2, it is easy to show that each insolvable arc-transitive group of Aut(K5 5) contains A5 x A5 (this is also easily checked by Magma [5]), and so |m/h| > 2 • 602. Noting that FM is semiregular on V(r), we have |Fm| | |K|. By Claim 7.2, |K : H| | 52, and hence |Fm : H| | 52. It follows that |M/Fm | = |M/H|/|Fm/H| > 2 • 602/52 > | PGL(2,5) |, a contradiction. Assume that l/Fm := soc(m/Fm) = PSL(2,11) with |M : F| = 12. Then M/Fm = PSL(2,11) or PGL(2,11), and |F/Fm| = |M/Fm|/|M : F| = 55 or 110. Moreover, L < M and K < L as |K| is odd and |M : L| < 2. Since 11 | |L/Fm|, Fm has at least three orbits on V(r), and by Proposition 2.2 FM is semiregular and rFM is a pentavalent F/FM-arc-transitive graph. Thus |FM| | |V(r)| and |V(rfm)| is even. Since |V(rFM)| = |V(r)|/|FM| = 2|K|/|Fm|, |Fm| is odd and |Fm| | |K|. Recall that H is the characteristic Hall {2, 5}'-subgroup of F by Claim 7.2. Set N = HnFM. Since FM has odd order, N is the characteristic Hall 5'-subgroup of FM, and since Fm < M, we have N < M. Hence FM/N is a 5-subgroup. By Claim 7.2, 53 f |K|, and since |Fm | | |K|, we have 53 f |Fm |, that is, |Fm/N| | 25. Thus Fm/N is abelian, and Aut(FM/N) is cyclic or Aut(FM/N) ^ GL(2,5). If Fm/N = CL/N(Fm/N), then PSL(2,11) = L/Fm = (L/N)/(Fm/N) < Aut(FM/N), which is impossible. Thus Fm/N is a proper subgroup of CL/N(FM/N), and since Mult(PSL(2,11)) = Z2, Lemma 2.3 implies that L/N = (L/N)' x FM/N with (L/N)' = PSL(2,11). Since |V(rw)| = |V(r)|/|N| = 2|K|/|N| with |K| odd, (L/N)' = PSL(2,11) cannot be semiregular on V(rN), implying that 5 | |(L/N)"| for a G V(rN). It follows from Proposition 2.2 that (L/N)' has at most two orbits on V(rN), and so |(L/N)|/|(L/N)'| = |V(rw)||(L/N)a|/(|V(rw)||(L/N)"|) = |(L/N)a|/|(L/N)"| or2|(L/N)a|/|(L/N)"|, implying that 5 \ (|L/N|/|(L/N)'|). Since |(L/N)/(L/N)'| = |fm/N| and fm/N is a 5-group, we have |fm/N| = 1, that is, L/N = (l/n)' = PSL(2,11) Since K ^ M and N < M, we have N = K, and since K < CL(N) and |N| is odd, Lemma 2.3 implies L = L' x N with L' = PSL(2,11). Note that L' < M. Since r has order twice an odd integer, L' cannot be semiregular on r, yielding 5 | |LV |. By Proposition 2.2, L' has at most two orbits, and so | PSL(2,11) | = |L'| = |V(r)||LV | or |V(r)||LV |/2. It implies that |V(r)| | 2| PSL(2,11)|, that is, |V(r)| | 23 • 3 • 11. Since | V(r) | = 2mpe+1 and it is not divided by 3 or 22 by Equation (7.1), we have | V(r) | = 22, contrary to the fact that e + 1 > 2. Assume that L/FM := soc(M/FM) = PSL(2,11) x PSL(2,11) with |M : F| = 144. Then there exists l^fm < l/fm such that L1/FM = PSL(2,11) and 11 | |L : L11. Since 11 | |L : FM |, FM has at least three orbits and so Tfm has order twice an odd integer. This implies that L/FM cannot be semiregular, and by Proposition 2.2, L/FM has one or two orbits. If L/FM has one orbit then L1/FM is semiregular on rfm as 11 | |L : L1| implies that L1 /FM has at least three orbits, and so 4 | |V(rfm)|, a contradiction. If L/FM has two orbits then rfm is bipartite and L/FM is edge-transitive on rFM. Furthermore, L1/FM fixes the bipartite sets setwise. Since 11 | |L : L1|, l1/fm has at least two orbits on each bipartite set, and by [20, Proposition 2.4], L1/FM is semiregular on rfm . Since L1/FM = PSL(2,11), again we have the contradiction that 4 | |v(rFM)|. □ References [1] J. A. Al-bar, A. N. Al-kenani, N. M. Muthana and C. E. Praeger, A normal quotient analysis for some families of oriented four-valent graphs, Ars Math. Contemp. 12 (2017), 361-381, doi:10.26493/1855-3974.1142.038. [2] I. Antoncic, A. Hujdurovic and K. Kutnar, A classification of pentavalent arc-transitive bicir-culants, J. Algebraic Combin. 41 (2015), 643-668, doi:10.1007/s10801-014-0548-z. [3] K. Bercic and P. Potocnik, Two-arc-transitive two-valent digraphs of certain orders, Ars Math. Contemp. 11 (2016), 127-146, doi:10.26493/1855-3974.747.2d3. [4] N. L. Biggs, Constructing 5-arc-transitive cubic graphs, J. London Math. Soc. 26 (1982), 193200, doi:10.1112/jlms/s2-26.2.193. [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235-265, doi:10.1006/jsco.1996.0125. [6] Y. Cheng and J. Oxley, On weakly symmetric graphs of order twice a prime, J. Comb. Theory Ser. B 42 (1987), 196-211, doi:10.1016/0095-8956(87)90040-2. [7] M. D. E. Conder and J. Ma, Arc-transitive abelian regular covers of cubic graphs, J. Algebra 387 (2013), 215-242, doi:10.1016/j.jalgebra.2013.02.035. [8] M. D. E. Conder and J. Ma, Arc-transitive abelian regular covers of the Heawood graph, J. Algebra 387 (2013), 243-267, doi:10.1016/j.jalgebra.2013.01.041. [9] J. D. Dixon and B. Mortimer, Permutation Groups, volume 163 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1996, doi:10.1007/978-1-4612-0731-3. [10] D. Z. Djokovic, Automorphisms of graphs and coverings, J. Comb. Theory Ser. B 16 (1974), 243-247, doi:10.1016/0095-8956(74)90070-7. [11] S. Du, A. Malnic and D. Marusic, Classification of 2-arc-transitive dihedrants, J. Comb. Theory Ser. B 98 (2008), 1349-1372, doi:10.1016/j.jctb.2008.02.007. [12] S.-F. Du, J. H. Kwak and M.-Y. Xu, Linear criteria for lifting automorphisms of elementary abelian regular coverings, Linear Algebra Appl. 373 (2003), doi:10.1016/s0024-3795(02) 00649-3. [13] S.-F. Du, J. H. Kwak and M.-Y. Xu, 2-arc-transitive regular covers of complete graphs having the covering transformation group Zp, J. Comb. Theory Ser. B 93 (2005), 73-93, doi:10.1016/ j.jctb.2003.09.003. [14] X. G. Fang and C. E. Praeger, Finite two-arc transitive graphs admitting a Suzuki simple group, Comm. Algebra 27 (1999), 3727-3754, doi:10.1080/00927879908826659. [15] Y.-Q. Feng and Y.-T. Li, One-regular graphs of square-free order of prime valency, European J. Combin. 32 (2011), 265-275, doi:10.1016/j.ejc.2010.10.002. [16] Y.-Q. Feng, J.-X. Zhou and Y.-T. Li, Pentavalent symmetric graphs of order twice a prime power, Discrete Math. 339 (2016), 2640-2651, doi:10.1016/j.disc.2016.05.008. [17] C. D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243-256, doi:10.1007/bf02579330. [18] J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1987. [19] S.-T. Guo and Y.-Q. Feng, A note on pentavalent s-transitive graphs, Discrete Math. 312 (2012), 2214-2216, doi:10.1016/j.disc.2012.04.015. [20] S.-T. Guo, H.-L. Hou and J.-T. Shi, Pentavalent symmetric graphs of order 16p, Acta Math. Appl. Sin. Engl. Ser. 33 (2017), 115-124, doi:10.1007/s10255-017-0642-9. [21] M. Hladnik, D. Marusic and T. Pisanski, Cyclic Haar graphs, Discrete Math. 244 (2002), 137152, doi:10.1016/s0012-365x(01)00064-4. [22] B. Huppert, Endliche Gruppen I, volume 134 of Die Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1967. [23] J. H. Kwak, Y. S. Kwon and J.-M. Oh, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, J. Comb. Theory Ser. B 98 (2008), 585-598, doi:10.1016/j. jctb.2007.09.005. [24] P. Lorimer, Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph Theory 8 (1984), 55-68, doi:10.1002/jgt.3190080107. [25] J. Ma, Arc-transitive dihedral regular covers of cubic graphs, Electron. J. Combin. 21 (2014), #P3.5, http://www.combinatorics.org/ojs/index.php/eljc/ article/view/v21i3p5. [26] A. Malnics, D. Marussics and P. Potocsnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004), 71-97, doi:10.1023/b:jaco.0000047294.42633.25. [27] A. Malnic, D. Marusic and P. Potocnik, On cubic graphs admitting an edge-transitive solvable group, J. Algebraic Combin. 20 (2004), 99-113, doi:10.1023/b:jaco.0000047284.73950.bc. [28] J. Pan, Z. Huang and Z. Liu, Arc-transitive regular cyclic covers of the complete bipartite graph Kp,p, J. Algebraic Combin. 42 (2015), 619-633, doi:10.1007/s10801-015-0594-1. [29] J. Pan, Z. Huang, F. Xu and S. Ding, On cyclic regular covers of complete graphs of small order, Discrete Math. 331 (2014), 36-42, doi:10.1016/j.disc.2014.04.023. [30] P. Potočnik, Census of pentavalent arc-transitive graphs (on up to 500 vertices admitting an arc-transitive group with faithful and solvable vertex stabiliser), http://www.fmf.uni-lj. si/~potocnik/work.htm. [31] C. E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. LondonMath. Soc. 47 (1993), 227-239, doi:10.1112/ jlms/s2-47.2.227. [32] W. Xu and S. Du, 2-arc-transitive cyclic covers of Kn,n — nK2, J. Algebraic Combin. 39 (2014), 883-902, doi:10.1007/s10801-013-0471-8. [33] W. Xu, Y. Zhu and S. Du, 2-arc-transitive regular covers of Kn,n — nK2 with the covering transformation group Zp, Ars Math. Contemp. 10 (2016), 269-280, doi:10.26493/1855-3974. 560.5be. [34] D.-W. Yang and Y.-Q. Feng, Pentavalent symmetric graphs of order 2p3, Sci. China Math. 59 (2016), 1851-1868, doi:10.1007/s11425-016-5146-1. [35] D.-W. Yang, Y.-Q. Feng and J.-X. Zhou, Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime, 2017, arXiv:1703.08316 [math.CO] . [36] J.-X. Zhou and Y.-Q. Feng, Cubic bi-Cayley graphs over abelian groups, European J. Combin. 36 (2014), 679-693, doi:10.1016/j.ejc.2013.10.005. [37] J.-X. Zhou and Y.-Q. Feng, Edge-transitive dihedral or cyclic covers of cubic symmetric graphs of order 2p, Combinatorica 34 (2014), 115-128, doi:10.1007/s00493-014-2834-8. [38] J.-X. Zhou and Y.-Q. Feng, The automorphisms of bi-Cayley graphs, J. Comb. Theory Ser. B 116 (2016), 504-532, doi:10.1016/j.jctb.2015.10.004. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 523-542 https://doi.org/10.26493/1855-3974.1062.ba8 (Also available at http://amc-journal.eu) Wonderful symmetric varieties and Schubert polynomials* Mahir Bilen Can, Michael Joyce Department of Mathematics, Tulane University, New Orleans, LA 70118, USA Benjamin Wyser Department ofMathematics, Oklahoma State University, Stillwater, OK 74078, USA Received 9 June 2016, accepted 25 October 2017, published online 11 September 2018 Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of the general linear group on the flag variety. Combining this with a slight extension of results of Can, Joyce and Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms. Keywords: Symmetric varieties, Schubert polynomials, wonderful compactification, equivariant co-homology, weak order, parabolic induction. Math. Subj. Class.: 14M27, 05E05, 14M15 1 Introduction Suppose that G is a connected reductive algebraic group over C. Suppose that B D T are a Borel subgroup and a maximal torus of G, respectively, W is the Weyl group, and let t denote the Lie algebra of T. By a classical theorem of Borel [1], the cohomology ring of G/B with rational coefficients is isomorphic to the coinvariant algebra Q[t*]/JW, where * We thank Michel Brion for many helpful conversations and suggestions. We thank the referee for their careful reading and many helpful suggestions. The first and second author were supported by NSA-AMS Mathematical Sciences Program grant H98230-14-1-142. The second is partially supported by the Louisiana Board of Regents Research and Development Grant 549941C1. The third author was supported by NSF International Research Fellowship 1159045 and hosted by Institut Fourier in Grenoble, France. E-mail address: mcan@tulane.edu (Mahir Bilen Can), mjoyce3@tulane.edu (Michael Joyce), bwyser@okstate.edu (Benjamin Wyser) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ IW denotes the ideal generated by homogeneous W-invariant polynomials of positive degree. Any subvariety Y of G/B defines a cohomology class [Y] in H*(G/B). It is then natural to ask for a polynomial in Q[t*] which represents [Y]. In this paper, for certain families of subvarieties of certain G/B, we approach and answer this question in two different ways. Relating the two answers leads in the end to our main result, Theorem 4.1, which, roughly stated, says that certain non-negative linear combinations of Schubert polynomials factor completely into linear forms. Our group of primary interest is G = GLn, with B its Borel subgroup of lower-triangular matrices, and T its maximal torus of diagonal matrices. In this case, there is a canonical basis xi,... ,xn of t* that correspond to the Chern classes of the tautological quotient line bundles on the variety of complete flags G/B. Let Zn denote the center of GLn, consisting of diagonal scalar matrices. Let On denote the orthogonal subgroup of GLn, and let Sp2n denote the symplectic subgroup of GL2n. Denote by GOn (resp. GSp2n) the central extension ZnOn (resp. Z2nSp2n). For any ordered sequence of positive integers m = (pi,... ,Ms) that sum to n, GLn has a Levi subgroup Lm := GLM1 x • • • x GLMs, as well as a parabolic subgroup PM = LM x UM containing B, where UM denotes the unipotent radical of PM. The subgroup Hm := (GOMi x---x GOMs) x UM of GLn is spherical, meaning that it acts on GLn/B with finitely many orbits. Moreover, there is a unique closed HM-orbit YM on GLn/B, which is our object of primary interest. The reason for our interest in this family of orbits is that they correspond to the closed B-orbits on the various G-orbits of the wonderful compactification of the homogeneous space GLn/GOn. This homogeneous space is affine and symmetric, and it is classically known as the space of smooth quadrics in Pn-i. Its wonderful compactification, classically known as the variety of complete quadrics [9, 13], is a G-equivariant projective embedding X which contains it as an open, dense G-orbit, and whose boundary has particularly nice properties. (We recall the definition of the wonderful compactification in Section 2.1.) It turns out that, with minor modifications, our techniques apply also to the wonderful compactification X' of the space GL2n/GSp2n, which parameterizes non-degenerate skew-symmetric bilinear forms on C2n, up to scalar. Letting G = GL2n in this case, the G-orbits on X' are again parametrized by compositions M = (Mi,..., Ms) of n; note that this is of course equivalent to parametrizing them by compositions of 2n with each part being even. Each G-orbit has the form G/H^, with H := (GSp2Mi x---x GSp2Ms) x UM, a spherical subgroup which again acts on GL2n/B with a unique closed orbit Y^. Let us consider two ways in which one might try to compute a polynomial representative of [Ym] (or [Y^]). For the first, note that YM, being an orbit of HM, also admits an action of a maximal torus SM of HM. Thus YM admits a class [YM]S^ in the SM-equivariant cohomology of GLn/B, denoted by HS (GLn/B). In brief, this is a cohomology theory which is sensitive to the geometry of the SM-action on GLn/B. It admits a similar Boreltype presentation, this time as a polynomial ring in two sets of variables (the usual set of x-variables referred to in the second paragraph, along with a second set which consists of y and z-variables) modulo an ideal. Moreover, the map HS (GLn/B) ^ H*(GLn/B) which sets all of the y and z-variables to 0 sends the equivariant class of any SM-invariant subvariety of GLn/B to its ordinary (non-equivariant) class. Thus if a polynomial representative of [Ym]sm can be computed, one obtains a polynomial representative of [YM] by specializing y, z ^ 0. In [15], this problem is solved for the case in which p has only one part, in which case Hm = GOn. Here, we extend the results of [15] to give a formula for the equivariant class [Y„l sm (and [Y' ]s' ) for an arbitrary composition p. The main general result is Proposition 3.4; it, together with Proposition 3.5, imply the case-specific equivariant formulas given in Corollaries 3.6 and 3.8. The formulas for [YM] and [Y^] obtained from these corollaries (by specializing y and z-variables to 0) are as follows: Corollary 1.1. The ordinary cohomology class of [YM] is represented in H* (G/B) by the formula ... _ _ . Vi=1 / i= 1 Vi+1 3.) So for m = (6,5) as above, we have that gl(x, z) = (xi + x2 - 2zi)(xi + x3 - 2zi)(xi + x4 - 2zi)(xi + x5 - 2zi) (x2 + x3 - 2zi)(x2 + x4 - 2zi), and g2 (x, z) = (xr + xg - 2z2)(xr + xg - 2z2)(xr + xio - 2z2)(x8 + xg - 2Z2). Finally, we define a third polynomial hM(x, y, z) in the x, y, and z-variables to simply be hM(x, p(y)), where p denotes restriction from the variables yl,..., yn corresponding to coordinates on the full torus T to the variables yitj, zi on the smaller torus SM. To be more explicit, for each i, j with 1 < i < j < s define hi,j (x, y, z) to be rifc=l nr=jl2(xVi+fc - y^- zj x^+k+y^- zj) if mis even, nr= l (xVi+k - zj) n[=l/2J (xVi+k - yj,i - zj)(xVi+k + yj,i - zj) if Mjis odd. So for the case n = 4, m = (2,2), we have hl,2(x, y, z) = (xi - y2,l - Z2)(xi + y2,l - Z2)(x2 - y2,l - Z2)(x2 + y2,l - Z2), while for the case n = 5, m = (2, 3), we have hl,2(x, y, z) = (xl - Z2)(x2 - Z2)(xl - y2,l - Z2)(xi + y2,l - Z2) (x2 - y2,l - Z2)(x2 + y2,l - Z2). Then we define hM(x, y, z) = JJ hj,j(x, y, z). l i. £=1 Then for each i = 1,..., n, define r(m, i) to be i) := £ Mj. B(m ,i) i the given x-variable appears in precisely Mj linear forms involving y, z terms associated with block j. The proof of Corollary 1.2 is almost identical, except simpler. □ 4 Factoring sums of Schubert polynomials We end by establishing explicit polynomial identities involving sums of Schubert polynomials, using the cohomological formulae of the preceding section together with the results of [5, 6] which were recalled in Section 2.4. Note that by Brion's formula (1.2) combined with the fact that the Schubert polynomial Sw is a representative of the class of the Schubert variety Xw in H*(G/B), we have the following two families of identities in H*(G/B): £ 2d(Y-w)6w = 2d(M) J]xf(^i)+5(^i) n ( n (xj + xk)Y (4.1) wEW(Ym) i=1 i=1 \ Vi + 1