ARS MATHEMATICA CONTEMPORANEA Volume 15, Number 1, Fall/Winter 2018, Pages 1-266 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 A New Strategy to Tackle the Backlog In 2008 when our journal was first launched, 20 papers were published. Ten years later, 60 papers will be published, producing a threefold increase. Also we started a new sister journal ADAM, with a somewhat broader scope, featuring 20 papers in 2018. And on top of this, there are still about 100 papers in the editorial process for AMC. Many of these papers will be eventually published in 2019, some of them even in 2020. This year we adopted a strict policy of controlling the number of papers filtered from the initial quick assessment into the actual refereeing phase. If we want to reduce the backlog, then only about five papers per month can be processed, and otherwise we will have no choice but to decline the rest, or divert them to ADAM. This policy should ensure that there will be no further increase in the backlog. But actually we would like to significantly reduce the backlog, and so we decided to increase the number of published papers from 15 to 20 per issue (and 60 to 80 in total) for the year 2019. In the meantime we expect that ADAM will also gain in reputation and popularity, and this should help us reduce the large influx of papers to the editorial system of AMC. Klavdija Kutnar, Dragan Marušic and Tomaž Pisanski Editors In Chief ARS MATHEMATICA CONTEMPORANEA Contents Linking rings structures and semisymmetric graphs: Combinatorial constructions Primož Potočnik, Stephen E. Wilson..................... 1 On Wiener inverse interval problem of trees Jelena Sedlar................................. 19 Groups in which every non-nilpotent subgroup is self-normalizing Costantino Delizia, Urban Jezernik, Primož Moravec, Chiara Nicotera ... 39 On prime-valent symmetric graphs of square-free order Jiangmin Pan, Bo Ling, Suyun Ding..................... 53 Rank 4 toroidal hypertopes Eric Ens....................................67 On 2-distance-balanced graphs Boštjan Frelih, Štefko Miklavic........................ 81 The arc-types of Cayley graphs Marston Conder, Nemanja Poznanovic....................97 Coloring properties of categorical product of general Kneser hypergraphs Roya Abyazi Sani, Meysam Alishahi, Ali Taherkhani............113 Coordinatizing n3 configurations William L. Kocay...............................127 Maximum cuts of graphs with forbidden cycles Qinghou Zeng, Jianfeng Hou.........................147 Circular chromatic number of induced subgraphs of Kneser graphs Meysam Alishahi, Ali Taherkhani......................161 Groups of symmetric crosscap number less than or equal to 17 Adrián Bacelo.................................173 Saturation number of lattice animals Niko Tratnik, Tomislav Došlic, Petra Žigert Pleteršek............191 Enumerating regular graph coverings whose covering transformation groups are Z2-extensions of a cyclic group Jian-Bing Liu, Jaeun Lee, Jin Ho Kwak...................205 Enumeration of hypermaps of a given genus Alain Giorgetti, Timothy R. S. Walsh.....................225 Volume 15, Number 1, Fall/Winter 2018, Pages 1-266 ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 1-17 https://doi.org/10.26493/1855-3974.1007.3ec (Also available at http://amc-journal.eu) Linking rings structures and semisymmetric graphs: Combinatorial constructions Primoz Potocnik * Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia affiliated also with: IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia Stephen E. Wilson Department of Mathematics and Statistics, Northern Arizona University, Box 5717, Flagstaff, AZ 86011, USA affiliated also with: FAMNIT, University ofPrimorska, Glagoljaska 8, SI-6000 Koper, Slovenia Received 30 December 2015, accepted 27 July 2017, published online 2 November 2017 This paper considers combinatorial methods of constructing LR structures: two isolated constructions, RC and SoP, two closely related constructions, CS(T, B, 0) and CS(T, B, 1) using cycle decompositions of tetravalent graphs, a generalization of those, CS(T, B, k) for k > 2, and finally a construction LDCS relating to cycle decompositions of graphs of higher even valence. This last construction is used to classify all LR structures of types {3, *} or {4, *}. Keywords: Graph, automorphism group, symmetry, locally arc-transitive graph, semisymmetric graph, cycle structure, linking ring structure. Math. Subj. Class.: 20B25, 05E18 1 Introduction 1.1 History An LR structure is a finite, simple, connected, tetravalent vertex-transitive graph together with a decomposition of its edge-set into cycles that satisfies certain symmetry conditions * Supported in part by Slovenian Research Agency, projects P1-0294. E-mail addresses: primoz.potocnik@fmf.uni-lj.si (Primož Potocnik), stephen.wilson@nau.edu (Stephen E. Wilson) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ Abstract (see Section 1.2 for details). It can also be seen as one of the possible symmetry types of the tetravalent vertex-transitive graphs, namely the one in which the stabiliser of a vertex in some vertex-transitive group of symmetries acts on the neighbourhood as the Klein 4-group in its intransitive action on four points (see [10, Section 1] for more on this topic). This paper is the third in a "trilogy" developing the theory of LR structures. In the first paper [10], we introduced LR structures and we explained their importance in the search for semisymetric graphs via the function P which creates a semisymmetric graph from such a structure. In that paper, we also introduced two related families of LR structures and discussed certain double covers of LR structures. All definitions from [10] appear in the next section; please consult [10] for more details. In the second paper, [9], we showed several purely algebraic constructions for such structures. We first gave a quite general approach to constructing an LR structure from a group having certain automorphisms. We then applied these techniques to several families of abelian groups and to dihedral groups. We noted in [9] that several of these constructions gave semisymmetric graphs having large vertex-stabilizers. It has been known for decades that the size of a vertex-stabilizer in a cubic edge-transitive graph is at most 48 in the dart transitive case (see [12]) and at most 384 in the semisymmetric case (see [4]). No such absolute bounds exist for tetravalent edge-transitive graphs. However, recently Spiga, Verret and the first mentioned author of this paper have discovered efficient bounds on the size of the vertex-stabiliser in terms of the number of the vertices for the case of the tetravalent dart-transitive graphs [7] and for the case of the tetravalent half-arc transitive graphs [11]. Semisymmetric graphs are thus the last remaining case of tetravalent edge-transitive graphs for which no good bounding behavior on the size of the vertex stabilizer is known. Perhaps the examples in [9] and especially the characterisation of the LR structures of type {4, q} in this paper will yield some insight into the phenomenon of the large vertex stabilizer in LR structures, and consequently, the tetravalent semisymmetric graphs of girth 4 (see Section 5). 1.2 Definitions Unless explicitly stated otherwise, all the graphs in this paper are finite, simple and connected. Let A be a regular tetravalent graph and C a partition of its edge-set E(A) into cycles. We shall call such a pair (A, C) a cycle decomposition. Two edges of A will be called opposite at vertex v, if they are both incident with v and belong to the same element of C. The partial line graph of a cycle decomposition (A, C) is the graph P(A, C) whose vertices are edges of A, and two edges of A are adjacent as vertices in P(A, C) whenever they share a vertex in A and are not opposite at that vertex. A symmetry of (A, C) is any permutation of the vertices of A which preserves C. The set of all such is called Aut(A, C). Because the two edges at v that belong to one cycle are connected to both of the edges in the other cycle containing v, the edges at each vertex of A form a 4-cycle in P(A, C). Thus, the girth of P(A, C) is usually 4 and never any larger. A cycle decomposition (A, C) is said to be flexible provided that for every vertex v and each edge e containing v, there is a symmetry which fixes pointwise the cycle D G C containing e and interchanges the other two neighbors of v. The edges joining v to those neighbors are in some other cycle C of C. The symmetry then reverses the cycle C and is called a C-swapper at v. A cycle decomposition (A, C) is called bipartite if C can be partitioned into two subsets G and R so that each vertex of A meets one cycle from G and one from R. Especially in constructions, we will refer to the edges of the cycles in G and those in R as green and red, respectively. The largest subgroup of Aut(A, C) preserving each of the sets G and R will be denoted by Aut+ (A, C), and we will think of it as the color-preserving group of (A, C). Definition 1.1. A cycle decomposition (A, C) is called a linking rings structure (or briefly, an LR structure) provided that it is bipartite, flexible and Aut+(A, C) is transitive on the vertices of A. Note that Aut+(A, C) acts transitively on the darts of each color class, and that (since A is assumed to be connected) its index in Aut(A, C) is at most 2. If there is a symmetry of A which preserves C but interchanges the edge color sets G and R (that is, if Aut+(A, C) = Aut(A, C)), then we say that (A, C) is self-dual. Since the color preserving group Aut+(A, C) of an LR structure (A, C) is transitive on R and on G, all cycles in R must have the same length, say p, and all cycles in G must be of the same length, say q. We then say that the LR structure (A, C) is of type {p, q}. For a self-dual structure, of course, p = q. Two LR structures (Ai, Ci) and (A2, C2) are isomorphic whenever there is a graph isomorphism from A1 to A2 which maps cycles in C1 to cycles in C2. We define the joining sequences of an LR structure to be Jr = [sr, dr, wr ] and Jg = [sg, dg, wg] where: sr is the least s such that some two red cycles are joined by two green paths of length s. The number dr is the least d such that two such green paths have starting points that are d apart on one of the red cycles. If the two paths are j apart on the other red cycle, a symmetry argument shows that d must divide j, and then we set wr = d; see Figure 1. In the case that no two red cycles are joined by two green paths of the same length, we declare Jr to be [0,0,0]. The numbers sg, dg, wg are defined similarly, with colors reversed. If (A, C) is self-dual, Jr structure is not self-dual. bo bi bd 1 Jg. More usefully, if Jr = Jg, then the bj ao aj a d Figure 1: Green paths of length s joining two red cycles. If (A, C) is a cycle decomposition, then a cycle C in A is said to be C-alternating if no two consecutive edges of C belong to the same element of C. If a C-alternating 4-cycle exists, then Jr = Jg = [1,1,1], and (A, C) is the partition of the edges of a toroidal map of type {4,4} into horizontal and vertical cycles. Definition 1.2. An LR structure (A, C) is called suitable provided that (1) (A, C) is not self-dual, and (2) A has no C-alternating 4-cycles. The primary result of [10] is that: Theorem 1.3. The partial line graph construction P induces a bijective correspondence between the set of suitable LR structures and the set of worthy tetravalent semisymmetric graphs of girth 4. The word worthy in this statement means that no two vertices of the graph have exactly the same neighbors. Every new suitable LR structure gives a new semisymmetric graph, and so we are interested in finding and creating LR structures. In the remainder of this paper, we show how varied examples can be, concentrating on combinatorial constructions. We first present two simple but non-trivial constructions to show some of the variety possible and to illustrate how the properties of LR structures enter into proofs. 2 Two constructions 2.1 Rows and columns We construct an LR structure RC(n, k) in the following way: the vertices are to be all ordered pairs (i, (r, j)) and ((i, r), j), where i and j are in Zn, and r is in Zk. Green edges join (i, (r, j)) to (i ± 1, (r, j)) and ((i, r), j) to ((i, r), j ± 1), while red edges join (i, (r, j)) to ((i, r ± 1), j) and so ((i, r), j) to (i, (r ± 1, j)). The function (i, (r, j)) ^ (i + 1, (r, j)) and ((i, r), j) ^ ((i +1, r), j) is a symmetry of the graph; we abbreviate it by saying i ^ i + 1. Similarly, each of the functions, j ^ j + 1, r ^ r + 1, i ^ —i, j ^ —j, r ^ —r acts as a symmetry of RC(n, k). These dihedral symmetries act transitively on the vertices of each kind, and the correspondance (i, (r, j)) ^ ((j, r), i) is a symmetry and interchanges the two sets. The green neighbors of (o, (0,0)) are (1, (0,0)) and ( — 1, (0,0)), while the red neighbors are ((0,1), 0) and ((0, —1), 0). Swappers at (0, (0,0)), then, are i ^ — i and r ^ —r. So RC(n, k) with the given coloring is an LR structure. It has 2n2k vertices, and its group has order at least 8n2k. The structure is of type {n, LCM(2, k)}. If k is even, the graph described above is disconnected; in this case, re-assign the name RC(n, k) to the component containing (0, (0,0)). Then the graph has only n2k vertices. It is easy to check that Jr = [1,2,1], while Jg = [2,1,1] and so this structure is always suitable. This LR structure is also described algebraically in [9]. 2.2 SoP In this section, we describe a family of LR structures whose symmetry groups have arbitrarily large vertex stabilizers. The structure is SoP(m, n), where m and n are multiples of 4. Let r = ^ + 1; then we have that r2 = 1 mod n. Further, if j is even, then rj = j, while if j is odd, rj = j + n. The vertex-set is Zm x Zn x Z2. Red edges join (i, j, k) to (i, j ± rk, k); for fixed i and j, green edges join the two vertices (2i, j, 0) and (2i, j, 1) to the two vertices (2i + 1, j, 0) and (2i + 1, j, 1) if j is even, to the two vertices (2i — 1, j, 0) and (2i — 1, j, 1) if j is odd. We claim that each of the following mappings p, a, t, y and S is a symmetry of the structure: (i, j, k)p = (i,j,k)M = (i,j,k)a = (i,j + 2, k) (i, —j, k) (i + 1,j + 1, k) Together, these symmetries show that the structure is vertex-transitive. The symmetry ^ acts as a red swapper at (0,0,0), and S acts as a green swapper there. Thus SoP(m, n) is an LR structure of order 2mn and type {4, n}. The conjugates of S by (a2) commute with each other and so form a subgroup of order 2m. We can see, then, that the order of a In this case, Jr = [1,2,1], while Jg = [2,2,1] and so this structure is always suitable. 3 LR structures from cycle structures 3.1 Voltage graphs and 2-coverings We wish to use the mechanism of voltage graphs to describe a family of LR structures. Let us first summarize the voltage construction and some related facts in the special case of 2-coverings: Let r be any connected graph or multigraph. A Z2-voltage assignment on r is a function Z: E(r) ^ Z2. The corresponding 2-fold covering Cov(r, Z) has V(r) x Z2 as its vertex-set. The edge-set is {{(u, i), (v, i + Z(e))} | e = {u,v} G E(r),i G Z2}. Two Z2-voltage assignments Z and Z' are equivalent provided there is an isomorphism between Cov(r, Z) and Cov(r, Z') which acts trivially on first coordinates. For any vertex v, define the function ^v on the set of all Z2-voltage assignments on r by letting Z^v be the assignment defined by We call such a function a "local reversal". Then Z is equivalent to Z^v, and any two equivalent assignments are related by a series of local reversals. It follows that if Z and Z' are equivalent, then there is a set U C V(r) such that Z(e) = Z '(e) exactly when both ends of e are in U or both not in U. Two voltage assignments Z and Z' are isomorphic provided that some isomorphism 7 of Cov(r, Z) onto Cov(r, Z') has the property that for each vertex v, (v, 0)7 and (v, 1)7 have the same first coordinate. Certainly Z and Z' are isomorphic if there is a symmetry P of r such that Z'(eP) = Z(e) for every e. We write Z' = ZP in this case, and then the function which sends (v, i) to (vP, i) is an isomorphism of Cov(r, Z) onto Cov(r, Z'). Finally, we say that a symmetry a of r lifts to a symmetry a of Cov(r, Z) provided that for each vertex of Cov(r, Z), (v, i)a = (va, j) for some j. Then, clearly, a lifts if and only if Za is equivalent to Z. 3.2 Voltage description of CS(r, B, i) (i, j, = (i,rj, 1 - k) = (-i,j + 1,k) m — 2 vertex-stabilizer is at least 2 2 . (CMv )(e) 1 + Z (e) v is an endvertex of e Z (e) v is not an endvertex of e The construction we wish to present here has to do with a kind of highly symmetric cycle decomposition called a cycle structure: Definition 3.1. A cycle structure in a tetravalent graph r is a cycle decomposition B of r such that Aut(r, B) acts transitively on the darts of r. Consider, for example, the graph of the octahedron O, shown in Figure 2. The set B of triangles induced by the triples {{1, 5, 6}, {1, 2, 3}, {2,4, 6}, {3,4, 5}} forms a cycle structure in O. (In what follows, we will refer to each of these triangles by naming the vertex-triple which induces it rather than specifying its edges.) The group Aut(O, B) is isomorphic to the symmetric group S4, is dart-transitive and acts as S4 on the four triangles. In particular, (O, B) is a cycle structure. Cycle structures were introduced in [5], where it was shown that a vast majority of dart-transitive 4-valent graphs admit a cycle structures—many have more than one. At the end of this section we will show all small cycle structures and see how they contribute semisymmetric graphs. 3.3 The multigraph r' and its symmetries We construct an LR structure from a cycle structure (r, B) in two steps: we form a multigraph and then 2-cover it. First form the multigraph r' from r by separating each vertex into a pair of vertices, so that the cycles from B remain cycles but become disjoint. Then connect the two vertices in each pair with two parallel edges. We will refer to these as "bridge" edges. In our example of the octahedron with cycles {1, 5, 6}, {1, 2,3}, {2,4,6}, {3,4,5}; Figure 3 shows the result. To be more specific, the vertices of r' are all (C, v) where C G B, and v G C. "Ordinary" edges join (C, u) to (C, v) where {u, v} is an edge in the cycle C. If v belongs to cycles C and D, the corresponding "bridge" edges ev,0 and eVj1 join (C, v) to (D, v). Continuing the example and setting A = {1,5,6}, B = {1,2, 3}, C = {2,4, 6}, D = {3,4,5}, the corresponding labels of vertices in the split graph are shown in Figure 4. If a is any symmetry in G = Aut(r, B), we choose the canonical representative a' of a to be the permutation which sends the vertex (C, v) to (Ca, va), the ordinary edge {(C, u), (C, v)} to the ordinary edge {(Ca, ua), (Ca, va)}, and the bridge edge eVji to eva,j. Then a' is clearly a symmetry of r'. If we let G' = {a' | a G G}, -(3) Figure 3: The octahedron, split. Figure 4: Labels in the octahedron. then G' = G. Also, for each v e V(r), let av interchange ev,0 and evj1 while fixing every vertex of r' and every edge other than those two. Clearly, each av is in Aut(r'). If we let K = (av : v e V(r)>, then Aut(r') is the inner semidirect product K x G'. For each C e B, define aC to be the product of all av for v e C, and let L = (aC : C e B> < K. Since the product of all aC for C e B involves each av twice, the product is trivial. On the other hand, if D is a proper non-empty subset of C, then an easy connectivity argument shows that the product of all aC for C e D is non-trivial. Therefore the group L has order 2|B|-1. Now, G' is transitive on the vertices of r' and is, in fact transitive on the darts of ordinary edges. So for any (C, v), some a' e G' acts as a swapper of ordinary edges there. And each non-trivial element of L acts as a swapper of bridge edges. Then L x G' is transitive on darts in ordinary edges and on darts in bridge edges as well. Thus the partition C of the edges of r' into cycles covering ordinary edges and cycles covering bridge edges is an LR coloring of this multigraph. In the following sections we will construct two covers of the graph r' and show that in both cases, L x G' is the group of symmetries that lifts. 3.4 The coverings of r' We now assign voltages 0,1 from Z2 to the edges of r' in two different ways; the assignments will be called Co and Z1. For bridge edges, let Zi(ev,o) = 0 and Ci(ev,i) = 1 for i = 0,1. We assign Zo(e) = 0 for each ordinary edge e. To define Z1, we choose one edge in each cycle C e B to receive the voltage 1, and assign 0 to the rest of the edges in C. The isomorphism class of the resulting 2-cover is independent of which edge in each cycle is chosen, as we show in the next paragraph. Let A(r, B, 0) and A(r, B, 1) be the 2-covers Cov(r', Zo) and Cov(r', Z1) of r' resulting from Zo and Z1, respectively. Let CS(r, B, 0) and CS(r, B, 1) be these graphs together with the decompositions into cycles covering those in C. In Section 3.5 we will show that CS(r, B, 0) and CS(r, B, 1) are, in fact, LR structures. To support our claim that the isomorphism class of Cov(r', Z1) does not depend on our choice of representatives in each cycle, it will suffice to show that for any C e B, two Z2-assignments which are identical except on two consecutive edges of C are isomorphic assignments. So suppose that vertices u, v, w are consecutive in C, and that one of the assignments is Z such that Z ({(C, u), (C, v)}) = 1, Z ({(C, v), (C, w)}) = 0, as in Figure 6. Then Z is isomorphic to , which in turn is equivalent to M(c,v) (where, ^(c,v) is a local reversal as described in Section 3.1) and this assignment is identical to Z except on the edges {u, v}, {v, w}, as required. Thus, by applying products such as av^(c,v) to the assignment at successive vertices v of C, we can move the edge bearing a 1 from any position in C to any other. By adjusting each cycle in turn, we can show isomorphism of any two such assignments. This in fact shows the following useful fact, which we will refer to later. 0 0 (a) Zo(O') (b) Zi(O') Figure 5: Voltage assignments. Remark 3.2. Let Z be an assignment on r' for which Z(ev,o) = 0 and Z(ev,1) = 1 for every vertex v of r, and let C e B. If the sum of all Z(e) for e e C is 0, then Z is isomorphic to Figure 6: Isomorphic voltage assignments. some assignment Z' with Z'(e) = 0 for all e G C. Similarly, if the above sum is 1, then Z is isomorphic to some assignment in which every edge of C except one has weight 0, and that one has weight 1. Consequently, every 2-cover of r' without multiple edges in which all cycles covering those in B have the same length must be isomorphic to CS(r, B, 0) or cs(r, b, 1). ♦ 3.5 The groups of CS(r, B, 0) and CS(r, B, 1) We will show in this section that the cycle decompositions CS(r, B, 0) and CS(r, B, 1), are LR structures, each admitting a group of order 2|B| |G|. Let G = Aut(r, B). Since (r, B) is a cycle structure, G is transitive on the darts of r. Further, let G', K and L be the groups of symmetries of r' as defined in Section 3.3, and recall that Aut(r') = K x G'. Observe that, since G' maps a cycle in B to another cycle in B and since L is generated by all oc, C G B, L is normalised by G' and hence normal in Aut(r'). In particular, (L, G') = L x G'. Fix i G {0,1} and let T be the group of those symmetries of r' that lift to a symmetry of CS(r, B, i). In view of Section 3.1, a symmetry P of r' is in T if and only if the voltage assignment ZiP is equivalent to Zi. We will prove that T = L x G' and that the lift of T contains the symmetries needed to show that CS(r, B, i) is an LR structure. Let us first show that for every a G G (and thus a' G G') there exists P G T such that P G a'K. In other words, we show that G' C TK, and since G'K = Aut(r'), that Aut(r') = G'K = TK. If i = 0, then Zia' = Zi, implying that a' lifts, and we can take P to be a' itself. Suppose now that i — 1. Then Zia' also has one edge in each C G B whose voltage is 1. Then as in Section 3.4, there is a (possibly empty) subpath v1, v2, v3,..., vr of C such that Z1a'aviM(c,Vi)... oVkM(c,Vr) coincides with Z1 on C. Denote O(a,C) = Ovi OV2 ••• &vr and M(a,C) = V(C,vi)V(C,V2) ••• M(C,vr) and observe that ^'s and the o's commute in their action on voltage assignments. Hence, by performing this adjustment for each C G B in turn, it follows that k k Z1 = Z1a^ O(a,Cj ) II ^(«,Cj ) j=1 j=1 and so, letting k =n a(a'°i) g j=i we see that Ci^ is equivalent to Qi and thus that fi G T. This completes the proof of the claim that Aut(r') = G'K = TK. Let us now show that TnK = L. This will then imply that G' = G'K/K = TK/K = T/(T n K) = T/L and hence that T = L x G', as claimed. First note that each element of L lifts, and hence L < T .To see this, let a G L and thus a = f] ceI) ac for some DC B. Since (Zi 2 . We describe a k-covering of r', and we call the covering structure CS(T, B, k). Give the weight 0 to each ordinary edge. Give each pair of bridge edges voltage 1 in opposite directions. Let CSI(r, B, k)) be one component of the resulting k-cover. If (r, B) is bipartite and k is even, then the k-covering has two components, while in all other cases, it has one. Thus if B has m cycles, each of length n, and so mn/2 vertices, then CSI(r, B, k)) is of type {n, LCM(2, k)} and has mnk or mnk/2 vertices. It is easy to check that Jr = [2,1,1], while Jg = [1, 2,1] and so this structure is always suitable. 4 Other constructions 4.1 Locally circular cycle structures Definition 4.1. Suppose C is a cycle decomposition of a graph r of valence 2q and let X be the set of all (C, v) such that C g C, v e V(r) and C passes through v. For a vertex v of r let Xv be the set of pairs with second coordinate v. If P is a permutation on X such that the orbits of (P} are the sets Xv for v e V(r), then we will say that (r, C, P) is locally circular. We will call such a P a locally circular ordering on (r, C). Definition 4.2. If a is a symmetry of (r, C), we will say that a respects P provided that, for each (C, v) e X, (C, v)Pa is either (Ca, va)P or (Ca, va)P-1. Let Aut(r, C, P) be the group of all symmetries of (r, C) which respect P. Definition 4.3. If (r, C, P) is locally circular and G < Aut(r), we will say it is G-locally dihedral provided that the following hold: (i) G acts transitively on darts, (ii) every element of G respects P, (iii) for every v e V(r), the stabiliser Gv acts dihedrally on the cycles through v and contains an element which fixes every cycle through v setwise and reverses at least one of them. A locally circular (r, C, P) is locally dihedral if it is G-locally dihedral for some G < Aut(r). While this definition appears to be very restrictive, notice that a large class of examples arises from reflexible maps: if M is a reflexible map of type {p, 2q}, we can consider two edges to be opposite at v provided that those edges are q apart in the cycle of edges incident to v. If M is proper (i.e., no two edges have the same endpoints), the edges fall into cycles in which each edge is joined to the edges opposite it at each end. Then the family of such cycles is a locally dihedral cycle structure. Construction 4.4. If (r, C, P) is locally circular, let LDCS(r, C, P) be the bipartite cycle decomposition (A, D) in which vertices of A are all (C, v) such that v is a vertex of cycle C e C, green edges are all {(C, u), (C, v)} such that {u, v} is an edge of cycle C e C, and red edges are all {(C, v), (C, v)P}. Theorem 4.5. If (T, C, P) is a locally dihedral cycle structure, then LDCS(T, C, P) is an LR structure which has no alternating 4-cycles. Proof. Every element of G = Aut(T, C, P) acts on LDCS(T, C, P) as a symmetry. Since G is transitive on darts, Aut(LDCS(T,C, P)) is transitive on vertices. To see that it is flexible, consider a vertex (C, v). Because (r, C, P) is locally dihedral, it has a symmetry p G G which fixes v, fixes each cycle at v setwise and reverses C. Then p acts as a green swapper at (C, v). Also, because Gv acts dihedrally on the cycles at v, it has a ^ which fixes C (setwise) and interchanges the neighboring cycles in the local order. Then ^ or ^p is a red swapper at (C, v). If there were an alternating 4-cycle in LDCS(r, C, P), the green edges would correspond to distinct edges in r with the same endpoints, which is forbidden in a graph. □ Our last results in this paper show that the constructions CS and LDCS generate all LR structures (A, C) for which C contains cycles of length 3 or 4. We begin by showing that LDCS covers all the cases in which sr > 2. Theorem 4.6. Let (A, C) be an LR structure of type {p, q} in which no two red cycles (of length p) are joined by more than one green edge (that is, if the joining sequence Jr is not of the form [1, *, *]). Then there is a locally dihedral cycle structure (r, D, P), where r is a graph of valence 2p and D is a partition of the edges of r into q-cycles, such that (A, C ) and LDCS(r, D) are isomorphic LR structures. Proof. Let r be the graph with the vertex set being the set of red cycles in (A, C) with two red cycles adjacent in r whenever they are joined by a green edge in A. For each vertex v of A, let n(v) be the red cycle to which v belongs. We can consider n to be a projection onto r of the subgraph of A induced by its green edges. Since two red cycles are joined by at most one green edge, this projection n induces a bijection between the green edges in (A, C) and the edges of r. Let D be the set of all cycles in r of the form n(D) where D is a green cycle in (A, C). Then r has valence 2p and D is a cycle decomposition of r in which every cycle has length q. Let X be the set of all (n(D), C) such that D is a green cycle in (A, C) and C is a red cycle in (A, C) contained (as a vertex of r) in the cycle n(D). Let us now define the permutation P on X yielding a locally dihedral (r, D, P). For each red cycle C in (A, C) choose one of the two possible orientations of C. We then let P map a pair (n(D), C) G X to the pair (n(D'), C) where D' is the green cycle through the next vertex (with respect to the chosen orientation of C) on C after the unique vertex of A that belongs to both C and D. Then all of Aut+ (A, C) respects this P. Since Aut+(A, C) is transitive on green darts, it acts dart-transitively on r. The set stabilizer of a red cycle acts dihedrally on the set of green cycles meeting it. Any green swapper fixes a red cycle pointwise and so fixes setwise each green cycle meeting it, and reverses at least one cycle. Thus (r, D, P) is a locally dihedral cycle structure. Finally, let ^ be the mapping which maps a vertex v of A to the vertex (n(D), C) of LDCS(r, D, P) where C and D are the red and the green cycle of (A, C) containing v, respectively. It is a matter of straightforward computation to verify that ^ is an isomorphism of the LR structures (A, C) and LDCS(r, D). □ With Theorem 4.6, we can now easily prove that all LR structures of types {3, q} or {4, q} without alternating 4-cycle arise by constructions in this paper. (We point out that the LR structures with alternating cycles have been characterised in [10, Lemma 6.3].) The first of the two corollaries below follows directly from Theorem 4.6 after observing that an alternating 4-cycle implies that two red cycles are joined by two green edges. The second one requires some additional work. Corollary 4.7. If (A, C) is a LR structure of type {3, q} without alternating 4-cycles, then there is a locally dihedral cycle structure (r, D, P), where r is a graph of valence 6 and D is a partition of the edges of r into q-cycles, such that (A, C) and LDCS(r, D) are isomorphic LR structures. Theorem 4.8. If (A, C) is an LR structure of type {4, q} without alternating 4-cycles, then one of the following happens: (1) there is a locally dihedral cycle structure (r, D, P), where r is a graph of valence 8 and D is a partition of the edges of r into q-cycles, such that (A, C) = LDCS(r, D); or (2) there is a cycle structure (r, B), where r has valence 4 and (A, C) = CS(r, B, i) for i = 0 or 1. Proof. Suppose that (A, C) is an LR structure, without alternating 4-cycles, in which the red cycles have length 4. Consider a green edge and the red cycles through its endvertices. If no other green edge joins those red cycles then Theorem 4.6 applies, and so (1) holds. If not, then, because (A, C) has no alternating 4-cycles, two green edges join two antipodal vertices on one red cycle with two antipodal vertices on the other red cycle. Call two green edges which are arranged in this way, mated edges. Then Jr = [1, 2,1]. Collapsing each red cycle to a single vertex, as in the proof of Theorem 4.6, identifies all pairs of mated green edge to form a tetravalent dart-transitive graph r. The green cycles of (A, C) are projected onto a cycle structure D in r. Since the projection is 2-to-1 on green edges, we see that if mated green edges come from different cycles, those two q-cycles project to a single q-cycle in r. If they are from the same cycle, then q must be even and that cycle projects onto a §-cycle. Consider now an intermediate projection in which we identify mated green edges, and within a red cycle identify antipodal vertices and opposite edges. This projects (A, C) onto a multigraph, in which the red "cycles" are actually 2-gons, i.e., each consists of a pair of parallel red edges. This is clearly isomorphic to the graph r' formed in the first step of the construction of CS(r, D, i). This presents (A, C) as a 2-covering of r'. Remark 3.2 shows that in the case where the green cycles of r are of length q, the LR structure (A, C) is isomorphic to CS(r, D, 0), and if the green cycles are of length §, then (A, C) is isomorphic to CS(r, D, 1). □ 5 Conclusion Though this paper and its predecessors [9, 10] have presented a number of constructions both agebraic and combinatorial, much remains to be done. Every new discovery of an LR structure gives us a new semisymmetric graph of girth and valence 4. Thus, finding LR structures and organizing them into parameterized families is important in the search for semisymmetric graphs. The smallest known LR structure not yet to be seen as part of a family of such has 36 vertices, and there are seven more with 72 vertices. Examples such as SoP, CS(r, B, 0) and CS(T, B, 1), whose vertex-stabilizers can grow without bound add to our growing knowledge about the structure of semisymmetric graphs. Our ultimate goal of the study of the LR structures is to develop the tools that would enable us to construct a complete list of all "small" LR structures. Such lists exist for both types of edge-transitive cubic graphs (see [1, 3] for the census of cubic edge-transitive graph of order at most 768 and [2] for the extension to order up to 10 000 in the case of dart-transitive graphs) and for cubic vertex-transitive graphs [6] for orders up to 1280. Moreover, lists of all dart-transitive and 1 -transitive tetravalent graphs of order up to 1000 have recently been compiled (see [6, 8]). The main ingredient of these results was always a theoretical result that bounded the order of the vertex-stabiliser in such a graph. While it has long been known that this order is bounded by a constant in the case of cubic edge-transitive graphs, this is not the case in the cubic vertex-transitive or tetravalent edge-transitive cases. What is more, for these cases, families of graphs where the order of the stabiliser grows exponentially with the order of the graph are known. The crucial point in the enumeration of these graphs was a result that identified the "problematic" families and proved that the order of the vertex-stabiliser in the "non-problematic" graphs is bounded by a tame (possibly sublinear) function of the order of the graph. As it happens, all the problematic graphs contain cycles of girth 4 (and there is a deep group theoretical reason for that). There is strong evidence that a similar result might hold in the case of the LR structures. This leads to the following question (we thank Gabriel Verret for a fruitful discussion on this topic): Question 5.1. Does there exist a polynomial function f such that for every LR structure (A, C) of type other than {4, q}, the symmetry group Aut(A, C) has order at most f (|V (A)|). This question complements Corollary 4.8 which reduces the classification of the LR structures of type {4, q} to the study of 8-valent locally dihedral cycle structures and cycle structures in tetravalent dart-transitive graphs. References [1] M. D. E. Conder and P. Dobcsanyi, Trivalent symmetric graphs on up to 768 vertices, J. Com-bin. Math. Combin. Comput. 40 (2002), 41-63. [2] M. D. E. Conder, C.-H. Li and P. Potocnik, On the orders of arc-transitive graphs, J. Algebra 421 (2015), 167-186, doi:10.1016/j.jalgebra.2014.08.025. [3] M. D. E. Conder, A. Malnic, D. Marusic and P. Potocnik, A census of semisymmetric cubic graphs on up to 768 vertices, J. Algebraic Combin. 23 (2006), 255-294, doi:10.1007/ s10801-006-7397-3. [4] D. M. Goldschmidt, Automorphisms of trivalent graphs, Ann. Math. 111 (1980), 377-406, doi:10.2307/1971203. [5] S. Miklavic, P. Potocnik and S. Wilson, Arc-transitive cycle decompositions of tetravalent graphs, J. Comb. Theory Ser. B 98 (2008), 1181-1192, doi:10.1016/j.jctb.2008.01.005. [6] P. Potocnik, P. Spiga and G. Verret, Cubic vertex-transitive graphs on up to 1280 vertices, J. Symbolic Comput. 50 (2013), 465-477, doi:10.1016/j.jsc.2012.09.002. [7] P. Potocnik, P. Spiga and G. Verret, Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs, J. Comb. Theory Ser. B 111 (2015), 148180, doi:10.1016/j.jctb.2014.10.002. [8] P. Potočnik, P. Spiga and G. Verret, A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two, Ars Math.. Contemp. 8 (2015), 133-148, doi:10.26493/ 1855-3974.559.c6c. [9] P. Potocnik and S. Wilson, Linking rings structures and semisymmetric graphs: Cayley constructions, Eur. J. Combin. 51 (2016), 84-98, doi:10.1016/j.ejc.2015.05.004. [10] P. Potocnik and S. E. Wilson, Linking rings structures and tetravalent semisymmetric graphs, Ars Math. Contemp. 7 (2014), 341-352, doi:10.26493/1855-3974.311.4a8. [11] P. Spiga and G. Verret, On the order of vertex-stabilisers in vertex-transitive graphs with local group Cp x Cp or Cp wr C2, J. Algebra 448 (2016), 174-209, doi:10.1016/j.jalgebra.2015.09. 033. [12] W. T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947), 459-474, doi:10.1017/s0305004100023720. [13] S.Wilson, Rose window graphs, Ars Math. Contemp. 1 (2008), 7-19, doi:10.26493/1855-3974. 13.5bb. /^creative ^commor ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 19-37 https://doi.org/10.26493/1855-3974.1376.7c2 (Also available at http://amc-journal.eu) ARS MATHEMATICA CONTEMPORANEA On Wiener inverse interval problem of trees Jelena Sedlar * University of Split, Faculty of civil engineering, architecture and geodesy, Matice hrvatske 15, 21000 Split, Croatia Received 6 April 2017, accepted 30 September 2017, published online 2 November 2017 Abstract The Wiener index W(G) of a simple connected graph G is defined as the sum of distances over all pairs of vertices in a graph. We denote by W [Tn] the set of all values of the Wiener index for a graph from the class Tn of trees on n vertices. The largest interval of consecutive integers (consecutive even integers in case of odd n) contained in W [Tn] is denoted by Wint [Tn]. In this paper we prove that both sets are of cardinality 1 n3 + O(n5/2) in the case of even n, while in the case of odd n we prove that the cardinality of both sets equals 12n3 + O(n5/2), which essentially solves two conjectures posed in the literature. Keywords: Wiener index, Wiener inverse interval problem, Tree. Math. Subj. Class.: 05C05, 05C90 1 Introduction The Wiener index of a connected graph G is defined as the sum of distances over all pairs of vertices, i.e. W (G) = ^ d(u,v). u,veV (G) It was first introduced in [13] and it was used for predicting the boiling points of paraffins. Since the index was very successful many other topological indices were introduced later which use the distance matrix of a graph. There is a recent survey by Gutman et al. [14] in which finding extremal values and extremal graphs for the Wiener index and several of its variations is nicely presented. Given the class of all simple connected graphs on n vertices it is easy to establish extremal graphs for the Wiener index, those are complete graph Kn and path Pn. The same holds for the class of tree graphs on n vertices in which the minimal * This work has been supported in part by Croatian Science Foundation under the project 8481 (BioAmpMode) and Croatian-Chinese bilateral project "Graph-theoretical methods for nanostructures and nanomaterials". Also, many thanks to the anonymous reviewers for careful reading of the paper and very valuable suggestions. E-mail address: jsedlar@gradst.hr (Jelena Sedlar) ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ tree is the star Sf and the maximal tree is the path Pf. Many other bounds on the Wiener index are also established in the literature. In [4] Gutman and Yeh proposed the inverse Wiener index problem, i.e. for a given value w the problem of finding a graph (or a tree) G for which W(G) = w. The first attempt at solving the problem was made in [7] where integers up to 1206 were checked and 49 integers were found that are not Wiener indices of trees. In [1] it was computationally proved that for all integers w between 103 and 108 there exists a tree with Wiener index w. The problem was finally fully solved in 2006 when two papers were published solving the problem independently. It was proved in [12] that for every integer w > 108 there is a caterpillar tree G such that W(G) = w. The other proof is from the paper [9] where it was proved that all integers except those 49 are Wiener indices of trees with diameter at most 4. Since the most interesting graphs to be considered are chemical trees (especially those in which maximum vertex degree is at most 3) and hexagon type graphs, in [11] this problem was further considered on classes of such graphs. A related question is to ask what value of the Wiener index can a graph (or a tree) G on n vertices have? In order to clarify further this problem one may also ask how many such values are there, how are they distributed along the related interval or how many of them are consecutive. In [6] this problem is named the Wiener inverse interval problem (see also a nice recent survey [5] which covers the topic). In that paper the set W[Gf] is defined as the set of all values of the Wiener index for graphs G G Gf, where Gn is the class of simple connected graphs on n vertices. Similarly, W[7f] is defined as the set of values W(T) for all trees on n vertices (7f denotes the class of trees on n vertices). Also, Wmi[G„] (or analogously Wint [7f ]) is defined to be the largest interval of consecutive integers such that Wint[Gn] C W[Gn] (or analogously Wint[7f] C W[7f]). In [6] the Wiener inverse interval problem on the class Gn was considered. First, the authors noted that obviously Wifi[£„] C W[£„] C [W(K„), W(P„)]. Since W(K„) and W(P„) are easily computed, the upper bound | Wifi[£„] | < |W[£„] | < f- - fr + f + 1 easily follows. Introducing dandelion and comet graphs and establishing how the values between the values of the Wiener index for dandelion a| nd comet |graph can be obtained, the authors obtain the following lower bound |W[£„]| > |Wifi[£„]| > f- - §n2 - ±n3/2 + 19n + 3n1/2. These bounds sandwich the value of | Wmi[£„] | and |W[Gf]| in terms of n3 tightly, therefore the result |Wift[Gf]| = |Wift[Gf]| = f- + O(n2) easily follows. The authors further conjecture that |W[Gf]| = f— fir + ©(n). Regarding the same problem on the class 7f the following two conjectures were made. Conjecture 1.1. The cardinality of W[7f ] equals 1 n3 + ©(n2). Conjecture 1.2. The cardinality of Wmi[7f ] equals ©(n3). In this paper we will consider these two conjectures. First, we will note that for a tree T on odd number of vertices n the value W (T) can be only an even number. That means that the Wiener inverse interval problem in that case has to be reformulated as the problem of finding the largest interval Wmi[7f] of consecutive even integers such that 1^,3 m2 , 11„ Wini[T„] c W[Tn]. Since |W[T„]| < W(P„) - W(S„) + 1 = 6n3 - n2 + n, we now conclude that the cardinality of W[7n] in the case of odd n can be at most 12n3 + O(n2). Given that reformulation, we will prove both conjectures to be true in terms of n3. Even more, we will prove the strongest possible version of Conjecture 1.2 in terms of n3 by 6n3 + O(n5/2) (i.e. ^? proving that |Wmi[7f]| also equals 6n3 + O(n5/2) (i.e. 12n3 + O(n5/2) in case of odd n) which is the best possible result given the upper bound on |W[7^]| derived from the difference between W(Pn) and W(Sn). These results will yield quite a strong result for the class Trf of chemical trees as a direct corollary. The present paper is organized as follows. In the next section basic definitions and preliminary results are given. In the third section the problem is solved for trees on even number of vertices, while in the fourth section the problem is solved for trees on odd number of vertices. In the fifth section we conclude the paper with several remarks and possible directions for further research. 2 Preliminaries Let G = (V(G),E(G)) be a simple connected graph having n = |V(G)| vertices and m = |E(G)| edges. For a pair of vertices u, v e V(G) we define the distance dG(u, v) as the length of the shortest path connecting u and v in G. For a vertex u e V (G) the degree dG(u) is defined as the number of neighbors of vertex u in graph G. When it doesn't lead to confusion we will use the abbreviated notation d(u, v) and d(u). Also, for a vertex u e V(G) and a set of vertices A C V(G) we will denote d(u, A) = J2veA d(u, v). Similarly, for two sets of vertices A, B C V(G) we will denote d(A, B) = ueA v£B d(u, v). We say that a vertex u e V(G) is a leaf if dG(u) = 1, otherwise we will say that u is an interior vertex of a graph G. A graph G which does not contain cycles is called a tree. A tree graph will usually be denoted by T throughout the rest of the paper. We say that a tree T is a caterpillar tree if all its interior vertices induce a path. Such a path will be called the interior path of a caterpillar. Let a and b be positive integers such that a < b. We say that the interval [a, b] is Wiener p-complete if there is a tree T in Tn such that W(T) = a + pi for every i = 0,..., . We say that the interval [a, b] is Wiener complete if it is Wiener 1 - complete. Let us now note that the value of the Wiener index for a tree T on odd number of vertices n is an even number. There are various ways to prove this fact, maybe the simplest one is to recall that for a tree T on n vertices it holds that W (T) = ^ n„ • nv uveE(T) where nu and nv are the number of vertices in the connected component of T\{uv} containing u and v respectively. Obviously, nu + nv = n and therefore in the case of odd n the product nu • nv must be an even number. I would like here to thank prof. Tomislav Doslic for suggesting this short proof to me and to the anonymous reviewers for referring me to the interesting survey [2] in which this fact is already explained and to several interesting papers ([3], [8] and [10]) in which one can read more on the subject. Before proceeding further, let us state this fact as a formal theorem which we can reference in further text. Theorem 2.1. Let T be a tree on odd number of vertices n > 3. Then W(T) is an even number. The main tool for obtaining our results throughout the paper will be a transformation of a tree which increases the value of the Wiener index by exactly four. We will call it Transformation A, but let us introduce its formal definition. Definition 2.2. Let T be a tree and u e V(T) a vertex of degree 4 such that neighbors vi and v2 of u are leaves, while neighbors w1 and w2 of u are not leaves. We say that a tree T' is obtained from T by Transformation A if T' is obtained from T by deleting edges uv1 and uv2, while adding edges w1v1 and w2v2. Theorem 2.3. Let T be a tree and let T' be a tree obtained from T by Transformation A. Then W(T') = W(T) + 4. Proof. For simplicity's sake we will use the notation d'(u,v) for dT> (u,v). Let TWi = (VWi ,EWi) be the connected component of T\{u} which contains vertex wi for i = 1, 2. Note that the only distances that change in Transformation A are distances from vertices v1 and v2. For every v G VW1 U VW2 we have d'(vi, v) — d(vi, v) + d/(v2, v) — d(v2, v) = 0. For the vertex u we have d'(vi, u) — d(vi, u) + d'(v2, u) — d(v2, u) = 2. Finally, we also have d'(v1,v2) — d(v1,v2) = 2. Therefore, W(T') — W(T) = 4 which proves the theorem. □ Although Transformation A can be applied on any tree graph, we will mainly apply it on caterpillar trees. Moreover, it is critical to find a kind of caterpillar tree on which Transformation A can be applied repeatedly as many times as possible. For that purpose, let us prove the following theorem. Theorem 2.4. Let T be a caterpillar tree and P = u1.. .ud its interior path. If there is a vertex ui G P of degree 4 such that ui±j is of degree 3 for every 1 < j < k — 1, then the interval [W(T), W(T) + 4k2] is Wiener 4—complete. Proof. Let us denote a caterpillar tree T from the statement of the lemma by Tk (since 1 < j < k — 1). Also, let us denote D = {ui±j : j = 0,... ,k — 1}. To obtain the desired result we will systematically apply Transformation A to vertices from D until there is no more vertices in D to which Transformation A can be applied. Let us now explain into greater detail by what system that is done. First, note that in Tk Transformation A can initially be applied only to ui. By applying transformation A to ui in Tk we will obtain a caterpillar tree in which Transformation A can be applied to vertices ui-1 and ui+1. By applying Transformation A to ui-1 and ui+1 consecutively we will obtain a caterpillar tree in which Transformation A can be applied to ui-2 and ui+2 (and ui but we will not further apply Transformation A to that vertex for the time being). By further applying transformation A to ui-2 and ui+2 consecutively and repeating this procedure we will reach a caterpillar tree in which Transformation A can be applied to vertices ui-(k-1) and ui+(k-1) and finally apply Transformation A to those two vertices. The caterpillar tree obtained after that last step we can denote by Tk-1 because of the following: in that tree vertex ui g P is of degree 4 and vertices ui±j are of degree 3 for every j = 1,... ,k — 2. Note that in the process of transforming Tk to Tk-1 we will have applied the Transformation A 2k — 1 times. Now, the same process can be repeated on Tk-1 to obtain Tk-2. The procedure stops when we reach T1 in which ui is the only vertex in D having degree greater than 2 (to be more precise, the degree of ui in T1 equals 4, so Transformation A can be applied to it one more time). Applying Transformation A on ui in T1 we finally obtain T0 in which Transformation A cannot be further applied to vertices from D. Therefore, in transforming Tk to T0 Transformation A was used J2j=1(2j — 1) = k2 times and each time the value of the Wiener index increased by 4. □ Note that the Transformation A in Theorem 2.4 is applied k2 times on a caterpillar in which interior path is of length d - 1. If we prove that there are ©(n) different values of d for which k = ©(n), we obtain roughly ©(n3) graphs with different values of the Wiener index which is exactly the result we aim at (of course, here one has to be careful to avoid significant overlapping of the values of the Wiener index for caterpillars with different values of d). So, that is what we are going to do in following sections, but in order to do that with sufficient mathematical precision we will have to construct four different special types of caterpillar trees. To easily construct those four types of caterpillar trees we first introduce two basic types of caterpillars from which those four types will be constructed by adding one or two vertices. Definition 2.5. Let n, d and x be positive integers such that n > 18 is even, r^i < d < and x < 4+42d-n. Caterpillar Bi(n, d, x) is a caterpillar on even number of vertices n obtained from path P = u_d ... m_1m0u1 ... ud by appending a leaf to vertices u_d_1+x and ud+1-x and by appending a leaf to 2k - 1 consecutive vertices u_(k-1),..., uk-1 where k = —2 '—. Caterpillar graph B1 (n, d, x) is illustrated by Figure 1 (vertex u of the interior path is in the images denoted just by i in order to make labels easier to see). Figure 1: Caterpillar graphs: a) B1(20,6, 2), b) B1(20,5,1). Lemma 2.6. Let n, d and x be integers such that B1 (n, d, x) is defined. Then n3 oj 5 13 W(B1(n,d,x)) = — + (-y - 5)n2 + (4d2 + 10d + y - 2x)n+ 2 8 ,o ,2 46d + 2x2 — d3 - 12d2---7. 3 3 Proof. Let k = "_(2d+1)_1 and x' = -d - 1 + x. Even though the structure of B1 is a bit complicated it is still regular enough so that the Wiener index can be computed exactly (as a function in variables n, d and x). Let us divide vertices of B1 into three sets A, B and C so that set A contains vertices u for i = - d,..., d, set B contains leaves attached to 2k - 1 consecutive vertices u_(k_1),..., uk_1 and set C contains two leaves attached to vertices u_d_1+x and ud+1_x. Note that we have d d k_1 k_1 d(A,A) = ££ (j - i),d(B,B)= £ £ (j - i + 2) i=_dj=i+1 i=_(k_1) j=i+1 d fc_1 d(C,C) = (2d +2 - 2(x - 1)),d(A,B) = £ £ (|i - j| + 1) i=_dj=_(k_1) d fc_1 d(A,C) = 2 £ (|i - x'| + 1),d(B,C) = 2 £ (|i - x'| +2) i=_d i=_(k_1) Noting that W(B1 (n, d,x)) = d(A, A) + d(B, B) + d(C, C) + d(A, B) + d(A, C) + d(B, C) and simplifying the obtained sum yields the formula from the statement of the lemma. □ Note that the caterpillar B1(n, d, x) is a caterpillar with relatively long interior path. Namely, the value d is roughly half of the length of the interior path and in the definition of B1 (n, d, x) the value of d is relatively large with respect to number of vertices n. We now introduce the formal definition of the second basic caterpillar which will have relatively short interior path. Definition 2.7. Let n, d and x be positive integers such that n > 18 is even, 4 < d < [^J and x < "_42d+2. Caterpillar B2 (n, d, x) is a caterpillar on even number of vertices n obtained from path P = u_d ... u_1u0u1.. .ud by appending a leaf to 2k - 1 consecutive vertices u_(k_1),..., uk_1 where k = d -1, by appending x leaves to each of the u_(d_1) and u(d_1), and by appending r leaves to each of the u_d and ud where r = "_4d_2x+2. Caterpillar graph B2 (n, d, x) is illustrated by Figure 2 (vertex ui of the interior path is in the images denoted just by i in order to make labels easier to see). ^ muiiiv b) mu iij. Figure 2: Caterpillar graphs: a) B2(20,4,1), b) B2(20,4, 3). Lemma 2.8. Let n, d and x be integers such that B2 (n, d, x) is defined. Then d 8d3 32d W(B2(n, d,x)) = (^ + 1)n2 + (-2d - 2)n - — + — - 5 + 8x - 8dx - 2x2. Proof. Let k = d -1 and r = "_4d^2x+2. To obtain the exact formula for W(B2(n,d,x)) we divide vertices from B2(n,d,x) into four sets: set A contains vertices ui for i = —d,..., d, set B contains leaves appended to 2k-1 consecutive vertices u_(fc_i),..., set C contains x leaves appended to each of the M_(d-1) and u(d_1), while finally set D contains r leaves appended to each of the u_d and ud. Note that d d k_1 k_1 d(A,A) = g g (j — i), d(B,B)= g g (j — i + 2)+ i=_dj=i+1 i= — ( k — 1) j=i+1 d(C,C) =4(2) + x2(2d), d(D,D) =4(2) + r2(2d +2) Also, we have d k_1 2d+1 d(A,B) = g g (|i — j| + 1), d(A,C)=2x(3 + g (i — 1)) i=_dj = _(k_1) i=3 2d+1 k_1 d(A,D) = 2r g i, d(B,C) = 2x g (i + d + 1) i=1 i=_(k_1) k_1 d(B,D) = 2r g (i + d +2), d(C,D) = 2(3xr + xr(2d + 1)). i=_(k_1) Noting that W(B1(n, d, x)) = d(A, A) + d(B, B) + d(C, C) + d(D, D)+ + d(A, B) + d(A, C) + d(A, D)+ + d(B, C) + d(B, D) + d(C, D) and simplifying the obtained sum yields the formula from the statement of the lemma. □ Finally, let us denote df" = [ 2 ] and xf^ = 4+4d:2'"_", while difax = [ f J . Note that B1(n,dmin,xmax) = B2(n,dfax, 1). (2.1) This equality will provide us with a nice transition from caterpillars based on B1(n, d, x) to caterpillars based on B2(n, d, x) in the following sections. 3 Even number of vertices In this section we will first introduce a special kind of caterpillar based on B1(n, d, x) which will have a longer interior path, then we will introduce a second special kind of caterpillar based on B2 (n, d, x) which will have a shorter interior path. For each of those two special kinds of caterpillars we will establish a bound on the value of d for which the interval between values of the Wiener index for two consecutive values of x and d is Wiener 4—complete. The equality (2.1) will then enable us to "glue" all those intervals into one big Wiener 4—complete interval. Definition 3.1. Let n, d and x be integers for which B1(n — 2, d, x) is defined. For s = — 1,0,1, 2 caterpillar T1 (n, d, x, s) is a caterpillar on even number of vertices n, obtained from B1 (n — 2, d, x) by appending a leaf to the vertex us and a leaf to the vertex ud of the path P = u_d ... u_1u0U1 ... Ud in B1(n — 2, d, x). a) b) Figure 3: Caterpillar graphs: a) ^(22,6,2,0), b) Ti(22,6, 2, 2). Caterpillar graph T1(n, d, x, s) is illustrated by Figure 3 (vertex ui of the path P is in the images denoted just by i in order to make labels easier to see). Lemma 3.2. Let n, d, x and s be integers for which T1 (n, d, x, s) is defined. Then n2 W(T1(n,d,x,s)) = W(B1(n - 2,d,x)) + — + — + 2d2 + 3d + 2s2 - s - 2x. Proof. Let k = ("-2)-(22d+1)-1, x' = -d - 1 + x. We define a function d k-1 f(v)= E (|v - i| + 1)+ E (|v - i| +2)+ i=-d i= — (k — 1) (|x' - v | + 2+ | —x' - v | +2) Now, the definition of T1(n,d,x, s) implies W(T1(n, d, x,s)) = W(B1(n - 2, d, x)) + f (s) + f (d) + d - s + 2. Plugging s and d into the formula for f and simplifying the obtained expression yields the result. □ As a direct consequence of Lemma 3.2 we obtain the following corollary. Corollary 3.3. It holds that W(T1(n, d, x, 1)) = W(T1(n, d,x, 0)) + 1, W(T1(n, d, x, 2)) = W(T1(n, d, x, 0)) + 6, W(T1(n, d, x, -1)) = W(T1(n, d, x, 0)) + 3. The main tool in proving the results will be Transformation A of the graph, which, for a given graph, finds another graph whose value of the Wiener index is greater by 4. Therefore, it is critical to find a graph on which Transformation A can be applied consecutively as many times as possible. That was the basic idea behind constructing graph T1(n, d, x, s) as we did, so that we can use Theorem 2.4 in filling the interval between values W(T1(n, d, x, s)) for consecutive values of x and d. So, let us first apply Theorem 2.4 (i.e. find the corresponding value of k) to the graph T1(n, d, x, s). Lemma 3.4. Let n, d, x and s be integers for which T1(n, d, x, s) is defined. For k = 1 n -d-4 the interval [W(T1(n, d, x, s)), W(T1(n, d, x, s)) + 4k2] is Wiener 4-complete. Proof. Let us denote ki = (n-2)-(22d+1)-1. Note that ki is half of the number of leaves appended to the vertices u±j of the interior path of T1(n, d, x, s) for j = 0,..., k - 1. Since s < 2, note that the definition of T1(n, d, x, s) and Theorem 2.4 imply the result for k = k1 - 2. □ So, let us now establish for which values of d the gap between W(T1(n, d, x, s)) and W(T1(n, d, x — 1, s)) is smaller than 4k2 which is the width of an interval which can be filled by repeatedly applying Transformation A on T1 (n, d, x, s) (i.e. by using Lemma 3.4). Lemma 3.5. Let n, d, x > 2 and s be integers for which T1(n, d, x, s) is defined. For d < 2 (n — a/2n — 8 — 8) the interval [W(T1(n, d, x, s)), W(T1(n, d, x — 1, s))] is Wiener 4—complete. Proof. First note that W(T1(n,d,x — 1,s)) — W(T1(n,d,x,s)) < W(T1(n,d, 2,s)) — W(T1(n,d, 1,s)) = = 2(n — 5)+ 2. Therefore, Lemma 3.4 implies it is sufficient to find integers n and d for which it holds that 4k2 > 2(n — 5) + 2 where k = 1 n — d — 4. By a simple calculation it is easy to establish that the inequality holds for d < 1 (n — a/2n — 8 — 8) so the lemma is proved. □ It is easy to show, using Lemma 3.2, that W(T1 (n, d, x — 1, s)) — W(T1 (n, d, x, s)) = 2n — 4x which is divisible by 4 since n is even. Therefore, Lemma 3.5 enables us to "glue" together Wiener 4—complete intervals [W(T1(n, d, x, s)), W(T1(n, d, x — 1, s))] into one bigger Wiener 4—complete interval [W(T1 (n, d, xmax, s)), W(T\(n, d, 1, s))] where x5"ax = 4+4d-("-2). Corollary 3.3 then implies that roughly the same interval will be Wiener complete when we take values for every s = —1,0,1,2. We say "roughly" because the difference W(T1(n, d, x, 2)) = W(T1(n, d, x, 0)) + 6 makes one point gap at W(T1 (n, d, xmax, 0)) + 2. We now want to "glue" together such bigger intervals into one interval on the border between d and d — 1 . The problem is that T1(n, d, x5"ax,s) = T1(n,d — 1,1,s), so we have to cover the gap in between. Moreover, it holds that W(T1(n, d, x5"ax,s)) — W(T1(n,d — 1,1,s)) = n — 3 which is not divisible by 4. Therefore, we have to find enough graphs whose values of the Wiener index will cover the gap of n — 3 plus the gap of 6 which arises from the "rough" edge of the interval for a given d. Lemma 3.6. Let n, d, xjiax = 4+4d 2(" 2) and s be integers for which T1(n, d, x™3*, s) and T1 (n, d — 1,1, s) are defined. For d < 2 (n — %/n + 3 — 6) the interval [W(T1(n, d — 1,1, s)), W(T1(n, d, x1^, s)) + 6] is Wiener 4—complete. Proof. Since W(T1(n, d, x5"ax, s)) + 6 — W(T1(n, d — 1,1, s)) = n — 3 + 6 = n + 3, Lemma 3.4 implies that it is sufficient to find for which d it holds that 4k2 > n + 3 where k = 1 n — (d — 1) — 4. By a simple calculation one obtains that inequality holds for d < 1 (n — a/n + 3 — 6) which proves the theorem. □ Note that the restriction on the maximum value of d is stricter in Lemma 3.5 then in Lemma 3.6 for every n > 4. Now we have taken out all we could from graph T\, but that covers only caterpillars with relatively large d. We can further expand the Wiener complete interval to the left side, i.e. to caterpillars with smaller d, using graph T2 which we will construct from the basic graph B2. Definition 3.7. Let n, d and x be integers for which B2(n — 2, d, x) is defined. For s = — 1,0,1, 2 caterpillar T2 (n, d, x, s) is a caterpillar on even number of vertices n, obtained from B2 (n — 2, d, x) by appending a leaf to the vertex us and a leaf to the vertex ud of the path P = u_d ... u_1UoU1 ... ud in B2(n — 2, d, x). Caterpillar graph T2 (n, d, x, s) is illustrated by Figure 4 (vertex u of the path P is in the images denoted just by i in order to make labels easier to see). a) MI I If! b) .Yuiym Figure 4: Caterpillar graphs: a) T2(22,4,3, —1), b) T2(22,4, 3,1). Lemma 3.8. Let n, d, x and s be integers for which T2 (n, d, x, s) is defined. Then W(T2(n, d,x, s)) = W(B2(n — 2,d,x)) + (2d + 4)n — 2d2 — 7d — 6 — 2x + 2s2 — s Proof. Let k = d — 1 and r = "-4d-2x. We define a function d k-1 f(v)= E (|v — i| + 1)+ E (|v — i| + 2) + i=-d i=-(k-1) + x(|v + (d — 1)| + 2) + x(|v — (d — 1)| + 2)+ + r(|v + d| + 2) + r(|v — d| +2). Now, the definition of T2 (n, d, x, s) implies W(T2(n, d, x, s)) = W(B2(n - 2, d, x)) + f (s) + f (d) + d - s + 2. Plugging s and d into the formula for f and simplifying the obtained expression yields the result. □ Again, as a direct consequence of Lemma 3.8 we obtain the following corollary. Corollary 3.9. It holds that W(T2(n, d, x, 1)) = W(T2(n, d, x, 0)) + 1, W(T2(n, d, x, 2)) = W(T2(n, d, x, 0)) + 6, W(T2(n, d, x, -1)) = W(T2(n, d, x, 0)) + 3. As in the case of large d, the main tool in obtaining the results will be the following lemma. Lemma 3.10. Let n, d, x and s be integers for which T2(n, d, x, s) is defined. For k = d—3 the interval [W(T2(n, d, x, s)), W(T2(n, d, x, s)) + 4k2] is Wiener 4—complete. Proof. Let us denote ki = d - 1. Note that ki is half of the number of leaves appended to the vertices of the interior path of T2 (n, d, x, s) for j = 0,..., k -1. Since s < 2, note that the definition of T2 (n, d, x, s) and Theorem 2.4 imply the result for k = k1 - 2. □ We will first use Lemma 3.10 to cover the interval between W(T2(n, d, x, s)) and W(T2(n, d, x - 1, s)), after that we will use it to cover the gap between W(T2(n, d -1,1,s)) and W(T2(n, d,x;fax, s)). Lemma 3.11. Let n, d, x > 2 and s be integers for which T2(n, d, x, s) is defined. For d > 1 (V2 n - 8 + 6) the interval [W(T2(n, d, x, s)), W(T2(n, d, x - 1, s))] is Wiener 4-complete. Proof. First note that for xmax = ("-2)-4d+2 it holds that W(T2(n,d,x - 1,s)) - W(T2(n,d,x,s)) < < W(T2 (n, d, xipax - 1,s)) - W(T2(n,d,xmax,s)) = = 2(n - 5) + 2. Therefore, Lemma 3.10 implies it is sufficient to find for which n and d it holds that 4k2 > 2(n - 5) + 2 where k = d - 3. By a simple calculation it is easy to establish that the inequality holds for d > 2 (%/2n - 8 + 6) so the theorem is proved. □ Again, it is easy to show that W(T2(n, d, x - 1, s)) - W(T2(n, d, x, s)) = 4d + 4x - 8 which is divisible by 4. Therefore, using Lemma 3.11 we can again "glue" the interval for different values of x into one bigger interval which will be "roughly" Wiener complete when taking values of W(T2(n, d, x, s)) for every s = -1,0,1,2. The next thing is to cover the gap between W(T2(n, d - 1,1, s)) and W(T2(n, d, xmax, s)) which equals n - 3 plus the gap of 6 which arises from the "rough" ends of the Wiener complete interval for given n and d. Lemma 3.12. Let n, d, x™3* = (n 2)2 4d+2 and s be integers for which T2(n, d, x™3*, s) and T2 (n, d — 1,1, s) is defined. For d > 2 (%/n + 3 + 8) the interval [W(T2(n, d — 1,1, s)), W(T2(n, d, x?ax, s)) + 6] is Wiener 4—complete. Proof. Since W(T2(n, d, x^, s)) + 6 — W(T2(n, d — 1,1, s)) = n + 3, Lemma 3.10 implies it is sufficient to find n and d for which it holds that 4k2 > n + 3 where k = (d — 1) — 3. By a simple calculation one obtains that the inequality holds for d > 1 (%/n + 3 + 8) which proves the theorem. □ Therefore, using graphs l\(n, d, x, s) and T2(n, d, x, s) we obtained two big Wiener complete intervals, which it would be nice if we could "glue" together into one big Wiener complete interval. In order to do that, note that the equality (2.1) implies T2(n, d?ax, 1, s) = Ti (n, df", x5"ax, s) for d?ax = [n42J, dfin = [nf4] and xf8* = 4+4d°'"2-(n-2). Now we can state the theorem which gives us the largest Wiener complete interval we have managed to obtain. Theorem 3.13. Let n > 30, d?in = [ 1 (V2n — 8 + 6)] , x?ax = (n-2)-24d°'"+2 and di"ax = L1 (n — V2n — 8 — 8)J . The interval [W(T2(n, df", x?8*, 2)), W(Ti(n, d?^, 1,0))] is Wiener complete. Now that we have obtained very large Wiener complete interval, we can finally prove the following theorem which is our main result and which proves Conjectures 1.1 and 1.2 in terms of n3. Theorem 3.14. For even n > 30 it holds that \ Wint[7n] | = |W[7^] | = 1 n3 + O(n5/2). Proof. Theorem 3.13 implies |W[Tn]| > \Wini[rn]\ > W(T1(n,d5nax, 1, 0)) — W(T2(n,dlnin,xmnax, 2)) wherediT" = 2(V2n — 8 + 6)+ p, x?^ = (n-2)-24d°'"+2 anddfax = 1 (n — V2n — 8 — 8) — 1 + q for p G [0,1} and q G (0,1]. From Lemmas 3.2 and 3.8 we further obtain that \Wint[Tn]\ > 1 n3 — 1 \/2n5 — 8n4 — 4n2 + 10v/2n3 — 8n2 + ^n + 21^2n — 8 — 51. 6 2 3 6 On the other hand, recall that \Wint[7n]\ < |W[7n]| < W(Pn) — W(Sn) + 1 = 1 n3 — n2 + 11 n, which proves the theorem. □ Note that caterpillar trees Ti(n, d, x, s) are chemical trees (i.e. trees in which the degree of every vertex is at most 4) for all possible values of its parameters and they remain chemical after repeated application of Transformation A. Therefore, half of these results hold for chemical trees and we obtain the following corollary. Corollary 3.15. Let Tn be a class of chemical trees on n vertices where n > 30 is even. Then |Wini[T„4]| = |W[T4]| = ©(n3). Proof. Note that Lemmas 3.5 and 3.6 imply that |W[T4]1 > |Wint[T44]1 > W(Ti(n, d5"ax, 1, 0)) - W(Ti(n,dmin,xmax, 2)) where dfax = 2(n - V2n - 8 - 8) - 1 + p, dfin = "f2 + q and xfax = 6+4dj°for p € (0,1] and q € [0,1}. From Lemma 3.2 we obtain |Wint[T4]| > 1 n3-1 V2n5 - 8n4-15n2 + 5^2n3 - 8n2+^-n+9^2^-8-134. 12 4 8 3 12 2 3 Note that this result for chemical trees is obtained using only chemical trees with relatively large diameter and the result is still the best possible with regard to the highest power n3 (just the power, not the coefficient). This means that this result is something that probably can be significantly improved by considering chemical trees with shorter diameter, but we leave that as an open problem for future research. 4 Odd number of vertices The strategy to prove the result in the case of odd number of vertices is the same as in the case of even number of vertices. The only difference is that in this case the value of the Wiener index can be only even number so we are aiming at the largest possible interval of consecutive even numbers which are values of the Wiener index for a tree. Definition 4.1. Let n, d and x be integers for which Bi(n - 1, d, x) is defined. For s = 0,1 caterpillar T3(n, d, x, s) is a caterpillar on odd number of vertices n, obtained from Bi (n-1, d, x) by appending a leaf to the vertex us of the path P = u-d ... w_iM0ui... ud in Bi(n - 1, d, x). Caterpillar tree T3 (n, d, x, s) is illustrated by Figure 5 (vertex u of the path P is in the images denoted just by i in order to make labels easier to see). Figure 5: Caterpillar graphs: a) T3(21,6,2,0), b) T3(21,6,2,1). Lemma 4.2. Let n, d, x and s be integers for which T3(n, d, x, s) is defined. Then n2 H W(T3(n, d,x, s)) = W(B1(n - 1,d,x)) + — - dn + 2d2 + 5d + — - 2x + 2s2. Proof. Let k = ("-1)-(22d+1)-1, x' = —d - 1 + x. The definition of T3(n, d, x, s) implies d W(Ts(n, d, x, s)) = W(Bi(n - 1, d, x)) + ^ (|s - i| + 1)+ i=-d k-1 + ^ (|s - i\ + 2) + (s - x' + 2) + i=-(k-1) + (-x' - s + 2). Simplifying this expression yields the result. □ As a direct consequence of Lemma 4.2 we obtain the following corollary. Corollary 4.3. It holds that W(T3(n, d, x, 1)) = W(T3(n, d, x, 0)) + 2. We now want to apply Theorem 2.4 to T3(n, d, x, s), i.e. we want to establish the value of k in the case of this special graph. Lemma 4.4. Let n, d, x and s be integers for which T3(n, d, x, s) is defined. For k = 2n-d- 5 the interval [W(T3(n, d,x,s)), W(T3(n, d, x, s))+4k2] is Wiener 4-complete. Proof. Let us denote k1 = ("-1)-(22d+1)-1. Note that k1 is half of the number of leaves appended to the vertices u±j of the interior path of T3(n, d,x,s) for j = 0,... ,k - 1. Since s < 1, note that the definition of T3(n, d, x, s) and Theorem 2.4 imply the result for k = k1 - 1. □ So, let us now establish for which values of d the gap between W(T3(n, d, x, s)) and W(T3(n, d,x - 1, s)) is smaller than 4k2 where k = 2n - d - |. Lemma 4.5. Let n,d, x > 2 and s be integers for which T3(n, d, x, s) is defined. For d < 2 (n -%/2n - 6 - 5) the interval [W(T3(n, d, x, s)), W(T3(n, d,x - 1, s))] is Wiener 4-complete. Proof. First note that T3(n, d,x - 1, s) - T3(n, d, x, s) < T3(n, d, 2, s) - T3(n, d, 1, s) = = 2(n - 4)+ 2. Therefore, Lemma 4.4 implies it is sufficient to find integers n and d for which it holds that 4k2 > 2(n - 4) + 2 where k = 1 n - d - |. By a simple calculation it is easy to establish that the inequality holds for d < 1 (n - sj2n - 6 - 5) so the lemma is proved. □ Using Lemma 4.2 it is easy to establish that W(T3(n, d,x - 1,s)) - W(T3(n,d,x,s)) = 2(n - 2x + 1) which is divisible by 4 since n is odd. Moreover, note that for xmax = 4+4d-("-1) it holds that T3(n, d, x^ax, s) = T3(n, d - 1,1, s). Therefore we can use Lemma 4.5 and "glue" together intervals both on the border between x and x - 1 and on the border of d and d - 1, so we will obtain one large interval which is Wiener 2-complete (because of Corollary 4.3). Again, here we have used T3(n, d, x, s) to the maximum, but we have covered thus only caterpillars with large d. Let us now use graph B2 (n, d, x) to create the fourth special kind of caterpillars which we will use to widen our interval to caterpillars with small d. Definition 4.6. Let n, d and x be integers for which B2(n - 1, d, x) is defined. For s = 0,1 caterpillar T4(n, d, x, s) is a caterpillar on odd number of vertices n, obtained from B2 (n-1, d, x) by appending a leaf to the vertex us of the path P = u-d ... m_1m0u1 ... ud in B2(n - 1, d, x). Caterpillar graph T3(n, d, x, s) is illustrated by Figure 6 (vertex u of the path P is in the images denoted just by i in order to make labels easier to see). ^ :vny i ij. b) muzm Figure 6: Caterpillar graphs: a) T4(21,4,3,0), b) T4(21,4,3,1). Lemma 4.7. Let n, d, x and s be integers for which T4(n, d, x, s) is defined. Then W(T4(n,d,x,s)) = W(B2(n - 1,d,x)) + (2 + d)n - 2d2 - 3d - 1 + 2s2 - 2x. Proof. Let k = d - 1 and r = (n-1)-4d-2x+2. The definition of T4(n, d, x, s) implies d W(T4(n, d, x, s)) = W(B2(n - 1,d,x)) + ^ (|s - i| + 1) + i=-d k-1 + ^ (|s - i| + 2) + (s - x' + 2)+ ¿=-(fc-1) + 2x(d +1) + 2r(d +2). Simplifying this expression yields the result. □ Corollary 4.8. It holds that W(T4(n, d, x, 1)) = W(T4(n, d, x, 0)) + 2. Let us now apply Theorem 2.4 to T4(n, d, x, s). Lemma 4.9. Let n, d, x and s be integers for which T4 (n, d, x, s) is defined. For k = d — 2 the interval [W(T4(n, d, x, s)), W(T4(n, d, x, s)) + 4k2] is Wiener 4—complete. Proof. Let us denote ki = d — 1. Note that ki is half of the number of leaves appended to the vertices j of the interior path of T3 (n, d, x, s) for j = 0,..., k — 1. Since s < 1, note that the definition of T4(n, d, x, s) and Theorem 2.4 imply the result for k = k1 — 1. □ Now we can establish the minimum value of d for which the difference between Wiener index of T4(n, d, x, s) and T4(n, d, x — 1, s) can be "covered" by Transformation A. Lemma 4.10. Let n, d, x > 2 and s be integers for which T4(n, d, x, s) is defined. For d > 2(V2 n — 6 + 4) the interval [W(T4(n, d, x, s)), W(T4(n, d, x — 1, s))] is Wiener 4—complete. Proof. First note that for xf3* = ("-1)-4d+2 it holds that W(T4(n, d, x — 1,s)) — W(T4(n, d,x, s)) < W(T4(n, d,xfax — 1,s)) — W(T4(n, d,xfax, s)) = = 2(n — 4)+ 2. Therefore, Lemma 4.9 implies it is sufficient to find integers n and d for which it holds that 4k2 > 2(n — 4) + 2 where k = d — 2. By a simple calculation it is easy to establish that the inequality holds for d > 2 (%/2n — 6 + 4) so the lemma is proved. □ Using Lemma 4.7 it is easy to establish that W(T4(n, d,x — 1,s)) — W(T4(n,d,x,s)) = 4(x + 2d — 2) which is divisible by 4. Moreover, note that for xmax = ("-1)-4d+2 it holds that T4(n, d, xfax, s) = T4(n, d — 1,1, s). Therefore we can use Lemma 4.10 and "glue" together intervals both on the border between x and x — 1 and on the border of d and d — 1, so we will obtain one large interval which is Wiener 2—complete (because of Corollary 4.8). Finally, noting that for d^" = [ ], xfax = 4+4d°'"2-("-1) and dfax = "f1 J it holds that Ts(n, dfin, xfax, s) = T4(n, dfax, 1, s), we conclude that we can "glue" together two large Wiener 2—complete intervals we obtained (one for large values of d and the other for small values of d), and thus we obtain the following theorem which gives us the largest Wiener 2—complete interval we manage to obtain. Theorem 4.11. Let n > 21, d= [1 (n - V2n - 6 - 5)J, 21 it holds that |Wini[7"]| = |W[7"]| = 12n3 + O(n5/2). Proof. Using Theorem 3.13 we obtain |W[7"]| > |Wint[7"]| > (W(T3(n,d4ax, 1,0)) - W(T4(n,d4in,x4ax, 1)))/2 where d4ax = 2 (n - V2n - 6 - 5) - 1 + p, d^ = 1 (V2n - 6 + 4) + q and ~max ("-1)-4d4i"+2 2 obtain 2 (n - v2n - 6 - 5) - 1 + p, «4 = 2(v2n - 6 + 4) + q and x4 for p € (0,1] and q G [0,1}. Now, using Lemmas 4.2 and 4.7 we further -I -I _ Q C __ |W[7;]| > |Wini[7;]| > — n3 - ^2n5 - 6n4 - 3n2 + -v/2n3 - 6n2 + 83 13 ^-- 253 --n--v2n — 6--. 12 12 12 On the other hand, Theorem 2.1 implies |Wint[7"]| < |W[7"]| < (W(P") - W(S") + 12n - 2n + 12? 1)/2 = n3 - 2n2 + 22n. ' ' □ Again, since caterpillars T3(n, d, x, s) are chemical trees for all possible values of parameters, and remain chemical after applying repeatedly Transformation A, half of these results hold for chemical trees, i.e. we obtain the following corollary. Corollary 4.13. Let 7"4 be a class of chemical trees on n vertices where n > 21 is odd. Then |Wini[r„4]| = |W[7"4]| = ©(n3). Proof. Using Lemma 4.5 we obtain |W[7"^ > |Wini[7;4]1 > (W(T3(n, dmax, 1, 0)) - W(T3(n, dmin, xmax, 1)))/2 where d^in = + p, x^ax = 4+4d°'"2_("_1) and d^ax = 2 (n - V2n - 6 - 5) - 1 + q for p G [0,1} and q G (0,1]. Using Lemma 4.2 we further obtain | 1 q 1 /—;-7 11 9 5 /—--- 19 61 ,- 685 |Wint[7"4]| > — n3--v^n5 - 6n4- — n2 + ^2n3 - 6n2+—n-—. 24 8 16 6 12 24 48 □ Again, this result for chemical trees is obtained by considering only chemical trees with large diameter so it probably can be significantly improved, but we leave that as an open problem for future research. 5 Conclusion In this paper we have proved that in the case of even n the cardinality of the largest interval Wint[Tn] of consecutive integers which are values of the Wiener index for a tree graph on n vertices equals 1 n3 + O(n5/2). In the case of odd n the value of the Wiener index for a tree on n vertices can be only even number, therefore the cardinality of the largest interval Wint[Tn ] of consecutive even integers which are values of the Wiener index for a tree graph on n vertices equals 112n3+O(n5/2). Since the set Wint[Tn] is a subset of the set W [Tn] of all values of the Wiener index for trees on n vertices, this immediately yields the same result on the cardinality of W[Tn]. The upper bound |W[T"]| < 1 n3 - n2 +11 n (i.e. | W [Tn] | < 12 n3 - 2n2 +11 n for odd n) is easily established by calculating the difference of the value of the Wiener index for maximal and minimal tree graphs (the path P" and the star Sn respectively). Comparing this bound with our results it is readily seen that our results are best possible with respect to n3. Yet, with respect to n2 the results are not so good, because we obtained |W [Tn ]| = |Wi"t[7"]| = 1 n3 + O(n5/2) (i.e. ^ n3 + O(n5/2) in the case of odd n). This may be due to the fact that in the paper we aimed at the bound for | Wint [Tn] | and we stopped with our search when the interval was interrupted (when the diameter of a tree became too small or too large). There is the possibility that the same approach extended to the caterpillars of all diameters would yield sufficient improvement on |W[Tn] | to reduce O(n5/2) to O(n2). But we leave that for future research. Furthermore, in our research we focused only on caterpillar trees, so the obvious corollary is that the same results hold in the narrower class of caterpillar trees. Half of the caterpillars we used are chemical trees, which yields relatively strong result for the class of chemical trees as a direct corollary. Also, we researched the caterpillars grouped by the length of the interior path (which is nearly the diameter), so the results for trees with given diameter would also follow easily though it is questionable how strong those results would be. Researching the same question in the classes of trees with other given parameters might also be interesting direction of future research. References [1] Y.-E. A. Ban, S. Bereg and N. H. Mustafa, A conjecture on Wiener indices in combinatorial chemistry, Algorithmica 40 (2004), 99-117, doi:10.1007/s00453-004-1097-y. [2] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, ActaAppl. Math. 66 (2001), 211-249, doi:10.1023/a:1010767517079. [3] I. Gutman, Frequency of even and odd numbers in distance matrixes of bipartite graphs, J. Chem. Inf. Comput. Sci. 34 (1994), 912-914, doi:10.1021/ci00020a027. [4] I. Gutman and Y.-N. Yeh, The sum of all distances in bipartite graphs, Math. Slovaca 45 (1995), 327-334, http://maslo.mat.savba.sk/paper.php?id_paper=14 3. [5] 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. [6] M. Krnc and R. Skrekovski, On Wiener inverse interval problem, MATCH Commun. Math. Comput. Chem. 75 (2016), 71-80, http://match.pmf.kg.ac.rs/electronic_ versions/Match75/n1/match75n1_71-80.pdf. [7] M. Lepovic and I. Gutman, A collective property of trees and chemical trees, J. Chem. Inf. Comput. Sci. 38 (1998), 823-826, doi:10.1021/ci980004b. [8] I. Lukovits, Frequency of even and odd numbers in distance matrices of trees, J. Chem. Inf. Comput. Sci. 33 (1993), 626-629, doi:10.1021/ci00014a016. [9] S. G. Wagner, A class of trees and its Wiener index, Acta Appl. Math. 91 (2006), 119-132, doi:10.1007/s10440-006-9026-5. [10] S. G. Wagner and H. Wang, On the parity of the Wiener index, European J. Combin. 30 (2009), 996-1004, doi:10.1016/j.ejc.2008.06.004. [11] S. G. Wagner, H. Wang and G. Yu, Molecular graphs and the inverse Wiener index problem, Discrete Appl. Math 157 (2009), 1544-1554, doi:10.1016/j.dam.2008.06.008. [12] H. Wang and G. Yu, All but 49 numbers are Wiener indices of trees, Acta Appl. Math. 92 (2006), 15-20, doi:10.1007/s10440-006-9037-2. [13] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), 17-20, doi:10.1021/ja01193a005. [14] K. Xu, M. Liu, K. Ch. Das, I. Gutman and B. Furtula, A survey on graphs extremal with respect to distance-based topological indices, MATCH Commun. Math. Com-put. Chem. 71 (2014), 461-508, http://match.pmf.kg.ac.rs/electronic_ versions/Match71/n3/match71n3_4 61-50 8.pdf. /^creative ^commor ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 39-51 https://doi.org/10.26493/1855-3974.1439.fdf (Also available at http://amc-journal.eu) Groups in which every non-nilpotent subgroup is self-normalizing Costantino Delizia University of Salerno, Italy Urban Jezernik, PrimoZ Moravec University ofLjubljana, Slovenia Chiara Nicotera University of Salerno, Italy Received 12 July 2017, accepted 29 August 2017, published online 2 November 2017 Abstract We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We describe soluble groups and finite perfect groups with the above property. Furthermore, we give some structural information in the infinite perfect case. Keywords: Normalizer, non-nilpotent subgroup, self-normalizing subgroup. Math. Subj. Class.: 20E34, 20D15, 20E32 1 Introduction A long standing problem posed by Y. Berkovich [3, Problem 9] is to study the finite p-groups in which every non-abelian subgroup contains its centralizer. In [6], the finite p-groups which have maximal class or exponent p and satisfy Berkovich's condition are characterized. Furthermore, the infinite supersoluble groups with the same condition are completely classified. Although it seems unlikely to be able to get a full classification of finite p-groups in which every non-abelian subgroup contains its cen-tralizer, Berkovich's problem has been the starting point for a series of papers investigating finite and infinite groups in which every subgroup belongs to a certain family or it contains E-mail addresses: cdelizia@unisa.it (Costantino Delizia), urban.jezernik@fmf.uni-lj.si (Urban Jezernik), primoz.moravec@fmf.uni-lj.si (Primoz Moravec), cnicoter@unisa.it (Chiara Nicotera) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ its centralizer. For instance, in [7] and [9] locally finite or infinite supersoluble groups in which every non-cyclic subgroup contains its centralizer are described. A more accessible version of Berkovich's problem has been proposed by P. Zalesskii, who asked to classify the finite groups in which every non-abelian subgroup equals its normalizer. This problem has been solved in [8]. In this paper we deal with the wider class S of groups in which every non-nilpotent subgroup equals to its normalizer. All nilpotent groups (and hence all finite p-groups) are in S. It is also easy to see that groups in S are either soluble or perfect. Further obvious examples of groups in S include the minimal non-nilpotent groups (that is, non-nilpotent groups in which every proper subgroup is nilpotent) and groups in which every subgroup is self-normalizing. Finite minimal non-nilpotent groups are soluble, and their structure is well known (see [18, 9.1.9]). Infinite minimal non-nilpotent groups have been first studied in [14] (see also [4] for more recent results). These groups are either finitely generated or locally finite p-groups (Cernikov groups or Heineken-Mohamed groups). Ol'shanskii and Rips (see [15]) showed that there exist finitely generated infinite simple groups all of whose proper non-trivial subgroups are cyclic of the same order (the so-called Tarski monsters). On the other side, groups whose non-trivial subgroups are self-normalizing are periodic and simple. Furthermore, in the locally finite case they are trivial or of prime order. Again, infinite examples are the Tarski p-groups. We describe soluble groups lying in the class S. It turns out that an infinite polycyclic group lies in the class S if and only if it is nilpotent (Proposition 3.3). We also prove that a non-periodic soluble group belongs to the class S if and only if it is nilpotent (Theorem 3.4). Moreover, a periodic soluble group which is not locally nilpotent lies in the class S if and only if it is a split extension of a nilpotent p'-group by a cyclic p-group whose structure is described in Theorem 3.5. In particular, this result characterizes non-nilpotent soluble finite groups in the class S. Furthermore, a locally nilpotent soluble group belongs to the class S if and only if it is either nilpotent or minimal non-nilpotent (Theorem 3.7). In the last part of the paper we prove that a finite perfect group lies in the class S if and only if it is either isomorphic to the group PSL2 (2n) where 2n — 1 is a prime number, or to the group SL2(5) (Theorem 4.8). Finally, we give some information on the structure of infinite perfect groups lying in the class S. Our notation is mostly standard (see for instance [3] and [18]). In particular, given any group G, we will denote by Z(G) the center of G, by ZTO(G) the hypercenter of G, by $(G) the Frattini subgroup of G, by G' the commutator subgroup of G, and, for all integers 1 > 1, by yj(G) the i-th term of the lower central series of G. 2 General properties of groups in S It is very easy to prove that the class S is subgroup and quotient closed. Furthermore, non-nilpotent groups in S are not products of two proper normal subgroups. Recall that a group G is said to be perfect if it equals its commutator subgroup G'. Clearly, if G G S then G is perfect or G' is nilpotent. Hence the groups in S are either perfect or soluble. Suppose now that a cyclic group (x) acts on a group H by means of an automorphism x. If a subgroup L of H is invariant with respect to (x), we will write L 1: p^(K) = 1 V (px(K)> = K). (*) Lemma 2.1. Let x act on H by means of an automorphism. Then for every K K' = Px(K)K'. Proof. Let h^h2 G K. Then [x, ] = [x, h2][x, h^[x, h^ h2]. It follows that Px(hih2) = Px(hi) Px(h2) (mod K/). □ The following easy observations are used in the sequel. Lemma 2.2. Let x act on H by means of an automorphism. 1. The action of x is fixed point free if and only if px is injective. 2. If px is injective and H is abelian, then (*) implies that px is an isomorphism. 3. If px is injective (or surjective) and H is finite, then px satisfies (*). Proof. (i) Note that px is injective if and only if whenever [x, h] = 1 it follows that h =1. This is precisely the same as x acting fixed point freely on H. (ii) Of course we can assume that H is non-trivial. If px is injective then there is no positive integer n with the property that p£(K) = 1, and so (*) implies that (px(H)> = H. If in addition H is abelian then px is a homomorphism, and hence (px(H)> = px(H). Therefore px is an isomorphism. (iii) If H is assumed to be finite, then px is injective if and only if it is surjective. In this case px is bijective, and we have that px(K) = K for all K H, where H is a nilpotent normal subgroup of G generated by a set Y. Suppose that there exists n > 1 such that p£(y) = 1 for every y G Y. Then G is nilpotent. Proof. By a theorem of Hall (see for instance [16, Theorem 2.27]) it suffices to show that G/H/ is nilpotent. The group H/H/ is generated by all yH/ with y G Y and px induces an endomorphism t of H/H/ such that r"(H/H/) = 1. Now G/H/ is nilpotent of class at most n since H/H/ C Z„(G/H/). □ Lemma 2.4. Let G = (x> x H be a non-nilpotent group where x has prime order p and H is nilpotent. Assume that px has property (*) and suppose that there exists a subgroup 1 = K x K is nilpotent by Lemma 2.3, it has a non-trivial center. Thus there exists an element 1 = h G CK(x). Now consider the group (x> x Z(H). By property (*), we either have px(Z(H)) = Z(H) or there is a positive integer n such that p£(Z(H)) = 1. In the latter case, we certainly have an element that belongs to Z(H) and commutes with x, so that Z(G) = 1. Suppose now that px(Z(H)) = Z(H) holds. By property (*) we have (Z(H), h> = px((Z(H), h>) = px(Z(H)) = Z(H), and hence h G Z(H). Thus we again have Z(G) = 1. □ The following proposition shows how property (*) is tightly related to the class S. Proposition 2.5. Let G = (x) x H be a group in S with xp acting trivially on a nilpotent subgroup H for some prime p. Then px has property (*). Proof. Let K 2. The dihedral group Dih(n) of order 2n belongs to S if and only if either n is a power of 2 or n is odd. Theorem 3.7. A locally nilpotent soluble group lies in the class S if and only if it is either nilpotent or minimal non-nilpotent. Proof. Clearly, nilpotent and minimal non-nilpotent groups belong to the class S. Let G G S be a periodic soluble group which is locally nilpotent, and assume that G is non-nilpotent. We will prove that G is minimal non-nilpotent. For the sake of contradiction, assume that there exists a proper non-nilpotent subgroup H of G. Let B be the last term of the derived series of G which is not contained in H. Then HB has the proper non-nilpotent subgroup H. Hence without loss of generality we may assume that G = HB. Put L = B n H. Then B' < L, so L is normal in B. Obviously L is normal in H, thus L is normal in G. The normal series L < B < G can be refined to a (general) principal series of G (see for instance [18, 12.4.1]). Let W/V be any factor of this principal series with W < B. As G is locally nilpotent, the principal factor W/V is central (see for instance [18, 12.1.6]). Hence [W, G] < V. This implies that W < NG(HV) = HV. Therefore w = w n hv = (W n h)v < lv = V. This means L = B, a contradiction, and that proves our result. □ Corollary 3.8. A locally nilpotent soluble group lying in the class S is nilpotent or a p-group for some prime p. Proof. Let G g S be a locally nilpotent soluble periodic group, and assume that G is non-nilpotent. Then by Theorem 3.7 the group G is minimal non-nilpotent, and the result follows by [14, Lemma 4.2]. □ 4 Perfect groups in the class S Lemma 4.1. Let G G S be a finite perfect group, and let F denote its Fitting subgroup. Then G/F is a non-abelian simple group. Proof. If there is a proper normal subgroup F < M < G, then M must be nilpotent since G g S, and so M = F. Thus G/F is simple. As G is also assumed to be perfect, G/F is non-abelian. □ We first classify the finite simple groups in S. This is done with the help of the following lemma. Lemma 4.2. Let G be a finite simple group. Then G belongs to S if and only if all of its maximal subgroups belong to S. Proof. Assume that all maximal subgroups of a finite simple group G belong to S, and let H be a non-nilpotent proper subgroup of G. As G is simple, we have NG (H) < G, and so there is a maximal subgroup M < G with NG (H) < M. Since M belongs to S, it follows that NG(H) = Nm(H) = H, as required. □ Lemma 4.3. The group PSL2(q) belongs to S if and only if q = 2n with q — 1 a prime, or q < 5. Proof. Suppose that PSL2(q) belongs to S with q > 5, and assume first that q is odd. This group contains dihedral subgroups of orders (q — 1)/2 and (q + 1)/2 by [10]. Unless q = 7, at least one of these does not belong to S by Corollary 3.6. Note that PSL2(7) has a subgroup isomorphic to Sym(4), so it does not belong to S. Whence we can assume that q = 2n for some n > 3. Now PSL2 (q) contains a diagonal torus of order q — 1 acting fixed point freely on the unipotent subgroup of order q. It follows from Lemma 3.1 that the torus must be simple, and so q — 1 is either trivial or a prime, as required. Finally, it follows from [8, Theorem 2.17] that such groups indeed belong to S. □ Proposition 4.4. A finite non-abelian simple group belongs to S if and only if it is isomorphic to PSL2(2n), where 2n — 1 is a prime. Proof. We reduce the situation to the case of Lemma 4.3 by using Lemma 4.2. • Alternating groups. It may be verified readily that Alt(n) belongs to S if and only n = 5, since Sym(4) is contained in Alt(n) for every n > 6. • Linear groups PSLn(q). If n = 2, this case is covered by Lemma 4.3. If n > 3, then there is a block embedding of SL2(q) into PSLn(q). The image of this subgroup is normalized by the class of a diagonal matrix of the form diag(a, ft, y,1,..., 1). As long as a = ft, this diagonal matrix does not belong to the image of the embedding of SL2 (q), and so PSLn (q) does not belong to S. The only exceptional case is when |F* | = 1, i.e., q = 2, in which case either n = 3 or PSLn(2) contains SL3(2) via a block diagonal embedding. Both of these groups quotient onto PSL3(2) = PSL2 (7), which does not belong to S. • Symplectic groups PSp2n(q). If n =1, then PSp2(q) = PSL2(q) and this is covered above. Now let n > 1. Letting W be a maximal isotropic subspace of the 2n-dimensional vector space on which Sp2n(q) acts, the stabilizer of the decomposition W © W^ is GLn(q) x C2, and so PSp2n(q) contains PGLn(q) x C2. Therefore these groups do not belong to S. • Unitary groups and orthogonal groups. Their associated root systems contain a subsystem of type A2, and so they contain subgroups that are isomorphic to either SL3(q) or PSL3(q). None of these belong to S by above. See [2]. • Exceptional Chevalley groups. We have an inclusion G2(q) C F4(q) C E6(q) C E7(q) C Eg(q), and the list of maximal subgroups of G2(q) in [20, p. 127] shows that G2(q), and hence all of the above groups, does not belong to S. • Steinberg groups 2£6(q2) and 3D4(q3). By [20, Theorem 4.3], the group 3D4(q3) has a maximal subgroup which is isomorphic to G2(q3), hence it is not in S by the above. Similarly, F4(q2) embeds into 2£6(q2) by [20, p. 173], hence the latter is not in S. • Suzuki groups Sz(q). By [20, Theorem 4.1], these contain Frobenius groups Cg+y^+i x C4 whose Fitting subgroups are of index 4. Such groups do not belong to S by Lemma 3.1. • Ree families. By [20, Theorem 4.2], 2 x PSL3(2n + 1) is a maximal subgroup of 2G2(32n+1), and Sz(22n+1) I 2 is a maximal subgroup of 2F4(22n+1) by [20, Theorem 4.5]. For the remaining case, 2F4(2)', we use ATLAS [5] to conclude that this group contains Sym(6). • Sporadic groups. Inspection of ATLAS reveals that each of 26 sporadic groups has a maximal subgroup which is clearly not in S. □ To deal with perfect finite groups in S, we make use of the theory of Schur covering groups. In particular, we will require the following. Theorem 4.5 (Hauptsatz 23.5 of [11]). Let G be a finite group and suppose there is an extension 1 ^ K ^ E ^ G ^ 1 with the property that K < Z(E) n E'. Then K embeds into the Schur multiplier M(G). Proposition 4.6. Let G G S be a perfect non-simple finite group, and let F denote its Fitting subgroup. Assume that the group G/F contains two elements a and b of distinct prime orders with the additional property that NG/F ((a)) D (a) and NG/F ((b)) D (b). Then the group Sp/$(Sp) embeds into the Schur multiplier M(G/F), for every p-Sylow subgroup Sp of F. Remark 4.7. It is easy to find such elements a, b for the simple groups PSL2(2n) that appear in Proposition 4.4. One can take a to be an involution (normalized by the Sylow 2-subgroup of order 2n) and b a diagonal matrix of order q — 1 (normalized by the class of the flip ("J)). Proof. The group F is nilpotent, so we can write F = f] Sq where Sq is a q-group. Now fix a prime p and consider Gi = G/ f]q=p Sq. The Fitting subgroup of Gi is isomorphic to Sp. Further, consider the group G2 = Gi /$(Sp). The Fitting subgroup F2 of G2 is an elementary abelian p-group, and G2 belongs to the class S. Write S = G/F. Then S is simple by Lemma 4.1. The group G2 acts on its subgroup F2 by conjugation. There is thus an induced homomorphism G2 ^ Aut(F2). This homomorphism factors through F2, so we get a homomorphism ^: S = G2/F2 ^ Aut(F2). As S is a simple group, we have that either ^ is injective or trivial. Let us show that ^ must be trivial. For the sake of contradiction, assume that ker ^ = 1. Since F2 is a p-group, at least one of the elements a, b from the statement of the proposition has order coprime to p. Without loss of generality, assume this element is a. Now consider the group H = (a, F2) < G2. By our assumption on the element a, the group H is not self-normalized in G2. But it is also not nilpotent. Indeed, the element a acts nontrivially on F2 because ^ is an embedding of S into Aut(F2). The order of a is coprime to p, so ^ restricted to (a) is a completely reducible representation of (a) on the GF (p)-vector space F2. This representation splits as a sum of 1-dimensional representations, and so a is a diagonalizable element in the image of Being non-trivial, we can not have that ^(a) - I is a nilpotent matrix, and so the group H can not be nilpotent. This leads to a contradiciton with the fact that G2 G S. We therefore have that ^ is trivial, and so S acts trivially on F2. This means that F2 is central in G2. Since G2 is also assumed to be perfect, the extension 1 ^ F2 ^ G2 ^ S ^ 1 has the property that F2 < Z(G2) n G2. It follows from Theorem 4.5 that F2 = Sp/$(Sp) embeds into M(S). □ Our next result, together with Proposition 4.4, gives a complete classification of all finite perfect groups in S. Theorem 4.8. A finite perfect group G belongs to the class S if and only if it is either isomorphic to PSL2(2n) where 2n — 1 is a prime, or to SL2(5). Proof. The Fitting quotient of G is a finite simple group belonging to S, so it must be one of the PSL's appearing in Proposition 4.4. Note that we have M(PSL2(4)) = C2 and all the other PSL's have trivial Schur multipliers. Therefore the only possibility for a non-simple perfect group G in S is a group whose Fitting quotient is PSL2(4). Such a group must have F a 2-group with cyclic Frattini quotient, so F itself is cyclic. But now as G/F acts trivially on the Frattini quotient of F, it follows that the image of the homomorphism G/F ^ Aut(F) is a p-group [12, Exercise 4.4]. Since G/F is a non-abelian simple group, this implies that G/F must act trivially even on F. Hence F is central in G. This implies that G is a Schur covering extensions of G/F by F, so it follows that |F| = |M(PSL2(4))| = 2 and G = SL2(5). □ Now we deal with infinite perfect groups in the class S. Lemma 4.9. Let G be a perfect group lying in the class S. Then G is simple if and only if its Fitting subgroup is trivial. Proof. Let F denote the Fitting subgroup of G. First suppose G is simple. If G = F then G is nilpotent, a contradiction since G is perfect. Therefore F =1. Now suppose F = 1, and let N be any proper normal subgroup of G. Since G e S, the subgroup N is nilpotent, so N < F. Therefore N = 1, and G is simple. □ Lemma 4.10. An infinite perfect group lying in the class S cannot be a Fitting group. Proof. Let G e S be an infinite perfect group, and suppose that G is a Fitting group. The group G cannot be minimal non-nilpotent by (see [4, Proposition 144] and [1, Corollary 1.4]), so there exists a proper non-nilpotent subgroup H of G. Choose x e G \ H. Since G is generated by its nilpotent normal subgroups, there exists a normal subgroup N of G such that N is nilpotent and x e N. Hence N % H. Let B be the last term of the derived series of N which is not contained in H. Put K = HB. Then K e S is locally nilpotent and non-nilpotent. Put L = B n H. Thus L is normal in K, and the normal series L < B < K can be refined to a (general) principal series of K. As in the proof of Theorem 3.7, all factors of this principal series which lie between L and B are trivial. This means L = B, a contradiction. □ Note that the above shows that the finiteness hypothesis in Lemma 4.1 may be omitted. Proposition 4.11. Let G e S be a perfect group, and let F denote its Fitting subgroup. Then G/F is a non-abelian simple group. Proof. By Lemma 4.1 we may assume that G is infinite. Moreover, by Lemmas 4.9 and 4.10 we may assume that F is a non-trivial proper subgroup of G. Clearly F is infinite and contains all proper normal subgroups of G. □ We leave it as an open problem whether or not there exist infinite perfect groups in S which are not simple. Note that, if such a group G is locally graded and finitely generated, then G/F is still locally graded (see for instance [13]), and hence it has to be finite. Therefore, by Proposition 4.4, G/F is isomorphic to PSL2 (2n), where 2n — 1 is a prime. References [1] A. O. Asar, Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov, J. London Math. Soc. 61 (2000), 412-422, doi:10.1112/s0024610799008479. [2] M. J. J. Barry and M. B. Ward, Simple groups contain minimal simple groups, Publ. Mat. 41 (1997), 411-415, doi:10.5565/publmat_41297_07. [3] Y. Berkovich, Groups of Prime Power Order, Volume 1, volume 46 of De Gruyter Expositions in Mathematics, Walter de Gruyter GmbH & Co., Kammergericht, Berlin, 2008, https: //www.degruyter.com/view/product/178 995. [4] C. Casolo, Groups with all subgroups subnormal, Note Mat. 28 (2008), 1-153, doi:10.1285/ i15900932v28n2supplp1. [5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985, http://brauer.maths.qmul.ac. uk/Atlas/v3/. [6] C. Delizia, H. Dietrich, P. Moravec and C. Nicotera, Groups in which every non-abelian subgroup is self-centralizing, J. 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Khukhro, p-Automorphisms of Finite p-Groups, volume 246 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1998, doi:10.1017/ cbo9780511526008. [13] P. Longobardi, M. Maj and H. Smith, A note on locally graded groups, Rend. Sem. Mat. Univ. Padova 94 (1995), 275-277, http://www.numdam.org/item?id=RSMUP_ 1995_94_275_0. [14] M. F. Newman and J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15 (1964), 241-250, doi:10.1007/bf01589192. [15] A. Y. Ol'shanskil, Geometriya opredelyayushchikh sootnoshenii v gruppakh, Sovremennaya Algebra, Nauka, Moscow, 1989. [16] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, volume 62 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1972. [17] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 2, volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1972. [18] D. J. S. Robinson, A Course in the Theory of Groups, volume 80 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2nd edition, 1996, doi:10.1007/978-1-4419-8594-1. [19] J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971), 373-391, doi:10.1017/s0305004100046818. [20] R. A. Wilson, The Finite Simple Groups, volume 251 of Graduate Texts in Mathematics, Springer-Verlag, London, 2009, doi:10.1007/978-1-84800-988-2. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 53-65 https://doi.org/10.26493/1855-3974.1161.3b9 (Also available at http://amc-journal.eu) On prime-valent symmetric graphs of square-free order* Jiangmin Pan t School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, P. R. China Bo Ling School ofMathematics and Computer Science, Yunnan Minzu University, Kunming, Yunnan, P. R. China Suyun Ding School ofMathematics and Statistics, Yunnan University, Kunming, Yunnan, P. R. China Received 16 July 2016, accepted 27 May 2017, published online 10 November 2017 Symmetric graphs of valencies 3, 4 and 5 and square-free order have been classified in the literature. In this paper, we will present a complete classification of symmetric graphs of square-free order and any prime valency which admit a soluble arc-transitive group, and a complete classification of 7-valent symmetric graphs of square-free order. Keywords: Symmetric graph, normal quotient graph, automorphism group. Math. Subj. Class.: 20B15, 20B30, 05C25 1 Introduction Throughout the paper, graphs considered are assumed to be undirected and simple with valency at least three. For a graph r, denote by Vr and Ar the vertex set and arc set of r respectively, denote by Autr the full automorphism group of r, and denote by r (a) the set of neighbors of a *The authors are very grateful to the referees for their valuable comments. This work was partially supported by the National Natural Science Foundation of China (11461007, 11231008). t Corresponding author. E-mail addresses: jmpan@ynu.edu.cn (Jiangmin Pan), bolinggxu@163.com (Bo Ling), 1328897542@qq.com (Suyun Ding) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ vertex a in r. Then r is called X-vertex-transitive or X-arc-transitive, with X < Autr, if X is transitive on Vr or Ar respectively. An arc-transitive graph is also called a symmetric graph. In particular, r is called arc-regular if Autr is regular on Ar. For a positive integer s, an s-arc of a graph r is a sequence v0, v\,..., vs of s + 1 vertices such that vj_i, vj are adjacent for 1 < i < s and vj_i = vi+1 for 1 < i < s - 1. If r has an s-arc and X < Autr is transitive on the set of s-arcs of r, then r is called (X, s)-arc-transitive. If r is (Autr, s)-arc-transitivebutnot (Autr, s + 1)-arc-transitive, then r is simply called s-transitive. Characterizing symmetric graphs was initiated by a nice result of Tutte (1949) which says that there exists no s-arc-transitive cubic graph with s > 6. This result was generalized by Weiss [27] who proved that there is no s-arc-transitive graph with s > 8 of valency at least 3. Since then, studying transitive graphs has been one of the main topics in algebraic graph theory, and numerous results have been obtained. In particular, transitive graphs of square-free order (not divisible by the square of a prime) have received considerable attention; for example, symmetric graphs of valencies 3, 4 and 5 and square-free order have been classified by [16, 17] and [6] respectively, and arc-regular graphs of square-free order and prime valency have been determined by [9]. The main purpose of this paper is to give a complete classification of symmetric graphs of square-free order and prime valency admitting a soluble arc-transitive group, and a complete classification of 7-valent symmetric graphs of square-free order. The terminology and notation used in this paper are standard. For example, we denote by J1 the Janko simple group, by HS the Higman-Sims simple group, and by Mn, with n = 11,12,22, 23, 24, the five Mathieu simple groups. For a positive integer m, denote by Am and Sm the alternating group and symmetric group of degree m, and by Zm, Fm and Dm (with m even) the cyclic group, Frobenius group and dihedral group of order m respectively. Given two groups N and H, denote by N x H the direct product of N and H, by N.H an extension of N by H, and if such an extension is split, then we write N :H instead of N.H. A graph r is called a Cayley graph if there exists a group G and a subset S C G \ {1} with S = S-1: = {s-1 | s € S} such that the vertex set V r = G andavertex x is adjacent to a vertex y if and only if yx-1 € S. This Cayley graph is denoted by Cay(G, S). The following Cayley graphs of dihedral groups give rise to an infinite family of prime-valent symmetric graphs, where the first two letters 'CD' of the name of the graph CD2m,p,k stand for 'Cayley graph of a dihedral group'. Example 1.1. Let G = (a, b | am = b2 = 1, ab = a-1) = D2m with m a positive integer, and let p be an odd prime and k a solution of the equation xp-1 + xp-2 +-----+ x +1 = 0 (mod m). Set CD2m,p,fc = Cay(G, {b, ab, afc+1b,..., afcP-2+fcP-3+-+fc+1b}). The following theorem determines the prime-valent symmetric graphs of square-free order which admit a soluble arc-transitive automorphism group. We remark that cubic graphs which admit a soluble edge-transitive or arc-transitive automorphism group have been characterized by [20] and [8], respectively. Theorem 1.2. Let r be a connected p-valent symmetric graph of square-free order n with p an odd prime, and suppose that r admits a soluble arc-transitive automorphism group G. Then either (1) r = Kpp, and G = (((Zp : Ztl) x (Zp : Z¡2)).Zr).Z2 < Sp I Z2, where lir | p — 1 for i = 1, 2; or (2) r = CDn p k, n = 2 • psp1p2 • • • pt and G = Autr = Dn:Zp, where 0 < s < 1, t > 1, and p1,p2,... ,pt are distinct primes such that p | pi — 1 for i = 1, 2,... ,t. Further, there are exactly (p — 1)t-1 non-isomorphic such graphs of order n. The next theorem present a complete classification of 7-valent symmetric graphs of square-free order, where the graph C330 in Table 1 is introduced in Example 3.2 for convenience. Theorem 1.3. Let r be a connected 7-valent symmetric graph of square-free order n. Then one of the following statements holds. (1) r = CD„i7jk, and the tuple (n, Autr) is as in part (2) of Theorem 1.2 with p = 7. (2) The triple (r, n, Autr) lies in Table 1. (3) Autr = PSL(2,p) or PGL(2,p), where p > 13 is a prime such that p(p2 — 1) | 225 • 34 • 52 • 7n. Table 1: Two 'sporadic' 7-valent symmetric graphs Row r n Autr (Autr )a Transitivity Remark 1 K7 7 14 S71Z2 S7.S6 3-transitive bipartite 2 C330 330 M22.Z2 Z4 : SL(3,2) 2-transitive not bipartite Remark 1.4. Graphs appearing in part (3) of Theorem 1.3 can be expressed as coset graphs of PSL(2,p) or PGL(2,p) (refer to [10] for the definition of the coset graph). However, it seems infeasible to determine all the possible values of p (and so the corresponding symmetric graphs r) for general square-free integer n. 2 Preliminaries In this section, we introduce some preliminary results that will be used later. For a group G with a subgroup H, let CG(H) and NG(H) denote the centralizer and normalizer of H in G, respectively. Lemma 2.1 ([14, Ch. I, Lemma 4.5]). Let G be a group and H a subgroup of G. Then Ng(H)/Cg(H) < Aut(H). For a group G, the largest nilpotent normal subgroup of G is called the Fitting subgroup of G. Clearly, the Fitting subgroup is a characteristic subgroup. The next lemma gives a property of the Fitting subgroup of soluble groups. Lemma 2.2 ([26, P. 30, Corollary]). Let F be the Fitting subgroup of a soluble group G. Then F = 1 and CG(F) < F. The maximal subgroups of the simple group PSL(2, q) are known, see [5, Section 239]. Lemma 2.3. Let T = PSL(2, q), where q = pn > 5 with p a prime. Then a maximal subgroup of T is isomorphic to one of the following groups, where d = (2,p — 1). (1) D 2(q—i), where q = 5,7,9,11; d (2) D 2(q+i), where q = 7, 9; d (3) Zjn : Z q—i; p d (4) A4, where q = 5, or q = p = 3,13, 27, 37 (mod 40); (5) S4, where q = p = ±1 (mod 8); (6) A5, where q = p = ±1 (mod 5), or q = p2 = —1 (mod 5) with p an odd prime; (7) PSL(2, r), where q = rm with m an odd prime; (8) PGL(2,r), where q = r2. By [2, Theorem 2], one may easily derive the maximal subgroups of PGL(2,p). Lemma 2.4. Let T = PGL(2,p) with p > 5 a prime. Then a maximal subgroup of T is isomorphic to one of the following groups: (1) Zp : Zp_i; (2) D2(p+1); (3) D2(p_i), wherep > 7; (4) S4, where p = ±3 (mod 8); (5) PSL(2,p). A group G is called perfect if G = G', the commutator subgroup; and an extension G = N.H is called a central extension if N C Z(G), the center of G. If a group G is perfect and G/Z(G) is isomorphic to a simple group T, then G is called a covering group of T. Schur [25] showed that a simple (and, more generally, perfect) group T possesses a universal covering group G with the property that every covering group of T is a homomor-phic image of G, in this case, the center Z(G) is called the Schur multiplier of T, denoted by Mult(T), see [12, P. 43]. The Schur multipliers of nonabelian simple groups are known (see [12, P. 302]), and the following lemma is easy to prove (see [23, Lemma 2.11]). Lemma 2.5. Let G = N.T, where N is a cyclic group and T is a nonabelian simple group. Then G = N.T is a central extension. Further, G = NG' and G' = M.T, where M is contained in G' n N and is isomorphic to a subgroup of Mult(T). The following lemma characterizes the vertex stabilizers of 7-valent symmetric graphs, see [13, Theorem 1.1]. Lemma 2.6. Let r be a connected 7-valent (X, s)-arc-transitive graph, where X < Autr and s > 1. Then one of the following holds, where a e Vr. (1) If Xa is soluble, then s < 3 and \Xa\ | 252. Further, the couple (s, Xa) is listed in Table 2. Table 2: Soluble vertex-stabilizers of 7-valent-symmetric graphs. s 1 2 3 Xa Z7, D14, F21, D14 X Z2, AGL(1,7), AGL(1, 7) X Z2, AGL(1, 7) x Z6 F21 X Z3 AGL(1, 7) x Z3 Table 3: Insoluble vertex-stabilizers of 7-valent symmetric graphs. s 2 3 Xa PSL(3, 2), ASL(3, 2), PSL(3, 2)XS4, A7XA6, ASL(3, 2)XZ2, A7, S7 S7XS6, (A7XA6):Z2, Z2:(SL(2, 2)XSL(3, 2)), Z20:(SL(2, 2)xSL(3, 2)) |Xa| 26^32^7, 26-34-52-7, 28^34^52^7, 27^34^52^7, 210^32^7, 224^32^7 (2) If Xa is insoluble, then \Xa\ | 224 • 34 • 52 • 7. Further, the couple (s,Xa) lies in Table 3. Analyzing a graph in terms of its normal quotients is a typical method for studying vertex-transitive graphs. Let r be an X-vertex-transitive graph with X < Autr, and suppose that X has a normal subgroup N which is intransitive on V r. Denote by VrN the set of all N-orbits on Vr. Then the normal quotient graph rN of r induced by N is defined as the graph with vertex set VrN, and B is adjacent to C in rN if and only if there exist vertices fi G B and 7 G C such that fi is adjacent to 7 in r .In particular, if for any adjacent vertices B and C in VrN, the induced subgraph [B, C] = mK2 is a perfect matching, where m = \B\ = \C\, then r is called a regular cover (or normal cover) of rN. The following theorem gives a basic reduction method for studying vertex-transitive locally primitive graphs (see [18, Lemma 2.5]), which slightly improves a nice result of Praeger [24, Theorem 4.1]. Recall that, a graph r is called X-locally primitive if the vertex stabilizer Xa acts primitively on the neighbour set r(a) for each a G Vr. Obviously, symmetric graphs with odd prime valency are locally primitive. Theorem 2.7. Let r be an X-vertex-transitive locally primitive graph, where X < Autr, and let N < X have at least three orbits on Vr. Then the following statements hold. (i) N is semi-regular on Vr, X/N < AutrN, and r is a regular N-cover of rN; (ii) Xa = (X/N)Y, where a G Vr and 7 G VrN; (iii) r is (X, s)-arc-transitive if and only if rN is (X/N, s)-arc-transitive, where 1 < s < 5 or s = 7. Symmetric graphs of prime-valency and order twice a prime are known, see [3]. Lemma 2.8. Let r be a connected symmetric graph of odd prime valency p and order 2r with r a prime. Then one of the following statements holds. (1) r = O2 and Autr = S5; (2) r = K2r with p = 2r — 1, and Autr = S2r; (3) r = Kr,r with p = r, and Autr = Sr I S2; (4) r = CD2r,p,k (which, up to isomorphism, is independent of the choice of k in this case), where p | r — 1, and one of the following statements holds. (i) (r,p) = (7, 3) and Autr = PGL(2, 7); (ii) (r,p) = (ll, 5) and Autr = PGL(2,11); (iii) (r,p) = (7,3) and (11,5), and Autr = D2r : Zp. Lemma 2.9 ([19, Theorem 1.1]). Let r be a connected 7-valent symmetric graph of order 2pq with p > q odd primes. Then one of the following holds: (1) Autr = PSL(2,p) withp > 13; (2) q = 7 or 7 | p — 1, 7 | q — 1, and r = CD2pq,7,k (as in Example 1.1). 3 A lemma and an example In this section, we give a technical lemma and introduce an example appearing in Theorem 1.3. The following is an assertion regarding simple groups, its proof depends on the classification of simple groups, see [12, P. 134-136]. Lemma 3.1. Let m be an odd square-free integer with at least three prime factors, and let T be a nonabelian simple group such that 28m | |T| and |T| | 225 • 34 • 52 • 7m. Then the couple (T, |T |) is listed in Table 4, where p in part 4 is the largest prime factor of m and p > 13. Proof. If T is a sporadic simple group, by [12, P. 135-136], T = M22, M23, M24, Ji, HS or Ru, as in part 1 of Table 4. If T = An is an alternating group, since 36 does not divide |T| and 36 | |Ai51, we have n < 14, it then easily follows that T = An, A12, A13 or A14, as in part 2 of Table 4. Now, suppose that T is a simple group of Lie type defined on the re-elements field GF(re), where r is a prime. If T is of exceptional Lie type, by [12, P. 135], T = Sz(512) or 3D4(2), as listed in part 3 of Table 4. Consider the case where T is of classical Lie type. Since re | |T|, we have that e = 1 if r > 7, and e < 2 if r = 7, by [12, P. 135], which give rise to examples T = PSL(2,p) with p > 13 a prime (noting that PSL(2,p) with p = 5, 7,11 does not satisfy the hypothesis of Lemma 3.1) and T = PSL(2,49). If r = 5, as 54 / |T|, we conclude from [12, P. 135] that T = PSL(2,125). For the case where r < 3, since 36, 54 and 73 do not divide |T|, by [12, P. 135] and with the help of Magma [1], we conclude that T is isomorphic to one of the groups listed in part 4 of Table 4. □ Given a permutation group G, a direct computation by Magma program [1] can determine all orbital graphs of G (see [7, P. 66] for the definition of orbital graph), or in other words, can determine all symmetric graphs which admit G as an arc-transitive automorphism group. It is then easy to have the following example. Example 3.2. There is a unique connected 7-valent symmetric graph of order 330, denoted by C330, which admits M22 or M22 .Z2 as an arc-transitive automorphism group. The graph C330 satisfies the conditions in Row 2 of Table 1. Table 4: Nonabelian simple groups T with 28m | |T| and |T| | 225 • 34 • 52 • 7m. Part T |T | 1 M22 M23 2^32^74L23 M24 210^33^74L23 J1 23^5^1L19 HS 29^32^53^741 Ru 214^33^53^743^29 2 A11 27^52^11 A12 29^35^52^741 A13 29-35-52-7-11-13 A14 210^35^52^724143 3 Sz(512) 218^743^73409 3D4 (2) 212^34^7243 4 PSL (2,p) p(p2 - 1)/2 PSL 2, 49) 24^52^72 PSL 2, 125) 22^32^53^31 PSL( 2, 26) 26-32-5-7-13 PSL( 2, 29) 29^33^749^73 PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSp PSp PSp ( PQ ( 7,4) PQ( 9, 2) PQ+ (10,2) PQ-( 8,2) PQ-( 8,4) 212) 215) 218) 221) 224) 8) 16) 64) 4) 212 32 5 7 13 17 241 215 218 221 224 ->9 711 31151 331 5^74349^37^73409 7243427^337^5419 5-7-13-17-97-241-257-673 29-32-72-73 212-32-52-7-13-17 218 212 210 220 215 221 218 216 212 218 216 220 212 5^724349^73 52-7-17 5^31 527111731 5^72^31 5^72^31427 53^74347 52^747 5^7243 53^74347 52^747 52-7-17-31 5 7 17 224-34-53-7-13-17-257 4 Proof of Theorem 1.2 In this section, we prove Theorem 1.2 which in particular gives a partial proof of Theorem 1.3. Let r be a connected symmetric graph of odd prime valency p and square-free order n, and let G be a soluble arc-transitive automorphism group of r. Since r is of odd valency, n is even. Set n = 2pip2 • • • pt, with pi, p2,..., pt distinct odd primes. If t = 1, by Lemma 2.8, r = CD2piiPjfc (as in part (2) of Theorem 1.2) or r = Kp,p. If r = Kp,p, then G < Autr = Sp IZ2, as G is arc-transitive on r, Z^ < G and G^S^, it is then routine to show that G = (((Zp : Zll) x (Zp : Z;2 )).Zr).Z2, where /¿r | p - 1 for i = 1, 2, as in part (1) of Theorem 1.2. Suppose t > 2 in the following. Let F be the Fitting subgroup of G. By Lemma 2.2, F = 1. As |Vr| = 2p1p2 • • • pt, G has no nontrivial normal Sylow a-subgroup, where a G {2,p1,p2,... ,pt} is aprime, hence F = 02(G) x Opi(G) x^x Opt(G), where 02(G) and 0pi(G) with i = 1,2,... ,t denote the largest normal 2- and p4-subgroups of G, respectively. For each prime q G {2,p1,p2,... ,pt}, since t > 2, 0q(G) has at least six orbits on Vr, by Theorem 2.7, 0q(G) is semi-regular on Vr, so is F and 0q(G) < Zq. Hence F < Z„ is cyclic and CG(F) = F by Lemma 2.2. If F is transitive on V r, then F = Zn is regular on V r and r is a Cayley graph of F. Set r = Cay(F, S), where S = S-1 C F \ {1} with size |S| = p. Since F < G, by [11, Lemma 2.9], G < F:Aut(F, S), so Gi < Aut(F, S) < Aut(F) is transitive on r(1) = S, where 1 denotes the vertex of r corresponding to the identity element of F, thus elements in S have the same order, say h. Clearly, h = 2 as F has a unique involution. If h > 2, as 5 = S-1, |S | is even, which is a contradiction. If F has at least three orbits on Vr, then Theorem 2.7 implies that the normal quotient graph rF is G/F-arc-transitive; however, by Lemma 2.2, G/F = G/Cg(F) < Aut(F) is abelian, it forces that G/F is regular on VrF, and so G/F is not transitive on Ar, also yielding a contradiction. Thus, F has exactly two orbits on Vr, and F = Zn. Because t > 2, F has a nontrivial normal subgroup K = Zp2p3...pt. Since K < G has 2p1 orbits on Vr, by Theorem 2.7, rK is a G/K-arc-transitive graph of valency p and order 2p1, and r is a regular K-cover of rK. Such covers have been classified by [21, Theorem 1.1], hence the triple (r, K, rK) (as (r, Zn, S) there) satisfies parts (1)-(5) of [21, Theorem 1.1]. Since |K| = 2, parts (1)-(3) are impossible. For part (4), since n is square-free, p1 / |K|, by [22, Theorem 1.1], r = CDnjPjfc. For part (5), noting that p1 / |K|, part (5)(ii) is not possible, we also have r = CDn p fc. Finally, the last statement in part (2) of Theorem 1.2 is true by [9, Theorem 3.1]. This completes the proof of Theorem 1.2. 5 Proof of Theorem 1.3 We will prove Theorem 1.3 in this final section. Let r be a connected 7-valent symmetric graph of square-free order n. Since r is of odd valency, n is even, so we may write n := 2m = 2p1p2 . . .pt, where... ,pt are distinct odd primes. Let A = Autr. Lemma 5.1. If t < 2, then Theorem 1.3 is true. Proof. If t =1, by Lemma 2.8, r = CD2pi,7,fc (as in part (1) of Theorem 1.3), or r = K7,7 (as in Row 1 of Table 1). If t = 2, by Lemma 2.9, r = CD2piP2,7,fc (as in part (1) of Theorem 1.3), or A = PSL(2,p) or PGL(2,p) with p > 13 aprime, satisfying part (3) of Theorem 1.3. □ Thus, assume t > 3 in the following, and assume inductively that Theorem 1.3 is true for the graph which satisfies assumption of Theorem 1.3 and is of order less than n. Let a € Vr. By Lemma 2.6, |Aa| | 224 • 34 • 52 • 7, hence |A| = |Aa||Vr| divides 225 • 34 • 52 • 7m. Let R be the soluble radical of A, that is, the largest soluble normal subgroup of A. Obviously, the soluble radical of A/R equals 1. The next lemma treats the case R =1. Lemma 5.2. Suppose R =1 and t > 3. Then either Autr = PSL(2,p) or PGL(2,p) with p > 13 a prime such that p(p2 — 1) | 225 • 34 • 52 • 7n, as in part (2) of Theorem 1.3; or r = C330 and Autr = M22.2, as in Row 2 of Table 1. Proof. Let N be a minimal normal subgroup of A, and let C = CA(N). Since R = 1, N = Td and |N | = |T |d divides 225 • 34 • 52 • 7m, where T is a nonabelian simple group and d > 1. Claim 1. C =1. Assume, on the contrary, C =1. Then C is insoluble as R = 1. If C is semi-regular on Vr, then |C| | n, so C is of square-free order and hence soluble, which is a contradiction. Thus Ca = 1. Since r is connected and C < A, we have 1 = C^(a) < A^(a), so 7 | |Ca|. Arguing similarly, one may have 7 | |Na|. Now, since N n C =1, (N, C} = N x C < A, so Na x Ca < Aa, hence 72 | |Aa|, which is a contradiction by Lemma 2.6. Therefore, C = 1. Claim 2. A is almost simple and the tuple (T, |T |) is listed in Table 4. As discussed above, 7 | |Na|. Then by Theorem 2.7, N has at most two orbits on Vr, hence m divides |N : Na|, we further conclude that 7m | |N|, 7 | |T| and m | |T|. Without a loss of generality, let pt be the largest prime dividing n. As t > 3, pt > 7, and as md = (p1p2 • • • pt)d divides 225 • 34 • 52 • 7p1p2 • • • pt, we have d < 2. If d =2, the only possibility is t = 3 and m = 3 • 5 • 7, so |T|2 | 225 • 35 • 53 • 72, hence |T| | 212 • 32 • 5 • 7; recall that m | |T|, by [15, Theorem III], T = A; with l = 7 or 8, and N = A2. By Claim 1, C =1, then Lemma 2.1 implies A = A/C < Aut(N) = S; I Z2, and as N = A2 is a minimal normal subgroup of A, we conclude that A = A; I Z2, (A; I Z2).Z2 or S; I Z2. Since | Aa | = ^, a direct computation by Magma [1] shows that no graph r exists in this case, a contradiction. Thus, d =1 and N = T, and by Lemma 2.1, A < Aut(T) is almost simple. Recall that |T| divides 225 • 34 • 52 • 7m and 7m divides |T|, and noting that 4 | |T| as T is nonabelian simple, we have 28m | |T|. By Lemma 3.1, the couple (T, |T|) is listed in Table 4. Now, we will analyse all the candidates of T in Table 4, thus proving Lemma 5.2. Recall that n = 2m and |T : Ta| = m or 2m. Denote by Out(T) the outer automorphism group of T. AssumeT = PSL(2,p) withp > 13 aprime. Thenp(p2 - 1) | 225 • 34 • 52 • 7n, andas Out(T) = Z2 (see [12, P. 135]), we have A = PSL(2,p) or PGL(2,p), the lemma is true. Assume T = M22. Then m = 3 • 5 • 11 = 165 and n = 330. Since Out(M22) = Z2, A = M22 or M22.Z2. By Example 3.2, r = C330, satisfying the conditions in Row 2 of Table 1. Assume T = M23. Then m = 3 • 11 • 23,5 • 11 • 23 or 3 • 5 • 11 • 23. Since Out(M23) = 1, A = T = M23 and so |T : Ta| = 2m = 1518,2530 or 7590. However, by [4], M23 has no subgroup with index 1518, 2530 or 7590, a contradiction. Assume T = Ji. Then m = 627,1045 or 3135. Since Out(Ji) = 1, we have A = T and |T : Ta| = 2m = 1254,2090 or 6270. By [4], Ji has no subgroup with index 1254,2090 or 6270, which is a contradiction. Suppose T = A12. Then m = 165 and |T : Ta| = 165 or 330. By [4], A12 has no subgroup with index 165 or 330, yielding a contradiction. Suppose T = PSL(2,49). Then m = 105 and |T : Ta| = 105 or 210, it follows |Ta| = 560 or 280 respectively. By Lemma 2.3, PSL(2,49) has no subgroup with order 560 or 280, a contradiction. Suppose T = PSL(2,224). Then Out(T) = Z24 by [12, P. 135], it follows A = PSL(2,224).Zr with r | 24, and |A| = 224-32-5-7-13-17-97-241-257-673r. Hence m = 343^ 17^97^241 ^257^673, 5-13-17-97-241-257-673 or 3^13T7^24L257^673. For the first case, |Aa | = 223 3 5 7r, which is impossible by Lemma 2.6. For the second case, |Aa| = 223^32 7r, by Lemma 2.6, the only possibility is r = 2 and Aa = Z20 : (SL(2, 2) x SL(3, 2)); for the last case, we have |Aa| = 223-3-7r, by Lemma 2.6, the only possibility is r = 6 and Aa = Z2° : (SL(2,2)xSL(3, 2)). However, by Lemma 2.3, both PSL(2,224).Z2 and PSL(2, 224).Ze have no subgroup isomorphic to Z20 : (SL(2, 2) x SL(3,2)), which is a contradiction. Suppose T = 3D4(2). Since Out(T) = Z3 (see [4]), A = 3D4(2) or 3D4(2).Z3, and so |A| = 212-34-72-13 or 212^7243 respectively, implying m = 3-7-13. Now, |Aa| = 2m = 211^33^7 or 2n-34-7, which is impossible by Lemma 2.6. Arguing similarly as above, one may prove that no graph r exists for all other candidates for T in Table 4 (the results have been checked by Magma [1]). □ We finally consider the case where A is insoluble and R = 1 by the following lemma. Lemma 5.3. Suppose that A is insoluble, R =1 and t > 3. Then no graph r exists. Proof. Let M be a minimal soluble normal subgroup of A. Then M = ZJ?, where r is a prime and d > 1. Since t > 3, M has at least 2 • 3 • 5 = 30 orbits on Vr, so, by Theorem 2.7, M is semi-regular on Vr (so d =1 and r G {2,p1,p2,... ,pt}), rM is a 7-valent A/M-arc-transitive graph of order 2m, and F is an arc-transitive regular Zr-cover of rM. r If r = 2, then rM is arc-transitive of odd order m and odd valency 7, which is impossible. Thus, r = pi with i G {1, 2,..., t}, and rM is a 7-valent A/M-arc-transitive graph of order . Recall that we assume by inductive hypothesis that Theorem 1.3 is true for all graphs which satisfy the assumptions of Theorem 1.3 and are of order less than n, so rM satisfies Theorem 1.3. Noting that A is insoluble and M is soluble, A/M is insoluble, so is Aut(rM). Then, checking the graphs in Theorem 1.3, we conclude that the soluble radical of Aut(rM) equals 1, and one of the following holds: (1) rM == C330 and Aut(rM) = M22.Z2; (2) Aut(rM) = PSL(2,p) or PGL(2,p) withp > 13 aprimeandt > 3. Since A/M acts arc-transitively on rM, we have 7|VrM | divides |A/M|. For case (1), then |VrM| = 330, and 7 • 330 = 2310 divides |A/M|, since A/M < M22.Z2, by [4], we have A/M = M22 or M22.Z2. Let X be a normal subgroup of A such that X = M.M22 = Zr.M22. Since (r, |VrM|) = (r, 330) = 1, we have r = 2 and 3. Then, as |Mult(M22)| = 12 (see [4]), Lemma 2.5 implies X = Zr x M22 and X' = M22. Since |X'| = |M221 does not divide |Vr|, X' < A is not semi-regular on Vr, and X' has at most two orbits on Vr by Theorem 2.7. Because r is connected and 1 = Xi < Aa, 1 = (Xa)r(a) < Ai(a), it follows 7 | X |. Hence r divides ^ = 27 • 32 • 5 • 11, which is a contradiction as (r, |VrM |) = (r, 330) = 1. We next consider case (2). Since A/M is insoluble and 7 | |A/M|, by Lemma 2.4, A/M = PSL(2,p) or PGL(2,p). Let B/M < A/M such that B/M ^ PSL(2,p). Since Mult(PSL(2,p)) = Z2 (see [12, P. 302]) and r > 3, Lemma 2.5 implies that B' = PSL(2,p) and B = M x B'. Since B,B' < A are insoluble, both B and B' have at most two orbits on Vr. In particular, m divides |B'|. If r > 7, since |A| divides 225 • 34 • 52 • 7m, and |B| = |M x B'| = r|B'| divides |A|, we have r / |B'|, which is a contradiction to m dividing |B'|. Assume r = 7. Since r is connected and 1 = B'a < Aa, we have 7 | |Bi |. Then, as |B':Bi | = m or 2m is divisible by 7, we further conclude that 72 divides |B'| = |B/M|. However, since |B/M:(B/M| = m or , which is not divisible by 7, we have 72 | |(B/M)a |, so 72 | |(a/m)s|, which is a contradiction by Lemma 2.6. Assume finally r = 3 or 5. Since B/M = B' has at most two orbits on VrN, and B' has at most two orbits on Vr, we have |B/M:(B/M)s | = m or ^, and |B':Bi | = m or 2m. It follows that r | |(B/M)s| and so r | |(A/M)s|. Also, as a/m acts arc-transitively on rM, 7 | |(A/M)s|, hence 7r | |(A/M)s|. Suppose r = 3. If (A/M)s is soluble, then (A/M )s is listed in part (1) of Lemma 2.6, and as 21 | |(A/M )s |, we have (A/M )s > F21; however, since A/M < PGL(2, p) and p = 7, by Lemma 2.4, PGL(2, p) has no soluble subgroup containing a subgroup isomorphic to F21, a contradiction. If (A/M)s is insoluble, noting that A/M < PGL(2,p), we have (A/M)s = PSL(2,p) or A5. For the first case, |VrM| = |A/M:(A/M)s| = 2, which is impossible. For the latter case, 7/ |(A/M)s|, also a contradiction. Suppose now r = 5. Then 35 | |(A/M)s|, and Lemma 2.6 implies that (A/M)s is insoluble, so (A/M)s = PSL(2,p) or A5 as A/M < PGL(2,p). Now, the same arguments as above draw a contradiction. □ Theorem 1.3 now follows directly from Theorem 1.2 and Lemmas 5.1-5.3. 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] P. J. Cameron, G. R. Omidi and B. Tayfeh-Rezaie, 3-designs from PGL(2,q), Electron. J. 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Suzuki, Group Theory II, volume 248 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, 1986, doi:10.1007/978-3-642-86885-6. [27] R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981), 309-311, doi: 10.1007/bf02579337. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 67-79 https://doi.org/10.26493/1855-3974.1319.375 (Also available at http://amc-journal.eu) Rank 4 toroidal hypertopes Eric Ens * Department of Mathematics, York University, Canada Received 8 February 2017, accepted 22 September 2017, published online 17 January 2018 We classify the regular toroidal hypertopes of rank 4. Their automorphism groups are the quotients of infinite irreducible Coxeter groups of euclidean type with 4 generators. We also prove that there are no toroidal chiral hypertopes of rank 4. Keywords: Regularity, chirality, toroidal, thin geometries, hypermaps, abstract polytopes. Math. Subj. Class.: 17B37, 15A21 1 Introduction A toroidal polytope is an abstract polytope that can be seen as a tessellation on a torus. By abstract polytope we mean a combinatorial structure resembling a classical polytope described by incidence relationships. Highly symmetric types of these polytopes are well known and understood, in particular the regular and chiral toroidal polytopes have been classified for rank 3 by Coxeter in 1948 [5], see also [6], and for any rank by McMullen and Schulte [10]. Regular toroidal polytopes and also regular toroidal hypertopes, which we define below, are strongly related to a special class of Coxeter groups, the infinite irreducible Coxeter groups of euclidean type which are also known as affine Coxeter groups (see, for example [11, page 73]). The symmetry groups of regular tessellations of euclidean space are precisely the affine Coxeter groups with string diagrams (see [11, Theorem 3B5]). When we talk about a tessellation we mean, informally, a locally finite collection of polytopes which cover En in a face-to-face manner. A toroidal polytope can then be seen as a "quotient" of a tessellation by linearly independent translations. For a precise definition of a toroidal polytope see [8]. The concept of a hypertope has recently been introduced by Fernandes, Leemans and Weiss (see [7]). A hypertope can be seen as a generalization of *Here I would like to acknowledge Maria-Elisa Fernandes, Dimitri Leemans, Egon Schulte and Asia Ivic Weiss for introducing the problem to me during the BIRS "Discrete Geometry and Symmetry" workshop on February 8-13, 2015. I also want to acknowledge Dimitri Leemans for his help with Magma [1], which was used to confirm the intersection conditions for Theorems 4.2 and 5.1. E-mail address: ericens@mathstat.yorku.ca (Eric Ens) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ a polytope. Or, from another perspective, as a generalization of a hypermap. For more infomation on hypermaps see [4]. In this paper we will classify the rank 4 regular toroidal hypertopes. Each affine Coxeter group in rank 4 (which are usually denoted by C3, B3 and A3), as we shall see, can be associated with the group C3 = [4,3,4], the symmetry group of the cubic tessellation of E3. The Coxeter Complex, denoted by C, of C3 can be seen as the simplicial complex obtained by the barycentric subdivision of the cubic tessellation {4, 3,4}. The Coxeter complex for the other two rank 4 affine Coxeter groups can be obtained by doubling the rank 3 simplicies for B3 and quadrupling them for A3. For details on the construction of C see [9, Section 6.5] or [11, Section 3B]. We note that C partitions E3. A regular toroidal hypertope (see Section 2 for a precise definition) can be seen as a quotient C/A/ by a normal subgroup of translations, denoted A/ where I represents a generating set identifying the normal subgroup. In particular the quotient induced by a normal subgroup of translations in the string affine Coxeter group C3 yields the three families of regular rank 4 toroidal polytopes, while the other two affine Coxeter groups with non-string diagrams do not yield regular polytopes, but as we shall see below, regular hypertopes. 2 C-groups and hypertopes Details of the concepts we review here are given in [7] and [11]. A C-group of rank p is a pair (G, S) such that G is a group and S = {r0,..., rp-1} is a generating set of involutions of G that satisfy the following property: VI, J C {0,... ,p - 1}, (r* : i e I) n (rj : j e J) = (rk : k e I n J). This is known as the intersection property which will be referred to later. A subgroup of G generated by a subset of S is called a parabolic subgroup. A parabolic subgroup generated by a single element of S is called minimal and a parabolic subgroup generated by all but one element of S is called maximal. For J C {0,..., p -1}, we define Gj := (rj: j e J) and Gi := (rj: rj e S, rj = r*). A C-group is a string C-Group if (rirj-)2 = 1G for all i, j with |i - j| > 1. A Coxeter diagram C(G, S) of a C-group (G, S) is a graph whose vertex set is S and two vertices, r* and rj are joined by an edge labelled by o(rirj), the order of rirj. We use the convention that if an edge is labeled 2 it is dropped and not labeled if the order of the product of the corresponding generators is 3. Thus the Coxeter diagram of a string C-group is a string. Affine Coxeter groups are C-groups and those with string diagrams are associated with toroidal polytopes. Hypertopes are generalizations of polytopes and we can, however, find toroidal hypertopes whose automorphism groups are quotients of any affine Coxeter group. We start with the definition of an incidence system. Definition 2.1. An incidence system r := (X, *, t, I) is a 4-tuple such that • X is a set whose elements are called elements of r; • I is a set whose elements are called types of r; • t: X ^ I is a type function that associates to each element x e X of r a type t(x) e I; • * is a binary relation of X called incidence, that is reflexive, symmetric and such that for all x, y G X, if x * y and t(x) = t(y) then x = y. A flag is a set of pairwise incident elements of r and the type of a flag F is {t(x): x G F}. A chamber is a flag of type I. An element x is said to be incident to a flag F when x is incident to all elements of F and we write x * F. Definition 2.2. An incidence geometry is an incidence system r where every flag is contained in a chamber. The rank of r is the cardinality of I. Let r := (X, *, t, I) be an incidence system and F a flag of r. The residue of F in r is the incidence system rF := (XF, *F,tF,IF) where • XF := (x G F: x * F, x / F); • If := I\t(F); • tF and *F are the restrictions of t and * to XF and IF. If each residue of rank at least 2 of r has a connected incidence graph then r is said to be residually connected. r is thin when every residue of rank 1 contains exactly 2 elements. Furthermore, r is chamber-connected when for each pair of chambers C and C', there exists a sequence of chambers C =: C0, Ci,..., Cn := C such that |C n Ci+11 = |I| — 1 (here we say that C and Ci+1 are adjacent). An incidence system is strongly chamber-connected when all of its residues of rank at least 2 are chamber-connected. Proposition 2.3 ([7, Proposition 2.1]). Let r be a thin incidence geometry. Then r is residually connected if and only if r is strongly chamber-connected. A hypertope is a strongly chamber-connected thin incidence geometry. To reinforce the relationship between polytopes and hypertopes we will sometimes refer to the elements of a hypertope r as hyperfaces of r, and elements of type I as hyperfaces of type I. Let r := (X, *,t, I) be an incidence system. An automorphism of r is a mapping a: (X, I) ^ (X, I) : (x,t(x)) ^ (a(x),t(x)) where • a is a bijection on X inducing a bijection on I; • for each x, y G X, x * y if and only if a(x) * a(y); • for each x,y G X, t(x) = t(y) if and only if t(a(x)) = t(a(y)). An automorphism a is type-preserving when, for each x G X, t(a(x)) = t(x). We denote by Aut(r) the group of automorphisms of r and by Aut/(r) is the group of type-preserving automorphisms of r. An incidence system r is flag transitive if Aut/(r) is transitive on all flags of type J for each J C I .It is chamber-transitive if Aut/(r) is transitive on all chambers of r. Furthermore, it is regular if the action of Aut/(r) is semi-regular and transitive. Proposition 2.4 ([7, Proposition 2.2]). Let r be an incidence geometry. r is chamber-transitive if and only if it is flag-transtive. A regular hypertope is a flag transitive hypertope. We note that every abstract regular polytope is a regular hypertope. The last concept we introduce here before we construct all rank 4 regular toroidal hypertopes is that of coset geometries. Proposition 2.5 ([14]). Let p be a positive integer and I := {1,... ,p} a finite set. Let G be a group together with a family of subgroups (Gi)ieI, X the set consisting of all cosets Gig, g G G,i g I and t: X ^ I defined by t(Gig) = i. Define an incidence relation * on X x X by: Gig2 * Gj g2 if and only if Gig1 n Gj g2 is non-empty in G. Then the 4-tuple r := (X, *, t, I) is an incidence system having a chamber. Moreover, the group G acts by right multiplication as a group of type-preserving automorphisms of r. Finally, the group G is transitive on the flags of rank less than 3. Whenever r is constructed as in the above proposition it is written as r(G; (Gi)ieI) and if it is an incidence geometry it is called a coset geometry. If G acts transitively on all chambers of r (thus also flags of any type) we say that G is flag transitive on r or that r is flag transitive. Now we note that we can construct a coset geometry r(G; (Gi)ieI) using a C-group (G, S) or rank p by setting Gi := {rj : rj G S,j G I\{i}) for all i G I := {0,... ,p - 1}. We introduce the following proposition which lets us know that constructions we use produce regular hypertopes. Proposition 2.6 ([7, Theorem 4.6]). Let (G, {r0,..., rp-1}) be a C-group of rank p and let r := r(G;(Gi)i£/) with Gi := {rj : rj G S,j G I\{i}) for all i G I := {0,... ,p-1}. If r is flag transitive, then r is a regular hypertope. Henceforth, we restrict our considerations to rank 4. Let G = {r0,r1,r2,r3) be an affine Coxeter group where each ri is a reflection through an associated affine hyperplane, Hi. Let C be the Coxeter complex of G formed from the hyperplanes His. Here r1,r2 and r3 will stabilize a point which, without loss of generality, can be assumed to be the origin o in E3. Then the maximal parabolic subgroup G0 is a finite crystallographic subgroup, which is a group that leaves a central point fixed. For details, see [3, pages 108-109]. The normal vectors to the reflection planes of the generators of G0 are called the fundamental roots. The images of the fundamental roots under G0 form a root system for G0. The lattice, A, generated by the root system is called the root lattice, and the fundamental roots form the integral basis for A. The region enclosed by the fundamental roots is called the fundamental region. This lattice gives us (and can be identified with) the translation subgroup T < G generated by the root lattice of G0, note that G = G0 x T [3]. For convenience we identify the translations with its vectors in addition a lattice also corresponds with its generating translation. If I is a set of linearly independent translations in T, and let TI < T be the subgroup generated by I. Then the sublattice A/ < A is the lattice induced by oTi, the orbit of the origin under TI. We note that C is a regular hypertope and each simplex in C represents a chamber where each vertex of the simplex is an element of a different type. In rank 4, when the quotient C/AI is a regular hypertope, we say it is a regular toroidal hypertope of rank 4. C/ AI is a regular hypertope (and thus a regular toroidal hypertope) when AI is large enough to ensure the corresponding group satisfies the intersection condition and when AI invariant under G0, i.e. riAIri = AI for i = 1,2,3. It is important to note that, in addition to denoting a lattice, AI is also denotes a set of vectors as well as a translation subgroup of G along those vectors. If I consists of all permutations and changes in sign of the coordinates of some vector s then we will write As. 3 Toroidal polytopes constructed from the group C3 = [4, 3,4] We begin, necessarily, with generating regular toroidal hypertopes (or, in this case, poly-topes) whose automorphism groups are quotients of the group C3, the affine Coxeter Group [4, 3, 4]. As generators of [4, 3, 4] we take pi, p2, p3 to be reflections in the hyperplanes with normal vectors (1, -1, 0), (0,1, -1), (0, 0,1) respectively, and po the reflection in the plane through (1/2, 0, 0) with normal vector (1, 0, 0) (see Figure 1). Then, (x,y,z)Po = (1 - x,y,z) (x,y,z)pi = (y,x, z), (3.1) (x,y,z)P2 = O^y^ (x,y,z)P3 = O^ -z). Figure 1: Fundamental simplex of [4, 3, 4]. In this case, the construction described in Section 2 will yield the regular polytopes since [4, 3, 4] is a string group. We denote by t the corresponding tessellation {4, 3, 4} of the Euclidean plane by cubes and by T it's full translation subgroup, where T is generated by the usual basis vectors, T = ((1,0,0), (0,1,0), (0,0,1)). Let Hi be the planes fixed by pi. The simplex bounded by the reflection planes Hi is a fundamental simplex of [4,3,4] and is denoted e, it is a simplex in the Coxeter complex of C3. Let Fi be the vertex of the fundamental simplex not on Hi then {F0, F, F2, F3} represents a flag of t, and the set of all j-faces of t = {4,3, 4} is represented by the orbit of Fj under C3. The regular polytope which results from factoring the regular tesselation t = {4, 3, 4} by a subgroup A of T which is normal in [4, 3, 4], is denoted by t/A (as above). We let As be the translation subgroup (or lattice) generated by the vector s and its images under the stabilization of the origin in [4, 3, 4] and hence under permutations and changes of sign of its coordinates. The regular polytope t/As is denoted by {4, 3, 4}s := {4, 3, 4}/As and the corresponding group [4, 3, 4]/As is written as [4, 3, 4]s. The following Lemma lists all possible such subgroups of T. Lemma 3.1. Let A be a subgroup of T, and if for every a e A, the image of a under all changes of sign and permutations of coordinates (which is conjugation of a by the stabilization of the origin in [4, 3,4]) is also in A, then A = ((x, 0,0), (0, x, 0), (0,0, x)), ((x, x, 0), (— x, x, 0), (0, —x, x)) or ((x, x, x), (2x, 0, 0), (0, 2x, 0)). Proof. As adapted from page 165 from Abstract Regular Polytopes [11]. Let s be the smallest positive integer from all coordinates of vectors in A, then we can assume that (s, s2, S3) G A. Then (—s, s2, S3) € A and thus 2se1 G A and so too are each 2sej. By adding and subtracting multiples of these we can find a vector all of whose coordinates are values between — s and +s. It then follows that A is generated by the all permutations of (sk, 03-k) with all changes of sign for some k G {1,2,3}. (Note that in rank n, k can be only 1,2 or n. Since otherwise (sk, 0n-k) — (0, sk, 0n-k-1) g A and so (s, s, 0n-2) G A if k is odd or (s, 0n-1) is if k is even. Though n = 3 in rank 4.) If k = 1 then we have the first basis mentioned in the Lemma, the second if k = 2 and the third when k = 3. □ It follows that As = sA(1fcj0n-'), and thus, as can be seen in [11, Theorem 6D1], we have the following theorem. Theorem 3.2. The only regular toroidal polytopes constructed from [4, 3,4] are {4, 3,4}s where s = (s, 0,0), (s, s, 0) or (s, s, s) and s > 2. Proof. Since conjugation of vectors in A by p1, p2 and p3 are precisely all permutations of coordinates and changes of sign, this theorem follows directly from Lemma 3.1. □ The following theorem also appears in [11] along with its proof. This theorem describes the group of each toroid. To arrive at the following result (and subsequent related results in sections 4 and 5) we note that the mirror of reflection p0 is x = 1/2 while the mirrors for p1, p2 are x = y and y = z respectively and the mirror for p3 is z = 0. Theorem 3.3 ([11, Theorem 6D4]). Let s = (sk, 03-k), with s > 2 and k =1,2,3. Then the group [4,3,4]s is the Coxeter group [4,3,4] = (p0, p1, p2, p3), where the generators are specified in (3.1), factored out by the single extra relation which is (PoP1P2P3P2P1)S = id, if k = 1, (P0P1 P2P3P2)2s = id, if k = 2, (p0P1P2P3)3s = id, if k = 3. As explained in [11], a geometric argument can be used to verify the intersection property for these groups when s > 2. However, note that [4, 3,4]s does not satisfy the intersection condition when s < 1 and thus is not a C-Group. We show the breakdown of the intersection condition for s = 1 by way of example for k = 1 where cases for k = 2,3 follow similar arguments. When s = 1, the identity P0P1P2P3P2P1 = id tells us that p0 G (pi, p2, p3) so G does not satisfy the intersection property. 4 Toroidal hypertopes whose automorphism group is bB3 (= Let {p0, p1, p2, p3} be the set of generators of [4,3,4] as in the previous section and e the corresponding fundamental simplex. We can double this fundamental simplex by replacing the generator p0 with p0 = poPiPo- Then p0 is a reflection through the hyperplane through the point (1,0,0) with normal vector (1,1,0). The transformation of a general vector by po is (x,y, z)p0 = (1 - y, 1 - x,z). (4.1) Then [p0, p1,p2, p3} generates B3, a subgroup of [4, 3,4] of index 2. The Coxeter diagram for this group is the non-linear diagram in Figure 2. In this section we let G(= B3) := (p0, p1,p2, p3) and let C(B3) be the Coxeter complex of G = B3. Figure 2: Coxeter diagram for B3. The fundamental simplex of B3 is the simplex in Figure 3 bounded by the planes H1,H2, H3 (fixed by pi, p2, p3 respectively) and Ho (now fixed by po). Let, as above, Fi be the vertices of the fundamental simplex opposite to Hj. The orbit of each vertex, Fj of the fundamental simplex of B3 represents the set of hyperfaces of type j. Since this fundamental simplex shares vertices F0, F2 and F3 with the fundamental simplex of (p0, pi, p2, p3) we will use the same names for hyperfaces as the names in Section 3, namely, vertices, faces and facets. Though the orbit or Fi (which is isomorphic to the orbit of F0 since the maximal parabolic subgroups generated by excluding pi or p0 are isomorphic) will be called hyperedges. Now the translation subgroup of G is different from the one translation subgroup in the previous section since the set of vertices of {4,3,4} now represent vertices and hyperedges (hyperfaces of type 0 and 1 respectively). The translation subgroup associated with this fundamental simplex is T = ((1,1,0), (-1,1,0), (0, -1,1)). We then note that the translation by vector (1, 1, 0) is the transformation (by right multiplication) Wi = pop2p3p2pip2p3p2, (-1, 1, 0) is W2 = pip2p3p2p0p2p3p2 and (0, -1, 1) is W3 = p2p3p2pip2pipop2p3pi. Now, to form a root lattice A we have the freedom to choose the crystollographic subgroup G0 by fixing either a vertex or a hyperedge (see [3, pages 108-109]). We choose to leave out po since this reflection does not fix F0. Doing so leaves [3,4] as the subgroup we are conjugating with, which is the same as was for [4,3,4]. We also note that if we chose pi rather than p0 then the result is functionally the same since we are still conjugating by [3,4] = (po, p2, p3) and this corresponds to forming a torus with its corners at hyper-edges. We now note that although the same conditions are satisfied as in Lemma 3.1, T is now a different set. So instead we have the following lemma. Figure 3: Fundamental simplex of B3. Lemma 4.1. If T = ((1,1,0), (-1,1,0), (0, -1,1)), A < T a subgroup and if for every a G A, the image of a under changes of sign and permutations of coordinates is also in A, then A = ((2x, 0, 0), (0, 2x, 0), (0,0, 2x)), ((x, x, 0), (-x, x, 0), (0, -x, x)) or ((2x, 2x, 2x), (4x, 0,0), (0,4x, 0)). Proof. We will only modify the proof of Lemma 3.1. In that proof we arrive at a generating set (sk, 03-k) for each k G {1,2,3}, given that T is different than the translation subgroup of Section 3. Similar arguments to the ones used in the proof to Lemma 3.1 can now be used to show that for k = 1 or k = 3, s is even. For k = 2, A is generated by permutations and changes of sign of (s, s, 0). This needs no further examination since it is clearly in T. □ As in the previous section, we describe the groups that will be used to construct each of the toroids. We denote by Gs the quotient B3/As. Earlier we noted wi as the translation (1,1,0) while (poP2P3P2P1)2 is the translation (2,0,0) and (PoP^P^^)3 is the translation (2,2,2). And now that the the mirror for P0 is y =1 - x. Theorem 4.2. Let s = (2s, 0,0), (s, s, 0) with s > 2 or (2s, 2s, 2s) with s > 1. Then the group Gs = B3/As is the Coxeter group B3 = (po, p1, p2, p3) with Coxeter diagram in Figure 2, factored out by the single extra relation which is (poP2P3P2Pi)2s = id ifs = (2s, 0,0), (P0P2P3P2PlP2P3P2)S = id ifs = (s, s, 0), (PoP2P3PiP2P3)3s = id ifs = (2s, 2s, 2s). Here, as in Section 3, we have that Gs fails the intersection property for small enough s. However, because the fundamental simplex is doubled, this time when s = (2s, 2s, 2s), Gs satisfies the intersection condition for s > 1 while s > 2 is still necessary for the other two vectors. Verifying that Gs fails the intersection condition for s = 1 when s = (2s, 0,0) and (s, s, 0) follows similar calculations as those done in Section 3. Namely, when s = 1 for the first vector, we arrive at the identity p0p2p3p2p0 = P1P2P3P2P1 and for the second vector we have the identity p0p2p3p2 = p2p3p2p1. Which violates the intersection condition. Magma [1] can be used to verify that Gs satisfies the intersection condition when s = (4,0,0) = (2s, 0,0), (2, 2,0) = (s, s, 0) or (2,2, 2). To see that the it also satisfies the intersection condition for greater values of s can be seen with a geometric argument. The orbit of a base chamber of each parabolic subgroup of Gs can be seen as a collection of chambers which are duplicated at each of the 8 corners of the boundaries of As. For instance, the subgroup (p1, p2, p3) consists of chambers forming octahedra centred around corner vertex. Given the collection of chambers in two such subgroups, there will always be some intersection between the collections occurring at the same corner (someones it's just the base chamber itself). However, If Gs fails the intersection condition, then there will be an intersection with the chambers of one subgroup centred around one corner that intersect with the chambers of the other subgroup on another corner. So, given a particular s where Gs satisfies the intersection condition, by increasing s, the corners of As get further and further apart. So if there are no such intersections for some s, then for larger s there will not be either. Adopting a similar notation as in the previous section and using As as defined in Section 2, we now have the following theorem. Theorem 4.3. The regular toroidal hypertopes of rank 4 constructed from G(= B3) = (po, p1, p2, p3), where the generators are specified in (3.1) and (4.1), are C(B3)/As where C(B3) is the Coxeter complex of B3 and s = (2s, 0,0), (s, s, 0) with s > 2 or (2s, 2s, 2s) with s > 1. Proof. To begin we need to find an s and corresponding As that is invariant under conjugation by a subgroup of G which is the symmetry group of "vertex"-figure (by vertex we mean, the element that the translations begin from). In this case our subgroup ends up being [3,4] as was described before Lemma 4.1. Now, since we are conjugating by [3,4] = (p 1, p2, p3), As must contain all permutations and changes of sign of any vector in As (which we discovered in the proof of Theorem 3.2 which is also on page 165 of [11]). Thus, by Lemma 4.1, s = (2s, 0,0), (s, s, 0) or (2s, 2s, 2s). However, we still do not know if this construction yields a regular hypertope. To do this, we start by noting that the Coxeter complex C(B3) formed from G is precisely the hypertope r(G; (Gj)ie/) (the construction of which follows from [7]). So we need to show that C(B3) is flag transitive (or, equivalently, chamber transitive). To do so we will note the rank 3 residue r- := r(G-; (G^g i})ie{12 3}). This is isomorphic to the cube, a regular polyhedron, which is flag transitive. So we pick to chambers in r(G; (Gj)ie/) = C(B3) which can be written as C1 = {G0go,G1g1,G2g2^3^3} and C2 = {Ggfeo, G^, G2h2,G3M for some g^h G G. Then, since G — G— x T and T acts transitively on elements of type 0 there is a translation t G G such that C1t = {G-h0,X, Y, Z} which is some chamber that shares the same element of type 0 as C2. Then the chambers C1t and C2 are both in some rank 3 residue which is isomorphic to Pg. Since this residue is flag transitive, there is some element, g G G such that C1tg = C2. Thus C(B3) is chamber transitive and thus flag transitive. So, by Proposition 4.6 from [7] this is a regular hypertope. So now we want to know if r(G'; (Gi)ie/) is a regular hypertope where G' is the group G/As where s > 2 (since otherwise G' fails the intersection condition and the resulting construction fails to be thin). Just as before, we take two chambers $ and ^ from r(G'; (G'i)ieI). Then to each of these chambers we can associate a family of chambers and in C(B3). Since C(B3) is chamber transitive for each $ e and e there exists j e G where $j = . In particular there exist chambers e and e in C(B3) where, since As is invariant under G, = and ^ e G'. We can see this by noting that and are the members of their respective families which lie inside the fundamental region of As. Thus r(G'; (G'i)ieI) is chamber transitive and thus face transitive, so is also a regular hypertope by Proposition 2.6. For the other two possibilities of A, we need only change the added relations, but because the relations were chosen specifically, they will also generate regular hypertopes. □ 5 Toroidal hypertopes whose automorphism group is A3 (= Pn) We can show that this group is, yet again a subgroup of [4, 3,4] by doubling the fundamental simplex a second time (this can be seen geometrically in Figure 5) and now defining p3 = p3p2p3 which is a reflection in the plane through (1,1, -1) with normal vector (0,1,1). Transformation of a general vector by p3 is (x,y,z)p3 = (x, -z, -y). (5.1) Now we let G(= A3) := {p0, p1, p2,p3} andC(A3) be the Coxeter complex of G. The defining relations for G are implicit in the Coxeter diagram in Figure 4. Figure 4: Coxeter diagram for A3. Here the fundamental simplex of A3 is a tetrahedron bounded by the planes Hi (fixed by pi). This fundamental simplex shares the planes fixed by p0,p1, p2 with the fundamental simplex of B3 as well as the corresponding vertices. The stabilizers of each vertex of the fundamental simplex are also isomorphic since all maximal parabolic subgroups of A3 are pairwise isomorphic. This implies that the set of hyperfaces of types i and j are isomorphic for each i,j e {0,1,2, 3}. This fundamental simplex gives us the same translation subgroup as we had in the previous Section. Though now we must use the new generators to find the translations. We define W1 = p0P2P1P3P1P2 = (1,1,0), w2 = P1P2P3P0P3P2 = (-1,1,0) and W3 = P2P1P0P3P0P1 = (0, -1,1). / / /1 K A / R / Figure 5: Fundamental simplex of A3. Using these translations, for a translation (a, b, c) G A, we have that pi(a, b, c)pi = (b, a, c). In a similar way, conjugating by p2 yields (a, c, b) and conjugating by p3 yields (a, -c, -b). So if we conjugate by p1p2p1 then we get (c, b, a) and so A must have all permutations. Now, from the previous we know A must also contain (a, -b, -c) and adding this to (a, b, c) gives (2a, 0,0), which then subtracted from (a, b, c) is (-a, b, c) and so with all permutations means that A must have all permutations and changes of sign. With this group, we leave out p0 to form the crystollographic subgroup G0. Though a curiosity of this group is that we use any generator of B3 to form a crystollographic subgroup and still finish with the same objects. With each choice simply changing where we draw the boundary of the torus. This leaves p1, p2 and p3 with which to conjugate A. As in the regular case, p1 and p2 show us that A must consist of all permutations of the coordinates of vectors. If (a, b, c) is a general vector in A then p3 tells us that (-c, b, -a) must also be in A and then so also must (-a, b, -c). Adding that to our original general vector tells us that (0,2b, 0) is also included. So, subtracting that from the general vector finally gives us (a, -b, c). Note that this can just as easily be done with either a or c with some simple permutations. As in the previous section, we describe the groups of each of the toroids. Earlier we noted w1 as the translation (1,1,0) while (pOp^pO2 is the translation (2,0,0) and (yOop2p1^)3s is the translation (2, 2,2). And now that the the mirror for p0 is y =1 - x while the mirror for p3 is y = - z. Theorem 5.1. Let s = (2s, 0,0), (s, s, 0) with s > 2 or (2s, 2s, 2s) with s > 1. Then the group Gs = A3/As is the Coxeter group A3 = (p0, p1, p2, p3) (with Coxeter group specifed in Figure 4), factored out by the single extra relation which is (P0P2P3p1)2s = id if s = (2s, 0, 0), (poP2P1P3P1P2)s = id ifs = (s, s, 0), (P0P2P1p3)3s = id if s = (2s, 2s, 2s). For the same reasons as in Section 4, the intersection condition is satisfied for s = (2s, 2s, 2s) when s > 1. Theorem 5.2. The regular toroidal hypertopes of rank 4 induced by G(= A3) = (p0, Pi,Pi, P3) (where the generators are specified in (3.1), (4.1) and (5.1)) are C(A3)/As where C(A3) is the Coxeter complex of A3 and s = (2s, 0,0), (s, s, 0) for s > 2 or (2s, 2s, 2s) with s > 1 . Proof. We first show that C (A3) is a regular hypertope, which requires showing that it is flag transitive. In the same manner as the proof of Theorem 4.3 we need only show each rank 3 residue is flag transitive, since all rank 3 residues are regular tetrahedra C (A3) is flag transitive. The translation subgroup is the same as in the previous Section and conjugating A by p\,p2,p3 gives all permutations and changes in sign of a general vector in A, the same arguments for Lemma 4.1 and Theorem 4.3 will prove this theorem. □ 6 Non-existence of rank 4 chiral hypertopes Here we recall that for an abstract polytope to be chiral its automorphism group must have two orbits when acting on flags and that adjacent flags are in different orbits. Chiral polytopes of any rank are examined in depth in [13]. The existence of these objects in any rank was proved in [12]. There is also a notion of chirality in hypermaps as well, see for example, [2]. Similarly we say for a hypertope to be chiral if its automorphism group action has two chamber orbits and adjacent chambers are in different orbits [7]. As in Section 2, given an affine Coxeter group G and associated Coxeter complex C, we define a subgroup G0 < G as the maximal parabolic subgroup fixing the origin. Then, given a set I of linearly independent translations in G and T/, the translation subgroup generated by I then we call the lattice A/ the lattice induced by the orbit of the origin under T/. When A/ is invariant under the rotation subgroup G0+ but there is no automorphism group of G that interchanges adjacent chambers, then in rank 4 we say that the quotient C/A/ is a chiral toroidal hypertope of rank 4. The proof that there are no chiral toroids of rank 4 for the group [4,3,4] comes from page 178 from [11] and the same proof can adapted for the other two rank 4 affine Coxeter groups. The basic idea for the proof is that since C/A is chiral, A is invariant under the rotation group [3,4]+, so A contains vectors that are compositions of an even number of permutations with an even number of sign changes or all compositions of an odd number of permutations with an odd number of sign changes. It then goes to show that if (a, b, c) G A then (b, a, c) G A, which is the image of (a, b, c) under an odd permutation, which is a contradiction. Therefore no such A can exist. We will use the same method to show the same is true for the other two groups. Theorem 6.1. There are no rank 4 chiral toroidal hypertopes. Proof. In [11] it was shown that there are no rank 4 hypertopes constructed from [4,3,4], so we show for constructions from B3 and A3. In previous sections we found that if A is a subgroup of thejranslations that is invariant under conjugation by the stabilizer of the origin in B3 and A3 with (a, b, c) G A, then A contains all permutations and changes of sign of (a, b, c), just as it did with the stabilizer in [4, 3,4]. Thus conjugation of A by the stabilizer of the rotation subgroup of each of these groups is all compositions of even permutations with an even number of sign changes or all compositions of odd permutations with an odd number of sign changes, just as for [4,3,4]. So the same arguments and calculations from page 178 in [11] still hold and show that (b, a, c) G A and we develop the same contradiction. □ 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] A. Breda D'Azevedo, G. A. Jones, R. Nedela and M. Skoviera, Chirality groups of maps and hypermaps, J. Algebraic Combin. 29 (2009), 337-355, doi:10.1007/s10801-008-0138-z. [3] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, volume 290 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1988, doi: 10.1007/978-1-4757-2016-7. [4] D. Corn and D. Singerman, Regular hypermaps, European J. Combin. 9 (1988), 337-351, doi: 10.1016/s0195-6698(88)80064-7. [5] H. S. M. Coxeter, Configurations and maps, Rep. Math. Colloq. 8 (1948), 18-38. [6] 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, Berlin-New York, 4th edition, 1980. [7] M. E. Fernandes, D. Leemans and A. Ivic Weiss, Highly symmetric hypertopes, Aequat. Math. 90 (2016), 1045-1067, doi:10.1007/s00010-016-0431-1. [8] I. Hubard, A. Orbanic, D. Pellicer and A. Ivic Weiss, Symmetries of equivelar 4-toroids, Discrete Comput. Geom. 48 (2012), 1110-1136, doi:10.1007/s00454-012-9444-2. [9] J. E. Humphreys, Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990, doi:10.1017/ cbo9780511623646. [10] P. McMullen, and E. Schulte, Higher toroidal regular polytopes, Adv. Math. 117 (1996), 17-51, doi:10.1006/aima.1996.0002. [11] P. McMullen and E. Schulte, Abstract regular polytopes, volume 92 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2002, doi:10.1017/ cbo9780511546686. [12] D. Pellicer, A construction of higher rank chiral polytopes, Discrete Math. 310 (2010), 12221237, doi:10.1016/j.disc.2009.11.034. [13] E. Schulte and A. Ivic Weiss, Chiral polytopes, in: Applied Geometry And Discrete Mathematics, Proceedings of a DIMACS Workshop, Providence, Rhode Island, USA, September 18, 1990, 1990 pp. 493-516. [14] J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in: Colloque d'Algèbre supérieure, Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Etablissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1975 pp. 261-289. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 81-95 https://doi.org/10.26493/1855-3974.1382.dee (Also available at http://amc-journal.eu) On 2-distance-balanced graphs* Boštjan Frelih t, Stefko Miklavic * University of Primorska, FAMNIT and IAM, Muzejski trg 2, 6000 Koper, Slovenia Received 12 April 2017, accepted 11 September 2017, published online 26 January 2018 Let n denote a positive integer. A graph r of diameter at least n is said to be n-distance-balanced whenever for any pair of vertices u, v of r at distance n, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. In this article we consider n =2 (e.g. we consider 2-distance-balanced graphs). We show that there exist 2-distance-balanced graphs that are not 1-distance-balanced (e.g. distance-balanced). We characterize all connected 2-distance-balanced graphs that are not 2-connected. We also characterize 2-distance-balanced graphs that can be obtained as cartesian product or lexicographic product of two graphs. Keywords: n-distance-balanced graph, cartesian product, lexicographic product. Math. Subj. Class.: 05C12, 05C76 1 Introductory remarks A graph r is distance-balanced if for each pair u, v of adjacent vertices of r the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Although these graphs are interesting from the purely graph-theoretical point of view, they also have applications in other areas of research, such as mathematical chemistry and communication networks. It is for that reason that they have been studied from various different points of view in the literature. Distance-balanced graphs were first studied by Handa [9] in 1999. The name distance-balanced, however, was introduced nine years later by Jerebic, Klavžar and Rall [12]. The * We thank the two anonymous referees for many useful comments and suggestions that have greatly improved our initial manuscript, especially for pointing out the mistake in Theorem 5.4. t This work is partially supported by Erasmus+ Programme of the European Union (project ECCUM - Establishment of Computing Centers and Curriculum Development in Mathematical Engineering Master Programme). t The author acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0285 and research projects N1-0032, N1-0038, J1-5433, J1-6720, J1-7051). E-mail addresses: bostjan.frelih@upr.si (Boštjan Frelih), stefko.miklavic@upr.si (Stefko Miklavic) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ family of distance-balanced graphs is very rich (for instance, every distance-regular graph as well as every vertex-transitive graph has this property [13]). In the literature these graphs were studied from various purely graph-theoretic aspects such as symmetry [13], connectivity [9, 16] or complexity aspects of algorithms related to such graphs [6], to name just a few. However, it turns out that these graphs have applications in other areas, such as mathematical chemistry (see for instance [3, 11, 12]) and communication networks (see for instance [3]). Another interesting fact is that these graphs can be characterized by properties that do not seem to have much in common with the original definition from [12]. For example, the distance-balanced graphs coincide with self-median graphs, that is graphs for which the sum of the distances from a given vertex to all other vertices is independent of the chosen vertex (see [4]). In [1], distance-balanced graphs are called transmission regular. Finally, even order distance-balanced graphs possess yet another nice property, making them what are called equal opportunity graphs (see [3] for the definition). In distance-balanced graphs one only considers pairs of adjacent vertices. However, it is very natural to extend the definition to the pairs of nonadjacent vertices. This generalized concept of n-distance-balanced graphs (see Section 2 for the definition) was first introduced by Bostjan Frelih in 2014 [8] (we point out that certain other generalizations of this concept, where one still focuses just on pairs of adjacent vertices, have also been considered in the recent years [10,14,15]). The n-distance-balanced graphs and their properties were extensively studied in [17]. They are also the main topic in the paper [7], but in this paper some of the stated results do not hold. We comment on one of these problems later (see Remark 5.1). In this article we consider 2-distance-balanced graphs. We now summarize our results. After some preliminaries in Section 2, we show in Section 3 that there exist 2-distance-balanced graphs that are not 1-distance-balanced (e.g. distance-balanced). It was shown in [9] that every distance-balanced graph is 2-connected. It turns out that not all 2-distance-balanced graphs are 2-connected. However, we characterize all connected 2-distance-balanced graphs that are not 2-connected. In [12] distance-balanced cartesian products and distance-balanced lexicographic products of two graphs were characterized. We characterize 2-distance-balanced cartesian products and 2-distance-balanced lexicographic products of two graphs in Section 4 and 5, respectively. 2 Preliminaries In this section we review some basic definitions that we will need later. Throughout this paper, all graphs are assumed to be finite, undirected, without loops and multiple edges. Given a graph r let V(T) and E(T) denote its vertex set and edge set, respectively. For v e V(r) we denote the set of vertices adjacent to v by Nr(v). If the number |Nr(v)| is independent of the choice of v e V(r), then we call this number the valency of r and we denote it by kr (or simply by k if the graph r is clear from the context). In this case we say that r is regular with valency k or k-regular. For u,v e V(r) we denote the distance between u and v by dr (u, v) (or simply by d(u,v) if the graph r is clear from the context). The diameter max{dr(u,v) | u,v e V(r)} of r will be denoted by Dr (or simply by D if the graph r is clear from the context). For any pair of vertices u,v e V(r) we let WUv be the set of vertices of r that are closer to u than to v, that is W„rv = {w e V(r) | dr(u,w) < dr(v,w)}. Let n denote a positive integer. A graph r of diameter at least n is said to be n-distance-balanced, if \WrV | = |WVr„| for any u, v e V(r) at distance n. The distance-balanced graphs are n-distance-balanced graphs for n = 1. For W C V(r) the subgraph of r induced by W is denoted by (W) (we abbreviate r - W = (V(r) \ W)). A vertex cut of a connected graph r is a set W C V(r) such that r - W is disconnected. A vertex cut of size k is called a k-cut. A graph is said to be k-connected if it has at least k + 1 vertices and the size of the smallest vertex cut is at least k. If a vertex cut consists of a single vertex v, then v is called the cut vertex. We complete this section by defining the cartesian product and the lexicographic product of graphs G and H. In both cases, the vertex set of the product is V(G) x V(H). Pick (gi,hi), (g2,h2) e V(G) x V(H). In the cartesian product of G and H, denoted by GDH, (gi, hi) and (g2, h2) are adjacent if and only if g1 = g2 and h1, h2 are adjacent in H, or h1 = h2 and g1, g2 are adjacent in G. Note that the cartesian product is commutative. In the lexicographic product of G and H, denoted by G[H], (g1, h1) and (g2, h2) are adjacent if and only if g1 = g2 and h1, h2 are adjacent in H, or g1, g2 are adjacent in G. 3 On the connectivity of 2-distance-balanced graphs In this section we characterize connected 2-distance-balanced graphs that are not 2-connect-ed (Corollary 3.4). As a consequence, using the well known fact that an arbitrary connected distance-balanced graph is at least 2-connected (see [9]), we construct an infinite family of 2-distance-balanced graphs that are not distance-balanced. Let G be an arbitrary (not necessary connected) graph, and let c be a vertex that does not belong to the set of vertices of G. We construct a graph, denoted by r(G, c), with the set of vertices V(r(G,c)) = V(G) U{c} and the set of edges E(r(G, c)) = E(G) U {cv | v e V(G)}. This graph is obviously connected. Next theorem follows directly from the construction of r(G, c). Theorem 3.1. G is not connected if and only if r(G, c) is not 2-connected. □ We show that regularity of G is a sufficient condition for r(G, c) to be 2-distance-balanced. Theorem 3.2. If G is a regular graph that is not a complete graph, then r = r(G, c) is 2-distance-balanced. Proof. Assume that G is a k-regular graph that is not a complete graph. Let Gi, G2,..., Gn be its connected components for some positive integer n. If G is connected, then n =1, otherwise G has at least two connected components. Since G is not a complete graph, it is clear that the diameter of r equals 2, which means that two arbitrary vertices of r are either adjacent or they are at distance 2. There are two different types of vertices at distance 2 in r. The first type is when both vertices at distance 2 belong to the same connected component of G. The second type is when vertices at distance 2 belong to different connected components of G. Let Gj be an arbitrary connected component of G. Let vi, v2 G V(Gj) be arbitrary vertices at distance 2 in r. We count vertices that are closer to vi than to v2 in r and vertices that are closer to v2 than to vi in r. We get WU = {vi} u (NGi(vi) \ (NGi(vi) n NGi(V2))). It follows that WU I = 1 + |NGi(vi)| - |NGi(vi) n NGi(v2)|. Changing the roles of vertices vi and v2 , we get |wV2v11 = 1 + |NGi(v2)| - |NGi(v2) n NGi(vi)|. Since G is regular, the number of vertices that are closer to vi than to v2 in r equals the number of vertices that are closer to v2 than to vi in r. Let now Gj and Gj be arbitrary different connected components of a disconnected graph G. Pick arbitrary vi G V(Gj) and v2 G V(Gj). Obviously these two vertices are at distance 2 in r. Observe that Wvriv2 = {vi} u NGi(vi) and W^ = {v2} U NGj (v2). Since every connected component of a k-regular graph is also a k-regular (induced) subgraph, it follows that |W„riJ=1 + k and |WV2 vi | = 1 + k, where k is the valency of G. So the number of vertices that are closer to vi than to v2 in r equals the number of vertices that are closer to v2 than to vi in r. Since this is true for an arbitrary pair of vertices at distance 2 in r, this graph is 2-distance-balanced. □ Next we prove that every connected 2-distance-balanced graph, that is not 2-connected, is isomorphic to r(G, c) for some regular graph G that is not connected. Theorem 3.3. Let r be a connected 2-distance-balanced graph that is not 2-connected. Then r is isomorphic to r(G, c) for some disconnected regular graph G. Proof. Since r is not 2-connected, there exists a cut vertex c G V(r). Let Gi, G2,..., Gn be connected components of G = r — {c}, n > 2. We want to prove that G is regular and that the cut vertex c is adjacent to every other vertex in r. To do this we will first prove some partial results. First we claim that the cut vertex c is adjacent to every vertex in a connected component Ge of G for at least one integer 1 < I < n. Suppose that this is not true. Let Gj and Gj be two different connected components of G. Then there exist v2 G V(Gj) and u2 G V(Gj), both at distance 2 from c in r. This means that there exists vi G V(Gj) that is adjacent to c and v2 in r, and there exists ui G V(Gj) that is adjacent to c and u2 in r. If we compare the set of vertices that are closer to c than to v2 in r and the set of vertices that are closer to v2 than to c in r, we get Wl D {c} u V(Gj) and W^ C V(G<) \ {vi}. It follows that 1 + |V(Gj)| < W r | cv 2 I and |Wr. < |V(Gi)|- 1. Since r is 2-distance-balanced, we get |V(Gj)| < |V(Gi)|- 2. Similarly as above (changing vertex v2 with u2) we get |V(Gi)| + 2 < |V(Gj)|. However, inequalities (3.1) and (3.2) imply (3.1) (3.2) |V(Gi)| + 2 < |V(Gj)| < |V(Gi)|- 2, a contradiction. It follows that the cut vertex c G V(r) is adjacent to every vertex in V(G^) for at least one integer 1 < I < n. Without loss of generality we may assume that I =1. Next we claim that the induced subgraph G1 of r is regular. Pick some u G V(G) \ V(G1) that is adjacent to the cut vertex c in r. Since c is adjacent to every vertex in V(G1), the distance between u and an arbitrary v G V(G1) equals 2 in r. Pick v G V(G1). Notice that It follows that |W1 WVu = {v}u (Nr(v) \{c}). 1 + |Nr(v)|- 1 = |NGl (v)| + 1. Pick w g V(G1). Since r is 2-distance-balanced and c is adjacent to every vertex of V(G1), we get the following sequence of equalities |NGi (v)| + 1 = |Nr (v)| = |Wvr | W1 | W1 So = |WWU| = |Nr(w)| = |Ngi (w)| + 1. |NGi (v)| = |Ngi (w) for arbitrary v, w G V(G1). From now on we may assume that the induced subgraph G1 of r is k-regular. This also means that every vertex in V(G1) has valency k + 1 in r. Our next step is to show that the cut vertex c G V(r) is adjacent to every vertex in V(G) \ V(G1). Suppose that this is not true. Then there exists some vertex u2 in a connected component Ge of G, 2 < I < n, that is at distance 2 from c in r. Without loss of generality we can take I = 2. Consequently there exists some u1 g V(G2) that is adjacent to both c and u2 in r. Pick an arbitrary v G V(G1). We have already proved that the valency of an arbitrary vertex in V(G1) is k + 1 in r. Now we count vertices that are closer to v than to u1 in r. Since wr {v}U (Nr(v) \{c}), we get |Wr„ J = 1 + k + 1 - 1 = k + 1. In addition, for vertices that are closer to c than to u2 in r, we have Wcru2 D V(Gi) U{c}. It follows that | | |Wcr„2|>|V(Gi)| + 1 > k + 2. (3.3) Consider the distance partition of r according to adjacent vertices c and u1 that is shown in Figure 1. The symbol Dj denotes the set of vertices that are at distance i from u1 and at distance j from c in r. Define a set U 2 Figure 1: The distance partition of r according to adjacent vertices c and u1. D U = U = (Di-1 U Di), i= i where D denotes the diameter of r. First we show that Wr2C C U. Recall that for u, v e V(r), d(u, v) denotes the distance between vertices u and v. Pick an arbitrary w e Wr2C. Since u1, u2 are, by the assumption, adjacent vertices in r, the triangle inequality tells us that d(ui, w) e {d(u2, w) — 1, d(u2, w), d(u2, w) + 1}. If we consider all three cases, we get d(ui,w) = d(u2,w) — 1 < d(c,w), d(ui,w) = d(u2,w) < d(c,w), d(ui, w) = d(u2, w) + 1 < d(c, w). Each considered case gives us that w e U and so c C U. Note also that U C V (G2). Now we show that U C WrlV (recall that v is an abitrary vertex in V(G1)). Let w be an arbitrary vertex in U, which means that w is also in V(G2). We get that d(ui,w) < d(c,w) < d(v,w), since vertices v and c are adjacent in r, and v is not in V(G2). It follows that w e W^v, and so U C W^1V. From relations W^2C C U C W^1V, we get that W^2C C W^1V and so (3.4) By taking into account inequalities (3.3) and (3.4), and since r is 2-distance-balanced, we get k + 2 <|Wcr„21 = |W„r2C|< k +1, which is a contradiction. This shows that the cut vertex c g V(T) is adjacent to all vertices in V(G). It remains to prove that the induced subgraph Ge (2 < I < n) of r is k-regular. Without loss of generality assume I = 2. Since we already know that the cut vertex c is adjacent to every vertex in r, an arbitrary vertex u in V(G2) is at distance 2 from an arbitrary vertex v in V(Gi) in r. Observe that WUV = {u} U (Nr(u) \ {c}) = {u} U NG2 (u) and = {v} U (Nr(v) \ {c}) = {v} U NGl (v). This means that |W„rv| = 1 + |NG2 (u)| and |W„ru|=1 + k. Since r is 2-distance balanced, it follows that | = |WVU| and so |NG2 (u)| = k for an arbitrary vertex u g V (G2). Therefore, G2 is regular and has the same valency k as the induced subgraph G1. It follows that G is regular and this completes the proof. □ The characterization of all connected 2-distance-balanced graphs that are not 2-co-nnected follows immediately from Theorems 3.1, 3.2 and 3.3. Corollary 3.4. Let r be a connected graph. Then r is 2-distance-balanced and not 2-connected if and only if it is isomorphic to r(G, c) for some disconnected regular graph G. □ 4 2-distance-balanced cartesian product Throughout this section let G and H be graphs and let r = GIUH be the cartesian product of G and H. We characterize connected 2-distance-balanced cartesian products of graphs G and H (see Theorem 4.4). It follows from the definition that the cartesian product r is connected if and only if G and H are both connected. In order to avoid trivialities we assume that |V(G)| > 2 and |V(H)| > 2. Recall that dr((gi, hi), (g2, h2)) = dG(gi, g2) + ¿H (hi, M (4.1) for arbitrary (gi, hi), (g2,h2) g V(r). Since we are dealing with 2-distance-balanced cartesian products of graphs, we are interested in vertices at distance 2. It follows from equality (4.1), that there exist three different types of vertices at distance 2 in r. We now state these three types and we will refer to them later. Let (gi, hi), (g2, h2) g V(r) be vertices at distance 2 in r. We say that these two vertices are of type • G2, if hi = h2 and <9G(gi, 02) = 2, • H2, if gi = g2 and dH(hi, h2) = 2, • GH2, if 5G(0i,02) = dH(hi,h2) = 1. Note that vertices of type G2 (H2, respectively) do not exist if G (H, respectively) is a complete graph. Denote the set of vertices that are at equal distance from g1 and g2 in G by EG S2, and the set of vertices that are at equal distance from h1 and h2 in H by E®^. We first prove three lemmas that we will need later in the proof of the main theorem of this section. W(gi ,h)(S2,h) |H| |wgifl21 and W(g2,h)(gi,h) tG | ' 3231 I Lemma 4.1. Let (gi, h) and (g2, h) be arbitrary vertices of type G2 in r = GDH. Then |H| |WSG Proof. Let (a, x) be an arbitrary vertex of r. It follows from the equality (4.1) that dr((gi,h), (a,x)) = dG(gi,a) + Oh(h,x) and dr((g2,h), (a, x)) = dG(g2,a) + Oh(h,x). So (a, x) is closer to (g1, h) than to (g2, h) in r if and only if a is closer to g1 than to g2 in G. Since (a, x) G V(r) was an arbitrary vertex, this means that W(3i,h)(32,h) lH IIWG32I Similarly we get that W(r32,h)(gi,h) = lHI IWG31 I . □ Lemma 4.2. Let (g, h1) and (g, h2) be arbitrary vertices of type H2 in r = GDH. Then W(3,hl )(3,h2) = |G| |wHh21 and Proof. Similar to the proof of Lemma 4.1. W(r,h2)(3,hi) = |G| |wHhi | . □ Lemma 4.3. Let (g1, h1) and (g2, h2) be arbitrary vertices of type GH 2 in r = GDH. Then and Wri,hi)(32,h2) W(32,h2)(3i,hi) |EHh2 ||W , + |wH | |WG u eg | 3i32 I 1 I hi"2I I 3i32 U ^3i32 I |E H | + |wHh h , ,W0G„ U EG„ | 2"i II 323i 3i32 I ^hih-2 I I '' 323i I Proof. Let (a, x) be an arbitrary vertex of r. It follows from the equality (4.1) that dr((g1,h1), (a,x)) = dG(g1,a) + Oh(hux) and dr((g2,h2), (a, x)) = dG(g2,a) + Oh(h2,x). (4.2) (4.3) There are three different cases according to the distance of h1 and h2 from x in H. In the first case let dH(h1, x) = dH(h2, x). From equalities (4.2) and (4.3) we get that dr((g1, h1), (a, x)) < dr((g2, h2), (a,x)) ^^ dG(g1,a) < 8g(g2, a). This is true for exactly those (a, x) G V(T), for which a G WGg2. Similarly we get that dr((g2, M, (a, x)) < dr((gi, hi), (a, x)) ^^ dG(g2,a) < dG(gi,a). And this is true for exactly those (a, x) G V(r), for which a G WGgi. In the second case let dH(h1, x) < dH(h2, x). Since h1 and h2 are adjacent in H, it is obvious that dH(h2, x) = dH(h1, x) + 1. From equalities (4.2) and (4.3) we get that dr((gi, hi), (a, x)) < 3r((g2, ^-2), (a,x)) ^^ 5G(gi,a) < dG(g2, a) + 1. This is true for exactly those (a, x) G V(r), for which a G WgGS2 U EGS2. Similarly we get that dr((g2, h2), (a, x)) < dr((gi, hi), (a,x)) ^^ dG(g2,a) + 1 < dG(gi,a). But such vertices do not exist, since dG(gi, a) < dG(g2, a) + 1 by the triangle inequality. In the third case let dH(h2, x) < dH(hi, x). Similarly as above we get that (a, x) is closer to (g2, h2) that to (gi, hi) if and only if a G WGSl U E^92, and that (a, x) is never closer to (gi, hi) that to (g2, h2). It follows from the above comments that Next theorem gives the characterization of connected 2-distance-balanced cartesian products of graphs G and H. Theorem 4.4. The cartesian product r = GDH is a connected 2-distance-balanced graph if and only if each of G, H is either a connected 2-distance-balanced and 1-distance-balanced graph, or a complete graph. Proof. We first prove that if each of G, H is either a connected 2-distance-balanced and 1-distance-balanced graph or a complete graph, then r is a connected 2-distance-balanced graph. Let us assume that G and H are connected 2-distance-balanced and 1-distance-balanced graphs. The connectivity of r follows from the connectivity of G and H. In this case all three types of vertices at distance 2 are present in r. Let (gi, h) and (g2, h) be arbitrary vertices of type G2 in r. Since G is, by the assumption, 2-distance-balanced and since vertices gi, g2 are at distance 2 in G, it follows from Lemma 4.1 that WLhi)(S2,h2) = (EHh2 x WGS2) U Mh2 x (WGS2 u EGlS2)) and WLh2)(Si,M = (EHh2x WL) U (WHhi x (W2 „ u )). The result follows. □ So for arbitrary vertices (gi, h), (g2, h) G V(r) of type G2, the number of vertices that are closer to (gi, h) than to (g2, h) in r equals the number of vertices that are closer to (g2, h) than to (gi, h) in r. If (g, h1) and (g, h2) are arbitrary vertices of type H2 in r, then similarly as above (using Lemma 4.2 instead of Lemma 4.1) we find that the number of vertices that are closer to (g, h1) than to (g, h2) in r equals the number of vertices that are closer to (g, h2) than to (g, h1) in r. Let (g1, h1), (g2, h2) G V(r) be arbitrary vertices of type GH2 in r. Since G and H are both, by the assumption, 1-distance-balanced, and since g1, g2 are adjacent in G and h1, h2 are adjacent in H, we have \WG \WG and \W, H hih.2 \W H hohi It follows from Lemma 4.3 that W(g1,h1)(S2,h2) W(g2,h2 )(gi,hi) for arbitrary vertices of type GH2 in r. So we proved that if G and H are both connected 2-distance-balanced and 1-distance-balanced graphs, then the cartesian product r = GDH is a connected 2-distance-balanced graph. Note that since G and H are 1-distance-balanced graphs, it follows that the cartesian product r = GDH is also 1-distance-balanced (see [12, Proposition 4.1]). If one (or both) of G, H is a complete graph, then the proof that r = GDH is a connected 2-distance balanced graph is similar to the proof above. The only diference is that we do not have to consider vertices of type G2 (H2, respectively). Assume now that r = GDH is a connected 2-distance-balanced graph. The connectivity of G and H follows from the connectivity of r. If G and H are complete graphs, then we are done. Therefore we assume that at least one of G or H is not a complete graph. First we show that in this case G and H are 2-distance-balanced graphs provided they are not complete. Assume that G is not a complete graph. For an arbitrary h G V (H) and arbitrary gi, 02 G V (G) that are at distance 2 in G, consider (g^ h), (g2, h) G V (r). Note that dr((gi, h), (32, h)) = 2 by (4.1) and that W(gi,h)(s2 ,h) |HI \WG and W(s2,h)(si,h) |H| \WG by Lemma 4.1. Since r is 2-distance-balanced, it follows that | WgGg21 = | WgGgi |, so also G is a 2-distance-balanced graph. Due to commutativity of the cartesian product, if H is not a complete graph, we can similarly show that H is a 2-distance-balanced graph. Finally we show that G and H are also 1-distance-balanced graphs. Pick arbitrary adjacent vertices g1,g2 of G and arbitrary adjacent vertices h1, h2 of H, and note that (g1, h1), (g2, h2) g V(r) are at distance 2. Since r is 2-distance-balanced, it follows that W(ri,hi)(S2,h2) W(g2,h2)(gi,hi) From Lemma 4.3 we get that \ rp H \ \Ehi h2 \ (\W SiS2l I S2Si I = \WhHhi \ \ws2si u eSGs2 \ -\WHh2\ \ wG u \ I Si S2 u ^SiS2 I (4.4) Assume that G is not a 1-distance-balanced graph. Then we could choose gi, in such a ' I > |WG I 9192 I ^ I 9291 I way that \ W„G„2\ > IWGL |. As a consequence we also have that \WG U EG \ > \ WG U EG \ . i 9i92 gig2 \ ^ i g2gi w 9i92 i It follows from (4.4) that |WHh \ > \WH^ \. Consider now vertices (gi, h2), (g2, hi), which are also at distance 2 in r. Similar argument as above shows that jW^^ \ < \ WH^ \, which is a contradiction. So G is a 1-distance balanced graph. Since the cartesian product is commutative, the proof that H is a 1-distance balanced graph is analogous to the proof for G. □ 5 2-distance-balanced lexicographic product Throughout this section let G and H be graphs and let r = G[H] be the lexicographic product of G and H. It follows from the definition that the lexicographic product r is connected if and only if G is connected. In order to avoid trivialities we assume that | V(G) | > 2 and |V(H)| > 2. We characterize connected 2-distance-balanced lexicographic products of G and H (see Theorem 5.4). Remark 5.1. A more general result about the characterization of connected n-distance-balanced lexicographic products of G and H as in Theorem 5.4 is stated in [7, Theorem 3.4]. But the result is not correct for at least n = 2. As a counterexample, let both G and H be paths on 3 vertices, which are connected graphs. Observe that G is 2-distance-balanced, and that H is locally regular (in a sense that any non-adjacent vertices in H have the same number of neighbours). By [7, Theorem 3.4], G[H] is 2-distance-balanced. However, one can easily check that the G[H] is not 2-distance-balanced. Notice that there exist two different types of vertices at distance 2 in r. We now state these two types and we will refer to them in the proof of the Theorem 5.4. Let (g1, h1), (g2, h2) € V(r) be vertices at distance 2 in r. We say that this two vertices are of type • G2, if dG(gi,g2) =2, • H2, if g1 = g2 and dH(h1, h2) > 2. It follows from the definition that there exist vertices of type G2 in r if and only if G is connected non-complete graph. Similarly, there exist vertices of type H2 in r if and only if H is non-complete graph. The following two lemmas will be used in the proof of the main theorem of this section. Lemma 5.2. Let (g1, h1) and (g2, h2) be arbitrary vertices of type G2 in r = G[H]. Then 1 + |Nh(h1)| + (\WgGg2\- 1) |V(H)| W(gi,hi)(g2,h2) and W(g2,h2)(gi,hi) = 1 + |Nh(h2)| + (\WgGgi |- 1) \ V(H)|. Proof. Let (g1, h1) and (g2, h2) be arbitrary vertices of type G2 in r. Clearly, (g1, h1) is closer to itself than to (g2, h2). Now consider vertices of r of type (g1, h), where h = h1. Note that 3r((g1,h), (g2,h2)) = 2, and so (g1,h) G W(r9i,hi)(92jh2) if and only if h € NH(hi). Finally, consider vertices of r of type (g, h), where g = gi. Then ^((g^ hl), (g, h)) = dG(gi,g),andso (g, h) € W(rs1ih1)(S2,h2) ifandonlyifg € \ {gi}. It follows that Wto ,hi)(fl2,h2) = {(gi, hi)} U ({gi} x Nh (hi)) u ((W^ \ {gi}) x V (H)). Similarly we get W(rS2,h2)(gi,hi) = {(g2, h2)} u ({g2} x Nh(h2)) U ((WGgi \ {g2}) x V(H)). The result follows. □ Lemma 5.3. Let (g, hi) and (g, h2) be arbitrary vertices of type H2 in r = G[H]. Then 1 + |Nh(hi)|-|NH(hi) n Nh(h2)| W(g,hi)(g,h2) and 1 + |Nh(h2)|-|NH(hi) n Nh(fcs)|. W(g,h2)(g,hi) Proof. Let (g, hi) and (g, h2) be arbitrary vertices of type H2 in r, and let (g', h') be an arbitrary vertex of r. Note that if g = g' then dr((g, hi), (g', h')) = dr((g, h2), (g', h')). Assume therefore that g' = g. But it is clear that in this case (g, h') € W^ hi )(g h2) if and only if dH(hi, h') < 1 < dH(h2, h'). It follows that ' ' W(rg,hi)(g,h2) = {(g, hi)} U ({g} x (Nh (hi) \ (Nh (hi) n Nh (hi)))). Similarly we get W(rg,h2)(g,hi) = {(g, h2)} U ({g} x (Nh(h2) \ (Nh(hi) n Nh(hO))). The result follows. □ Next theorem gives the characterization of connected 2-distance-balanced lexicographic products of graphs G and H. Theorem 5.4. The lexicographic product r = G[H] is a connected 2-distance-balanced graph ifandonly if one of the following (i), (ii) holds: (i) G is a connected 2-distance-balanced graph and H is a regular graph. (ii) G is a complete graph, H is not a complete graph, and each connected component of the complement of H induces a regular subgraph of the complement of H. Proof. We first prove that if one of (i), (ii) holds, then r is a connected 2-distance-balanced graph. The connectivity of r follows from the connectivity of G. Assume that (i) holds. Take arbitrary (gi, hi), (g2, h2) € V(r) of type G2. Since G is a 2-distance-balanced graph and H is a regular graph, we have that | W^, | = | WgG 1 and |Nh(hi)| = |Nh(h2)|. It follows from Lemma 5.2 that W(Si,hi)(g2,h2) W(rg2,h2 )(gi,hi) for arbitrary vertices of type G2 in r. Take now arbitrary (g, hi), (g, h2) e V(r) of type H2. Since, by the assumption, H is a regular graph, we have that |NH(hi) | = |NH(h2) |. It follows from Lemma 5.3 that W(g,hi)(S,h2) W(s,h2)(s,hi) for arbitrary vertices of type H2 in r. So, if (i) holds then r is a connected 2-distance-balanced graph. Assume that (ii) holds. Then G is a complete graph and H is not a complete graph, so we only have vertices of type H2 in r. Let us denote the complement of H by H. Let (g, hi), (g, h2) e V(r) be arbitrary vertices of type H2. Note that this implies that hi, h2 are not adjacent in H, and so hi, h2 are adjacent in H. As a consequence, hi, h2 are contained in the same connected component of H. It follows that |Nh(hi)| = |N^(h2)|, and consequently also |NH(hi)| = |NH(h2)|. It follows from Lemma 5.3 that W(g,h1)(gM) W(g,h2)(s,hi) So, if (ii) holds then r is a connected 2-distance-balanced graph. Assume now that the lexicographic product r = G[H] is a connected 2-distance-balanced graph. The connectivity of G follows from the connectivity of r. In what follows we first treat the case where G is not a complete graph, and then the case when G is a complete graph. Suppose that G is not a complete graph. Take arbitrary gi, g2 e V(G) at distance 2 in G. Then (gi, h), (g2, h) e V(r) are of type G2 in r for an arbitrary h e V(H). Since r is, by the assumption, a 2-distance-balanced graph, it follows from Lemma 5.2 that | WflGS21 = | Wg | for arbitrary vertices at distance 2 in G. So, G is a connected 2-distance-balanced graph. For arbitrary hi,h2 e V(H) and arbitrary gi,g2 e V(G) at distance 2 in G, consider (gi, hi), (g2, h2) e V(r). These two vertices are of type G2 in r. Since r is, by the assumption, a 2-distance-balanced graph and we already know that G is also 2-distance-balanced graph, it follows from Lemma 5.2 that |NH(hi)| = |NH(h2)| for arbitrary two vertices in H. So, H is a regular graph and (i) holds. From now on let G be a complete graph. Since r is not a complete graph, it follows that also H is not a complete graph. This means that all vertices at distance 2 in r are of type H2. We want to show that in this case each connected component of the complement of H induces a regular subgraph of the complement of H. Let hi,h2 e V(H) be arbitrary vertices at distance greater or equal than 2 in H (that is, vertices hi, h2 are not adjacent in H). Observe that (g, hi), (g, h2) e V(r) are of type H2 for an arbitrary g e V(G). From Lemma 5.3 we get that |NH(hi)| = |NH(h2)|, and consequently also |N;H(hi)| = |N^(h2)|. This shows that any adjacent vertices of H have the same valency in H, and therefore each connected component of H induces a regular subgraph of H. □ We finish our paper with a suggestion for further research. A fullerene is a cubic planar graph having all faces 5- or 6-cycles. Examples include the dodecahedron and generalized Petersen graph GP(12,2). Dodecahedron is distance-regular, and so it is n-distance-balanced for every 1 < n < 5 (recall that the diameter of dodecahedron is 5). On the other hand, the diameter of GP(12, 2) is also 5, but GP(12, 2) is n-distance-balanced only for n = 5, see [17]. Therefore, it would be interesting to know, which fullerenes are n-distance-balanced at least for some values of n (for example, for n G {1, 2, D}, where D is the diameter of a fullerene in question). For more on fullerenes see [2, 5, 18]. References [1] A. Abiad, B. Brimkov, A. Erey, L. Leshock, X. Martinez-Rivera, S. O, S. Y. Song and J. Williford, On the Wiener index, distance cospectrality and transmission-regular graphs, Discrete Appl. Math. 230 (2017), 1-10, doi:10.1016/j.dam.2017.07.010. [2] 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. [3] K. Balakrishnan, B. Bresar, M. Changat, S. Klavzar, A. Vesel and P. Žigert Pletersek, Equal opportunity networks, distance-balanced graphs, and Wiener game, Discrete Optim. 12 (2014), 150-154, doi:10.1016/j.disopt.2014.01.002. [4] K. Balakrishnan, M. Changat, I. Peterin, S. Spacapan, P. Sparl and A. R. Subhamathi, Strongly distance-balanced graphs and graph products, European J. Combin. 30 (2009), 1048-1053, doi:10.1016/j.ejc.2008.09.018. [5] A. Behmaram, T. Doslic and S. Friedland, Matchings in m-generalized fullerene graphs, Ars Math. Contemp. 11 (2016), 301-313, doi:10.26493/1855-3974.882.539. [6] S. Cabello and P. Lukssics, The complexity of obtaining a distance-balanced graph, Electron. J. Combin. 18 (2011), #P49, http://www.combinatorics.org/ojs/index.php/ eljc/article/view/v18i1p4 9. [7] M. Faghani, E. Pourhadi and H. Kharazi, On the new extension of distance-balanced graphs, Trans. Comb. 5 (2016), 21-34, doi:10.22108/toc.2016.15048. [8] B. Frelih, Različni vidiki povezave regularnosti v grafih, Ph.D. thesis, University of Primorska, 2014, in Slovene. [9] K. Handa, Bipartite graphs with balanced (a, b)-partitions, Ars Combin. 51 (1999), 113-119, http://www.combinatorialmath.ca/arscombinatoria/vol51.html. [10] T. Hilado and K. Nomura, Distance degree regular graphs, J. Comb. Theory Ser. B 37 (1984), 96-100, doi:10.1016/0095-8956(84)90050-9. [11] A. Ilic, S. Klavzar and M. Milanovic, On distance-balanced graphs, European J. Combin. 31 (2010), 733-737, doi:10.1016/j.ejc.2009.10.006. [12] J. Jerebic, S. Klavzar and D. F. Rall, Distance-balanced graphs, Ann. Comb. 12 (2008), 71-79, doi:10.1007/s00026-008-0337-2. [13] K. Kutnar, A. Malnic, D. Marusic and S. Miklavic, Distance-balanced graphs: Symmetry conditions, Discrete Math. 306 (2006), 1881-1894, doi:10.1016/j.disc.2006.03.066. [14] K. Kutnar, A. Malnic, D. Marusic and S. Miklavic, Matchings in m-generalized fullerene graphs, Ars Math. Contemp. 2 (2009), 41-47, doi:10.26493/1855-3974.75.895. [15] K. Kutnar and S. Miklavic, Nicely distance-balanced graphs, European J. Combin. 39 (2014), 57-67, doi:10.1016/j.ejc.2013.12.002. [16] S. Miklavic and P. Sparl, On the connectivity of bipartite distance-balanced graphs, European J. Combin. 33 (2012), 237-247, doi:10.1016/j.ejc.2011.10.002. [17] S. Miklavic and P. Sparl, ¿-distance-balanced graphs, 2017, arXiv:1702.05257 [math.CO] . [18] N. Tratnik and P. Zigert Pletersek, Resonance graphs of fullerenes, Ars Math. Contemp. 11 (2016), 425-435, doi:10.26493/1855-3974.1000.8db. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 97-112 https://doi.org/10.26493/1855-3974.1327.6ee (Also available at http://amc-journal.eu) The arc-types of Cayley graphs * Marston D. E. Conder, Nemanja Poznanovic Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Received 20 February 2017, accepted 6 November 2017, published online 26 February 2018 Let X be a finite vertex-transitive graph of valency d, and let A be the full automorphism group of X. Then the arc-type of X is defined in terms of the sizes of the orbits of the action of the stabiliser Av of a given vertex v on the set of arcs incident with v. Specifically, the arc-type is the partition of d as the sum n1 + n2 + ••• + nt + (m1 + mi ) + (m2 + m2) + • • • + (ms + ms), where n1, n2,...,nt are the sizes of the self-paired orbits, and m1, m1, m2,m2,... ,ms,ms are the sizes of the non-self-paired orbits, in descending order. In a recent paper, it was shown by Conder, Pisanski and Zitnik that with the exception of the partitions 1 + 1 and (1 + 1) for valency 2, every such partition occurs as the arc-type of some vertex-transitive graph. In this paper, we extend this to show that in fact every partition other than 1, 1 + 1 and (1 + 1) occurs as the arc-type of infinitely many connected finite Cayley graphs with the given valency d. As a consequence, this also shows that for every d > 2, there are infinitely many finite zero-symmetric graphs (or GRRs) of valency d. Keywords: Symmetry type, vertex-transitive graph, arc-transitive graph, Cayley graph, zero-symmetric graph, Cartesian product, covering graph. Math. Subj. Class.: 05E18, 20B25, 05C75, 05C76 *This work was supported by the N.Z. Marsden Fund (via grant UOA1323), and we acknowledge the assistance of Magma [1] in finding and constructing several examples of Cayley graphs with a given arc-type and small valency, as well as in testing various cases. We would also like to thank Joy Morris for asking the question we addressed here, and Gabriel Verret for some helpful discussions about case (d) in Section 4. E-mail addresses: m.conder@auckland.ac.nz (Marston D. E. Conder), nempoznanovic@gmail.com (Nemanja Poznanovic) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ 1 Introduction Vertex-transitive graphs hold a significant place in mathematics, and within this, a major role is played by Cayley graphs, which represent groups in a very natural way. A Cayley graph can be defined as any graph that admits some group of automorphisms which acts regularly (sharply-transitively) on the vertices of the graph. Equivalently, a Cayley graph can be constructed from the regular permutation representation of a group G, with vertices taken as the elements of G and edges indicating the effect of a subset S C G (by left multiplication). The set S U S-1 consists of the elements of G that take the identity vertex to one of its neighbours. It often happens that the automorphism group of a connected finite Cayley graph itself acts regularly on vertices. Any Cayley graph with this property is called a zero-symmetric graph, or a graphical regular representation of the group G, or briefly, a GRR. But of course the automorphism group of a Cayley graph X may be much larger than the vertex-regular subgroup G, and can sometimes even be the full symmetric group on the vertex-set (when the graph is null or complete). Intermediate cases, with Aut(X) larger than G but smaller than Sym(G), as well as other kinds of vertex-transitive graphs, fall into a number of different and interesting classes of graphs, including those that are arc-transitive (or symmetric), and those that are half-arc-transitive (which are vertex-transitive and edge-transitive but not arc-transitive). A means of classifying vertex-transitive graphs was given in a recent paper by Conder, Pisanski and Zitnik [3], using what is known as the arc-type of the graph. This can be defined as follows. Let X be a d-valent vertex-transitive graph, with automorphism group A, let Av be the stabiliser in A of any vertex v of X, and consider the orbits of Av on the set of arcs (v, w) with initial vertex v. The Av-orbit of any arc (v, w) can be 'paired' with the Av-orbit of the arc (v, w') whenever (v, w') lies in the same orbit of A as the reverse arc (w, v), and if those two A-obits are the same, then we say the Av-orbit of (v, w) is 'self-paired'. Then the arc-type of X is the partition n of its valency d as the sum n = ni + n +----+ nt + (mi + mi) + (m2 + m2) +-----+ (ms + ms) (f) where n1, n2,..., nt are the sizes of the self-paired orbits of Av on arcs with initial vertex v, and m1, m1, m2, m2,..., ms, ms are the sizes of the non-self-paired orbits, in descending order. Similarly, the edge-type of X is the partition of d as the sum of the sizes of the orbits of Av on edges incident with v, and can be found by simply replacing each bracketed term (mj + mj) by 2mj, for 1 < j < s. For example, if X is arc-transitive, then its arc-type is simply d, while if X is half-arc-transitive, then its valency d is even and its arc-type is (| + |), and X is a GRR if and only if all the terms n and mj in its arc-type are 1. The authors of [3] also answered the natural question of which arc-types occur for a given valency d. Every vertex-transitive graph of valency 2 is a union of cycles and is therefore arc-transitive, with arc-type 2. Hence in particular, the partitions 1 + 1 and (1 + 1) of 2 do not occur as the arc-type of a vertex-transitive graph. It was shown in [3] that these are the only exceptional cases. Using a construction that takes Cartesian products of pairwise 'relatively prime' vertex-transitive graphs, Conder, Pisanski and Zitnik proved that in all other cases, every partition of d as given in (f) occurs as the arc-type of some vertex-transitive graph X of valency d. In this paper, we prove a much stronger theorem, namely that every such partition other than 1, 1 + 1 and (1 + 1) occurs as the arc-type of infinitely many connected finite Cayley graphs. (This answers a question posed by Joy Morris at the 2015 PhD Summer School in Discrete Mathematics, in Rogla, Slovenia.) As corollaries, we find that every standard partition of a positive integer d is realisable as the edge-type of infinitely many connected finite Cayley graphs of valency d, except for 1 and 1 + 1 (when d < 2), and that for every d > 2, there are infinitely many finite zero-symmetric graphs of valency d. To prove our main theorem, we adopt the same approach as taken in [3], but show there are infinitely many Cayley graphs that can be used in the construction as building blocks with the required basic type. In particular, we show that the half-arc-transitive Bouwer graphs B(m, k, n) and the 'thickened covers' used in [3] are Cayley graphs, and we construct some new families of Cayley graphs with various arc-types as well. We begin by setting notation and giving some further background in Section 2. Then in Section 3 we briefly summarise what has to be done to prove our main theorem, which we proceed to do in Section 4. We complete the paper with the consequence for zero-symmetric graphs in Section 5. 2 Preliminaries and further background 2.1 Notation All the graphs we consider in this paper are finite, simple, undirected and non-trivial (in the sense of containing at least one edge). Given a graph X, we denote by V(X), E(X) and A(X) the set of vertices, the set of edges, and the set of arcs of X, respectively. We denote an edge with vertices u and v by {u, v}, and an arc from u to v by (u, v). The automorphism group of X is denoted by Aut(X). Note that the action of Aut(X) on the vertex-set V(X) also induces an action of Aut(X) on the edge-set E(X) and one on the arc-set A(X). If the action of Aut(X) is transitive on the vertex-set, edge-set, or arc-set, then we say that X is vertex-transitive, edge-transitive or arc-transitive, and sometimes abbreviate this to 'VT', 'ET' or 'AT', respectively. Obviously, vertex-transitive graphs are always regular. Moreover, because a disconnected vertex-transitive graph consists of pairwise isomorphic connected components, we may restrict our attention here to connected graphs. An arc-transitive graph is often also called symmetric. A graph is called half-arc-transitive if it is vertex-transitive and edge-transitive, but not arc-transitive. The valency of every half-arc-transitive graph is necessarily even; see [11, p. 59]. Now let G be a group, and let S be a subset of G that is inverse-closed and does not contain the identity element. Then the Cayley graph Cay(G, S) is the graph with vertex-set G, and with vertices u and v being adjacent if and only if vu-1 G S (or equivalently, v = xu for some x G S). Since we require S to be inverse-closed, this Cayley graph is undirected, and since S does not contain the identity, the graph has no loops. Also Cay(G, S) is regular, with valency |S|, and is connected if and only if S generates G. Furthermore, it is easy to see that G acts as a group of automorphisms of Cay(G, S) by right multiplication, and this action is transitive on vertices, with trivial stabiliser, and hence sharply-transitive (or regular). In particular, Cay(G, S) is vertex-transitive. More generally, a graph X is a Cayley graph for the group G if and only if G acts regularly on V (X) as a group of automorphisms of X. This is very well known — see [10] for example. 2.2 Cartesian products and (relatively) prime graphs Given a pair of graphs X and Y (which might or might not be distinct), the Cartesian product X □ Y is a graph with vertex set V(X) x V(Y), such that two vertices (x, y) and (u, v) are adjacent in X □ Y if and only if x = u and y is adjacent with v in Y, or y = v and x is adjacent with u in X. This definition can be extended to the Cartesian product Xi □ • • • □ Xk of a larger number of graphs X1,... ,Xk, which are then called the factors. A graph X is called prime (with respect to the Cartesian product) if it is not isomorphic to the Cartesian product of a pair of smaller, non-trivial graphs. Every connected graph can be decomposed as a Cartesian product of prime graphs, in a way that is unique up to reordering and isomorphism of the factors; see [6, Theorem 4.9] for a proof. Then two graphs can be said to be relatively prime (with respect to the Cartesian product) if there is no non-trivial graph that is a factor of both. Note that two prime graphs are relatively prime unless they are isomorphic. For the construction in [3] and here, we need a number of other properties of the Cartesian product, and some ways in which we can tell if a given graph is prime with respect to the Cartesian product. We summarise these as follows: Proposition 2.1. (a) The Cartesian product operation □ is associative and commutative. (b) A Cartesian product graph is connected if and only if all its factors are connected. (c) If X1,... ,Xk are regular graphs with valencies d1,... ,dk, then their Cartesian product X1 □ • • • □ Xk is also regular, with valency d1 + • • • + dk. (d) The Cartesian product ofCayley graphs is a Cayley graph. (e) If X1,... ,Xk are connected graphs that are pairwise relatively prime, then Aut(X) = Aut(X1) x • • • x Aut(Xk). (f) A Cartesian product of connected graphs is vertex-transitive if and only if all its factors are vertex-transitive. (g) If X1,...,Xk are non-trivial connected vertex-transitive graphs with arc-types t1 ,... ,Tk, and X1,... ,Xk are pairwise relatively prime, then the arc-type of their Cartesian product X = X1 □ • • • □ Xk is t1 + • • • + Tk. Proof. Parts (a) to (c) are easy, and part (d) follows by induction from the fact that Cay(G,S) □ Cay(H,T) = Cay(G x H, (S x {1H}) U ({1G} x T)). Proofs of parts (e) and (f) can be found in [6], and part (g) was proved in [3]. □ Proposition 2.2. Let X be a Cartesian product of non-trivial connected graphs. Then: (a) Every edge of X lies in some 4-cycle in X. (b) All the edges in any cycle of length 3 in X belong to the same factor of X. (c) If (x, y, z, w) is any 4-cycle in X, then the edges {x, y} and {z, w} belong to the same factor of X, as do the edges {y, z} and {x, w}. (d) [The square property] If two edges are incident in X but do not belong to the same factor of X, then there exists a unique 4-cycle in X that contains both of these edges, and this 4-cycle has no diagonals. Proof. Part (a) is easy, and others were proved in [7], for example. □ Corollary 2.3. Let X be a connected graph. If some edge of X is not contained in any 4-cycle (and in particular, if X has no 4-cycles), then X is prime. 2.3 Thickened covers Let X be any simple graph, F any subset of the edge-set of X, and m any positive integer. Then the authors of [3] defined the thickened m-cover of X over F as the graph X(F, m) that has vertex-set V(X) x Zm, and edges of two types: (a) an edge from (u, i) to (v, i), for every i e zm and every {u, v} e E(X) \ F, (b) an edge from (u, i) to (v, j), for every (i, j) e Zm x Zm and every {u, v} e F. One can think of this graph as being obtained from X by replacing each vertex of X by m vertices, and each edge by the complete bipartite graph Km,m whenever the edge lies in F, or by mK2 (a set of m 'parallel' edges) whenever the edge does not lie in F. For example, the thickened 2-cover of the cycle graph C6 over one of its 1-factors is shown in Figure 1. Figure 1: A thickened 2-cover of C6 (over a 1-factor). It was shown in [3] that if X is a vertex-transitive graph, and F is a union of orbits of Aut(X) on edges of X, then X(F, m) is vertex-transitive for every m > 2. We can take this further, as follows: Proposition 2.4. If X = Cay(G, S) is a Cayley graph, and F is an orbit of G on edges of X, then the thickened cover Y = X (F, m) is a Cayley graph for G x Zm. Proof. We show that Y is exactly the same as the Cayley graph Cay(G x Zm, W), where multiplication in the group G x Zm is given by (g, i)(h, j) = (gh, i + j) for all g, h e G and i, j e Zm, and W is the union of the two sets Wi = {(s,0): s e S, {1G,s} e F} and W2 = {(t, i) : {1G,t} e F, i e Zm}. Take any edge of Y of the first kind, say from (u, i) to (v, i) where {u, v} e E(X) \ F. Then v = su for some s e S, and it follows that (v,i) = (su, i) = (s, 0)(u, i), with {1G, s} = {u, su}u-1 = {u, v}u-1 e F. Conversely, if s e S and {1G, s} e F then {u, su} = {1G, s}u e F, and so (s, 0)(u, i) = (su, i) is adjacent to (u, i), for all u and i. Similarly, for any edge of the second kind, from (u, i) to (v, j) with {u, v} G F, we have v = tu for some t G S and so (v, j) = (t, j— i)(u,i) with {1G,t} = {u,v}u-1 G F, and conversely, if {1G,t} G F (where t G S), then {u,tu} = {1G,t}u G F, and therefore (t,j — i)(u, i) = (tu, j) is adjacent to (u, i), for all u, i and j. □ Also we need some other information about thickened covers, taken from [3]. The fibre over a vertex u of X is the set {(u, i) : i G Zm} of vertices of X(F, m), and any element of this set is said to project onto u. Similarly the fibre over an edge {u, v} of X is the set {{(u, i), (v, i)} : i G Zm} of edges of X(F, m) when {u, v} G E(X) \ F, or the set {{(u,i), (v,j)} : i, j G Zm} when {u,v} G F, and any element of this set is said to project onto {u, v}. The fibre over an arc is defined similarly. Proposition 2.5. Let X be a vertex-transitive graph, and let F be a union of edge-orbits of X, with the property that every edge in F joins vertices from two different components of X \ F. Then for every two arcs (x, y) and (u, v) from the same arc-orbit of X, any two arcs of X (F, m) that project onto (x, y) and (u, v) respectively lie in the same arc-orbit of X(F,m), for all m > 2. Proof. See [3, Theorem 7.6]. □ 2.4 Bouwer graphs The first known infinite family of half-arc-transitive graphs of arbitrary even valency greater than 2 was constructed by Bouwer [2] in 1970. These graphs were a sub-family of a wider class of graphs, which we now denote by B(k, m, n), defined as follows. Let m and n be any integers such that 2m = 1 mod n, with m > 2 and n > 3, and also let k be any integer such that k > 2. Then the vertices of B(k, m, n) may be taken as the k-tuples (a, b) = (a, b2, b3,..., bk) with a G Zm and bj G Zn for 2 < j < k, with any two such vertices being adjacent if and only if they can be written as (a, b) and (a + 1, c) where either c = b, or c = (c2, c3,..., ck) differs from b = (b2, b3,..., bk) in just one position, say position j, where cj = bj + 2°. Bouwer himself proved in [2] that every such graph is connected, edge-transitive and vertex-transitive, with valency 2k. He also proved that the graphs B(k, 6,9) are half-arc-transitive, and his theorem was extended recently by Conder and Zitnik [4], who proved that B(k, m, n) is arc-transitive only when n = 3, or (k, n) = (2, 5), or (k, m, n) = (2, 3, 7) or (2,6, 7) or (2, 6, 21). In particular, it follows that B(k, m, n) is half-arc-transitive whenever m > 6 and n > 5. Moreover, as shown in [4], if m > 6 and n > 7, then B(k, m, n) has girth 6, and hence in that case, B(k, m, n) is prime. These prime graphs gave the infinite family of half-arc-transitive graphs with arc-type (k + k), for each k > 2, used in Lemma 8.2 of [3]. We can take this further, by proving the following (which a referee has also pointed out was proved very recently by Ramos Rivera and Sparl in [9]): Proposition 2.6. Every Bouwer graph B(k, m, n) is a Cayley graph. Proof. First note that n is odd, since 2m = 1 mod n. Now let G be the semi-direct product Zm k Z^1, where a generator of the complement Zm acts by conjugation from the right on the kernel Z^-1 in the same way as component-wise multiplication by 2. Also let R be the set of all elements of G of the form (1, b), where b is either the zero vector 0 in Z^-1, or one of the elementary basis vectors ej (with all its entries being 0 except for a 1 in position j). The k elements of R are non-involutions, whose inverses are the elements of the form (-1, d) where d = 0 or —2-1ej for some i. It follows that the 2k-valent Cayley graph Cay(G, R U R-1) is isomorphic to the Bouwer graph B(k, m, n), for if (a, b) = (a, b2, b3,..., bk) is any vertex, then (1, 0)(a, b) = (a+1, b), and (1, ej)(a, b) = (a +1, b + 2aej) for all i. □ 3 Main theorem and overview of the proof As indicated in the Introduction, our main theorem and its first immediate corollary are as follows: Theorem 3.1. For any positive integer d, let n be any partition of d as given in (f). Then n occurs as the arc-type of infinitely many connected finite Cayley graphs of valency d, except when n is one of the partitions 1, 1 + 1 and (1 + 1) in the cases with d < 2. Corollary 3.2. With the exception of 1 and 1 + 1 (in the cases with d < 2), every standard partition of a positive integer d is realisable as the edge-type of infinitely many connected finite Cayley graphs ofvalency d. Corollary 3.2 follows easily from Theorem 3.1. To prove Theorem 3.1, we use much of the proof of the theorem in [3] showing that every such partition is the arc-type of at least one vertex-transitive graph of valency d. In that proof, the given partition n was written as a sum of 'basic' partitions, each having one of a number of forms, and then a VT graph with arc-type n was constructed as a Cartesian product of pairwise relatively prime graphs with arc-types of the associated forms. This required a good supply of prime vertex-transitive graphs with particular arc-types as 'building blocks', and the following were sufficient. (a) Arc-type m: infinitely many prime connected VT graphs, for each integer m > 2; (b) Arc-type (m + m): infinitely many prime connected VT graphs, for each m > 2; (c) Arc-type m +1: infinitely many prime connected VT graphs, for each m > 2; (d) Arc-type 1 + (1 + 1): at least two prime connected VT graphs; (e) Arc-type m + (1 + 1): at least one prime connected VT graph, for each m > 2; (f) Arc-type 1 + (m + m): at least one prime connected VT graph, for each m > 2; (g) Arc-type (1 + 1) + (1 + 1): infinitely many prime connected VT graphs; (h) Arc-type (m + m) + (1 + 1): at least one prime connected VT graph, for each m > 2; (i) Arc-type 1 + 1 + 1: infinitely many prime connected VT graphs; (j) Arc-type 1 + 1 + (1 + 1):at least one prime connected VT graph; (k) Arc-type 1 + 1 + 1 + 1: at least one prime connected VT graph; (l) Arc-type (1 + 1) + (1 + 1) + (1 + 1):at least one prime connected VT graph. Now to extend this to a proof of our theorem, we need infinitely many connected finite Cayley graphs of each of the basic forms listed in cases (a) to (l) above. Such infinite families were provided explicitly for cases (g) and (i) in Lemmas 8.6 and 8.8 of [3]. Also in cases (d), (e) and (g), a single vertex-transitive graph was produced for each m in Lemmas 8.4, 8.5 and 8.7 of [3], as a thickened cover of a particular Cayley graph over an edge-orbit. These are Cayley graphs, by Proposition 2.4, but we have to produce infinitely many of them, for each m > 2. Hence it remains for us to find infinitely many connected finite Cayley graphs in the cases (a)-(f), (h), and (j)-(l) above. We do that in the next section. Specifically, we construct new families of Cayley graphs for cases (a), (d) and (j)-(l), we use the Bouwer graphs for case (b), we show that the thickened covers used in [3] for case (c) are Cayley graphs, and we show that thickened covers of the graphs in cases (d) and (g) provide infinitely many Cayley graphs for cases (e), (f) and (h). 4 Proof of main theorem As noted earlier, all we need to do to prove Theorem 3.1 is show that there exist infinitely many prime connected finite Cayley graphs with each of the arc-types in the cases listed in the previous section, and then the rest follows by the same argument as in [3, Section 9]. We do this case-by-case below. For completeness, we give a brief description of the Cayley graphs in the cases that do not require any further analysis, and we give more detailed arguments for the rest. Case (a): Arc-type m, for all m > 2. For m = 2, we can take the family of all cycle graphs Cn with n > 5. These graphs have arc-type 2, and since they contain no 4-cycles, by part (a) of Proposition 2.2 they are all prime (with respect to the Cartesian product). For m > 3, we construct an infinite family of arc-transitive prime connected finite Cayley graphs of valency m using the same groups as for this case in [3, Lemma 8.1]. We know by Macbeath's theorem [8] that for every prime p > m, the simple group G = PSL(2,p) is generated by elements x and y such that x2 = ym = (xy)m+4 = 1. Now take S to be the set {x, y-1xy, y-2xy2,..., y-(m-1)xym-1} of all conjugates of x by powers of y, and let X = Cay(G, S). The elements of S are distinct involutions (since G has trivial centre), and so X has valency |S| = m. Moreover, the subgroup generated by S is normal in (x, y) = G, because x G S and conjugation by y permutes the elements of S among themselves. Hence S generates G, and therefore X is connected. But also conjugation by y induces an automorphism of X that fixes the identity vertex and cyclically permutes its m neighbours among themselves, and so X is arc-transitive. Hence X has arc-type m. Finally, X is prime, for if it were the Cartesian product of two relatively prime graphs Y and Z, then its arc-type m would be the sum of the arc-types of Y and Z, and if it were the kth Cartesian power of some prime graph Y, then we would find that | V(Y) |k = |V(X)| = |G| = | PSL(2,p)| = p(p2 - 1)/2, which can occur only if k = 1. Case (b): Arc-type (m + m), for all m > 2. If n and r are any integers such that 2r = 1 mod n, with n > 7 and r > 6, then by Lemma 8.2 of [3], the Bouwer graph B(m, r, n) is a prime half-arc-transitive graph with arc-type (m + m), for every m > 2. Also by Proposition 2.6 above, this graph is a Cayley graph. Hence in particular, the Bouwer graph B(m, r, n) is a prime Cayley graph with arc-type (m + m), whenever m > 2, r > 6 and n > 7. Case (c): Arc-type m + 1, for all m > 2. By Theorem 7.5 of [3], for every integer m > 2 and every integer n > 3, the thickened m-cover of the n-cycle C2n over one of its 1-factors is a prime VT graph with arc-type m + 1 (that is, with two self-paired arc orbits of lengths m and 1). This thickened cover is also a Cayley graph, as we show below. Let a and b be canonical generators for the dihedral group Dmn of order 2mn, satisfying amn = b2 = (ab)2 = 1, and define Ymn = Cay(Dmn,S) where S is the set {b, ban, ba2n,..., ba(m-1)n, ba}, consisting of m+1 involutions. Now let n be the natural epimorphism from Dmn to Dn = Dmn/Cm with kernel (an} = Cm, and let S and x be the images of S and any x G Dmn under n. Then n induces a graph homomorphism from Ymn to Cay(Dn, S) = Cay(Dn, {b, ba}), which is clearly a cycle of length |Dn | = 2n. Moreover, the pre-image of an edge of the form {x, bx} is a complete bipartite subgraph of order 2m with m2 edges {xz, bxw} for z, w G (an} = Cm, while the pre-image of an edge of the form {x, bax} is a subgraph of order 2m with m parallel edges {xz, baz} for z G (an} = Cm. Hence Ymn is isomorphic to the m-thickened cover of C2n used in Theorem 7.5 of [3], and so is a prime Cayley graph with arc-type m + 1, for all m > 2 and all n > 3. Case (d): Arc-type 1 + (1 + 1). Let p be any prime such thatp = 1 mod 4, with p > 5, and let k be any integer such that k2 = -1 mod p. Now take G to be the semi-direct product Cp x k C4, which is generated by elements a and b such that ap = b4 = 1 and b-1ab = ak. Note that conjugation by b2 inverts a, while bab-1 = a-k. Now let X = Cay(G, {b, b-1, ab2}). This graph is 3-valent, since ab2 is an involution, and connected, since (b, ab2} = G. It is also non-bipartite, because if it were bipartite, then its parts would be preserved by the only subgroup of index 2 in G, namely the subgroup generated by a and b2, but that cannot happen since there is an edge from 1 to ab2. We will show that X is prime and has arc-type 1 + (1 + 1), for all p and k. First, note that the arcs (1, b), (1, b-1) and (1, ab2) can each be extended to a path of length 2 in two ways, namely to (1, b, b2), (1,b, ab3), (1,b-1,b2), (1, b-1,ab), (1, ab2, a-kb3) and (1, ab2, akb). It follows that the edge {1, ab2} lies in no 4-cycle, and in particular, that X is prime. Moreover, the edges {1,b} and {1,b-1} lie in a single 4-cycle up to reversal, namely (1, b, b2, b-1), and so {1, ab2} lies in a different edge orbit from {1, b} and {1, b-1}. On the other hand, the edges {1, b} and {1, b-1} lie in the same orbit of Aut(X), since right multiplication by b-1 takes the former to the latter. Hence the edge-type of X is 2 + 1, and the arc-type of X must be 1 + (1 + 1) or 2 + 1. Next, we consider the stabiliser A1 in A = Aut(X) of the identity vertex 1. By the above observations, A1 fixes the vertex ab2, as well as b2 (the only other common neighbour of b and b-1) and a-1 (the vertex opposite ab2 in the 4-cycle (ab2, a-kb-1, a-1, akb) containing ab2). By induction and connectedness, A1 fixes every vertex in the orbit of the subgroup H of G generated by a and b2. The latter subgroup has index 2 in G, with coset representatives 1 and b, and if fi is any element of A1 that also fixes the vertex b, then by vertex-transitivity, fi fixes every vertex in the orbit of the coset bH, and hence fixes every vertex, so fi is trivial. It follows that |A1| < 2, and furthermore, since A = GA1 (with G n A1 = {1}), we find that G has index 1 or 2 in A. Hence in particular, G is normal in A (a fact which also follows from a theorem by Zhou and Feng [12, Theorem 2.3] on 3-valent Cayley graphs of order 4p, for p prime). Now suppose that A1 is non-trivial. Then there exists an automorphism 0 G A1 such that A = (G, 0}, and moreover, 0 has order 2 and must swap the neighbours b and b-1 of 1, and fix ab2. Hence conjugation of G by 0 fixes ab2 and swaps b with b-1 (as elements of G). It follows that 0 fixes b2 and hence also fixes (ab2)b2 = a, but then ak = (ak) = (b 1a6)e = bab 1 = a k, and so ak has order 2, contradiction. Thus Ai is trivial, and Aut(X) = A = G, so X is a GRR, with arc-type 1 + (1 + 1). Case (e): Arc-type m + (1 + 1), for all m > 2. For any prime p = 1 mod 4 with p > 5, and any square root k of —1 mod p, let X be the Cayley graph for Cp x k C4 produced in case (d) above. This graph has two edge-orbits, one of length 4p containing the edge {1, b}, and the other of length 2p containing the edge {1, ab2}, where a and b are generators for G = Cp xk C4 satisfying the relations ap = b4 = 1 and b - 1ab = ak. Now let Ym = X(F, m) be the thickened m-cover of X over F, where F is the smaller of the two edge-orbits of X. Then Ym is regular of valency m+2, and is a Cayley graph, by Proposition 2.4, so all we have to do is show that Ym is prime and has arc-type m + (1 +1). We do this in much the same way as was done for the single example (for each m) in [3, Lemma 8.4]. First, we note that X \ F is a union of quadrangles (unordered 4-cycles), and every edge of F joins vertices from different quadrangles. Hence by Proposition 2.5, we find that all edges in a fibre over an edge in E(X) \ F lie in the same edge-orbit of Ym, and all edges in a fibre over an edge in F lie in the same edge-orbit. In particular, all edges of the form {(1,0), (ab2,i)} for i G Zm lie in the same edge-orbit of Ym. Also multiplication by (b, 0) puts {(1,0), (b -1, 0)} in the same edge-orbit as {(1,0), (b, 0)}. On the other hand, up to reversal the edge {(1,0), (b, 0)} lies in just one 4-cycle, namely ((1,0), (b, 0), (b2,0), (b-1,0)), while the edge {(1,0), (ab2,0)} lies in (m — 1)2 distinct 4-cycles, namely ((1,0), (ab2,0), (1,j), (abV)) for j,^ G Zm \ {0}. Hence if m > 2 then {(1,0), (b, 0)} cannot lie in the same orbit as {(1,0), (ab2,0)}. Similarly, when m = 2, up to reversal the edge {(1,0), (b, 0)} lies in precisely four 6-cycles, namely ((1,0), (b, 0), (ab-1,j), (b, 1), (1,1), (abV)) for j,^ G {0,1}, while the edge {(1,0), (ab2,0)} lies in eight 6-cycles, viz. ((1,0), (ab2,0), (1,1), (be, 1), (ab-e,j), (be, 0)) and ((1, 0), (ab2,0), (aekbe, 0), (a1-ekb-e, j), (aekbe, 1), (ab2,1)) for e = ±1 and j G {0,1}, and again we find that the edge {(1,0), (b, 0)} cannot lie in the same orbit as the edge {(1,0), (ab2,0)}. Hence the edge-type of Ym is m + 2, and its arc-type is m + 2 or m + (1 + 1). Next, consider the stabiliser A(10) in A = Aut(Ym) of the vertex (1,0). We know that A(10) preserves the set of m neighbours of (1,0) of the form (ab2, i) for i G Zm, and as a consequence, A(10) must preserve the set of all paths of length 2 of the form ((1,0), (ab2, i), y). For any such i, the third vertex y is either (a-kb-1, i), or (akb, i), or (1, j) for some j G Zm \ {0}. Moreover, if y = (a-kb-1, i) or (akb, i), then there is just one path of length 2 from (1,0) to y, while if y = (1, j) for some j, then there are m distinct paths of length 2 from (1,0) to y. Hence A(10) must preserve the set of all vertices (1, j) with j g Zm \ {0}, and so A(10) preserves the fibre over (1,0). By vertex-transitivity, the same thing holds for every vertex, and so Aut(Ym) permutes the fibres over vertices of X. Hence every automorphism of Ym can be projected to an automorphism of X .In particular, since X has arc-type 1 + (1 + 1), no automorphism can take the arc ((1, 0), (b, 0)) to the arc ((1,0), (b-1, 0)), and thus Ym has arc-type m+(1 +1). Finally, we show that Ym is prime. To do this, consider any decomposition of Y into Cartesian factors, which are connected and vertex-transitive, by Proposition 2.1. The edge {(1,0), (b, 0)} does not lie in a 4-cycle with any of the m edges of the form {(1,0), (ab2,i)} for i G Zm, and so by part (d) of Proposition 2.2, all of those m edges must lie in the same factor of Ym as {(1,0), (b, 0)}, say Z. The same argument holds for the edge {(1,0), (b-1,0)}, and so this edge must lie in Z as well. Hence Z contains all m + 2 edges incident with the vertex (1,0). By vertex-transitivity and connectedness, all edges of Ym lie in Z, so Z = Ym, and therefore Ym is prime. Case (f): Arc-type 1 + (m + m), for all m > 2. This case is similar to the previous one, except that we let Ym = X(F, m) be the thickened m-cover of X where this time F is the larger of the two edge-orbits of X. Again, Ym is a Cayley graph, by Proposition 2.4, but of valency 2m + 1, and all we have to do is show that Ym is prime and has arc-type 1 + (m + m). The neighbours of the vertex (1,0) are the 2m vertices of the form (b, i) or (b-1, i) where i G Zm, plus the single vertex (ab2,0). It is easy to see that every edge of the form {(1,0), (b±1, i)} lies in many different 4-cycles, while the edge {(1,0), (ab2,0)} lies in no 4-cycles at all. In particular, this shows that Ym is prime, and that the vertex (ab2,0) is fixed by the stabiliser A(10) of (1,0) in A = Aut(Ym). Moreover, X \ F is a union of non-incident edges, and so by Proposition 2.5, all arcs of the form ((1,0), (b, i)) lie in the same arc-orbit of Ym, and the same holds for all arcs of the form ((1,0), (b-1, i)). Hence the arc-type of Ym is either 2m + 1 or 1 + (m + m). To prove that the arc-type is 1 + (m + m), again we consider the local effect of the stabiliser A(10) on vertices at short distance from the vertex (1,0). We know that A(10) preserves the set of 2m neighbours of (1,0) of the form (b±1, i) for i G Zm, and fixes the neighbour (ab2,0). In particular, A(10) must preserve the set of all paths of length 2 of the form ((1,0), (b±1, i), y). This time the third vertex y is either (ab3, i), or (ab, i), or (1, or (b2, for some I G Zm, and in the first two cases, there is just one such path of length 2 from (1,0) to y, while if y = (1, or (b2, for some A then there are 2m such paths. Also each vertex v of the form (1, or (b2, lies at distance 3 from the vertex (ab2,0) fixed by A(1j0), via the 2m paths (v, (be, j), (1,0), (ab2, 0)) with e = ±1 and j G Zm. Moreover, if v is one of the vertices of the form (1, ¿), then there are 2m additional paths, namely ((1, ¿), (ab2, (aefcbe, j), (ab2,0)) for e = ±1 and j G Zm, but there are no such additional paths from a vertex of the form (b2, It follows that no element of A(10) can take a vertex of the form (1, to one of the form (ab3, i) or (ab, i) or (b2, ¿"), and therefore A(10) preserves the fibre over (1,0). By vertex-transitivity, the same thing holds for every vertex, and hence as before, every automorphism of Ym can be projected to an automorphism of X. In particular, since X has arc-type 1 + (1 + 1), no automorphism can take the arc ((1,0), (b, 0)) to the arc ((1,0), (b-1,0)), and thus Ym has arc-type 1 + (m + m). Case (g): Arc-type (1 + 1) + (1 + 1). By Lemma 8.6 of [3], if p is any prime number with p = 1 mod 6, if k is a primitive 6th root of 1 mod p, and G is the semi-direct product Cp xfc C6, generated by two elements a and b of orders p and 6 such that b-1ab = ak, then the Cayley graph Cay(G, {b, b-1, ab2, (ab2)-1}) is prime and has arc-type (1 + 1) + (1 + 1). In fact, the edges {1, ab2} and {1, (ab2)-1} lie in 3-cycles, but the edges {1, b} and {1, b-1} do not. Case (h): Arc-type (m + m) + (1 + 1), for all m > 2. For any prime p = 1 mod 6, and any primitive 6th root k of 1 mod p, let X be the Cayley graph produced in case (g) above. This graph has arc-type (1 + 1) + (1 + 1), and its two edge-orbits both have length 4p, with representatives {1, b} and {1, ab2}, where a and b are generators for G = Cp xfc C6 satisfying ap = b6 = 1 and b-1ab = ak. Now let Ym = X(m, F) be the thickened m-cover of X over F, where F is the edge-orbit containing {1, b}, or equivalently, the set of edges that lie in no 3-cycle. This graph is regular, with valency 2m + 2, and by Proposition 2.4 is a Cayley graph, so all we have to do is show it is prime and has arc-type (m + m) + (1 + 1). We do this in the same way as was done for the single example (for each m) in [3, Lemma 8.7]. First, X \ F is a union of triangles (unordered 3-cycles), and every edge of F joins vertices from different triangles, and it follows that every automorphism of Ym induces a permutation of the fibres over the edges in E(X) \ F, and also a permutation of the fibres over the edges in F. On the other hand, Proposition 2.5 tells us that all edges in a fibre over an edge in E(X) \ F lie in the same edge-orbit, and all edges in a fibre over an edge in F lie in the same edge-orbit. Hence the edge-type of Ym is 2m + 2. Next, from vertex (u, i) in Ym there are precisely 2m paths from (u, i) to any other vertex (u, in the fibre over (u, i), namely those of the form ((u, i), (bu, j), (u, ¿)) and ((u, i), (b-1u, j), (u, ¿)) for each j G Zm, while on the other hand, there are only one, two or m paths from (u, i) to any other vertex v at distance 2 from (u, i). Hence the stabiliser in Aut(Ym) of the vertex (u, i) preserves the fibre over the vertex u, and it follows that Aut(Ym) permutes the fibres over the vertices of X. Thus every automorphism of Ym can be projected to an automorphism of X, and the arc-type of Ym is (m + m) + (1 + 1), as required. Finally, we show that Ym is prime. If Ym were the Cartesian product of two relatively prime graphs, then one of them would have arc-type (1 + 1), which is impossible. On the other hand, if Ym were a proper Cartesian power of some prime graph Z, say Ym = Zr with r > 2, then by part (b) of Proposition 2.2, all edges in a 3-cycle of Ym would lie in the same factor of Ym, so Z would contain a 3-cycle, but in that case a vertex of Ym = Zr would lie in at least two triangles, contradiction. Thus Ym is prime. Case (i): Arc-type 1 + 1 + 1. By Lemma 8.8 of [3], if n is any odd integer greater than 11, and G is the dihedral group Dn, generated by two elements x and y satisfying x2 = yn = 1 and xyx = y-1, then Cay(G, {x, xy, xy3}) is prime and has arc-type 1 + 1 + 1. Case (j): Arc-type 1 + 1 + (1 + 1). This is similar to case (d). Let p be any prime such that p = 1 mod 4, with p > 5, let k be any integer such that k2 = — 1 mod p, and let G be the semi-direct product Cp x k C4, generated by two elements a and b such that ap = b4 = 1 and b-1ab = ak. Now take S = {b, b-1, ab2, a2 b2}, which consists of an inverse pair of elements of order 4 and two involutions (as conjugation by b2 inverts a), and let X = Cay(G, {b, b-1, ab2, a2b2}). Then X is 4-valent and connected, since (b, ab2} = G, and is also non-bipartite, just as in case (d). We will show that X is prime and has arc-type 1 + 1 + (1 + 1). First, by considering the vertices at distance 2 from the identity we see that up to reversal, the edges {1, b} and {1, b-1} lie in a single 4-cycle, namely (1, b, b2, b-1), while each of the edges {1, ab2} and {1, a2b2} lies in no 4-cycle. In particular, it follows from the latter observation that X is prime. Also as before, the edges {1,b} and {1,b-1} lie in the same edge-orbit. On the other hand, the edges {1,ab2} and {1,a2 b2} lie in different edge orbits, because up to reversal the edge {1, ab2} lies in four 5-cycles, namely those of the form (1, ab2, u, v, w) with (u,v,w) = (a, b2,b), (a, b2,b-1), (akb, a-1, a2b2) and (a-kb-1, a-1, a2b2), while the edge {1, a2b2} lies in only two 5-cycles, namely those of the form (1, a2b2, u, v, w) with (u,v,w) = (a \akb, ab2) and (a 1,a kb 1,a62). Hence the edge-type of X is 2 + 1 + 1. This also implies that every automorphism of X preserves the set T = {x, a2b2x} of all edges corresponding to left multiplication by the element a2b2 G S, and hence induces an automorphism of the subgraph obtained by removing those edges, namely Cay(G, S \ T) = Cay(G, {b, b-1, ab2}). By case (d), however, the latter subgraph is a GRR, with automorphism group G, and so every such automorphism is given by right multiplication by some element of G. It follows that G = Aut(X), and hence X is also a GRR, and has arc-type 1 + 1 + (1 + 1). Case (k): Arc-type 1 + 1 + 1 + 1. For any integer n > 15, let G be the dihedral group Dn, generated by elements a and b such that an = b2 = (ab)2 = 1, and take S = {b, ba, ba2, ba5}. Then since S consists of four involutions and G is generated by b and ba, the graph X = Cay(G, S) is 4-valent and connected. We show that X is prime and has arc-type 1 + 1 + 1 + 1. First, the paths of length 2 in X starting at the identity vertex 1 are (1, b, a?) for j G {-1, -2, -5}, and (1, ba, a?) for j G {1, -1, -4}, and (1, ba2, a?) for j G {2,1, -3}, and (1, ba5, a?) for j G {5,4,3}. By considering the final vertex of each of these, we see that the vertex 1 lies in only two 4-cycles up to reversal, namely (1, b, a-1, ba) and (1, ba, a, ba2). Hence the edges {1, b} and {1, ba2} lie in just one 4-cycle, while {1, ba} lies in two 4-cycles, and {1, ba5 } lies in no 4-cycles at all. In particular, X is prime, and also X has edge-type 1 + 1 + 1 + 1 or 2+1 + 1, with each of {1, ba} and {1, ba5} lying in different orbits from each other and from {1, b} and {1, ba2}. Next, multiplying by b, we find that ba5b = a-5 plays the same role for the vertex b as ba5 does for the vertex 1, namely that {b, a-5} is the only edge incident with b that lies in no 4-cycle. Now consider the cycles of length 6 containing one of the paths (ba5,1, b), (a-5, b, 1) and (ba5,1, ba2). An easy calculation shows there are precisely three 6-cycles of the form (ba5,1, b, u, v,w), namely with (u,v,w) = (a-1, ba4, a3), (a-1,ba4,a4) and (a-1, ba3, a3), and similarly, there are three 6-cycles of the form (a-5, b, 1, u, v, w), namely with (u, v, w) = (ba, a-4, ba-3), (ba, a-4, ba-4) and (ba2, a-3, ba-3), but there are seven 6-cycles of the form (ba5,1, ba2, u, v, w), namely with (u, v, w) = (a, ba3, a3), (a, ba6, a4), (a, ba6, a5), (a2, ba3, a3), (a2, ba4, a3), (a2, ba4, a4) and (a2, ba7, a5). In fact, up to reversal the edge {1, b} lies in 16 different 6-cycles altogether, while the edge {1, ba2} lies in 20 different 6-cycles, but this takes more work to verify. Both calculations show that the edge {1, ba2} cannot lie in the same orbit as {1, b} under Aut(X), and it follows that X has edge-type and arc-type 1 + 1 + 1 + 1. Case (l): Arc-type (1 + 1) + (1 + 1) + (1 + 1). This is somewhat similar to case (g). Let p be any prime with p = 1 mod 6, but this time where p > 7, let k be a primitive 6th root of 1 mod p, with k3 = -1 mod p, and let G be the semi-direct product Cp x k C6, generated by two elements a and b of orders p and 6 such that b-1ab = ak. Now take S = {b, ab2, a2b2, b-1, a-k b4, a-2k b4}, which consists of the elements b, ab2 and a2b2 and their inverses, and let X = Cay(G, S). Then clearly X is 6-valent and connected. We will show that X is prime, and has arc-type (1 + 1)+ (1 + 1)+ (1 + 1), for all p. First we note that {1, s} and {1, s}s-1 = {1, s-1} lie in the same edge orbit of X, for each s g S. Hence X has at most three distinct edge orbits. Next, up to reversal the edge {1, b} lies in just two 4-cycles, namely (1, b, ab3, a-k b4) and (1, b, a2b3, a-2k 2b4), and multiplying by b-1 gives the two 4-cycles containing the edge {1, b-1} as (1, b-1, a-k b3, ab2) and (1, b-1, a-k b3, a2b2). Each of the other four edges incident with the vertex 1 is contained in only one 4-cycle (up to reversal), namely one of the four just listed for {1, b} and {1, b-1}. Hence the orbit of the edges {1, b±1} under Aut(X) is different from the orbit(s) of {1, s±1} for s = ab2 and s = a2b2. Also the edge {1, ab2} lies in five 5-cycles, viz. those of the form (1, ab2, u, v, w) with (u,v,w) = (afcb,afc-1,a2b2), (akb,a-kb-1,a-fc2b4), (a-k 2b3,ab,b-1), (a-fc+1, b2, b), and (a-k b4, ab3, b), while the edge {1, a2b2} lies in only four 5-cycles, namely those of the form (1, a2b2, u, v, w) with (u,v,w) = (ak-1,akb,ab2), (a2kb,a-2kb-1,a-2fc2b4), (a-2k b3, a2b, b-1) and (a-2k b4, a2b3, b). Hence the orbit of the edges {1, (ab2)±1} is different from the orbit of {1, (a2b2)±1}, and so the edge-type of X is 2 + 2 + 2. But furthermore, if there exists an automorphism of X that fixes the vertex 1 and swaps b with b-1, then that automorphism must swap the two 4-cycles (1, b, ab3, a-k b4) and (1, b, a2b3, a-2fc2b4) with the two 4-cycles (1, b-1, a-fc2b3, ab2) and (1, b-1, a-fc2 b3, a2b2), and hence must swap ab2 with a-k b4 = (ab2)-1 and swap a2b2 with a-2k b4 = (a2b2)-1. Similarly, if if there exists an automorphism that fixes 1 and swaps ab2 with a-k b4, then it must swap the 4-cycle (1, b-1, a-k b3, ab2) with the 4-cycle (1, b, ab3, a-k b4), and hence must swap b with b-1, and the same holds for a2b2 and a-2k b4. It follows that any automorphism that fixes the vertex 1 must either fix all its six neighbours, or induce the triple transposition (b, b-1)(ab2, a-k b4)(a2b2, a-2k b4) on them. By vertex-transitivity, the analogous thing happens at every vertex, and an easy argument then shows that the stabiliser A1 in A = Aut(X) of the vertex 1 acts faithfully on its neighbourhood, and therefore | A11 = 1 or 2. Now suppose that |A1| = 2. Then |A| = |GA1| = 2|G|, and so G is normal in A. Hence if 0 is any non-trivial element of A1, then 0 normalises G, and so induces an automorphism of G = (a, b). Moreover, as 0 fixes the vertex 1 and acts non-trivially on its neighbourhood, we find that 0 swaps b with b-1, and ab2 with a-k b4 = (ab2)-1. In turn, this implies that 0 swaps a = (ab2)b-2 with a-k b4b2 = a-k , but then we find that ak = (ak)e = (b-1ab)e = ba-k b-1 = a-k, and so ak has order 2, contradiction. Thus A1 is trivial, and Aut(X) = G, so X has arc-type (1 + 1) + (1 + 1) + (1 + 1). Finally, X cannot be the Cartesian product of two smaller graphs that are relatively prime, since those would have to be connected and vertex-transitive, and one of them would have arc-type (1 + 1), which is impossible. Also X cannot be a Cartesian power of some smaller VT graph, since its order 6p is not a non-trivial power of any integer. Hence X is prime, as required. Accordingly, we have infinitely many connected finite Cayley graphs with each of the basic arc-types, and this completes the proof of Theorem 3.1 and Corollary 3.2. For the benefit of the reader (and for possible future reference), we summarise some of the details of the basic arc-types used here, in Table 1. 5 A consequence for zero-symmetric graphs Another consequence of Theorem 3.1 is the following: Corollary 5.1. For every integer d > 2, there exist infinitely many finite zero-symmetric graphs (or GRRs) of valency d. Table 1: Summary of some Cayley graphs with the basic arc-types. Arc-type Cayley graphs m Cycle graphs Cn (n > 5) for m = 2, and Cayley graphs for PSL(2,p) via m conjugate involutions for m > 3 (m+m) Bouwer graphs B(m, r, n) with n > 7 and r > 6 m+1 Thickened m-cover of C2n over a 1-factor 1 + (1 + 1) 3-valent Cayley graph for Cp x k C4 (for prime p) m+(1 + 1) Thickened cover of Cayley graph for Cp x k C4 1 + (m+m) Thickened cover of Cayley graph for Cp x k C4 (1 + 1) + (1 + 1) 4-valent Cayley graph for Cp x k C6 (for prime p) (m+m) + (1 + 1) Thickened cover of Cayley graph for Cp x k C6 1 + 1 + 1 3-valent Cayley graph for dihedral groups Dn 1 + 1 + (1 + 1) 4-valent Cayley graph for Cp x k C4 (for prime p) 1+1+1+1 4-valent Cayley graph for dihedral groups Dn (1 + 1) + (1 + 1) + (1 + 1) 6-valent Cayley graph for Cp x k C6 (for prime p) This is not at all surprising, but appears to be new, in the sense that we cannot find the statement or something similar in the literature on GRRs or zero-symmetric graphs. It is shown in [5, Theorem 3.10.4] that there exists a GRR of valency d for the symmetric group Sd+1 whenever the latter group can be generated by an 'asymmetric' set of d transpositions. The latter happens for all d > 5, but gives only finitely many GRRs with given valency d. On the other hand, it is clear that larger sets of involutory generators for dihedral or symmetric or other groups will give GRRs, even if this does not appear to have been explicitly proved elsewhere. 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] I. Z. Bouwer, Vertex and edge transitive, but not 1-transitive, graphs, Canad. Math. Bull. 13 (1970), 231-237, doi:10.4153/cmb-1970-047-8. [3] M. D. E. Conder, T. Pisanski and A. Zitnik, Vertex-transitive graphs and their arc-types, Ars Math. Contemp. 12 (2017), 383-413, doi:10.26493/1855-3974.1146.f96. [4] M. D. E. Conder and A. 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Sparl, The classification of half-arc-transitive generalizations of Bouwer graphs, European J. Combin. 64 (2017), 88-112, doi:10.1016/j.ejc.2017.04.003. [10] G. Sabidussi, On a class of fixed-point-free graphs, Proc. Amer. Math. Soc. 9 (1958), 800-804, doi:10.2307/2033090. [11] W. T. Tutte, Connectivity in Graphs, volume 15 of Mathematical Expositions, University of Toronto Press, Toronto, Ontario, 1966. [12] C. Zhou and Y.-Q. Feng, Automorphism groups of connected cubic Cayley graphs of order 4p, Algebra Colloq. 14 (2007), 351-359, doi:10.1142/s100538670700034x. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 113-126 https://doi.org/10.26493/1855-3974.1414.58b (Also available at http://amc-journal.eu) Coloring properties of categorical product of general Kneser hypergraphs* Roya Abyazi Sani, Meysam Alishahi t School of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran Ali Taherkhani Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran Received 26 May 2017, accepted 9 October 2017, published online 6 March 2018 More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture was generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier, in 2016, introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful type results for the coloring of categorical product of general Kneser hypergraphs, which generalize the Hajiabolhassan-Meunier result. Also, we present a new lower bound for the chromatic number of categorical product of general Kneser hypergraphs which can be much better than the Hajiabolhassan-Meunier lower bound. Using this lower bound, we enrich the family of hypergraphs satisfying Zhu's conjecture. Keywords: Categorical product, chromatic number, Hedetniemi's conjecture, general Kneser hypergraph. Math. Subj. Class.: 05C15 * We thank the two anonymous referees for the carefully reading our paper and for all their useful remarks that helped in improving the presentation of the paper. t Corresponding author. E-mail address: roya.abyazisani@shahroodut.ac.ir (Roya Abyazi Sani), meysam_alishahi@shahroodut.ac.ir (Meysam Alishahi), ali.taherkhani@iasbs.ac.ir (Ali Taherkhani) ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ Abstract 1 Introduction and main results For two graphs G and H, their categorical product G x H is the graph defined on the vertex set V(G) x V(H) such that two vertices (g, h) and (g', h') are adjacent whenever gg' G E(G) and hh' G E(H). The categorical product is the product involved in the famous long-standing conjecture posed by Hedetniemi which states that the chromatic number of G x H is equal to the minimum of x(G) and x(H). It was shown that the conjecture is true for several families of graphs, but it is wide open in general (see Tardif [21] and Zhu [23]). In spite of being investigated in several articles, there is no substantial progress in solving this conjecture. This conjecture was generalized to the case of hypergraphs by Zhu [22]. A hypergraph H is an ordered pair (V(H), E(H)) where V(H) is a set of vertices, and E(H) is a family of nonempty subsets of V(H). The elements of E(H) are called edges. All hypergraphs considered in the paper have no multiple edges and E(H) is thus a usual set. For a subset S C V(H), the subhypergraph induced by S, denoted by H[S], is a hypergraph with vertex set S and edge set {e G E(H) : e C S}. A hypergraph H is said to be r-uniform if E(H) is a family of r-subsets of V(H). In particular, a 2-uniform hypergraph is called a simple graph. From now on, by a graph we mean a simple graph. An r-uniform hypergraph H is called r-partite if V(H) can be written as a union of r pairwise disjoint subsets (parts) Ui,..., Ur such that each edge of H intersects each part U in one vertex. An r-partite hypergraph is called complete if it contains all possible edges. Also, it is said to be balanced if | U | - | Uj | < 1 for each i, j G [r]. Let H be a hypergraph and r be an integer, where r > 2. For pairwise disjoint subsets Ui,...,Ur C V (H), the hypergraph H[Ui,..., Ur ] is defined to be a subhypergraph of H whose vertex set is ur=iUj and whose edge set consists of all edges of H which are contained in ur=iUj and have exactly one element in each Uj. Note that H[Ui,..., Ur] is an r-uniform r-partite hypergraph. A proper coloring of a hypergraph H is an assignment of colors to the vertices of H such that there is no monochromatic edge. The chromatic number of a hypergraph H, denoted by x(H), is the smallest number k such that there exists a proper coloring of H with k colors. If there is no such a k, we define the chromatic number to be infinite. Let c be a proper coloring of a complete r-partite hypergraph H with parts Ui,..., Ur. The hypergraph H is colorful (with respect to the coloring c) whenever for each i G [r], the vertices in Uj receive different colors, that is, |c(Uj)| = |Uj| for each i G [r]. Let Hi = (Vi,Ei) and H2 = (V2, E2) be two hypergraphs. For i = 1, 2, the projection nj is defined by nj: (vi, v2) ^ vj. The categorical product of two hypergraphs Hi and H2, defined by Dörfler and Waller in 1980 [10], is the hypergraph Hi x H2 with vertex set Vi x V2 and edge set {e C Vi x V2 : ni(e) G Ei,nj(e) G E2}. In 1992, Zhu [22] proposed the following conjecture as a generalization of Hedetniemi's conjecture. Conjecture 1.1 ([22]). Let Hi = (Vi, Ei) andH2 = (V2, E2) be two hypergraphs. Then x(Hi xH2)=min{x(Hi),x(H2)}. One can easily derive a proper coloring of Hi x H2 from a proper coloring of Hi or of H2. Therefore the hard part is to show that x(Hi x H2) > min{x(Hi), x(H2)}. Let F be a subhypergraph of Hi x H2 with the same vertex set and whose edge set consists of minimal edges of H1 x H2. It is clear that any proper coloring of F is also a proper coloring of H1 x H2. This observation shows that Conjecture 1.1 is a generalization of Hedetniemi's conjecture. For an integer r and a hypergraph H, the r-colorability defect of H, denoted by cdr (H), is the minimum number of vertices that shall be removed from H so that the hypergraph induced by the remaining vertices admits a proper coloring with r colors. Let Zr = {w, w2,..., wr } be a multiplicative cyclic group of order r with generator w. For X = (x1,..., xn) € (Zr U {0})n, a sequence xil,..., xim with 1 < i1 < • • • < im < n is called an alternating subsequence of X if xij = 0 for each j € [m] and xij = xij+1 for each j € [m - 1]. The alternation number of X, denoted by alt(X), is the length of the longest alternating subsequence of X. We set 0 = (0,..., 0) and define alt(0) = 0. Also, for an X = (x1,..., xn) € (Zr U {0})n and for e € Zr, define Xe = {i : xi = e} . Note that the r-tuple (Xe)£eZ uniquely determines X and vice versa. Therefore, with abuse of notations, we can write X = (Xe)£eZ . The notation |X| stands for the number of nonzero coordinates of X, i.e., |X| = £ez |Xe|. For two vectors X, Y € (Zr U {0})n, we write X C Y whenever Xe C Ye for each e € Zr . For a hypergraph H and a bijection a: [n] ^ V(H), the r-alternation number of H with respect to the permutation a is defined as follows: alt; (H) = max {alt(X) : E(H[a(Xe)]) = 0 for all e € Zr} . The r-alternation number of H, denoted by altr (H), is equal to min; alt; (H) where the minimum is taken over all bijections a: [n] ^ V(H) (for more details see [3]). For any hypergraph H = (V(H), E(H)) and positive integer r > 2, the general Kneser hypergraph KGr (H) is an r-uniform hypergraph whose vertex set is E(H) and whose edge set is the set of all r-subsets of E(H) containing r pairwise disjoint edges of H. Note that by this notation the well-known Kneser hypergraph KGr (n, k) is the Kneser hypergraph KGr ([n], (M)) .For r = 2, we will rather use KG(H) than KGr (H). Lovisz in 1978, by using tools from algebraic topology, proved that x(KG(n, k)) = n - 2k + 2. His paper showed an inspired and deep application of algebraic topology in combinatorics [15]. As a generalization of this result and to confirm a conjecture of Erdos [11], Alon, Frankl, and Lovisz [5] proved that the chromatic number of KGr (n, k) is equal to (k- 1)r A different kind of generalization of Lovisz's theorem has been 1 obtained by Dol'nikov [9]. He proved that X(KG(H)) > cd2(H). Then, in 1992, Krfz [13] extended the both latter results by proving that " cdr (H)" x(KGr(H)) > r1 Alishahi and Hajiabolhassan [3] introduced the alternation number as an improvement of colorability defect. Using the Zp-Tucker lemma, they proved that x(KGr(H)) > |V(H)| - altr (H) r1 It can be verified that |V(H)| - altr (H) > cdr (H) and the inequality is often strict [3]. Therefore, the preceding lower bound for chromatic number surpasses the Dol'nikov-Kriz lower bound. Recently, by an innovative use of the Zp-Tucker lemma, Hajiabolhassan and Meunier [12] extended the Alishahi-Hajiabolhassan result (as well as the Dol'nikov-Kriz result) to the categorical product of general Kneser hypergraphs as follows. Theorem A ([12]). Let Hi,..., Ht be hypergraphs and r be an integer, where r > 2. Then x(KGr(Hi) x ••• x KGr(Hi)) > 1 r — 1 ie[t] min(|V (Hi )| — altr (Hi)) Using Theorem A, Hajiabolhassan and Meunier introduced new families of hypergraphs satisfying Zhu's conjecture. From another point of view, Simonyi and Tardos [20] generalized the Dol'nikov result. Indeed, they proved that for any hypergraph H, if t = cd2(H), then any proper coloring of KG(H) contains a complete bipartite subgraph K^t/2j, \t/2-\ such that all vertices of this subgraph receive different colors and these different colors occur alternating on the two parts of the bipartite graph with respect to their natural order. Then, this result as well as the Dol'nikov-Kriz result was extended to Kneser hypergraphs by Meunier [19] as the following theorem. Theorem B ([19]). Let H be a hypergraph and p be a prime number. Any proper coloring of KGp(H) contains a colorful, balanced, and complete p-partite subhypergraph F with cdp(H) vertices. It should be mentioned that, in his paper [19], Meunier also generalized Theorem B and proved that this theorem remains true by replacing cdp(H) with |V(H)| — altp(H). In his proof, Meunier used a Zq-generalization of a theorem by Ky Fan which is stated in terms of chain maps. Later, by introducing an appropriate generalization of the Zp-Tucker lemma, the present second author [2] gave a simple proof for Meunier's result. Moreover, several extensions of Meunier's result can be found in [2]. Another common generalization of the Simonyi-Tardos result and a result by Chen [7, Theorem 7] can be found in [4]. As an improvement of r-colorability defect, the equitable r-colorability defect was introduced in [1]. For a hypergraph H, the equitable r-colorability defect of H, denoted by ecdr (H), is the minimum number of vertices that shall be removed so that the sub-hypergraph induced by the remaining vertices admits a proper equitable r-coloring, i.e., a proper r-coloring in which the sizes of color classes differ by at most one. Clearly, ecdr (H) > cdr (H). As a generalization of Theorem B, it was proved [1] that any proper coloring of KGp (H) contains a colorful, balanced, and complete p-partite subhypergraph F with ecdp(H) vertices. It is not difficult to construct a hypergraph H for which ecdr (H) — cdr (H) is arbitrary large. Surpassing the Dol'nikov-Kriz lower bound, Abyazi Sani and Alishahi [1] proved x(KGr(H)) > ecdr (H) r1 It is worth mentioning that they indeed proved a more general result which in particular implies the prior lower bound. To be more specific, they gave a new lower bound for the chromatic number of a generalization of Kneser hypergraphs introduced by Ziegler which improves substantially Ziegler's lower bound [24, 25]. Furthermore, they compared their lower bound with the Dol'nikov-Krfz lower bound and the Alishahi-Hajiabolhassan lower bound. In this regard, it was shown that there is a family of hypergraphs H such that for each hypergraph H G H, x(KGr (H)) ecdr (H) r — 1 |V (H)|-altr (H) r-1 are both unbounded while x(KGr(H)) - £drJTi and x(KGr(H)) -for the hypergraphs H in H. Although there are hypergraphs H for which ecdr (H) -(|V(H)| - altr(H)) is arbitrary large, one can construct some hypergraphs H making (|v(h)| - altr (H)) - ecdr (H) arbitrary large, see [1]. As the main results of this paper, motivated by the preceding discussion, we simultaneously extend the results by Abyazi Sani and Alishahi [1] and by Hajiabolhassan and Meunier [12] to the following theorems. Theorem 1.2. Let H1;..., Ht be hypergraphs. Let p be a prime number and n = max(minecdp(Hi), min (|V(H,)| - altp(H,)U. L ¿e [t] ¿e [t] J Any proper coloring of KGp(H1) x • • • x KGp(Ht) contains a colorful, balanced, and complete p-partite subhypergraph F with n vertices. Remark. The question of whether Theorem 1.2 holds for an arbitrary positive integer r instead of a prime number p is an interesting open question. Let c be the proper coloring with color set [C]. Let F be the colorful, balanced, and complete p-partite subhypergraph whose existence is ensured by Theorem 1.2. Clearly, any color appears in at most p - 1 vertices of F. Consequently, the previous theorem implies x(KGp(Hi) x ••• x KGp(Ht)) > which can be extended for an arbitrary r > 2 as follows. " n " > p - 1 1 p - 1 ¿e[t] min ecdp(Hi) Theorem 1.3. Let H1;..., Ht be hypergraphs and r be a positive integer, where r > 2. Then x(KGr(H1) x • • • x KGr(Ht)) > minecdr(Hi) r - 1 ¿e[t] Example. In what follows, by introducing some hypergraphs, we compare the two lower bounds presented in Theorems A and 1.3. Let n, k, r and a be positive integers, where n > rk, n > a and r > 2. Define H(n, k, a) to be a hypergraph with vertex set [n] and edge set {B C [n] : |B| = k and B % [a]}. Let KGr (n, k, a) denote the hypergraph KGr (H(n, k, a)). It was proved [1, Proposition 7] that if either a < 2k - 2 or a > rk - 1, then x (KGr(n, k, a)) = Indeed, for a > rk - 1 , it was proved that X (KGr (H(n, k, a))) n-max{a,r(k-1)} ecdr(H(n, k, a)) n - a r - 1 r - 1 One should notice that the chromatic number of KGr (H(n, k, a)) was left open for several values of a with 2k — 1 < a < rk — 2. Note that Theorem 1.3 implies the validity of Zhu's conjecture for the family of hypergraphs KGr (n, k, a) provided that a > rk — 1. What is interesting about the hypergraph KGr (H(n, k, a)) is the fact that for r > 4 and a > rk — 1, the value of ecdr(H(n, k, a)) — (n — altr(H(n, k, a))) is unbounded. Thus, by the lower bound presented in Theorem A, we cannot derive that the family of hyper-graphs KGr (n, k, a) satisfies Zhu's conjecture. On the other hand, there is a family H of hypergraphs (see [1]) such that for H G H, the value of (n — altr(H(n, k, a))) — ecdr (H(n, k, a)) is unbounded. Hence, Theorem A and Theorem 1.3 introduce two somehow complementary lower bounds. 2 Proofs This section is devoted to the proofs of Theorem 1.2 and Theorem 1.3. In the first subsection, we define some necessary tools which will be needed in the rest of the paper. We assume basic knowledge in topological combinatorics. For more details, see [16]. 2.1 Notations and tools A simplicial complex is a pair (V, K) where V is a finite nonempty set and K is a family of nonempty subsets of V such that for each A G K, if 0 = B C A, then B G K. Respectively, the set V and the family K are called vertex set and simplex set of the simplicial complex (V, K). For simplicity of notation and since we can assume that V = U AeK A, with no ambiguity, we can point to a simplicial complex (V, K) just by its simplex set K. The barycentric subdivision of K, denoted by sd K, is a simplicial complex whose vertices are the simplices of K and whose simplices are the chains of simplices of K ordered by inclusion. Let V and W be two sets. We write V I W for the set V x {1} U W x {2}. Let K and L be two simplicial complexes with vertex sets V and W, respectively. We define K * L, the join of K and L, to be a simplicial complex with vertex set V I W and simplex set {A I B : A G K, B G L}. The join operation is obviously associative: if K, L, M are simplicial complexes, then the simplicial complexes K * (L * M) and (K * L) * M are the same up to a natural relabeling of their vertices. This allows us, if we do not care about the names of the vertices, to use K * L * M for both of K * (L * M) and (K * L) * M. The n-fold join of K, denoted by K*", is a simplicial complex obtained by joining n copies of K. By relabeling the vertices of K*", we assume that K*n has vertex set V(K) x [n] where for each vertex (v, i) g V(K) x [n], the index i indicates that the vertex v is considered as a vertex of the ith copy of K. For a prime number p, we also consider Zp as a simplicial complex with vertex set Zp and simplex set {{w}, {w2},..., {wp}}. Clearly Zpn is a simplicial complex whose vertex set is Zp x [n] and whose simplices are all nonempty subsets A C Zp x [n] such that for each i g [n], the number of e's for which (e, i) G A is at most one. This observation implies that the simplex set of Zpn can be identified with the set (Zp U {0})n \ {0}, i.e., for each simplex A in Zpn, define A ^ (xi,..., xn) where xj = e if (e, i) G A and xj =0 otherwise. Also, the simplicial complex ap-1 is a simplicial complex with vertex set Zp and with simplex set consisting of all nonempty proper subsets of Zp. Note that (ap- J is a simplicial complex with vertex set Zp x [n] and 0 = t C Zp x [n] is a simplex of / _ \ *n r _. \ *n (ap_2j if and only if |t n (Zp x {i}) | < p -1 foreach i G [n]. It is clear that (0^-2) is a free simplicial complex where for each e G Zp and (e', i) G Zp x [n], the action is i - *n defined by e • (e', i) = (e • e', i). Let t g (opP-2 J be a simplex. For each e G Zp, define te = {(e, j) : (e, j) G t}. Also, define ¿(t)= p • h(T) + |{e G Zp : |te| > h(T)}|, where h(T) = min£eZp |re|. As stated above, each X G (Zp U {0})n \ {0} represents a / _ j\ tn simplex in Z*vn C i j and vice versa. Therefore, speaking about h(X) and ¿(X) is meaningful. Indeed, we have h(X) = min |Xe| and ¿(X)= p • h(X) + |{e G Zp : |Xe| > h(X)}|. Note that Zp acts freely on (Zp U {0})n \ {0} by the action e • X = (e • x2,..., e • xn), where e • 0 is defined to be 0 for each e G Zp. Now, we are ready to present the proof of Theorem 1.2. For simplicity, we first assume that n = minie[t] ecdP(Hj) and then, in Subsection 2.2.2, we sketch the proof for n = minie[t] (|V(Hi)| - altP(Hj)). The proof will follow by applying Dold's theorem on a Zp-equivariant simplicial map A: sd(Zpn) Zpm X (s(X), v(X)) with n = J2¡= 1 |V(Hi)| and m as small as possible. Indeed, Dold's theorem implies that if there is such a map A, then m > n. It is worth noting that the idea of using Dold's theorem or some of it specializations such as the Zp-Tucker lemma has been used in several articles initiated by a fascinating paper of Matousek [17]. For instance, see [1, 4, 6, 7, 12, 18, 19, 24]. Usually, the most challenging task in using Dold's theorem is how to define the map A, especially the sign part s(X). In what follows, we show that some of the techniques used in these works can be fruitfully mixed and extended to get a common generalization. However, some additional tricks are introduced to make these techniques work together. In particular, in our approach, we use a different way to define the sign map s(X) and also we appropriately modify the value function v(X). Being more specific, to define the map A, we partition sd(Zpn) = (Zp U {0})n \ {0} into two subsets S2 and £2, where S2 is the set of vectors X g sd(Zpn) such that for each j g [t] and e G Zp, the set {i G [nj]: xi+y>j-1 n ,, = e} contains some edge of Hj = ([nj], Ej), and hence j = 1 j Xe somehow contains a vertex of the hypergraph KGp(H2) x • • • x KGp(Ht). For each X G £2, we define v(X) g {a + 1,..., m}, where a = n - n + p -1, according to a given proper coloring of KGp(H2) x • • • x KGp(Ht) and we define s(X) G Zp with the help of an auxiliary sign map s3(-). Defining A(X) for the remaining vectors X, i.e., X G £2, is even more difficult and technical which will be done by the use of two auxiliary sign maps s2 (-) and s2 (-). A larger value of n will allow us to make a smaller and consequently m smaller, giving a better bound in the end. 2.2 Proof of Theorem 1.2 When n = 0, there is nothing to prove. If 1 < n < p - 1, then consider pairwise disjoint sets Ui,..., Up C V(KGp(Hi) x • • • x KGp(Ht)) such that |Ui| = 1 for i < n and \Ui \ = 0 otherwise. Note that for at least one i, we have U = 0. In view of the definitions, the subhypergraph KGp(Hi) x • • • x KGp(Ht)[U^..., Up] which has no edge is clearly balanced and p-partite. Furthermore, for any proper coloring of KGp(H1)x^ • xKGp(Ht), this subhypergraph is colorful which is desired. Henceforth, we assume that n > p. For simplicity of notation, assume that H1 = ([n1], E1),..., Ht = ([nt], Et) and moreover, set n = J2¿=1 ni. For each X = (x1,..., xn) G (Zp U {0})n \ {0}, let X(1) G (Zp U {0})ni t»e the first n1 coordinates of X, X(2) G (Zp U {0})n be the next n2 coordinates of X, and so on, up to X(t) G (Zp U {0})nt be the last nt coordinates of X. Also, for each j g [t], define Aj (X) to be the set of signs e G Zp such that X(j)e contains at least one edge of Hj. We remind that X(j)e is the set of all i G [nj] such that j-1 n = e. Define E1 = {X G (Zp U {0})n \ {0} : Aj (X) = Zp for at least one j G [t]| and £2 = {X G (Zp U {0})n \ {0} : Aj (X) = Zp for all j G [t]}. Note that for an X g (Zp U{0})n\{0} and for each j g [t],if we set S = U£eZp X (j )e, then X (j) = (X (j)e) Z can be thought of as a partition of vertices of Hj [S] into p color classes, i.e., the vertices in X(j)e receive the color e. Intuitively, the value h(X(j)) is then the size of the smallest color class, ¿(X(j)) is the maximum possible number of vertices colored by an equitable sub-coloring (not necessarily proper), while Aj (X) is the set of colors e G Zp for which there is an e-monochromatic edge in Hj [S]. 2.2.1 Proof of Theorem 1.2 when n = minie[t] ecdp(Hi) In what follows, we first define two sign maps s1 and s2 playing important roles in the proof. These two maps will help us to define s(X) for each X g E1. Definition of si(-). Let X g E1 be a vector such that Aj(X) g {0, Zp} for each j g [t]. Define fX(j) if Aj (X) = Zp, Bj(X) = I {£^X(jr = 0} if Aj (X) = 0 and h(X(j)) = 0, iXj) if Aj (X) = 0 and h(X (j)) > 0, where X(j) G (Zp U {0})nj \ {0} and for each e G Zp, we have Xj)e = |x(j)£ if \X(j)e\ = h(X(j)), 0 otherwise. Note that Bj (X) may be of two different natures: a vector in (Zp U{0})nj \{0} or a proper subset of Zp. Now, set B(X) = fB1(X),..., Bt(X)) and L1 = { B(X) : X G E1 and Aj (X) G {0,Zp} for all j G [t^. Note that L1 is a subset of (fZp U{0})ni U f2Zp \ {Zp})) x^^^x (fZp U {0})nt U f2Zp \ {Zp})) \ ({0, 0} x • • • x {0,0}). For an e G Zp and a vector B = (Bi,..., Bt) G Li, we define £ • B = (e • B1,...,e • Bt), where „ j (e • xi,...,e • i„,) if Bi = (xi,...,x„i) G (Zp U{0})n \{0}, £ • Bi — < , . i [{e • z : z G Bj if Bi C Zp. With respect to this action, one can simply check that L1 is closed and free and furthermore, B(-) is a Zp-equivariant map, i.e., B(e • X) = e • B(X) for each e G Zp and for each X g E i such that Aj (X) g {0,Zp} for each j g [t].Now,let si: Li ^ Zp be an arbitrary Zp-equivariant map. Note that such a map can be defined by choosing one representative in each orbit and defining the value of the map arbitrarily on this representative. Definition of s2(-). Clearly Zp acts freely on L2 = 2Zp x • • • x 2Zp \ ({0, Zp} x • • • x {0, Zp}) by the action e • (Ci,..., Ct) = (e • Ci,..., e • Ct), where e • Ci = {e • z : z G Ci}. Similar to the definition of si(-), let s2: L2 ^ Zp be an arbitrary Zp-equivariant map. Set a = n - mini£[t] ecdp(Hi) + p — 1. Note that since mini£[t] ecdp(Hi) > p, we have a < n. For every j G [t], define the function Vj : (Zp U {0})n \ {0} ^ N as follows: f|X (j)| if Aj (X )= Zp, Vj(X) = 1 A(X )| + maX {¿(Z) : Z C X (j) and , if Aj (X) = Zp. [ E(Hj [Ze]) = 0 for all e G Zp j j ( )= p We remind the reader that |X(j)| denotes the number of nonzero coordinates in X(j). Now, let v(X) = Ej=i Vj(X). Defining the map Ai. Set a = n — mineeZp ecdp(Hi) + p — 1. Define the map Ai : Ei —> Zp x {1,..., a} X —^ (s(X), v(X)). For defining s(X), we consider the following different cases. • If for each j g [t], we have Aj (X) g {0, Zp}, then s(X) = si (B(X)). • If for some j G [t], we have Aj (X) G {0,Zp}, then s(X) = S2(Ai(X),..., At(X)). Lemma 2.1. The map Ai is a Zp-equivariant map with no X, Y G Ei such that X C Y, v(X) = v(Y) and s(X) = s(Y). Proof. Clearly, Ai is a Zp-equivariant map since the two maps si( —) and s2(—) are Zp-equivariant and v(e • X) = v(X) for all e G Zp. For a contradiction, suppose that X and Y are two vectors in Ei such that X C Y, v(X) = v(Y) and s(X) = s(Y). Since X C Y, we have X(j) C Y(j) and consequently, Aj (X) C Aj (Y) for each j g [t]. Additionally, X(j) C Y(j) implies that {¿(Z) : Z C X(j) and E(Hj[Ze]) = 0 Ve G Zp} C {¿(Z) : Z C Y(j) and E(Hj[Ze]) = 0 Ve G Zp}. Thus, Vj(X) < Vj(Y) for each j e [t]. Therefore, the equality v(X) = v(Y) implies Vj (X) = Vj(Y). This equality together with above discussion results in Aj(X) = Aj(Y) for each j e [t]. This observation leads us to the following cases. (I) Aj(X) e {0, Zp} for each j. Therefore, s(X) = si (B(X)). Since Aj(X) = Aj(Y) for each j, we have s(Y) = s1 (B(Y)), Consequently, the fact that s(X) = s(Y) implies that B(X) = B(Y). Now, let j0 be the smallest integer for which Bj0 (X) = Bj0 (Y). We consider the following different cases. (1) When Aj0 (X) = Aj0 (Y) = Zp. In view of the definition of Bj0 (-), we have X(j0) C Y(j0). Therefore, the definition of vj0 implies that vj0 (X) < vj0 (Y), which is not possible. (2) When Aj0 (X) = Aj0 (Y) = 0. Using j (X) = j (Y), we have ¿(X(jo)) = ¿(Y(j0)). Therefore, p ■ h(X(jo)) + |{e : |X(jo)e| > h(X(jo))}| = j0 j0 which clearly implies that h(X(j0)) = h(Y(j0)) and |{e : |X(jo)e| > h(X(jo))}| = |{e : |Y(jo)£| > h(Y(jo))}|. The fact that X (j0) C Y (j0) results in P ■ h(Y(jo)) + |{e : |Y(jo)e| > h(Y(jo))}|, {£ : |X(jo)e| > h(X(jo))} = {£ : |Y(jo)£| > h(Y(jo))}. Therefore, in view of the definition of B(-), we have Bj0 (X) = Bj0(Y) which is a contradiction. (II) Aj (X) e {0, Zp} for some j e [t]. Since s(X) = s(Y), we have s2(Ai(X),..., At(X)) = s2(Ai(Y),..., At(Y)). Consequently, we must have (Ai(X),..., At(X)) = (Ai(Y),..., At(Y)). Therefore, there is at least one j for which Aj (X) = Aj (Y) which is not possible. □ In what follows, we will define some new notations needed in the rest of proof. Let c be a proper coloring of KGp(Hi) x ■ ■ ■ x KGp(Ht) with color set [C]. For each X e S2 and each e e Zp, define E e(X) = j(ei,..., et) e Ei x ■ ■ ■ x E : ej C X (j)e for each j e [t^. Note that, in view of the definition of S2, for each e e Zp, we have Ee(X) = 0. Now, set tx to be defined as follows: tx = |(e, c(u)) : e e Zp and u = (ei,..., et) e Ee Note that if we choose u£ e Ee(X) for each e e Zp, then {u£ : e e Zp} is an edge of KGp(Hi) x ■ ■ ■ x KGp(Ht). Consequently, since c is a proper coloring of KGp(Hi) x ■ ■ ■ x KGp(Ht), for each i e [C], there is at least one e e Zp for which (e, i) e tx. This observation indicates that tx is a simplex of . Furthermore, since Ee(X) = 0 for each e e Zp, we have ¿(tx ) > p. Definition of s3(-). For a positive integer b e [C], let Ub be the set consisting of all simplices t e (^-2) such that |te| e {0, b} for each e e Zp. Define U = ug^U^. Choose an arbitrary Zp-equivariant map s3 : U ^ Zp. Also, for each t e (^-2) with h = h(T) = min |te|, define r = U t^. e : |t e | = h Note that r is a sub-simplex of t which is in U. Therefore, s3(r) is defined. Defining the map A2. Define the map A2 : £2 —> Zp x {a + 1, ...,a - p + 1 + maxxes2 ¿(tx )} X (s(X ),v (X)), where s(X) = s3(TX) and v(X) = a -p + 1 + ¿(tx). Lemma 2.2. The map A2 is a Zp-equivariant map with no X, Y e £2 such that X C Y, v(X) = v(Y) and s(X) = s(Y). Proof. Obviously, A2 is a Zp-equivariant map. Suppose for a contradiction that X and Y are two vectors in £2 such that X C Y, v(X) = v(Y) and s(X) = s(Y). In view of the definition of A2, we must have ¿(tx) = ¿(ty). Using the definition of ¿( —), it implies that h(TX) = h(TY). From the last equality and tx C ty, we deduce that TX = TY and consequently, s(X) = s3(TX) = s3(TY) = s(Y), which is a contradiction. □ In the following lemma, we show that how the existence of an X with large ¿(X) completes the proof. Lemma 2.3. If there is an X e £2 with ¿(tx ) > q, then KGp(Hi) x • • • x KGp(Ht) contains a colorful, balanced, and complete p-partite subhypergraph with q vertices. Proof. Let X e £2 be a vector for which we have ¿(tx ) > q. Let t C tx be a subsimplex such that ¿(t) = |t| = q. For each i e [p], set Si = {j e [C] : (wJ, j) e t}. First note that |_ p J < |Si | < [ p ] for each i e [p]. Moreover, it is clear that J]P=2 |Si | = q. For each i e [p] and s e Si, in view of the definitions of t (X) and Si, there is a ef,i,..., ei,t) vertex ui,s = (ef 2,..., ef t) of KGp(H2) x • • • x KGp(Ht) such that c(ui,s) = s and e| j C X(j)w for each j e [t]. Now, for i e [p], set Ui = {ui,s : s e Si}. Clearly, kGp(Hi) x • • • x KGp(Ht)[Ui,..., Up] is the desired subhypergraph. □ Completing the proof of Theorem 1.2 when n = minie[t] ecdp(Hi). For completing the proof of Theorem 1.2, we need to use a generalization of the Borsuk-Ulam theorem by Dold, see [8, 16]. Indeed, Dold's theorem implies that if there is a Zp-equivariant simplicial map from a simplicial Zp-complex K2 to a free simplicial Zp-complex K2, then the dimension of K2 should be strictly larger than the connectivity of K2. For simplicity of notation, let m = a — p +1+ max ¿(t(X)). X6E2 In view of Lemma 2.3, it suffices to show that max ¿(tx) > minecdp(?{A xeS2 ie[t] To this end, define A: (Zp U {0})n \ {0} ^ Zp x [m] such that for each X G (Zp U {0})n \ {0}, if X G Ei, then A(X) = Ai(X), otherwise A(X) = A2(X). In view of Lemma 2.1 and Lemma 2.2, A(-) is a Zp-equivariant simplicial map from sd(Zpn) to Zpm. Consequently, according to Dold's theorem, the dimension of Zpm should be strictly larger than the connectivity of sd(Zpn), that is m - 1 > n - 2 as desired. □ 2.2.2 Proof of Theorem 1.2 when n = minie[t](|V (Hi) | - altp(Hi)) In this subsection, we sketch the proof of Theorem 1.2 for the n = mini£[t](|V(H)| -altp(Hj)) case. To this end, we need to slightly modify the proof of Theorem 1.2 in the case of n = minie[t] ecdp(Hi) as follows. • Throughout Subsection 2.2.1, replace mini£[t] ecdp(Hi) by mini£[t](|V(Hi)| -altp(Hi)). • Use alt(-) instead of function ¿(-) to define each Vj (X). • For any X such that Aj(X) G {0, Zp} for each j G [t], in the definition of A1(X), set s(X) to be the first nonzero entry of X. With the same approach as in Subsection 2.2.1, it is straightforward to check that Lemmas 2.1, 2.2, and 2.3 are still valid with the preceding modifications. Therefore, again applying Dold's theorem leads us to the desired assertion. 2.3 Proof of Theorem 1.3 To prove Theorem 1.3, we reduce this theorem to the prime case of r which is known to be true by the discussion right after Theorem 1.2. One should notice that this reduction is a refinement of the well-known reduction originally due to Kriz [14], which has been used in some other papers as well, for instance see [3, 12, 24, 25]. In what follows, we use a similar approach as in [12]. Lemma 2.4. Let r' and r" be two positive integers. If Theorem 1.3 holds for both r' and r", then it holds also for r = r'r''. For two positive integers s and C and a hypergraph H, define a new hypergraph Tn,c,s as follows: V(Tn,c,s) = v (H) E(Tu,c,s) = {A C V(H) : ecds(H[A]) > (s - 1)c}. The following lemma can be proved with a similar approach as in [12, Lemma 3]. Lemma 2.5. Let r and s be two positive integers. Then ecdrs(H) < r(s - 1)C + ecdr(Tn,c,s). Proof of Lemma 2.4. Using Lemma 2.5 instead of Lemma 3 in the proof of Lemma 1 in [12] leads us to the proof. □ References [1] R. Abyazi Sani and M. Alishahi, A new lower bound for the chromatic number of general Kneser hypergraphs, 2017, arXiv:1704.07052v1 [math.CO]. [2] M. Alishahi, Colorful subhypergraphs in uniform hypergraphs, Electron. J. Combin. 24 (2017), #P1.23, http://www.combinatorics.org/ojs/index.php/eljc/ article/view/v2 4i1p2 3. [3] M. Alishahi and H. 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A 118 (2011), 1820-1828, doi:10.1016/j.jcta.2011.02.010. [19] F. Meunier, Colorful subhypergraphs in Kneser hypergraphs, Electron. J. Combin. 21 (2014), #P1.8, http://www.combinatorics.org/ojs/index.php/eljc/ article/view/v21i1p8. [20] G. Simonyi and G. Tardos, Colorful subgraphs in Kneser-like graphs, European J. Combin. 28 (2007), 2188-2200, doi:10.1016/j.ejc.2007.04.015. [21] C. Tardif, Hedetniemi's conjecture, 40 years later, Graph Theory Notes N. Y. 54 (2008), 46-57, http://gtn.kazlow.info/GTN54.pdf. [22] X. Zhu, On the chromatic number of the products of hypergraphs, Ars Combin. 34 (1992), 25-31. [23] X. Zhu, A survey on Hedetniemi's conjecture, Taiwanese J. Math. 2 (1998), 1-24, doi:10. 11650/twjm/1500406890. [24] G. M. Ziegler, Generalized Kneser coloring theorems with combinatorial proofs, Invent. Math. 147 (2002), 671-691, doi:10.1007/s002220100188. [25] G. M. Ziegler, Erratum: "Generalized Kneser coloring theorems with combinatorial proofs" [Invent. Math. 147 (2002), 671-691], Invent. Math. 163 (2006), 227-228, doi:10.1007/ s00222-005-0466-8. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 127-145 https://doi.org/10.26493/1855-3974.1059.4be (Also available at http://amc-journal.eu) Coordinatizing n3 configurations William L. Kocay * Department of Computer Science and St. Pauls College, University of Manitoba, Winnipeg, Manitoba, Canada Received 14 March 2016, accepted 7 January 2018, published online 13 June 2018 Given an n3 configuration, a one-point extension is a technique that constructs (n +1)3 configurations from it. A configuration is geometric if it can be realized by a collection of points and straight lines in the plane. Given a geometric n3 configuration with a planar co-ordinatization of its points and lines, a method is presented that uses a one-point extension to produce (n +1)3 configurations from it, and then constructs geometric realizations of the (n + 1)3 configurations. It is shown that this can be done using only a homogeneous cubic polynomial in just three variables, independent of n. This transforms a computationally intractable problem into a computationally practical one. Keywords: (n, 3) -configuration, geometric configuration, anti-Pappian, rational coordinatization, elliptic curve. Math. Subj. Class.: 51E20, 51E30 1 Projective configurations A projective configuration consists of a set S of points and lines, and an incidence relation n, such that two lines intersect in at most one point. We denote this by (S, n). For example, a triangle with points A, B,C and lines a, b, c can be represented by the pair ({A, B, C, a, b, c}, {Ab, Ac, Ba, Bc, Ca, Cb}). A configuration (S, n) can also be viewed as a bipartite incidence graph of points versus lines. We will always assume that the incidence graph of a configuration is connected. Excellent references on configurations are the recent books by Griinbaum [10], and by Pisanski and Servatius [18]. An n3-configuration is a projective configuration with n points and n lines such that every line is incident with 3 points, and every point is incident with 3 lines. There is a unique *This work is partially funded by a discovery grant from the Natural Sciences and Engineering Research Council of Canada. E-mail address: bkocay@cs.umanitoba.ca (William L. Kocay) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ 73-configuration, the Fano configuration, and a unique 83-configuration, the Mobius-Kantor configuration. An n3 configuration which can be represented by a collection of points and straight lines in the real or rational plane, such that all incidences are respected, and no two points or two lines coincide is termed a geometric n3 configuration. In order to show that an n3 configuration is geometric, the usual method is to assign suitable homogeneous coordinates to its points and lines. We call this a coordinatization of the configuration. A central problem [10] is to characterize which n3 configurations are geometric configurations, and to find rational coordinatizations [4, 10, 21, 22, 23] of those that are geometric. Griinbaum [9], and [10] (p. 151) has conjectured that an (n3) configuration that admits a real coordinatization also admits a rational coordinatization. He considers this the single most important outstanding problem in n3 configurations [11]. Sturmfels and White [22, 23] have shown that all (113) and (123) configurations have rational coordinatizations. These configurations were originally discovered by Martinetti [17], and Daublebsky von Sterneck [6, 7]. Sturmfels and White and Bokowski [4, 22, 23] found rational coordinatizations by constructing systems of diophantine equations, and then using methods of computer algebra to solve them, in particular, Grassmannian algebras and Grobner bases. A coordinatization of an n3 configuration is usually represented by homogeneous coordinates in the plane, e.g., let P1 = (x1,y1, z1) and P2 = (x2,y2, z2) be the homogeneous coordinates of two points, and let L = (a, b, c) be the homogeneous coordinates of a line. Then P1 and P2 are incident with L if and only if Pi • L = P2 • L = 0. Equivalently, L is a multiple of P1 x P2. Consequently there is an exterior algebra that the homogeneous coordinates generate. If there are n points and n lines, with 3n incidences, there are 6n variables, and numerous algebraic constraints that the coordinates must satisfy. Bokowski and Sturmfels [4] used computer-aided algebra to search for rational solutions to these algebraic constraints. Eventually the constraints can be manipulated to produce a homogeneous polynomial with at most 3n variables whose zeros characterize the coordinatizations. The polynomial has degree bounded by n. The difficulty of this work led Sturmfels and White [23] to suggest that the problem of finding rational coordinatizations of n3 configurations may be recursively undecidable. A simpler method of finding a coordinatizing polynomial, without the need of Grobner bases and the exterior algebra, was presented in Kocay-Szypowski [15]. The degree of the polynomial is still bounded by n. This method was used in Kocay [13] to find a rational coordinatization of the Georges configuration, which is a (253) configuration. In Sturmfels and White [22, 23], ad-hoc methods were used to find rational roots of the coordinatizing polynomials for each of the (113) and (123) configurations. There are 31 (113) and 229 (123) configurations. A note on homogeneous polynomials and their zeros: Homogeneous quadratic polynomials are well understood, see Conway [5]. It is the theory of quadratic forms. Cubic homogeneous polynomials are much more difficult. When there are three variables, they include the class of elliptic curves [20]. The rational points on an elliptic curve form a group. If there are one or more known rational points on the curve, then others can be found by combining them using the group operation. This generates a countable number of points. Mordell's theorem says that these groups are finitely generated, i.e., a finite number of starting points is needed to find the entire group. It does not say what the group is, or whether there are any rational points on the curve. And it does not provide a method to determine if there are any rational points on the curve. Because it is relatively easy to do computation in these groups, but simultaneously, there are theoretical difficulties in characterizing them, these groups are used in elliptic curve cryptographic systems [20]. Homogeneous polynomials of degree four or more are much more difficult, apparently not amenable to the same techniques. Thus the degree of the polynomial is important. The purpose of this paper is to present an algorithm which can be used to construct real or rational coordinatizations of (n + ^-configurations from coordinatizations of n3-configurations, by finding the roots (real or rational) of a cubic homogeneous polynomial in three variables. The use of a cubic homogeneous polynomial in three variables makes the formerly intractable problem of finding rational coordinatizations computationally practical and efficient. Some of the techniques are similar to methods used in the theory of elliptic curves [3, 20]. An elliptic curve is a cubic polynomial that can be expressed in the form y2 = ax3 + bx2 + cx + d The rational points on an elliptic curve form a group. See [20] for further information on these groups. Theorem 1.1 (Mordell's theorem). If a non-singular plane cubic curve has a rational point, then the group of rational points is finitely generated. Methods that originated with Diophantus [1] are used to find the rational roots of elliptic curves [20]. We use similar methods to construct coordinatizations of n3-configurations. As there can be very many rational points on an elliptic curve, there can be also be very many different rational coordinatizations of an n3 configuration. They are related in a way that is similar to the group operation of an elliptic curve. In general, it seems to be difficult to characterize when a rational coordinatization is possible. However the method presented here is very fast in practice, and can be automated. We begin with a 1-point extension [14] in an n3 configuration, which extends it to an (n +1)3 configuration, and which leads to the coordinatization algorithm. This extension is different from Martinetti's extension [17], which is described in Griinbaum [10] (p. 89). As pointed out in [10], it is in general quite difficult to characterize exactly which configurations are generated by an inductive construction which produces an (n + 1)3 configuration from an n3 configuration. This is true even if the construction can easily be described. In [14] the configurations that can be built using a 1-point extension are characterized. Theorem 1.2 (1-Point Extension). Let (E, n) be an n3-configuration. Let a\,a2,a3 be 3 distinct points in E, and let (i,(2, t3 be 3 distinct lines in E such that a i = (i n t2, a2 = C2 n C3 and a3 G t3, where a3 G £i. We can represent this in tabular form as (E, n) h Í2 Í3 ••• ai ai a2 • • • bi a2 a3 • • • b2 b3 b4 • • • where the dots indicate other points of the configuration. Here the points in each column are incident with the line at the top of the column. Let C be the third line containing ai. Suppose further that if ( n C3 = 0, then ( n t3 = a3. Construct a new configuration (E', n') as follows. E' = E U jao, C0} where a0 is a new point and C0 is a new line. Define the new incidences as n' = n — {ai(i, a2(2, a3(3} U {ai(3, a2(0, a3(0, a0(0, a0(i,a0(2}. We can represent this in tabular form as (E',no ic ii 12 13 ••• a2 ao ai ai • • • «3 bi ac a2 ■ ■ ■ ac 62 63 64 ■■■ Here the dots represent exactly the same points as in the previous table. Then (E', n') is an (n + 1)3-configuration. (Refer to Figure 1.) initial points and lines after 1-point extension Figure 1: A 1-point extension with 3 points, before (a), after (b). Example. The Fano configuration can be represented by the following table. Fano li l2 l3 l4 l5 4 l7 1 2 3 4 5 6 7 2 3 4 5 6 7 1 4 5 6 7 1 2 3 Choose l1, l2, l3 as indicated, and choose a1 = 2, a2 = 3, a3 = 6, and let a0 = 8. Notice that the third line containing a1 is l' = l6, which intersects l3 in a3 = 6. Then by Theorem 1.2, the following table represents an 83-configuration, which is known to be unique. 83-config lo li I2 I3 l4 I5 le lr 3 1 2 2 4 5 6 7 6 4 5 3 5 6 7 1 8 8 8 4 7 1 2 3 The diagram of Figure 1 illustrates the 1-point extension schematically, showing the incidences altered by the extension. The method uses three points a1; a2, a3 and three lines l1, l2, l3 sequentially incident, with a new point ac and line lc added. It can be generalized to m points a1; a2,..., am and m lines l1, l2,..., sequentially incident, see Kocay [14] for more details. This is indicated in Figure 2 for m = 4. When m = 4, the 1-point extension theorem has the following abridged form. Theorem 1.3 (1-Point Extension with 4 points and 4 lines). Let (E, n) be an ^-configuration. Let a1, a2, a3, a4 be 4 distinct points in E, and let l1, l2, l3, l4 be 4 distinct lines in E such that a1 = l1 n l2, a2 = l2 n l3, a3 = l3 n l4, and a4 G l4, where a3, a4 ^ l1, l2, and a1 G l4. Let ¿1 be the third line containing ai, and ¿2 be the third line containing a2. Suppose further that if ¿1 n ¿3 = 0, then ¿1 n ¿3 = a3; and if ¿2 n ¿4 = 0, then ¿2 n ¿4 = a4. Construct a new configuration (E', n') as follows. E' = E U {a0, ¿0} where a0 is a new point and ¿0 is a new line. Define the new incidences as n' = n — {ai¿1, a^2, a3i3, a4¿4} U {ai¿3, a2¿4, a3¿o, a4¿o, ao¿o, ao¿l, ao¿2}. Then (E', n') is an (n + l)3-configuration. (Refer to Figure 2.) When one point extensions are generated by computer, it is necessary to name them, so that the extensions generated can be identified. We have used the following naming convention. Here a configuration (E, n) is assumed, but is not explicitly indicated in the notation, as this will be clear from the context. Definition 1.4. A 1-point extension using three lines ¿1, ¿2, ¿3 and three points ai, a2, a3 is denoted Ext^i, ¿2, ¿3; ai, a2, a3). A 1-point extension using four lines ¿1 , ¿2, ¿3, ¿4 and four points a1,a2,a3, a4 is denoted Ext^i, ¿2, ¿3, ¿4; a1,a2,a3, a4), and so forth. When the starting n3 configuration has a real or rational coordinatization, we want to use its coordinatization to find a real or rational coordinatization of the resulting (n + 1)3 configuration. Both Theorems 1.2 and 1.3 are needed for the extension algorithm. initial points and lines after 1-point extension Figure 2: A 1-point extension with 4 points, before (a), after (b). 2 The coordinatization algorithm Let the points of a geometric n3 configuration (E, n) be {a^ a2,..., an} and let the lines be {¿1, ¿2,..., ¿n}. Let the homogeneous coordinates of a4 be Pi, and the homogeneous coordinates of ¿i be Li. These can be either real or rational. Then point ai is incident on line ¿j if and only if Pi • Lj = 0. Suppose that a 1-point extension is applied to (E, n) to obtain an (n + 1)3 configuration (E', n'), using three points and lines of (E, n), as in Figure 1. We can assume that the points and lines are labelled so that the extension uses points ai, a2, a3 and lines ¿1, ¿2, ¿3 as in Figure 1, and adds a0 and ¿0. Let G denote the incidence graph, also known as the Levi graph, of (E, n). The subgraph induced by {ai, a2, a3, ¿1, ¿2, ¿3} is a path of length five, since a3 ^ ¿1, and because the girth of the incidence graph must be at least six. After the extension, a0 and ¿0 are added. Let G' be the new incidence graph. The subgraph now induced is illustrated in Figure 3(a), since the girth of the incidence graph must be at least six. The significant feature of this subgraph is the hexagon induced by {a0, a^ a2, ¿0, ¿2, ¿3}. We now look for a shortest path Q in the incidence graph, not using any edges of the hexagon, from any one of {a0, ai, a2} to any one of {¿0, ¿2,¿3}. This is easy to do using a breadth-first search of the incidence graph. Note that the shortest path may possibly contain a3 and/or ¿1. Q must contain at least two internal vertices, i.e., one point and one line. Let the endpoints of Q be ai and ¿j .If u is an internal vertex of Q, then u is not incident with the other vertices ¿k on the hexagon (where k = j), or there would either be a shorter path than Q, or else the girth requirement would not be satisfied. Similarly, u is not incident with the other vertices am on the hexagon (where m = i). ao ¿0 ¿1 O b a3 induced subgraph for Figure 1 (b) ao ¿0 ¿1 w a4 induced subgraph for Figure 2 (b) 2 Figure 3: An induced subgraph of the incidence graph of (£', n') of Figures 1 and 2. We now have a theta subgraph in the incidence graph, that is, two vertices (ai and ¿j), connected by three internally disjoint paths. When m = 4, the situation is similar. The vertices a1, a2, a3, a4, ¿1, ¿2, ¿3, ¿4 of Figure 2(b) determine a path of length 7 in the incidence graph G. After the extension, the subgraph of G' determined by Figure 2(b) is illustrated in Figure 3(b). It is necessary that this be an induced subgraph for the coordi-natization algorithm. We now look for a shortest path Q in the incidence graph, not using any edges of the octagon, from any one of {a0, a1, a2, a3} to any one of {¿0, ¿2, ¿3, ¿4}. Let the endpoints of Q be ai and ¿j. Once again we find that Q must contain at least two internal vertices, and again we have a theta-subgraph, ©. The algorithm requires that this be an induced theta subgraph. The incidence graph is 3-regular, so that vertices ai and ¿j are adjacent only to vertices of ©. All other vertices of © are adjacent to exactly one vertex not in ©. We now look for a coordinatization of (£', n') such that all points and lines have the same coordinates as in (£, n), except for the points and lines of ©. Let the homogeneous coordinates of ai be (x, y, z), where x, y, z are real or rational indeterminates, according to whether the coordinatization of (£, n) is real or rational. Then © contains three internally disjoint paths Q1, Q2, Q3 from ai to ¿j. We follow each path, and execute the following statements, assigning coordinates to its vertices in terms of x, y, z. For each vertex not in ©, its homogeneous coordinates are those of (£, n). These are known constants. The algorithm below constructs coordinates for the vertices of © in terms of x, y, z, by starting at ai, and successively following each path Qm of © to ¿j. Note that if L and L' are homogeneous coordinates of lines, then the cross product L x L' gives the homogeneous coordinates of the unique point which is the intersection of the two lines. Similarly P x P' gives the homogeneous coordinates of the unique line containing points with coordinates P and P'. procedure FOLLOWPATH(a,, ij, Qm) comment: follow a path Qm of © from a, to ij, assigning coordinates u ^ a, v ^ first vertex on path Qm after a, while v = ij 'if v is a point " let i be the unique adjacent line not in © let L be the known coordinates of i let L' be the assigned coordinates of u P ^ L x L' assign P as the coordinates of v do ^ let a be the unique adjacent point not in © let P be the known coordinates of a else let P' be the assigned coordinates of u L ^ P x P' assign L as the coordinates of v u ^ v ^ v ^ next vertex on path Qm after u comment: every vertex of Qm except for ij now has coordinates assigned then Observation. Once the algorithm FOLLOWPATH() has been executed for each path of ©, all vertices of © except for ij have homogeneous coordinates assigned such that each coordinate is a linear homogeneous function of x, y, z. There are three vertices of © adjacent to ij. Let their coordinates be P, P' and P''. Define the polynomial p(x, y, z) = P • P' x P''. Observation. p(x, y, z) is a cubic homogeneous polynomial in x, y, z. Note that by projective duality we could equally well follow the paths in the other direction, from ij to a,, starting with (x, y, z) as the coordinates of ij. Theorem 2.1. If there is a coordinatization of (£', n') such that all points and lines not in © have the same coordinates as in (£, n), then the values of x, y, z must satisfy p(x,y, z) = 0. Proof. The three points incident on ij all belong to ©, with coordinates P, P', P''. Therefore P • P' x P'' = p(x, y, z) = 0. Note that the coordinates of ij can be taken as any one of P x P', P x P'' or P' x P''. □ Thus, if there is a coordinatization of (£', n') of the type we are looking for, we can find it by solving p(x, y, z) =0 for x, y, z. In general, there will be many values (x, y, z) with p(x, y, z) = 0. They do not all give valid coordinatizations. According to the current coordinatization of (£, n), we want the values to be either real or rational. We will use a method that originated with Diophantus (see [1]), as frequently used in the theory of elliptic curves [3,20]. Now the groups defined by elliptic curves are used for cryptography, because it is relatively easy to calculate with them, but a characterization of the groups appears to be algorithmically intractable. A similar situation exists in the search for coordinatizations of n3 configurations. But if we can find suitable values of x, y, z such that p(x, y, z) = 0, then a real or rational coordinatization of (Y', n') can be relatively easy to find. The method described below works very effectively. Lemma 2.2. Let t be any one of the three lines adjacent to a in O, and let its coordinates be L. Let a be the unique point not in O adjacent to t, and let its coordinates be P. If (x, y, z) is set equal to P, then p(x, y, z) = 0. Proof. If (x, y, z) = P, then L = P x (x,y,z) = (0,0,0). Each subsequent vertex on this path in O will have coordinates (0,0,0), so that tj will also have coordinates (0,0,0). Therefore p(x, y,z) = 0. □ As there are three lines in O adjacent to a4, this gives three different points (x, y, z) with p(x, y, z) = 0. None of these give coordinatizations of (£', n'), because (0,0,0) is not a valid homogeneous coordinate. However, we can now proceed as follows. Suppose that p(x,y,z) = 0, for some value (x,y,z) = (u,v,w). The equation p(x, y, z) = 0 defines a cubic curve in the projective plane. The tangent line at point (u, v, w) has the equation xdp/dx + ydp/dy + zdp/dz = 0, where the partial derivatives are evaluated at (u, v, w). This is a linear equation in (x, y, z). As long as at least one partial derivative is non-zero, say dp/dz, we can solve for the associated variable, and obtain z = -[xdp/dx + ydp/dy]/[dp/dz] along the tangent line. This is substituted into the cubic homogeneous polynomial p(x, y,z) = 0 to obtain q(x, y) = 0, where q(x, y) is a cubic homogeneous polynomial in x, y. At this point, we can divide by y3 to obtain the cubic polynomial q(x/y, 1) = 0 in one variable x/y. Now q(x/y, 1) = 0 has three roots, of which one, x/y = u/v, is already known (note: if v = 0, use y/x = v/u instead). The tangent line has double contact (see [3]) with the curve p(x, y, z) = 0 at (x, y, z) = (u, v, w). Therefore we can divide q(x, y) by vx - uy twice to obtain a linear homogenous equation h(x, y) = 0. The single root of h(x, y) is then easy to find, even over the rational numbers. Combining this with the expression for z, we obtain another root (x, y, z) = (u', v', w') of p(x, y, z) = 0. This new value for (x, y, z) is now substituted into the coordinates for the vertices of O, and the coordinates (which are linear homogeneous functions of x, y, z) of all vertices of O are evaluated. It is then quickly determined whether this produces a valid coordinatization of (£', n'). The conditions that must be satisfied are: 1. All points must have inequivalent homogeneous coordinates; 2. All lines must have inequivalent homogeneous coordinates; 3. P • L = 0 only if point P is incident with line L. If some points or lines coincide, or if unwanted incidences are produced, then the method can be repeated, starting from (x, y, z) = (u', v', w'). Either anew point (u'', v'', w'') will be found, or else a value previously found will recur, and so forth. This can be done for each of the three lines in O adjacent to a4, which frequently produces a number of valid coordinatizations of (Y', n'). There is still another possibility. The coordinates of any two of the three lines in O adjacent to a4 determine a line in the projective plane, intersecting the curve p(x, y,z) = 0 in two known points. The third point of intersection is then easy to find. This calculation allows a sequence of points satisfying p(x, y,z) = 0 to be found. We can then continue with the tangents from these points, or take any two known roots on the curve to find another. The number of points on the curve that can be generated from the starting values can be either finite or countably infinite, as this is the situation that holds for rational points on elliptic curves (see [20]). We summarize this method as two theorems. Theorem 2.3. Let (x, y, z) = (u, v, w) be a rational solution of the cubic homogeneous polynomial p(x,y,z) = 0. If at least one of dp/dx,dp/dy,dp/dz evaluated at (x,y,z) = (u, v, w) is non-zero, then the tangent line xdp/dx + ydp/dy + zdp/dz = 0 intersects the curve in another rational point. Theorem 2.4. Let (x,y,z) = (ui,vi,w1) and (x,y,z) = (u2,v2,w2) be two rational solutions of the cubic homogeneous polynomial p(x, y, z) = 0. Then the line containing (ui, vi, wi) and (u2,v2, w2) intersects the curve in another rational point. In practice, we want at least two of the partial derivatives to be non-zero at (x, y, z) = (u, v, w). For if two of them are zero, then solving for the third variable forces one of x, y, z to be zero. This invariably leads to a solution which does not give a valid coordinatization. (However, it can then be used to find another rational solution.) Once a valid coordinatization of (Y', n') has been found for a suitable value (x, y, z) = (u, v,w), this process can be repeated, and more coordinatizations can be found. In general, numerous coordinatizations for a given configuration can be found in this way. They are inter-related through tangents to the cubic polynomialp(x, y, z), and through lines containing pairs of rational solutions, similar to the relation between points of the group of rational points on an elliptic curve. Example. We begin with a rational coordinatization of a (93) configuration, shown in Figure 4. This is the (93) configuration listed as (93)2 in Figure 2.2.1 of [10], and as 9.2 in [2]. It is cyclic and self-dual, with an automorphism group of order 9. The two "parallel" lines i4 and i8 meet in point a9 at infinity. Similarly i5 and i7 meet in a8 at infinity, and lines £i and l3 meet in a2 at infinity. These three points at infinity are all contained in the line £q, which is the "line at infinity". The drawing is based on the rational coordinatization of the configuration given by the coordinates shown in Table 1. Table 1: Rational coordinates of the 93 configuration of Figure 4. Pi = (2, 4, -3) Li = (1 , 1, 2) P2 = (-1,1,0) L2 = (2, -1,0) P3 = (1, 2, -3) L3 = (1,1,1) P4 = (1,1,-1) L4 = (0, 3, 2) P5 = (0, 0, 1) L5 = (3, 0, 2) Pe = (1, 0, -1) Le = (0, 0, 1) P7 = (2, 2, 3) L7 = (1, 0, 1) Ps = (0,1,0) Ls = (0,1,0) P9 = (1, 0, 0) L9 = (1 , -1 , 0) A 1-point extension using four points Ext(£1,£9,£4, £e; a4, a7, a9,as), as in Figure 2, is then done. (This example using 4 points was chosen instead of one using 3 points, 17 ¿5 (oq) © (as) Figure 4: A 93 configuration. because the resulting (IO3) configuration has a "nice" drawing.) Observe that a4 = £1 n ¿9, a7 = i9 n ¿4, a9 = ¿4 n ¿6, and that a8 e ¿6. The third line through a4 is t7. It intersects ¿6 in a8, as required for the 1-point extension. The result of the extension is the 103 configuration shown in Figure 5. It is (103)6 in Griinbaum [10]. Lines ¿1, ¿3 and ¿6 in Figure 5 meet in point a2 at infinity. Points and lines whose coordinates did not change from (93) are drawn in heavier lines. (But note that the scaling of the two diagrams may be slightly different.) In order to find a rational coordinatization of it, we first find a theta subgraph by searching for a shortest path from one of a4, a7, a9, a10 to one of ¿9, ¿4, ¿6, ¿10, where a10 and ¿40 are the new point and line that were added. The theta subgraph is shown in Figure 6. It consists of the octagon of Figure 3(b) and the shortest path just found. This is partly indicated in Figure 5. The "corners" of the theta subgraph, a4 and ¿10, are shaded light grey. With the aid of Figure 6, the paths can be traced out in Figure 5. We now assign homogeneous coordinates (x, y, z) to ¿10, as it is one of the "corner" vertices of the theta subgraph, and using the coordinates of Table 1 for the points and lines not in the theta subgraph, we calculate coordinates for those of the theta subgraph in terms of (x, y, z). Each point or line of the theta subgraph (except for the "corner" vertices) is adjacent to exactly one line or point not in the theta subgraph. The adjacent vertices can be determined from Figure 5. The calculated homogeneous coordinates are linear homogeneous forms, shown in Table 2. Note that homogeneous coordinates can be multiplied by a constant without changing the configuration. Therefore sometimes the coordinates in Table 2. were multiplied by -1, or a common factor was removed from the individual coordinates in order to simplify them. We then find thatp(x, y, z) = L9 L7 x L4, which is expanded to 099 9 9 o p(x, y, z) = —4x + 4x y + 4x z + 7xz — 16xyz + 11yz — 6z where a common factor of six has been removed from each term. The partial derivatives ¿5 Figure 5: The extended IO3 configuration. a 10 ¿9 Figure 6: A theta subgraph in the 103 configuration. are dp/dx = — 12x2 + 8xy + 8xz — 16yz + 7z2 dp/dy = 4x2 — 16xz + 11z2 dp/dz = 4x2 — 16xy + 14xz + 22yz — 18z2 We have three known solutions to p(x, y, z) = 0, namely (x, y, z) = Li = (1,1,2) (which makes P10 = (0,0,0)), (x, y, z) = L5 = (3,0,2) (which makes Pg = (0,0,0)), (x, y, z) = L6 = (0,0,1) (which makes P9 = (0,0,0)). The tangent line at (x, y, z) = L1 = (1,1, 2) has equation 2x + 4y — 3z = 0. Solving for 2x = —4y + 3z, substituting this into p(x, y, z) = 0, and removing common factors gives q(y, z) = 4y3 — 4y2z + yz2 The point L1 on the tangent line has (y, z) = (1, 2) so that q(y, z) is divisible twice by 2y — z. We find that q(y, z) = 6y(2y — z)2 Therefore the third point of intersection of the tangent with p(x, y, z) =0 occurs when y = 0. Then since 2x + 4y — 3z = 0, we can take z = 2, and obtain 2x = —4y + 3z = 6, giving (x, y, z) = (3,0,2). This does not give a valid solution, as it makes P8 = (0,0,0). Table 2: Homogeneous coordinates for the theta subgraph. L10 = (x,y,z) P10 = L10 X L1 = (2y — z, z — 2x, x — y) Pg = ¿10 X L5 = (2y, 3z — 2x, —3y) P9 = ¿10 X ¿8 = (—z, 0, x) ¿9 = P10 X P5 = (z — 2x,z — 2y, 0) ¿7 = P8 X P6 = (2x — 3z, —y, 2x — 3z) ¿6 = P9 X P2 = (x, x, z) P7 = ¿6 X ¿5 = (—2x, 2x — 3z, 3x) ¿4 = P7 X P3 = (4x — 3z, x, 2x — z ) We then try the tangent line at (x, y, z) = L5 = (3,0, 2), which has equation 2x + y — 3z = 0. Solve for y = 3z — 2x and substitute this into p(x, y, z) to obtain q(x, z) = 4x3 — 16x2z + 21xz2 — 9z3 The known solution is (x, z) = (3, 2), so that this is divisible twice by 2x — 3z, giving q(x, z) = (x — z)(2x — 3z)2 We find that the third intersection point with the curve p(x, y, z) = 0 occurs when x = z. Without loss of generality, we take (x, y, z) = (1,1,1). If we then calculate the coordinates, we find that L10 and L6 both have coordinates (1,1,1), which is not acceptable. However, this gives another rational point on the curve, so we find the tangent line at (x, y, z) = (1,1,1). It is —5x — y + 6z = 0. We substitute y = 6z — 5x into p(x, y, z) to obtain q(x, z) = 2x3 — 9x2z + 12xz2 — 5z3 The known solution is (x, z) = (1,1), so that this is divisible twice by x — z, giving q(x, z) = (2x — 5z)(x — z)2. The third point of intersection is therefore (x,y, z) = (5, —13,2). This value of (x, y, z) is then found to give a valid coordinatization of the 103 configuration found. The coordinates that result are shown in Table 3. At this point, the algorithm could continue, and find the tangent line at (x, y, z) = (5, —13,2) to look for more rational coordinatizations. Or the known rational points on the curve could be taken two at a time, as the line containing two points intersects the curve in a third rational point, and so forth. In practice, very many rational coordinatizations can be found in this way from a single theta subgraph of a single one-point extension of a geometric configuration. Table 3: Rational coordinates of the 103 configuration of Figure 5. P1 = (2,4, -3) P2 = (-1,1, 0) P3 = (1, 2, -3) P4 = (-14, -4, 27) P5 = (0,0,1) Pe = (1,0,-1) P7 = (-10,4,15) P8 = (-26, -4, 39) P9 = (-2,0, 5) Li = (1,1, 2) ¿2 = (2, -1,0) ¿3 = (1,1,1) L4 = (14, 5, 8) (3, 0, 2) (5, 5, 2) (4, 13, 4) (0,1, 0) ¿5 Le ¿7 ¿8 ¿9 = (-2, 7,0) P10 = (-14, -4, 9) L10 = (-5,13, -2) We now start from the 103 configuration of Figure 5, with the rational coordinatization given in Table 3. There is a one-pont extension Ext(^10,¿6,¿3; a9, a2, ae) that can be done, resulting in an 113 configuration. Its incidence table is given in Table 4. This configuration is isomorphic to configuration (113)X in Martinetti [17]. The new point and line added are a11 and ^11. We use a theta subgraph to find a rational coordinatization of it. The theta subgraph consists of the three paths [a9, ¿3, a2, ^11], [a9,¿6, a11, ¿n], [a9,4, ae, ¿nj. There are many rational coordinatizations that result. One of them is shown in Table 5. We see that the integer coordinates are getting bigger. This is the single greatest obstacle that the algorithm has to deal with. One of the questions that needs to be addressed is how to limit the number of digits in the integers that arise. It is very easy for integer overflow to occur after several successive extensions have been done. The Desargues configuration cannot be obtained by a 1-point extension (see [14]). The "anti-Pappian" (see [8, 16]) is the only non-geometric 103 configuration. Rational coordinatizations of all the other (103) configurations, can be easily found using one-point extensions of the (93) configurations in this way. Then rational coordinatizations of all the (113) configurations can be found from the (103 ) configurations, which then extend to coordinatizations of all the (123) configurations. The author has written a computer program to generate coordinatizations from a theta subgraph in a one point extension. It produces thousands of them very quickly. Currently the program has to be run individually for each starting configuration, and the resulting output files must be individually collated and then tested for isomorphisms. Table 4: The 113 configuration extended from Figure 5. ¿1 ¿2 ¿3 ¿4 ¿5 ¿6 ¿7 ¿8 ¿9 ¿10 ¿11 1 1 2 3 1 7 4 5 4 8 2 2 3 3 4 7 9 6 6 5 10 6 10 5 9 7 8 11 8 9 10 11 11 Table 5: Rational coordinates of the 113 configuration extended from Table 3. Pi = 2, 4, -3) Li = 1 , 1 , 2) P2 = 16, -34, 9) ¿2 = ( 2,-1, 0) P3 = 1, 2, -3) ¿3 = (28,19, 22) P4 = -14, -4, 27) ¿4 = (14, 5, 8) P5 = 0, 0,1) ¿5 = (3, 0, 2) P6 = -136,4,123) ¿6 = (27,-15, 22) P7 = -10, 4,15) ¿7 = (4,13,4) P8 = -26, -4, 39) ¿8 = (1, 34,0) P9 = -748, 22, 933) ¿9 = (-2, 7, 0) Pi0 = (-14, -4, 9) ¿10 = = (-5,13, -2) Pli = (-64,-14, 69) ¿11 = = (37, 28, 40) 3 In practice Given an n3 configuration (E, n), it is relatively easy to write a computer algorithm that searches for all possible one-point extensions Ext^, ¿2,¿3; a^ a2, a3) or Ext(^i, ¿2, ¿3, ¿4; ai, a2, a3, a4 ), and extends (E, n) to an (n + 1)3 configuration (E', n'), in all possible ways. We also want a coordinatization of (E', n') when (E, n) is geometric. For each extension (E', n') found, the coordinatization algorithm of the previous section can be used to look for a coordinatization of (E', n'). There are various situations that one has to be aware of when programming this. 1. The polynomial p(x, y, z) is a cubic homogeneous polynomial in three variables. Sometimes a cubic polynomial will factor into the product of three linear homogeneous polynomials, or a linear and quadratic polynomial. In these cases the algorithm will not succeed. This happens occasionally in practice. It will usually be detected when the tangent is found. Not every extension (E', n') has a coordinatization extended using a given theta subgraph. However, another theta subgraph can be chosen in this case. 2. The tangent at (x, y, z) = (u, v, w) is a linear homogeneous polynomial. It may be identically 0. In this case the extension does not succeed. 3. The tangent at (x, y, z) = (u, v, w) may be be a monomial, e.g., x = 0. This does not tend to produce valid coordinatizations. 4. Suppose that the tangent at (x, y, z) = (u, v, w) is ax + by + cz = 0. Solving for one variable, e.g., cz = -ax - by and substituting this into p(x, y, z) gives the reduced polynomial q(x, y) = 0, which is divisible twice by vx - uy. It can happen that q(x,y)/(vx - uy)2 is amonomial, e.g., q(x,y) = x(vx — uy)2. This gives x = 0, from which we find the solution (x, y, z) = (0, -c, b). This frequently occurs as a special case. 5. The general case is when q(x, y) factors into (vx - uy)2(rx + sy). In this case the solution is cx = cs, cy = -cr and cz = -as + br, or equivalently, (cs, -cr, -as + br) is taken as the solution. The majority of solutions fall into this case. 6. The algorithm stores an array of solutions (x, y, z) = (u, v, w) to p(x, y, z) = 0. Initially there are three such points (u, v, w) known, and they are known not to give valid coordinatizations of (£', n'). They are placed on the array of solutions. For each (u, v, w) on the array, the tangent is used to find another possible solution, which is appended to the array. The solutions on the array are then taken in pairs (ui, vi, wi) and (u2, v2, w2), to find more solutions, which are also appended to the array. The algorithm proceeds to build an array of all solutions (u, v, w) that can be obtained by these methods. This is similar to generating the elements of a group. Typically a potentially infinite number of solutions will be found, so that a limit must be placed on the maximum number allowed. The algorithm can stop with the first valid coordinatization found, or it can look for some maximum number of valid coordinatizations. It can easily find thousands of valid integer coordinatiza-tions. However, the values of the integers u, v, w rapidly become enormous if a large number of coordinatizations is required, causing integer overflow even when 64-bit integers are used. The author has programmed it to find a maximum of three valid coordinatizations for each extension (£', n') found, using 64-bit integers, and using only one theta subgraph. More theta subgraphs could be chosen. If (£, n) is an n3 configuration, there will be various (n+1)3 configurations that can be produced from it by one-point extensions. If (£', n') is such an (n+1)3 configuration, then there are usually very many different extensions of (£, n) that give rise to an isomorphic (£', n'). Each extension will have up to three coordinatizations found. And this same (£', n') may also arise by a one-point extension from another n3 configuration, which will also produce numerous coordinatizations of (£', n'). The result is thousands of integer coordinatizations for (£', n') when n =10,11 or 12. Graph isomorphism software is used to distinguish and recognize the various configurations that are produced. The author has used the software of [12], although others could also be used. The configuration is represented by its Levi graph, with an initial partition of vertices into points and lines. This method of finding coordinatizations is much simpler than that of [22, 23] because it only requires finding the roots of cubic homogeneous polynomials with three variables, whereas [23] states that solving their general diophantine equations for (n3) configurations is likely to be recursively undecidable. So far, the author has used this method to produce integer coordinatizations of all the geometric (103), (113) and (123) configurations. As n increases, the integer coordinates rapidly tend to have more and more digits, so that it is necessary to filter them somewhat to limit the number of digits in the coordinates. If fixed size integers are used (e.g. 64 bits), overflow can soon occur, which limits the number of coordinatizations found. It is advantageous to choose a coordinatization of (£, n) to extend from, whose coordinates are "small" integers. Very many coordinatizations of (£, n) are then obtained. This is the case with n = 10,11,12,13, where thousands of coordinatizations are easily found. If multi-precision integer arithmetic is used, it is likely that coordinatizations can be found for nearly any fixed n. The number of distinct (133) configurations is 2036 (see [10], p. 69). One of these is a Fano-type configuration, as described in [14], and therefore does not arise as a one point extension. Using ad-hoc methods, the author has shown that it is geometric, and in fact has a rational coordinatization. The other 2035 (133) configurations can all be constructed as one point extensions of (123) configurations. All of them are geometric, and all have rational coordinatizations. The coordinatization algorithm finds many integer coordinatizations of them. One of them was much more difficult than all the others, requiring integer coordinates with up to 22 digits in the intermediate calculations, and 13 in the final coordinates. For this one configuration, the algorithm was carried out by hand using Maple [24] as a calculator with unlimited precision. Maple was also used for constructing a coordinatiza-tion of the Fano-type configuration. The description of the coordinatizations is too long to include here. An article containing the details is currently in preparation. 4 Additional coordinatizations Suppose that (£, n) is an n3 configuration for which an integer coordinatization is known. We would like to find more integer coordinatizations. One method is this. 1. Find an induced theta subgraph © in the incidence graph of (£, n). This is most easily done by finding an induced cycle of reasonable length, and then finding a suitable path across the cycle. The path must have odd length. 2. The vertices not in © are to keep their current coordinates. One of the vertices of degree three in © is chosen to have coordinates (x, y, z), with values to be determined. 3. The algorithm FollowPath() is used to assign coordinates that are homogeneous linear forms to the vertices of © of degree two. A polynomial p(x, y, z) is constructed using the second vertex of degree three of ©. Solutions of p(x, y, z) = 0 are found as in the previous section. This allows us to find "related" coordinatizations of (£, n). The author has used this method to produce many rational coordinatizations of the (93) configurations, which can then be used as starting points for the generation and coordinatization of the (103) configurations and beyond. A given © may not produce any additional coordinatizations. In general, different choices of © will produce different results. This method is less reliable that the extension method of the previous section. The reason seems to be that the poly-nomialp(x, y, z) frequently has large integer coefficients, resulting in solutions which lead to integer overflow. For some configurations (£, n), no additional coordinatizations are found like this. For others, it gives dozens of new coordinatizations. 5 Real coordinatizations - the anti-Pappian The previous sections are concerned with using one-point extensions to find rational coordinatizations of n3 configurations. Theorems 2.3 and 2.4 also apply to real coordinatizations. The anti-Pappian [8, 16, 19] is the only (103) configuration that is not geometric. It cannot be coordinatized over any field, as shown in [8, 16]. However, it can be coordinatized over the quaternions [16]. The anti-Pappian can be obtained by a one-point extension from a geometric (93) configuration (it is (93)3 in [10] and 9.1 in [2], a self-dual configuration with an automorphism group of order 12). When the extension algorithm is applied to find a coordinatization, it is necessary to divide polynomials. It is easy to divide polynomials with integer coefficients, as the division is always exact. However, when a computer works with real numbers, they are represented as floating point numbers, and round-off error is always present. Consequently division will always leave a non-zero remainder, which is usually very small, even when the division is theoretically exact. A suitably small number is then replaced by zero, e.g. 10-9. When Pi • Lj is evaluated to test for incidence of a point and line, the result will usually not be exactly zero, due to round-off error, even if they are incident. So if Pi • Lj is sufficiently close to zero, it must be considered to be zero. Thus, it is possible to have a point and line not exactly incident, but very, very close to incident, for example, |Pi • Lj | < 10-9. Thus, a near-coordinatization can be found. Every real coordinatization found using floating point numbers is in fact a near-coordinatization. When the coordinatization algorithm is applied to the extension that produces the anti-Pappian, several near-coordinatizations are found, even though the anti-Pappian cannot be coordinatized over the reals. One of them is shown in Figure 7. Question. Let e be a small positive real value, and let A be a fixed positive real value, e.g., A = 1. How small can e be chosen so that there is a near-coordinatization of the Figure 7: A near-coordinatization of the anti-Pappian configuration. anti-Pappian configuration such that p • Lj | < e for all points Pj and lines Lj which are incident, and p • Lj | > A if Pj and Lj are non-incident? GrUnbaum [10] (p. 151) also asks whether there are any n3 configurations with n > 10 which are non-geometric? One place to look for them is the Fano-type configurations of [14], as they cannot be constructed using a one point extension, and so are not accessible to the cubic-polynomial-based coordinatization algorithm. The smallest Fano-type configuration is the unique (73). The next one is a (133) configuration (which is geometric). Then (143). References [1] I. G. Bashmakova, Diophantus and Diophantine Equations, volume 20 of The Dolciani Mathematical Expositions, Mathematical Association of America, Washington, DC, 1997. [2] A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v3, Discrete Appl. Math. 99 (2000), 331-338, doi:10.1016/s0166-218x(99)00143-2. [3] R. Bix, Conics and Cubics, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1998, doi:10.1007/978-1-4757-2975-7. [4] J. Bokowski and B. Sturmfels, Computational Synthetic Geometry, volume 1355 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1989, doi:10.1007/bfb0089253. [5] J. H. Conway, The Sensual (Quadratic) Form, volume 26 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1997. [6] R. Daublebsky von Sterneck, Die Configurationen II3, Monatsh. Math. 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Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992, doi:10.1007/978-1-4757-4252-7. [21] B. Sturmfels, Aspects of computational synthetic geometry, I. Algorithmic coordinatization of matroids, in: H. Crapo (ed.), Computer-Aided Geometric Reasoning, INRIA, Rocquencourt, pp. 57-86, 1987, papers from the INRIA Workshop held in Sophia-Antipolis, June 22 - 26, 1987. [22] B. Sturmfels and N. White, Rational realizations of 113 and 123 configurations, in: H. Crapo et al. (ed.), Symbolic Computations in Geometry, volume 389 of IMA Preprint Series, 1988, http://hdl.handle.net/112 9 9/4 7 62. [23] B. Sturmfels and N. White, All 113 and 123-configurations are rational, Aequationes Math. 39 (1990), 254-260, doi:10.1007/bf01833153. [24] Waterloo Maple Inc., Maple, https://www.maplesoft.com/. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 147-160 https://doi.org/10.26493/1855-3974.1218.5ed (Also available at http://amc-journal.eu) Maximum cuts of graphs with forbidden cycles* Qinghou Zeng School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, P. R. China, and Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian, P. R. China Jianfeng Hou f Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian, P. R. China Received 18 October 2016, accepted 2 November 2017, published online 13 June 2018 For a graph G, let f (G) denote the maximum number of edges in a bipartite subgraph of G. For an integer m > 1 and for a set H of graphs, let f (m, H) denote the minimum possible cardinality of f (G), as G ranges over all graphs on m edges that contain no member of H as a subgraph. In particular, for a given graph H, we simply write f (m, H) for f (m, H) when H = {H}. Let r > 4 be a fixed even integer. Alon et al. (2003) conjectured that there exists a positive constant c(r) such that f (m,Cr_i) > m/2 + c(r)mr/(r+1) for all m. In the present article, we show that f (m, Cr-1) > m/2 + c(r)(mr log4 m)1/(r+2) for some positive constant c(r) and all m. For any fixed integer s > 2, we also study the function f (m, H) for H = {K2,s, C5} and H = {C4, C5,..., Cr-1}, both of which improve the results of Alon et al. Keywords: H-free graph, partition, maximum cut. Math. Subj. Class.: 05C35, 05C70 1 Introduction All graphs considered here are finite, undirected and have no loops and no parallel edges, unless otherwise specified. All logarithms in this paper are with the natural base e. For a graph G, let f (G) denote the maximum number of edges in a cut of G, that is, the *This work is supported by NSFC (Grant No. 11671087). The authors are indebted to the anonymous referees for many valuable comments and constructive suggestions. Qinghou Zeng would like to thank his advisors Genghua Fan and Jianfeng Hou for getting him interested in graph theory. t Corresponding author. E-mail addresses: zengqh@ustc.edu.cn (Qinghou Zeng), jfhou@fzu.edu.cn (Jianfeng Hou) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ maximum number of edges in a bipartite subgraph of G. For an integer m, let f (m) denote the minimum value of f (G), as G ranges over all graphs with m edges. Thus, f (m) is the largest integer f such that any graph with m edges contains a bipartite subgraph with at least f edges. It is easy to show that f (m) > m/2 by considering a random bipartition of a graph with m edges. Edwards [10, 11] proved that for every m f (m) > m + 4(V2m +4 - 2) ' (U) and noticed that this is tight when m = (2) for odd integers k. For more information on f (m) and some related topics, we refer the reader to [1, 3, 5,6,8,14,15,16,21,26,27,28]. For survey articles, see [7, 23]. Suppose that H is a set of graphs. Let f (m, H) denote the minimum possible cardinality of f (G), as G ranges over all graphs on m edges that contain no member of H. In particular, for a given graph H, we simply write f (m, H) for f (m, H) when H = {H}. It is noted (see, e.g., [2]) that for every fixed graph H there exist positive constants e = e(H) and c = c(H) such that f (m, H) > m/2 + cm1/2+e for all m. However, the problem of estimating the error term more precisely is not easy, even for relatively simple graphs H. For example, let r > 4 be an integer and let H be the cycle Cr-1. The case r = 4 has been studied extensively. After a series of papers by various researchers [12, 22, 24], Alon [1] proved that f (m, C3) = m/2 + ©(m4/5) for all m. For general r > 4, Alon et al. [4] proposed the following conjecture. Conjecture 1.1. For every integer r > 4, there is a positive constant c(r) such that f (m, Cr_i) > m + c(r)m^ (1.2) for all m. This is tight, up to the value of c(r), for all r > 4. The authors confirmed (1.2) for all odd r > 4. In this paper, we consider the conjecture for every even integer r > 4 and establish the following theorem. Theorem 1.2. For every even integer r > 4, there is a positive constant c(r) such that f (m,Cr-1) > m + c(r) (mr log4 m) ^ for all m. Alon et al. [4] also studied the function f (m, H) when H is the complete bipartite graph K2,s. It is proved that, for every s > 2, there is a positive constant c(s) such that f(m,K2,s) > m + c(s)m5/6 for all m, and this is tight up to the value of c(s). Now, we consider the function f (m, H) for H = {K2,s, C5}, which improves the above lower bound as follows. Theorem 1.3. For each s > 2, let G be a {K2,s, C5}-free graph with m edges. Then there exists a positive constant c(s) such that f(G) > m + c(s)m6/7 for all m. Moreover, Alon et al. [2] considered the function f (m, H) for H = {C3,..., Cr-1}, and proved that f (m, H) > + c(r)m for all m. In the following, we allow the occurence of triangles and get a stronger result. Theorem 1.4. Let r > 4 be a fixed even integer and Hr = {C4,..., Cr-1}. Then there exists a positive constant c(r) such that f (m, Hr) > m + c(r)m^fr for all m. 2 Maximum cuts of C2k+1-free graphs In this section, we give a proof of Theorem 1.2. The goal is to prove that the chromatic number of a C2k+1-free graph is relatively small, since graphs with small chromatic number must have large bipartite subgraphs. For a graph G, let x(G) and a(G) denote the chromatic number and independence number of G, respectively. We need the following lemma, whose easy proof can be found in [1] (see also [2, 12, 21]). Lemma 2.1. Let G be a graph with m edges and chromatic number at most x. Then f (G) > m. 2X To find an upper bound on the chromatic number of a C2k+1-free graph, we require a lemma of Jensen and Toft [17] (see also [18]), which is a general lemma on monotone properties. Note that a graph property is called monotone if it holds for all subgraphs of a graph which has this property, i.e., is preserved under deletion of edges and vertices. Lemma 2.2 (Jensen and Toft [17, §7.3]). For s > 1, let ^: [s, to) ^ (0, to) be a positive continuous non-decreasing function. Suppose that P is a family of graphs with monotone properties such that a(G) > ^(\V (G)|) for every G G P with \V (G)| > s. Then for every such G with \V(G)\ > s, r |V(G)| x x(G) < s + / -— dx. Js ^(x) In order to bound x(G) by Lemma 2.2, we need bound a(G) of a C2k+1-free graph G in terms of \V(G)\. The following well-known Turan's lower bound (see, e.g., [25]) and another two lemmas from [19] and [20] will be used to bound a(G). Lemma 2.3 (Turan's Lower Bound). Let G be a graph on n vertices with average degree at most d. Then n a(G) > TTd- Lemma 2.4 (Li et al. [19]). Let G be a graph on n vertices with average degree at most d. If the average degree of the subgraph induced by the neighborhood of any vertex is at most a, then a(G) > nFa+1(d), where rw n f1 (1 - t)1/a , log(x/a) - 1 , Fa(x) = (, - ) u dt > g( 1 )-, (x > 0). Jo a + (x — a)t x Lemma 2.5 (Li and Zang [20]). For a fixed integer k > 2, let G be a C2k+1-free graph with degree sequence d1,d2,... ,dn. Then 1 -L. ^ k—1 ' x v 1 k — 1 "(G) > ^ 1)' i=1 Next, we shall also use the following upper bound, proved by Erdos and Gallai [13], on the maximum number of edges in Pt-free graphs, where Pt stands for a simple path with t vertices. Lemma 2.6 (Erdos and Gallai [13]). Let G be a Pt+1-free graph with n vertices. Then G contains at most (t — 1)n/2 edges. Finally, we give a simple inequality, which is used frequently in our proofs of the following several theorems. We omit the proof details. Lemma 2.7. For any real number x > 0,we have x > max j log(x + 3) — — ,e log x^ (2.1) and that the function g(x) = log x/x is monotonically increasing over the interval (0, e] and decreasing over the interval (e, ). Having finished all the necessary preparations, we are ready to give lower bounds of the independence number of a C2fc+1-free graph. Theorem 2.8. For any fixed integer k > 2, let G = (V, E) be a C2k+1-free graph on n vertices with average degree at most d. Then ^ (nlog d 1 . 1 a(G) > max{-^-^, —(nk logn) . Proof. First, we prove that n log d a(G) > W. Case 1. d < e2(2k — 1). By inequality (2.1), we have 5 , , 1 log d (1 + d) log d 2k > log(2k — 1) + - > log d + - > log d + -2- = v ; s . 2 e d d This together with Lemma 2.3 implies that n n log d a(G) > TTd > "2k^. Case 2. d> e2(2k — 1). It follows from inequality (2.1) that 2k — 1 > 1 + log(2k — 1). This together with d > e2(2k — 1) yields that 2k log d > 2 + log(2k — 1) > --- (1 + log(2k — 1)), 2k 1 which gives that log d - (l+log(2k - 1)) > iOgp (2.2) Since G is C2k+1-free, the subgraph induced by the neighborhood of any vertex of G is P2k-free. By Lemma 2.6, the average degree of any P2k-free graph is at most 2(k - 1). It follows from Lemma 2.4 and inequality (2.2) that TT f,n ^ nl0g e(2fc-i) ^ nlogd a(G) > nF2k-i(d) > --- > d - 2kd ' as desired. Now, we show that *(G) > TTÖ log n) . 5k2 Let vi,... ,vn be the vertices of G such that d(vj) = di for 1 < i < n. Set S = { vi € V : di > (n logk n) k+r j. If |S| > 2n/5, then, by Lemma 2.5, we have 1 n r k-l i 2 fc-l I a(G) > E dr) k > i • (n logk n) ^ k > (nk log n) ^. i=i Suppose that |S| < 2n/5. Consider the graph H induced by V\S. Clearly, the number of vertices contained in H is at least 3n/5, and the average degree d(H) of H is at most (n logk n)l/(k+l). If d(H) < e, then the desired result follows immediately from Lemma 2.3. Otherwise, by the preceding result, we obtain ^ / tt\ ___ 3n log d(H) 1 . a(G) > a(H) > 35- • -2^ > (nk log n) , where the last inequality holds because the function g(x) = log x/x is monotonically decreasing over the interval [e, (n logk n)l/(k+l)] by Lemma 2.7. This completes the proof of Theorem 2.8. □ With the help of Lemma 2.2 and Theorem 2.8, we establish the following theorem, which plays a key role in our proof of Theorem 1.2. The approach we take is an extension of that by Poljak and Tuza [22]. Theorem 2.9. For any fixed integer k > 2, let G be a C2k+l-free graph with m > 1 edges. Then i \3 ( m \ k+2 x(G) < 32(k +1) .—2 log m Proof. Let G be a C2k+l-free graph on n vertices with m > 1 edges. If G is bipartite, then x(G) = 2 and the claim follows. Suppose that x(G) > 3. Without loss of generality, we may assume that G is vertex-critical. Note that each vertex-critical graph has minimal degree at least x(G) - 1. It follows that the minimal degree of G is at least 2. Thus, we have m > n. Now, we end the proof by showing the following series of claims. i x(G) < 15k3' " ^ Claim 1. X(G) < 15k3/ vlog Uj This is trivial for u < e2 as x(G) < u < e2, hence we may assume that u > e2. For x > e2, define the functions Y(x) = 1 — log-1 x and ^(x) = -^-(xfc log x) k+r. 5k2 Clearly, 7(x) > 1/2 for x > e2, and y(x), ^(x) are positive continuous and non-decreasing. By Theorem 2.8, we have a(G) > ^(n). Thus, Lemma 2.2 gives that n 1 — - 5k2 r y(x) dx n 1 ^ 1 , q / n \ k+T < 15k3' * + Vlog n i'n 1 5k2 f X(G) < e2 + —— dx < e2 + t ^ Hx) Y(e2 ) ./e2 (xk log x) k+T = e2 + 10k2 n*. Otherwise, assume that n < n*. By Lemma 2.7, we know the function g(x) = x/ log x is monotonically increasing over the interval (e, to) and log x > e log log x for each x > 1. Note that m > 1 (which implies n > 3 > e). It follows from Claim 1 that ( ) 1 ( ) 1 ( ) 1 Xo according to the following procedure, which we will call the G algorithm. Set i = 0, G0 = G and n0 = |V(G0)|. Repeat the following steps until ni < n*. • Choose Si to be a maximum independent set of Gj. • Set Gi+1 = Gi\Si andni = |V(Gi)|. Increment i. Let I +1 be the length of the resulting sequence G. By the G algorithm, we immediately have n < n* and that G can be colored by at most x(G^) + I colors. Clearly, we may assume that Ge is vertex-critical. Thus, by Claim 1, for u > 3, we have ( ) 1 ( ) 1 ( ) 1 X(G,) < 15k3 () kTT < 15k3 () kTT < 32k3 (-m-) kT2. (2.3) Vlogn^/ Vlog nv Mog2 m/ Note that x(G^) clearly satisfies the above inequality for u < 2. In the following, we aim to bound the value of Firstly, we give a lower bound of |S^. Let t = 1. It follows from Claim 2 that t — 2. Let I = {0,1,..., I - 1} and J = {2,3,..., t}. Note that n > n* — n/t for each i e I by the G algorithm and the definition of t. Let v 1,..., vn0 be a labelling of the vertices of G0 such that Si = {vp : ni+1 < p _ n4} for each i e I. Denote S the union of Si for all i e I. Thus, for each j e J, we can define {n n ^ ( n ^ Vp G S : -

-j. j j - u j L j Note that S\S£_i C ujejVj C S and I2 C I3 C ... C It. Therefore, for each x e Vj, there exists an i e Ij such that x e Si. In addition, we have V l< Claim 3. For each i G Ij = 0, n j - 1 j (2.4) 2 1 2 jm n2 log —— |Si| — g2 n . 4fcj2m Let di denote the average degree of Gi for each i e I. Clearly, for each i e Ij, we have di _ 2m _ 2jnm. Suppose that di > e. Recall that the function g(x) = log x/x is decreasing over the interval (e, to). By Theorem 2.8, we have niiogdi > n2 log ^ 1 i| - 2kdi - 4kj2m . Otherwise, di _ e. It follows from Lemma 2.3 that |Si| — nk - 2nj, which together with the fact that x > log x implies the required result as well. This completes the proof of Claim 3. Then, for each x e Si and i e I, define w(x) = |Si | 1. Therefore, for each x e Si and i e Ij , it follows from Claim 3 that 1 4kj 2m 4kj2 mn-2 w(x) =|Si|- _ n^iop-m _ log j + iogm. By the definition of w(x) and the above inequality, we immediately have * -1= E E w(x) ie/\{£-1> xeSi 4kj2|Vj|mn-2 ^^ 16kmn-1 w(x) _ Z^ iog j + iog m _ Z^ iog j + iog m . (2.5) 4kj2|Vj|mn 2 ^^ 16kmn 1 PV . 2 iog j+iog mm _ ^ iog j+iog m' jGJ xEVj j=2 n j=2 n The last inequality follows from (2.4) and the fact j — 2. Finally, we give the following upper bound of Claim 4. t - 1 < 64(k +1)2( ™ ) k+2. v log m / n By the definition of n*, we have — ■ — = — = (m logk m) k+2. (2.6) n* n n* i It follows that max{m/n, n/n*} > m2(k+2>, and then i m n ^ 1 maxllog nlog w > 2^log m. (2.7) Suppose that n/n* < m/n. Note that t — 1 < n/n* by the definition of t. Then, we delete the first term of the denominator of (2.5) and obtain ^^ 16kmn-1 16k(t — 1)m 16km RA12( m ) - - log mm - n log mm < n* log mm - v log2 m , where the last inequality follows from (2.6) and (2.7); as desired. Otherwise, n/n* > m/n. Recall that t — 1 < n/n* - t. It follows that f* 1 , 2(t — 1) 2n --dx - -i--—---. J2 log x log t n* log ^T Deleting the second term of the denominator in (2.5), we have „ 16km v^ 1 16km f* 1 , 32km m ) k+2 i — 1 - -V--- --dx - —--- - 64(k + 1)2' ^ + n log j n log x n* log j=2 n ^ logj ~ n ,/2 logx™ " n* log ^Tr " ' "' Vlog2 m Again, the last inequality follows from (2.6) and (2.7). This completes the proof of Claim 4. Now, it follows from (2.3) and Claim 4 that 1 x(G) - x(G,) + i - (32k3 + 64(k + 1)2) k+2 + 1 log m log2 m - 32(k + 1)3(2 1 k + 2 log2 m Thus, we get the desired result and complete the proof of Theorem 2.9. □ We are now in a position to establish Theorem 1.2. Proof of Theorem 1.2. Let r > 4 be a fixed integer and let G be a Cr-1-free graph with m edges. The desired result follows immediately for m =1. Suppose that m > 1. Set r — 1 = 2k + 1 and c(r) = 1/(8r3). By Theorem 2.9, we have 2 2X(G) - 8r3 (-m-) r+2. log2 m This together with Lemma 2.1 yields that mm 1 f (G) > y + c(r) (mr log4 m) r+2. Thus, we complete the proof of Theorem 1.2. □ 3 Maximum cuts of H-free graphs In this section, we obtain lower bounds on the size of the maximum cuts of H-free graphs. Let G = (V,, E) be a graph. For a subset U c V, denote E(U) the set of edges of G spanned by U. We need the following simple lemma from [1,4, 8]. Lemma 3.1. Let G = (V, E) be a graph with m edges. Suppose that U c V and let G' be the induced subgraph of G on U. If G' has m' edges, then f (G) > f (G') + . Next, we need another result from [4], which provides a very useful lower bound on the size of a maximum cut in an H-free graph for a certain class of graphs H. Lemma 3.2 (Alon et al. [4]). There exists an absolute positive constant e such that for every positive constant C there is a S = 5(C) > 0 with the following property. Let G be a graph with n vertices (with positive degrees), m edges, and degree sequence d1,d2,... ,dn. Suppose, further, that the induced subgraph on any set of d > C vertices, all of which have a common neighbour, contains at most ed3/2 edges. Then n f (G) > 2 + S . i=1 A graph is r-degenerate if every one of its subgraphs contains a vertex of degree at most r. We need the following easy and well-known fact. See, e.g., [1, 2, 4] for a proof. Lemma 3.3. Let H be an r-degenerate graph on h vertices. Then there is an ordering v1,... ,vh of the vertices of H such that for every 1 < i < h the vertex vi has at most r neighbours vj with j < i. Finally, we shall also use the following lower bound in extremal set theory, proved by Corradi [9], on the size of a set Q from which we can draw N subsets of size at least q such that any two of them share at most A elements. Lemma 3.4 (Corradi [9]). Let Q1,... ,QN be N sets with |Qi | > q for each i = 1,..., N, and let Q be their union. If |Qi n Qj | < A for all i = j, then q2N IQI> qN q + (N - 1)A' Having finished all the necessary preparations, we are ready to give proofs of Theorems 1.3 and 1.4. Our proofs combine combinatorial and probabilistic techniques, including extensions of ideas that appear in [1, 2, 4]. Proof of Theorem 1.3. For each s > 2, let G = (V, E) be a {K2,s, C5}-free graph on n vertices with m edges. Define I = |_4sm2/7J. The proof proceeds by considering two possible cases depending on the existence of dense subgraphs in G. Case 1. G is (£ - 1)-degenerate, that is, it contains no subgraph with minimum degree at least I. In this case, we use Lemma 3.2 to bound f (G). By Lemma 3.3, we can get a labelling vi, v2,..., vn of the vertices of G such that d+ < I for every i, where d+ denotes the number of neighbors vj of v. with j < i. Note that J2n=1 d+ = m. Let d. be the degree of vj for each 1 < i < n. Then n n--J+ -i E ^ > E ^ * m6" j=1 j=1 v v Now, we check the condition of Lemma 3.2. For each v G V, let N(v) be the neighborhood of v in G and Nd(v) be any subset of cardinality d of N(v). Denote Gd the subgraph induced by Nd(v). Since G is C5-free, we know that Gvd contains no path of length 3. It follows from Lemma 2.6 that Gvd contains at most d edges, which is smaller than ed3/2 for all d > e-2. Thus, by Lemma 3.2, we have / (G) > * ± ^ > m+2^, j=1 where * = *(e) is a constant, as required. Case 2. There exists a subset Q of q vertices of G such that the induced subgraph G[Q] has minimum degree at least i. Now, we prove that Q contains a subset Q' such that the induced subgraph G[Q'] spans at least qi/4 edges and is 3t-colorable for t = |"4sq/i2]. For fixed x G Q, denote by S(x) the set of vertices in Q which are at distance exactly 2 from x and denote by sx the size of S(x). We bound sx by Lemma 3.4. Claim 5. sx > i2/(2s) for each x G Q. For each x G Q, let Nq (x) be the neighborhood of x in G[Q]. For each v G Nq (x), let Qv = Nq(v) n S(x) . Since G is K2,s-free, we conclude that |Qu n Qv| < s - 1 for each pair of vertices u, v G Nq(x) and that v is adjacent to at most s - 1 vertices in Nq (x). It follows that |Qv | > i - (s - 1) - 1 = i - s. Note that S(x) = (J Qv vENq(x) and |NQ(x)| > ^ > 4s. By Lemma 3.4, we obtain U Qv vENq(x) > _ — s) + (|Nq(x)| — 1)(s - 1) > 2s. '- s)2|nq(x) This completes the proof of Claim 5. sx Let T be a random subset of Q obtained by picking uniformly at random, with repetitions, t vertices of Q. Let Q' be the set of all vertices x in Q such that S(x) n T = 0 and let G[Q'] be the induced subgraph of G on Q'. Claim 6. There exists a set T such that G[Q'] spans at least qi/4 edges. By the definition of Q', for each x G Q, we have where the second inequality follows from Claim 5, and the last inequality holds by noting that t - 4sq/t2. Thus, for each edge xy G E(Q), we obtain P(xy G E(Q')) = P(x G Q') • P(y G Q') > (l - 1) • (l - 4) > 1. By linearity of expectation, and noting that |E (Q)| > qt/2, we have ll P(xy G E(Q')) > ' xy€E(Q) E(|E(Q')|) = £ P(xy G E(Q')) > -|E(Q)| > -qt. Hence, there exists a set T of at most t vertices so that the corresponding graph G[Q'] has at least qf/4 edges. Thus, we complete the proof of Claim 6. Fix such sets T and Q', let G' = G[Q'] and T = {ui,..., uv}, where 1 < t' < t. Now we show G' is 3t-colorable. Define a coloring c of G' in t' colors by coloring each vertex x G Q' with the smallest index of a vertex from T which belongs to S(x). For each 1 < i < t', let Hi be the subgraph of G' induced by the vertices of Q' with color i. Claim 7. For each 1 < i < t', Hi i's 3-colorable. For each ui G T and v G N(ui), let HV be the subgraph induced by the neighbors of v with color i in G'. By the above definition and the fact that G is C5-free, we have the following properties: • For each v G N(ui), HV is P4-free; • For each v1, v2 G N(ui) and u G V(H^1) n V(H^2), u is an isolated vertex in both HV1 and HV2; • For each x G V(H^1) and y G V(HiV2), x and y are nonadjacent in Hi. Note that Hi is induced by the union of V(HV) over all v G N(ui). This together with the above three properties implies that Hi is P4-free, i.e., 3-colorable. Thus, we complete the proof of Claim 7. By the definition of c and Claim 7, we conclude that G' is 3t-colorable. According to Lemma 2.1, it follows that f(GO > |E(Q')| + |E(Q')U |E(Q')| + qt f ( ) " 2 + 6t " 2 + 24 4sq I2" ^ |E(Q')| + |E(Q')| + m6/7 - 2 + 144s - 2+9 . The second inequality follows from Claim 6, and the third inequality holds because q -sx - t2/(2s) by Claim 5. The above inequality together with Lemma 3.1 gives that f(G) - m -|E(Q')| + "EMI + 4s!m6/r = m + 4s!m6/r. JK ' > 2 2 9 2 9 Therefore, the desired result follows immediately from Cases 1 and 2 by setting c(s) = min{ , 4|- }, completing the proof of Theorem 1.3. □ i The proof of Theorem 1.4 is similar to that of Theorem 1.3. Proof of Theorem 1.4. Let G be an Hr -free graph on n vertices with m edges. Define ¿ = |_2m2/(r+1)J and proceed as before, by considering two possible cases. Case 1. G contains no subgraph with minimum degree at least In this case, we proceed as in the previous proof. Similarly, the induced subgraph of G on any set of common neighbors of a vertex can span only a linear number of edges, as it contains no copy of C4. Thus, we can apply, again, Lemma 3.2 and conclude, as in the proof of Theorem 1.3, that ^ m m m 5 f (G) >--+ 5>--+ -m r+1, f () > 2+V2I > 2+2 , where 5 = 5(e) is also a constant, as needed. Case 2. There exists a subset Q of q vertices of G such that the induced subgraph G[Q] has minimum degree at least 2£ Here, too, we prove that there exists Q' c Q such that the induced subgraph G[Q'] spans at least q^/2 edges and is 2t-colorable for t = [q/^fc]. Let r = 2k + 2. Denote by Sk (x) the set of vertices in Q which are at distance exactly k from x and denote by sx the size of Sk(x). Since the minimal degree of G[Q] is at least 2^ and G[Q] contains no cycle of length from 4 to 2k + 1, it can easily be seen that sx > 2^(2^ - 2)k-1 > 2^fc for each x G Q. Let T be a random subset of Q obtained by picking, with repetitions, t vertices of Q, each chosen randomly with uniform probability. This together with the fact sx > 2^fc yields that the probability that Sk(x) n T is empty is at most sx\ , 2^fc\« , ( 2^fct l 1 1 - i) — I1 - TJ -/< 4 An argument similar to the one used in the proof of Claim 6, the details of which we omit, shows that there exists a set T of at most t vertices so that the corresponding graph G[Q'] has at least q^/2 edges. Fix such sets T and Q'. Now, we define a coloring c of G' and the induced subgraphs Hi of G' for 1 — i — |T| as in the proof of Theorem 1.3. Claim 8. For each 1 — i — |T |, Hi is the disjoint union of edges modulo isolated vertices. For fixed ui G T and for each v G Sk_1(ui), let HV be the subgraph induced by the neighbors of v with color i in G'. By the above definition, and recalling that G contains no cycle of length from 4 to 2k + 1, we have the following properties: (i) for each v G Sfc_i(ui), HV is P3-free; (ii) for each vi, V2 G Sfc_i(ui), V(Hf) n V(Hf) = 0; (iii) for each x G V(H^1) and y G V(H^2), x and y are nonadjacent in Hi. It follows from (ii) and (iii) that Hi is the disjoint union of HV over all v G Sk_1(ui). Thus, by (i), Hi is the disjoint union of edges modulo isolated vertices. This completes the proof of Claim 8. By the definition of c and Claim 8, we know that G' is 2t-colorable. Using Lemma 2.1, we conclude that |E(Q')| , |E(Q')| |E(Q')| , q^ r q f (G') > ' n + ' ^ n > ' n + — f () > 2 + 4t > 2 +8 ¿k |E (Q')| + ¿k+1 |E(Q')| + (\/2)r The second inequality follows from the fact |E (Q')| > qi/2, and the third inequality holds because q > sx > 2ik. Taking Lemma 3.1 into consideration, we obtain f(G)> m - |E(Q')| + |E(Q')| + (V2)r _r_ m + (V2)r f(G) >-2-+ ""T" + mr+ =12+~L2Tmr+1. 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Comput. 20 (2011), 631-640, doi:10.1017/s0963548311000204. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 161-172 https://doi.org/10.26493/1855-3974.1296.5c7 (Also available at http://amc-journal.eu) Circular chromatic number of induced subgraphs of Kneser graphs* Meysam Alishahi School of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran Ali Taherkhani t Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran Received 17 January 2017, accepted 11 September 2017, published online 13 June 2018 Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Hajiabolhassan and Zhu in 2003 proved that if n is sufficiently large with respect to k, then the Schrijver graph SG(n, k) has the same chromatic and circular chromatic number. Later, Meunier in 2005 and independently, Simonyi and Tardos in 2006 proved that x(SG(n, k)) = xc(SG(n, k)) if n is even. In this paper, we study the circular chromatic number of induced subgraphs of Kneser graphs. In this regard, we shall first generalize the preceding result to s-stable Kneser graphs for large even n and even s. Furthermore, as a generalization of the Hajiabolhassan-Zhu result, we prove that if n is large enough with respect to k, then any sufficiently large induced subgraph of the Kneser graph KG(n, k) has the same chromatic number and circular chromatic number. Keywords: Chromatic number, circular chromatic number, Kneser graph, stable Kneser graph. Math. Subj. Class.: 05C15 *The authors would like to acknowledge Professor Hossein Hajiabolhassan for his invaluable comments and suggestions. They also thank the anonymous referees for all their remarks that helped in improving the presentation of the paper. t Corresponding author E-mail addresses: meysam_alishahi@shahroodut.ac.ir (Meysam Alishahi), ali.taherkhani@iasbs.ac.ir (Ali Taherkhani) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ 1 Introduction Throughout the paper, the symbol [n] stands for the set {1,..., n}. Let n and d be two positive integers. The circular complete graph Kn is a graph with vertex set [n] and two vertices i and j are adjacent whenever d < |i - j| < n - d. For a graph G, the circular chromatic number of G, denoted by Xc(G), is defined as follows: Xc(G) =f inf j n : there is a homomorphism from G to Kn j . It is known that the infimum can be replaced by minimum. Moreover, one can see that x(G) - 1 < Xc(G) < x(G), see [36]. Therefore, the circular chromatic number is a refinement of the chromatic number. It is a natural question to ask for which graphs G, we have Xc(G) = x(G). However, it turns out to be a difficult question. Hell [19] proved that the problem of determining whether a graph has the circular chromatic number at most n is NP-Hard. Hatami and Tusserkani [18] showed that the problem of determining whether or not Xc(G) = x(G) is NP-Hard even if the chromatic number is known. Therefore, finding necessary conditions for graphs to have the same chromatic and circular chromatic number turns out to be an interesting problem. This problem has received significant attention, for instance see [1, 17, 36, 37]. For two positive integers n and k, where n > 2k, the Kneser graph KG(n, k) is a graph with vertex set ('k'), that is, the family of all k-subsets of [n], and two vertices are adjacent if their corresponding k-subsets are disjoint. Kneser in 1955 [23] conjectured that the chromatic number of KG(n, k) is n — 2k + 2. In 1978, Lovisz [26] gave an affirmative answer to Kneser's conjecture. He used algebraic topological tools, giving birth to the field of topological combinatorics. For a positive integer s, a nonempty subset S of [n] is said to be s-stable if for any two different elements i and j in S, we have s <|i - j | < n - s. Throughout the paper, the family of all s-stable k-subsets of [n] is denoted by ('k') s. The subgraph of KG(n, k) induced by all s-stable k-subsets of [n] is called the s-stable Kneser graph and is denoted by KGs (n, k). The 2-stable Kneser graph KG2 (n, k) is known as the Schrijver graph SG(n, k). Schrijver [31] proved that Schrijver graphs are vertex critical subgraphs of Kneser graphs with the same chromatic number. Meunier [30] showed that for any two positive integers n and k, where n > sk, the s-stable Kneser graph KGs (n, k) can be colored by n - s(k -1) colors and conjectured that the chromatic number is n - s (k -1). Jonsson [22] proved that this conjecture is true provided that s > 4 and n is sufficiently large with respect to k and s. Also, Chen [12] confirmed Meunier's conjecture for even values of s. Lovisz's theorem [26] has been generalized in several aspects. For a hypergraph H, the general Kneser graph KG(H) is a graph with vertex set E(H) and two vertices are adjacent if their corresponding edges are vertex disjoint. Dol'nikov [13] generalized Lovisz's result and proved that the chromatic number of KG(H) is at least the colorability defect of H, denoted by cd(H), where the colorability defect of H is the minimum number of vertices which should be excluded from H so that the induced subhypergraph on the remaining vertices is 2-colorable. For a vector X = (xi,...,xn) G {-, 0, +}n, a sequence xHl , xi2 , ...,xit (ii < • • • < it) is called an alternating subsequence of X with length t if Xj = 0 for each j G {1,..., t} and Xj. = xij+1 for each j G {1,..., t - 1}. The maximum length of an alternating subsequence of X is called the alternation number of X, denoted by alt(X). For 0 =f (0,..., 0), we define alt(0) =f 0. Also, we define X + and X- to be respectively the indices of positive and negative coordinates of X, i.e., X + =f {i : xj = +} and X- =f {i : xj = -}. Note that both X + and X- are subsets of [n] and by abuse of notation, we can write X = (X +, X-). For two vectors X, Y e {-, 0, +}n, by X C Y, we mean X + C Y+ and X - C Y -. Let H = (V, E) be a hypergraph and a : [n] —> V(H) be a bijection. The alternation number of H with respect to a, denoted by altCT(H), is the maximum possible value of an alt(X) such that E(H[a(X +)]) = E(H[a(X-)]) = 0. Also, the strong alternation number of H with respect to a, denoted by saltCT (H), is the maximum possible value of an alt(X) such that E(H[a(X +)]) = 0 or E(H[a(X-)]) = 0. The alternation number of H, denoted by alt(H), and the strong alternation number of H, denoted by salt(H), are respectively the minimum values of altCT (H) and saltCT(H), where the minimum is taken over all bijections a : [n] —> V(H). The present first author and Hajiabolhassan [4] proved the following theorem. Theorem A. For any hypergraph H, we have x(KG(H)) > max{|V(H)| - alt(H), |V(H)| - salt(H) + 1} . One can simply see that this result improves the aforementioned Dol'nikov's result [13]. Using this lower bound, the chromatic number of several families of graphs is computed, for instance see [2, 3, 5, 6, 8]. In 1997, Johnson, Holroyd, and Stahl [21] proved that Xc(KG(n, k)) = x(KG(n, k)) provided that 2k +1 < n < 2k + 2 or k = 2. They also conjectured that the circular chromatic number of Kneser graphs is always equal to their chromatic number. This conjecture has been studied in several articles. Hajiabolhassan and Zhu [17] using a combinatorial method proved that if n is large enough with respect to k, then xc(KG(n, k)) = x(KG(n, k)). Later, using algebraic topology, Meunier [29] and Simonyi and Tardos [33] independently confirmed this conjecture for the case of even n. It should be mentioned that the results by Hajiabolhassan and Zhu [17], Meunier [29], and Simonyi and Tardos [33] are indeed proved for the Schrijver graph SG(n, k). Eventually in 2011, Chen [11] confirmed the Johnson-Holroyd-Stahl conjecture. Chen's proof was simplified by Chang, Liu and Zhu in [10] and by Liu and Zhu in [25]. The present first author, Hajiabolhassan, and Meunier [8] generalized Chen's result to a larger family of graphs. They introduced a sufficient condition for a hypergraph H having x(KG(H)) = Xc(KG(H)). Plan. The paper contains two main sections. In Section 2, we shall investigate the coloring properties of stable Kneser graphs. In this regard, we prove the equality of the chromatic number and the circular chromatic number of s-stable Kneser graph KGs(n, k) provided that n > (s + 2)k - 2 and both n and s are even. In the last section, we study the circular chromatic number of large induced subgraphs of Kneser graphs. Indeed, it is proved that, for large enough n, any sufficiently large induced subgraph of the Kneser graph KG(n, k) has the same chromatic number and circular chromatic number. In particular, giving a partial answer to a question posed by Lih and Liu [24], we present a threshold n(k, s) such that for any n > n(k, s), the chromatic number and circular chromatic number of KGs(n, k) are equal. 2 Chromatic number of stable Kneser graphs As it is mentioned in the previous section, the chromatic number of s-stable Kneser graph KGs(n, k) is determined provided that k and s > 4 are fixed and n is large enough [22] or s is even [12]. In this section, we first present a generalization of Theorem A. Using this generalization, for any even s, we prove that any proper coloring of s-stable Kneser graph KGs(n, k) contains a large colorful complete bipartite subgraph, which immediately gives solutions to the chromatic number of s-stable Kneser graphs KGs(n, k). Also, this result concludes that the circular chromatic number of s-stable Kneser graph KGs(n, k) equals to its chromatic number provided that n > (s + 2)k - 2 and both n and s are even. Tucker's lemma is a combinatorial counterpart of the Borsuk-Ulam theorem with several useful applications, for instance, see [27, 28]. Lemma A (Tucker's lemma [35]). Let A: {-, 0, +}n \ {0} —> {±1,..., ±m} be a map satisfying the following properties: • it is antipodal: A(—X) = —A(X) for each X g {—, 0, +}n \ {0}, and • it has no complementary edges: there are no X and Y in {-, 0, +}n \ {0} such that X C Y and A(X) = —A(Y). Then m > n. There are several results concerning the existence of a large complete bipartite multicolored subgraph in any proper coloring of a graph G, see [4, 11, 32, 33, 34]. In what follows, we present a similar type of result with a combinatorial proof. Note that since there is a purely combinatorial proof [28] for Tucker's lemma, any proof using Tucker's lemma with combinatorial approach can be considered as a purely combinatorial proof. Theorem 2.1. Let H be a hypergraph and set t = max {|V(H)| — alt(H), |V(H)| — salt(H) +1}. For any proper coloring c: V(KG(H)) —> [C], there exists a complete bipartite subgraph Kyt/2j,[t/2] of KG(H) whose vertices receive different colors and moreover, these different colors occur alternating on the two parts of the bipartite graph with respect to their natural order. Proof. Let V(H) be two bijections for which we have alt(H) = altCTl (H) and salt(H) = saltCT2 (H). Now, we shall proceed the proof with two different cases, t = n—alt(H) and t = n—salt(H) + 1. Assume thatt = n—alt(H) (resp. t = n—salt(H) + 1). For simplicity of notation, by identifying the set V(H) and [n] via the bijection o1 (resp. 0-2), we may assume that V(H) = [n]. For each X = (X +, X-) g {—, 0, +}n \ {0}, define c (X) = (c(X +) , c(X )) g {—, 0, +}C to be a signed vector, where c(X +) d=f {c(e) : e G E(H) and e C X +} and c(X-) d=f {c(e) : e G E(H) and e C X"} . For each X g {—, 0, +}n \ {0}, define A(X) as follows. • If alt(X) < altCTl (H) (resp. alt(X) < saltCT2 (H)), then define A(X) = ± alt(X), where the sign is positive if the first nonzero term of X is positive and is negative otherwise. • If alt(X) > altff1 (H) + 1 (resp. alt(X) > saltff2 (H) + 1), then define A(X) = ±(altff1 (H) + alt(c(X))) (resp. A(X) = ±(saltff2 (H) + alt(c(X)) -1)), where the sign is positive if the first nonzero term of c(X) is positive and is negative otherwise. One can simply check that A satisfies the conditions of Lemma A. Consequently, there should be an X G {-, 0, +}n \ {0} such that |A(X)| = A(X) > n. Clearly, we should have alt(X) > altCT1 (H) + 1 (resp. alt(X) > saltCT2 (H) + 1). Therefore, the definition of A(X) implies that alt(c(X)) > n - altCT( (H) (resp. alt(c(X)) > n - saltCT2(H) + 1). Let Z = (Z+, Z-) C c(X) be a signed vector such that alt(Z) = |Z| = t, as alt(c(X)) > t. Note that if Z+ U Z- = {n, i2,..., it}, where 1 < i1 < • • • < it < C, then we should have Z + = {ij : j G [t] is odd} and Z- = {ij : j g [t] is even}. For an j g [t], if j is odd (resp. even), then according to the definition of c(X), there is an edge e g E(H) such that e C X + (resp. e C X-) with c(e) = ij. Note that the induced subgraph of KG(H) on the vertices {e1,..., et} contains the desired complete bipartite graph. □ Note that the complete bipartite graph whose existence is guaranteed by Theorem 2.1 is not necessarily an induced subgraph. Also, it is worth mentioning that we here used Tucker's lemma though, in case t = |V(H)| - alt(H), the previous theorem was proved in [4] by use of Ky Fan's lemma [14]. Let n, k, and s be positive integers, where n > sk and s is even. It is not difficult to see that if n is large enough (with respect to s and k), then any 2-stable (f (k - 1) + 1)-subset of [n] contains an s-stable k-subset of [n]. In the following two lemmas, we shall prove that n > (s + 2)k - 2 would be sufficient for this observation. Lemma 2.2. Let s be an even positive integer and let n = 2s + 2. If S is a 2-stable subset of [n] of cardinality f + 1, then there are a, a' G S such that a - a' G {s, s + 1, s + 2}. Proof. Without loss of generality, we may assume that 1 G S and 2s + 2 G S .If s + 1 G S, then there is nothing to prove. Therefore, let s + 1 G S. For 1 < i < f, define B = {2i - 1,2i, 2i + s, 2i + s + 1}. Therefore, for some i, 1 < i < §, |Bj n S| = 2. Let a, a' G Bi n S, since S is 2-stable, we have a - a' G {s, s + 1, s + 2}. □ Lemma 2.3. Let k and n be two positive integers and let s be an even positive integer, where n > (s + 2)k - 2. If S is a 2-stable subset of [n] of cardinality f(k - 1) + 1, then there is an s-stable k-subset of S. In particular, salt ([n], (k)§) = s(k - 1) + 1. Proof. First note that for given k and s, if the statement is true for some n > k(s + 2) - 2, then it is true for all integers n' > n. Therefore it is enough to prove the lemma for n = k(s + 2) - 2. We use induction on k to prove the lemma. The validity of the lemma when k = 1 is trivial and when k = 2 it was shown in Lemma 2.2. Thus, we may assume that k > 3. If for each i G S, we have {i + s, i + s + 1, i + s + 2}nS = 0 (where addition is modulo n), then we can greedily find an s-stable k-subset, and there is nothing to prove. Otherwise, without loss of generality, assume that n — s — 1 G S and n — 1, n, 1 G S. Set An-s-1 = {n - s - 1, n - s,..., n}. Note that since n - 1, n G S, we have |An-s-1 nS| = § - p, for some 0 < £ < §. Now, consider [n] \ and S\ . Set n = n - (s + 2) and S = S \ An_s_ 1. Note that [n] and [n] \ An_s_ 1 are equal and since 1 G S, S is a 2-stable subset of [n] of cardinality § (k - 2) + p +1. Define the s-subset B of [n] by B = {n - 2s - 1, n - 2s,. .., n - s - 2}. By induction, we may consider the following two cases: (i) There is an s-stable (k - 1)-subset of S, say D, which has no element of B. In this case, it is readily verified that D = D U {n - s - 1} is an s-stable k-subset of [n], completing the proof in this case. (ii) There are at least fi + 1 s-stable (k - 1)-subsets of S, say Di, D2,..., Dsuch that each Dj has exactly one distinct element of B, say bj. Now, consider the 2-stable subset {b1, b2,..., b^+1} U (Sn ), by Lemma 2.2, there exist two elements a, b such that a - b e {s, s + 1, s + 2}. Since n - 1, n e S, both a, b are not in . Hence, we may assume that a e An-s-1 and b = bj for some i, 1 < i < fi + 1. Let d be the smallest element of DDj. Since Dj is an s-stable (k - 1)-subset of [n], therefore we have s < b - d < n - s = n - (2s + 2). On the other hand, s < a - b < s + 2. Therefore, 2s < a - d < n - s. Therefore, Dj U {a} is an s-stable k-subset of [n] as desired. Note that for an X e {-, 0, +}n \ {0} with alt(X) > s(k - 1) + 2, both X + and X- contain 2-stable subsets of size at least § (k - 1) + 1, which implies that both X+ and X- contain s-stable subsets of size at least k. This concludes that salt ([n], (k)s) = s(k - 1) + 1. S □ We remind the reader that Meunier [30] showed that KGs (n, k) has a proper coloring with n - s(k - 1) colors. Note that if we set H = ([n], ([k])s), then KG(H) = KGs(n, k). Clearly, using these observations, Lemma 2.3, and Theorem 2.1, we have the next theorem. Theorem 2.4. Let k and n be two positive integers and let s be an even positive integer, where n > (s + 2)k - 2. Any properly colored KGs(n, k) contains a complete bipartite subgraph K^i/2j,pi/2], where t = n - s(k - 1) such that all vertices of this subgraph receive different colors and these different colors occur alternating on the two parts of the bipartite graph with respect to their natural order. In particular, we have x(KGs(n, k)) = n - s(k - 1). Let r be a positive integer. For an r-coloring c of a given graph G, a cycle C = v1, v2,..., vm, v1 is called tight if for each i e [m], we have c(vj+1) = c(vj) +1 (mod r). It is known [36] that Xc(G) = r if and only if the graph G is r-colorable and every r-coloring of G contains a tight cycle. In view of this result, to prove the next theorem, it suffices to show that any proper (n - s(k - 1))-coloring of KGs(n, k) contains a tight cycle. Theorem 2.5. Let n, k, and s be positive integers, where n and s are even and n > (s + 2)k - 2. Then, we have Xc(KGs(n,k)) = n - s(k - 1). Proof. For simplicity of notation, we set t = n - s (k -1). In view of the former discussion, to prove the assertion, let c be a proper t-coloring of KGs(n, k). Consider the complete bipartite subgraph Kt/2,t/2 of KGs(n, k), whose existence is ensured by Theorem 2.4. Clearly, this subgraph contains a tight cycle, which completes the proof. □ The original proof of Lovisz of Kneser's conjecture is rather long and complicated [26]. B^riny [9], using Gale's lemma [15], presented a short proof of this result. For n > 2k, Gale [15] proved that the set [n] can be identified with a subset of Sn-2k in such a way that any open hemisphere contains at least one k-subset of [n] (a vertex of KG(n, k)). Schrijver [31] generalized Gale's lemma to 2-stable k-subsets of [n]. He also used this generalization to prove that x (SG(n, k)) = n - 2k + 2. For an interesting proof of Gale's lemma, see [16]. Moreover, the present first author and Hajiabolhassan [7] generalized Gale's lemma. For any hypergraph H = (V, E), they introduce a lower bound for the maximum possible value of m for which there is a subset X of Sm and a suitable identification of V with X such that any open hemisphere of Sm contains an edge of H. The next lemma can be obtained directly from this result. However, for the sake of completeness, we prove it here with a different approach. Lemma 2.6. Let k and n be two positive integers and let s be an even positive integer, where n > (s + 2)k — 2. There exists an n- subset X of S"-s(fc-i)-2 and a suitable identification between X and [n] such that every open hemisphere of Sn-s(k-1)-2 contains an s-stable k-subset of [n]. Proof. Set p = 2 (k — 1) + 1. In view of the generalization of Gale's lemma by Schrijver [31], there exists an n-subset X of Sn-2p and an identification of X with [n] such that any open hemisphere of Sn-2p contains a 2-stable p-subset of [n]. Now, by Lemma 2.3, any 2-stable p-subset contains an s-stable k-subset. This implies that any open hemisphere of Sn-s(k-1)-2 contains an s-stable k-subset of [n] as desired. □ Simonyi and Tardos [34], using the Tucker-Bacon lemma (Lemma B), proved that if the chromatic number of a graph G equals to a topological lower bound for chromatic number, then for any optimal coloring of G with colors [C] and for any partition L i±i M of [C], there is a multi-colored complete bipartite subgraph K|L|,|M| of G such that all colors in L are assigned to the vertices of one side of K|L|,|M| and all colors in M are assigned to the vertices of the other side. These kinds of results are known as m type theorems, see [32, 34]. Lemma B (Tucker-Bacon lemma). Let U1, U2,..., Ud+2 be open subsets of the d-sphere Sd such that for any 1 < i < d + 2, U n — U = 0 and also, U1 U • • • U Ud+2 = Sd. Then for any partition A U B = {1, 2,..., d + 2} for which A = 0 and B = 0, there is an x G Sd such that x G nieAUj and — x G nj£B Uj. In what follows, similar to the Simonyi-Tardos result, using the Tucker-Bacon lemma, we prove a m type theorem for s-stable Kneser graphs provided that n is large and s is even. Theorem 2.7. Let n, k, and s be positive integers, where s is even and n > (s + 2)k — 2. Also, let c be a proper coloring of KGs(n, k) with colors {1, 2,..., n — s(k — 1)} and assume that A and B form a partition of {1,2,..., n — s(k — 1)}. Then there exists a complete bipartite subgraph Klm of KGs(n, k) with parts L and M such that |L| = l = | A |, |M | = m = |B| and the vertices in L and M receive different colors from A and B, respectively. Proof. By Lemma 2.6, we can identify [n] with a subset of Sn-s(k-1)-2 such that every open hemisphere of Sn-s(k-1)-2 contains an s-stable k-subset of [n]. For 1 < i < n — s(k — 1), define Ui = jx € Sn-s(k-1)-2 : H(x) contains a vertex with color i j . One can see that each Ui is an open set, U1, U2,..., Un-s(k-1) covers sn-s(k-1)-2 and also none of them contains a pair of antipodal points. Thus, the Tucker-Bacon lemma implies that there is an x € gn-s(fc-1)-2 such that x € nieAUi and —x € njeB Uj. Therefore, in view of the definition of Ui's, for each i € A (resp. j € B), there is an s-stable k-subset Li (resp. Mj) of [n] such that c(Li) = i and Li C H(x) (resp. c(Mj) = j and Mj C H(—x)). Note that since H(x) n H(—x) = 0, for each i € A and j € B, Li is adjacent to Mj in KGs (n, k), which completes the proof. □ We would like to mention that the idea of our proof is close to the Biriny's proof of Kneser conjecture [9]. 3 Circular coloring of induced subgraphs of Kneser graphs The concept of free coloring of graphs was introduced in [1] by the present first author and Hajiabolhassan as a tool for studying the circular chromatic number of graphs. Indeed, they proved that if the free chromatic number of a graph G is at least twice of its chromatic number, then x(G) = Xc(G). An independent set in a graph G is called a free independent set if it can be extended to at least two distinct maximal independent sets in G. Clearly, one can see that an independent set F in G is a free independent set if and only if there exists an edge uv € E (G) such that (N(u) U N(v)) n F = 0. The maximum possible size of a free independent set in G is denoted by a(G). Furthermore, a vertex of a graph G is contained in a free independent set if and only if the graph obtained by deleting the closed neighborhood of this vertex has at least one edge (for more details, see [1]). As a natural extension of the chromatic number, we can define the free chromatic number of graphs as follows. Definition 3.1. The free chromatic number of a graph G, denote ^>(G), is the minimum size of a partition of V(G) into free independent sets. If G does not have such a partition, then we set ^(G) = to. The next lemma plays a key role in the rest of the paper. Lemma C ([1, Lemma 2]). Let G be a graph such that Xc(G) = d with gcd(n, d) = 1. If d > 2, or equivalently, if Xc(G) = x(G), then ^(G) < 2x(g) — 1. Let G be a graph with at least one free independent set. By definition, we have ^(G) > |V(G)|/a(G). It was proved by Hilton and Milner [20] that if T is an independent set of KG(n, k) of size at least then n — 1\ in — k — 1 k — 1 — k — 1 n a=iii, AeT for some i £ [n]. By using this result of Hilton and Milner, it was proved by Hajiabolhassan and Zhu in [17] that if n > 2k2(k - 1), then xc(KG(n, k)) = x(KG(n, k)). This result was improved in [1] by proving that we have xc(KG(n,k)) = x(KG(n, k)) for n > 2k2(k - 1) - 2k + 3. It was also showed in [17] that there is a threshold n(k) such that for n > n(k), we have xc(SG(n, k)) = x(SG(n, k)). This gave a positive answer to a question of Lih and Liu [24]. Lih and Liu [24] also posed the question of what is the smallest value of n(k). They proved that n(k) > 2k + 2. One should note that in [17] only the existence of the threshold n(k) is ensured and the authors did not present any upper bound for it. Using the Hilton-Milner theorem, one can simply see that, for n > 2k, the size of any free independent set in the Kneser graph KG(n, k) is at most - ("—) < k(n-2), see [1]. In view of this observation, we generalize the result by Hajiabolhassan and Zhu [17] to the following theorem. Theorem 3.2. Let n and k be two positive integers, where n > 2k2 (k - 1). Let H be an induced subgraph of KG(n, k) with at least 2k (k) vertices. Then H has the same chromatic number and circular chromatic number. Proof. Obviously, the assertion holds for k = 1. So, let k > 2. Assume that H is an induced subgraph of KG(n, k) with at least 2k2(nk-1) (k) vertices. According to Lemma C, it is enough to show that ¿(H) > 2x(H). To this end, note that |V (H )| a(H) |V (H )| a(KG(n,k)) 2k2(k-1) (n) n ) k) k(n-2) 2k2(k - 1)n(n - 1) nk2 (k - 1) therefore ¿(H) > 2n - 2 > 2x(KG(n, k)) > 2x(H) as desired. □ In the rest of this section, we will return to the study of s-stable Kneser graphs from Section 2, KGs(n, k), but this time we consider KGs(n, k) as an induced subgraph of KG(n, k). We focus on the chromatic number and the circular chromatic number of the s-stable Kneser graph KGs (n, k). As a special case of the previous theorem, we introduce a threshold n(k, s) such that for any n > n(k, s), we have x(KGs(n, k)) = xc(KGs(n, k)). In this regard, we first need to count the number of vertices of KGs(n, k). Let Ni be the number of vertices of KGs (n, k) containing i. It is obvious that Ni = Nj for all i, j £ [n]. Also, let A = {x1,..., xk} be a vertex of KGs(n, k), where 1 = x1 < x2 < • • • < xk < n. Define yi = xi+1 - xj for all 1 < i < k - 1 and yk = n - xk + 1. Since A g V(KGs(n, k)) and 1 £ A, we have y» > s for all i £ [k]. Also, since y1 + y2 + • • • + yk = n, any vertex A of KGs (n, k) with 1 £ A leads us to a solution of the following system: Z1 + Z2 +-----+ Zk = n Zi > s for each i £ [n] ¿(H) > > > and vise versa. Note that the number of solutions of the preceding system is (n • Consequently, for each i e [n], we have N, = N1 = i"-^-^-1) for all i e [n]. By an easy double counting, one can see that |V(KG.(n,k))| = k ± N, = n (n - 'k8;/' - 1 i= 1 V Theorem3.3. Ifn > 2k2(k-1)+(s-1)k(k-1)+1, thenxc(KGs(n,k)) = x(KGs(n,k)). Proof. Let X be the number of (k-1)-subsets B of the set [n-1] such that B n[(s-1)k] = 0, i.e., X = # {B : B C [n - 1] and B n [(s - 1)k] = 0} . Obviously, we have (k-1) = ("-(sfcl11)fc-1) + X. 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ARS MATHEMATICA CONTEMPORANEA 15 (2018) 173-190 https://doi.org/10.26493/1855-3974.1341.5a3 (Also available at http://amc-journal.eu) Groups of symmetric crosscap number less than or equal to 17 Adrián Bacelo * Departamento de Algebra, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain Received 3 March 2017, accepted 21 March 2018, published online 13 June 2018 * Abstract Every finite group G acts on some non-orientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of G. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain the groups with symmetric crosscap number less than or equal to 17. Also, we obtain six infinite families with symmetric crosscap number of the form 12k + 3. Keywords: Symmetric crosscap number, Klein surfaces. Math. Subj. Class.: 57M60, 20F05, 20H10, 30F50 1 Introduction A Klein surface X is a compact surface endowed with a dianalytic structure [1]. Klein surfaces may be seen as a generalization of Riemann surfaces including bordered and non-orientable surfaces. An orientable unbordered Klein surface is a Riemann surface. Given a Klein surface X of topological genus g with k boundary components the number p = rig + k — 1 is called the algebraic genus of X, where n = 2 if X is an orientable surface and n = 1 otherwise. In the study of Klein surfaces and their automorphism groups the non-euclidean crys-tallographic (NEC) groups play an essential role. An NEC group r is a discrete subgroup of G (the full group of isometries of the hyperbolic plane H) with compact quotient H/r. For a Klein surface X with p > 2 there exists an NEC group r, such that X = H/r, [27]. *The author wish to express his deep gratitude to the referees for their careful checking of the manuscript and for their very useful suggestions concerning both the style and the precision of arguments. The author is partially supported by UCM910444 and MTM2014-55565. E-mail address: abacelo@ucm.es (Adrian Bacelo) ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ A finite group G of order N is a subgroup of the automorphism group of a Klein surface X = H/r if and only if there exists an NEC group A such that r is a normal subgroup of A with index N and G = A/r. Every finite group G acts as a subgroup of the automorphism group of some non-orientable surface without boundary, see [7]. The minimum topological genus of these surfaces is called the symmetric crosscap number of G and it is denoted by a(G). Such a surface of topological genus g > 3 has at most 84(g - 2) automorphisms. Hence, for each g there is a finite number of groups acting on surfaces of genus g. The systematic study of the symmetric crosscap number was begun by May in [23], although previous results from other authors are also to be noted, see for instance [7, 14, 19]. Four types of inter-related problems arise naturally when dealing with the symmetric crosscap number a(G). First of all, to obtain a(G) for any given group G, and for the groups belonging to a given infinite family. Second, to obtain a(G) for all groups G with o(G) < n for a given (small) value of n. Third, for a given value of g, to obtain all groups G such that a(G) = g. Evidently this question is feasible only for low values of g. Finally, to determine which values of g are in fact a(G) = g for a group G. The set of such values is called the symmetric crosscap spectrum and there exists a conjecture according to which g = 3 is the unique positive integer not belonging to the spectrum. In this paper we deal with the third question. We will study which groups have symmetric crosscap number less than or equal to 17. First, we will indicate all the results we know and then we will make a study of each group with symmetric crosscap number g < 17 that has not been studied in detail. Also, results on the spectrum are given. The contents of this paper form part of the doctoral thesis of the author, [3]. 2 Preliminaries An NEC group r is a discrete subgroup of isometries of the hyperbolic plane H, including orientation-reversing elements, with compact quotient X = H/r. Each NEC group r has associated a signature [22]: a(r) = (g, ±, [mi, .. ., mr], |(niji, .. ., nijSi), i = 1, ..., k}), (2.1) where g, k, r, m4, n,j are integers satisfying g, k, r > 0, > 2, n> 2. We will denote by [-], (-) and {-} the cases when r = 0, sj = 0 and k = 0, respectively. The signature determines a presentation of r, see [30], by generators xj (i = 1,..., r); e (i = 1,..., k); cj,j (i = 1,..., k; j = 0,..., Sj); aj, b (i = 1,..., g) if a has sign '+'; and dj (i = 1,..., g) if a has sign '-'. These generators satisfy the following relations: mi _ 1 2 _ 2 _ ( )ni,j _ 1- —1 _ 1 Xj 1; cj,j — 1 cj,j (cj,j—1cj,j) ' 1; ej cj,0ejcj,si 1 and nr=i xi nk=i ntiMiO—V) =1 if a has sign'+' nr=i xi nti ei ng=i % =1 ifa has sign -' The isometries xj are elliptic, ej, aj, bj are hyperbolic, Cj,j are reflections and dj are glide-reflections. Every NEC group r with signature (2.1) has associated a fundamental region whose area "(r), called area of the group, is: "(r) = 4g+k - 2+1 (l- + 1(l- ¿j))' <"> with n = 2 or 1 depending on the sign '+' or '-' in the signature. An NEC group with signature (2.1) actually exists if and only if the right-hand side of (2.2) is greater than 0. We denote by |r| the expression "(r)/2n and call it the reduced area of r. If r is a subgroup of an NEC group A of finite index N, then also r is an NEC group and the Riemann-Hurwitz formula holds, |r| = N|A|. Let X be a non-orientable Klein surface of topological genus g > 3 without boundary. Then by [28] there exists an NEC group r with signature: a(r) = (g, -, [-], {-}), (2.3) such that X = H/r. A group r with this signature is called a surface NEC group. If G acts as an automorphism group of X = H/r, then there exists another NEC group A such that G = A/r. From the Riemann-Hurwitz relation we have g - 2 = o(G)|A|, where o(G) denotes the order of G. Then ¿KG) < g = 2 + o(G)|A|, and so to obtain the symmetric crosscap number of G is equivalent to find a group A and an epimorphism 0: A ^ G, such that r = ker 0 is a surface NEC group (and so, without elements with finite order) and G = 0(A+), where A+ is the subgroup consisting of the orientation-preserving elements of A, see [28], and minimal |A|. The groups having symmetric crosscap numbers 1 and 2 have been classified by T. W. Tucker, [29]. The groups of symmetric crosscap number 1 are Cn, Dn, A4, S4 and A5. We have two families of groups of symmetric crosscap number 2, C2 x Cn, n > 2 even, and C2 x Dn, n even. It is known that there exists no group of symmetric crosscap number 3, [23]. The groups with symmetric crosscap number 4 and 5 were obtained in [8]. M. D. E. Conder at a conference in Castro-Urdiales in 2010 announced that using computing software, he had obtained the groups of symmetric crosscap number up to 65, in terms of their "SmallGroupLibrary" description. The result of this research is available in his webpage, [9]. The list contains the GAP reference of each group, its symmetric crosscap number and the corresponding NEC group A. However, this list gives information neither on the algebraic structure of the groups nor on the epimorphism 0 which determines the action of the NEC group A over the group G. Throughout the paper, we use extensively this fundamental work by Conder, in order to study which are the concerned groups. For each group G we have described its algebraic structure, its presentation and the corresponding epimorphism, but here we will only show the algebraic structure and its presentation. In the most complicated cases, we will show also the epimorphism. In the presentations we skip the abelian relations. The full details are to be found in [2] and [3]. For groups of order 32 and 64 we use the notation given by Hall and Senior in [20]. The algebraic identification allows us to know the subgroups structure of the involved groups, and this is essential to determine all the groups that act on a surface of a given genus. Along the article Cn, Dn, DCn and QAn denote, respectively, the cyclic, dihedral, dicyclic and quasiabelian groups, for more details see [12, 13]. 3 Groups of symmetric crosscap number 6 to 9 In symmetric crosscap number 6 some groups stand out: 1. The group [80,46]: Coxeter described this group of order 80 in [12], where he named it as (2, 5, 5; 2) with presentation and algebraic structure as shown in the table. 2. The group [160, 234]: This group contains the previous one of order 80. In [12] it is denoted as (4, 5 | 2,4). Table 1: Groups of symmetric crosscap number 6. GAP G Relations [+ Generators] Reference [8, 4] DC2 ~ Q a4 a2b2,b !aba [23] [16, 3] (4, 4 | 2, 2) a4 b4, (ab)2, (a-1b)2 [15] [16, 6] QA4 a8 b2, baba3 [15] [16, 8] ¿4 a8 b2 , baba5 [15] [16, 13] (2, 2, 2)2 a2 b2, c2, abcacb, abcbac, bcabac [15] [16, 14] C2 x C2 x C2 x C2 a2 b2,c2,d2 [19] [32, 27] T4ai a2 b2, c2, d2, e2, cecae, dedbe [32, 43] Tea! a8 b2, c2, (ab)2, aca3c [80, 46] (2, 5, 5; 2) a2 b5, (ab)5, (a-1b-1ab)2 [120, 35] C2 x A5 a2 [+ (1 2 34 5), (1 2 3)] [l60, 234] (4, 5 | 2,4) a4 b5, (ab)2, (a-1b)4 Attending to symmetric crosscap number 7, we must analyze the group [72,15], which contains the group of order 36 that appears in the table below (see [26]) and so that the algebraic structure is ((C2 x C2) x C9) x C2. In this case, we are going to give the epimorphism. This group has a presentation given by generators a, b, c and relations a4 = b9 = c2 = (ac)2 = (cb)2 = (ab)2 = cb-iab-ia-2 = 1. An associated NEC group is A with signature (0; +; [-]; {(2,4,9)}) and reduced area 752 and an epimorphism 0: A ^ G is 0(ci,o) = cb, 0(ci,i) = ac, 0(ci^) = c, 0(ci^) = cb. The image of ci;icij2 is the generator a, the image of cij2cij3 is the generator b, and finally, c is the image of the element (ciiicij2)2cij2cij3(cijicij2)3cij2ci 3. So we have the generators as images of orientation-preserving elements, and so that the group acts on a non-orientable surface. For symmetric crosscap number 8 we just have to emphasize the group of order 504, that is PSL(2,8), whose symmetric crosscap number was firstly studied in detail by Wendy Hall in [21]. To end this section, we comment some groups with symmetric crosscap number 9, where we find: 1. The group [42,1], which we call (7, 6,5), according to the Coxeter-Moser notation in [13]. It contains G2i, which is also a group of this symmetric crosscap number, and so that its algebraic structure is G2i x C2. Its presentation can be expressed in terms of permutations taking a =(1 23456 7) and b =(1 54 6 2 3). Table 2: Groups of symmetric crosscap number 7. GAP G Relations [+ Generators] Reference [12,1] DC3 a6, a3b2, b-1aba [23] [24, 8] (4, 6 | 2, 2) a4,b6, (ab)2, (a-1b)2 [15] [36, 3] (C2 x C2) x C9 a2, b2, c9, [a, b], c-1acb, c-1bcba [72,15] ((C2 x C2) X C9) x C2 a4, b9, c2, (ac)2, (cb)2, (ab)2, cb-1ab-1a-2 Table 3: Groups of symmetric crosscap number 8. GAP G Relations [+ Generators] Reference [24, 5] C4 x D3 a4, b2, c2,(bc)3 [16] [24,10] C3 x D4 a3, b2, c2,(bc)4 [16] [48, 38] D3 x D4 a2, b2, c2, d2, (ab)3, (cd)4 [17] [56,11] (C2 x C2 x C2) x C7 a7, b2, c2, d2, badca-1, caba-1, daca-1 [504,156] PSL(2, 8) a2, b3, (ab)7, ([a, b]4b)2 [21] 2. The group [168,42] is PSL(2, 7). In this case, the presentation given in the table can be expressed by permutations b = (234)(576) and a = (1 2 3)(4 5 6) and relations a3 = b3 = (ab)4 = (a-1b)4 = 1, see [12]. Two more presentations for this group are useful: (a) R4,S4, (RS)2, (R-1S)3 (b) R2, S3, (RS)7, (R-1S-1RS)4 Studying this group, there are actions given by NEC groups with two different signatures: (i) For an NEC group A with signature (0; +; [-]; {(3, 3,4)}) and reduced area 24, we take the presentation given by permutations. So an associated epimorphism 0: A ^ G is: 0(c1jO) = (baba2)2, 0(cM) = (a2b)2, 0(c1j2) = (ba2)2, 0(c1j3) = (baba2)2 Consider the image of c1,0c1,1 and the image of c1,1c1,2. Then the image of the element (c1j0c1;1 )2c1j1c1i2c1i0c1i1(c1i1c1j2)2 is (1 5 4 3 6 2 7), a permutation of order 7. This element, together with the elements of order 3 and order 4, 0(c1,Oc1,1) and 0(c1j2c1j3), generate a group of order 84 at least, but PSL(2, 7) is simple, so it is the full group. So the group G is generated by images of orientation-preserving elements and the group acts on a non-orientable surface. (ii) For an NEC group A with signature (0; +; [3]; {(4)}) and reduced area 24, we use the presentation (b). An associated epimorphism 0: A ^ G is: 0(xi) = S, 0(ei) = S2, 0(c1jO) = R, 0(01,1) = SRS-1 It is clear that 9 is an epimorphism. The element ci o^i is orientation-reversing, its seventh power is also orientation-reversing and the image of (c1j0x1)7 is the identity element, so the group acts on a non-orientable surface. 3. The group [336,208] has order 336 = 168 • 2. Then we can guess its algebraic structure is PSL(2, 7) x C2. We can find a presentation of this group in [10], and an epimorphism 9 does exist. Hence this is the group we are looking for. Table 4: Groups of symmetric crosscap number 9. GAP G Relations [+ Generators] Reference [21,1] G21 a7, b3, aba3b-1 [15] [30,1] C5 x D3 a5, 62, c2, (bc)3 [16] [30, 2] C3 x D5 a3, b2, c2, (bc)5 [16] [42,1] (7, 6, 5) a7,b6,b-1aba2 [60, 8] D3 x D5 a2, b2, c2, d2, (ab)3, (cd)5 [17] [168,42] PSL(2, 7) a3, b3, (ab)4, (a-1b)4 [336, 208] PSL(2,7) x C2 a3, b8, c2, (ac)2, (cb)2, (ab)2, cb-1(ab-2)3a-1 4 Groups of symmetric crosscap number 10 to 17 Firstly, we analyze the groups with symmetric crosscap number 10, where we can find 30 different groups, most of them of order 32, 48 and 64. We just emphasize: 1. For the group [48, 29] we use two presentations, the one given in the table (generators a, b, c and relations a2,b3, c3, (bc)4, (ab)2, (ac)2, [b, c](bc)2) and another one given by generators R, S and relations R8, S3, (RS)2, R4SR4S-1. For this case, three signatures of NEC groups are given: (i) For an NEC group A with signature (0; +; [-]; {(2, 2, 3,3)}) and reduced area 6 we take the presentation given in the table and an epimorphism 9: A ^ G given by 9(ci,o) = ac, 9(ci,i) = (bc)2, 9(ci^) = ba, 9(ci^) = a, 9(ci^) = ac The group acts on a non-orientable surface, because the image of the element cij2cij3 is the generator b, the image of the element ci 3ci 4 is the generator c and the image of the element cij3ciji(cij2ci 4)2 is the generator a, so these three images generate the group, and they are images of orientation-preserving elements. (ii) For an NEC group A with signature (0; +; [3]; {(2, 2)}) and reduced area i we take the second presentation and so an associated epimorphism is 9: A ^ G given by 9(xi) = S, 9(ei) = S-\ 9(ci0) = RS, 9(cM) = R4, 9(cij2) = SR The images of the elements (ciiicij0ei)5 and x1 are the generators R and S respectively and both are orientation-preserving elements, so it is a group acting on a non-orientable surface. 6(x1) = RS, 0(x2) = S, 6(e1) = SR-1, 0(cM) = R4 The quotient gives a non-orientable surface because the images of the elements x1x2 and x2 are the generators R and S respectively and both elements are orientation-preserving. 2. The group [96,70] can be expressed in terms of permutations, by means of the generators a = (1 2)(3 4)(5 8)(6 7) and b = (1 5)(2 8 3 64 7). 3. We can find the group [96,193] in [24], called G|8, but in the presentation given there, one relation is missing. We have added it, as can be seen in Table 5. In symmetric crosscap number 11 we have to stand out two things: One is that the presentation of group [108,15] can be expressed in terms of permutations of S18 as a = (4 7)(5 8)(6 9)(13 16)(14 17)(15 18) and b = (1 17 5 14 2 18 6 15 3 16 4 13) (7 12 9 118 10); and the other is that the group [108,17] is G3'6'6 in the notation of [12]. For symmetric crosscap number 12 and 13, we have nothing to remark. In symmetric crosscap number 14 we find several groups of order 48, and the following groups stand out: 1. The presentation of the group [72,43] has been deduced from its algebraic structure (C3 x A4) x C2. We have taken d as the generator of C2 and we have determined how d acts on the other generators. 2. The same argument has been applied to the group [96, 89], where its algebraic structure (D2 x D6) x C2 determine its presentation. In this case, e is the generator to add. The presentation is given by generators a, b, c, d, e and relations a2, b2, c2, d2, e2, (ab)2, (cd)6, eabea, ecdec. Let A be an associated NEC group with signature (0; +; [-]; {(2, 2, 2,4)}) and reduced area |, so an epimorphism is 0: A ^ G given by 8 0(c1,o) = e, 0(c1,1 ) = b, 0(c1,2) = a, 0(c^) = c, 0(cM) = e The elements c1,0, c1,2, c1,3, c1,1 and (c1,4c1,3)2 have as images the generators e, a, c, b, d respectively and generate the group. On the other hand the element (c1'0c1,2)2 c1,1 has as image the identity element and it is orientation-reversing. Thus, the group acts on a non-orientable surface. 3. The same happens for [96,115] and its algebraic structure is (C2 x D12) x C2, where d is the generator of C2 and so that we have to determine its relations with the other generators. In symmetric crosscap number 15, we just note that the group [1092, 25] was obtained in [21] by Wendy Hall, who proved that PSL(2,13) is a group of 84(g - 2) automorphisms of a surface of genus g, and so g = 15. Nothing stands out in symmetric crosscap number 16. But in symmetric crosscap number 17 we have again the same situation that in symmetric crosscap number 14. For the group [72,23] we have deduced the presentation from its algebraic structure (G6xD3)xG2, taking d as the generator of C2 and obtaining its action on the other generators. Table 5: Groups of symmetric crosscap number 10. GAP G Relations [+ Generators] Ref. [16, 2] C4 x C4 a4,b4 [19] [16,4] C4 X C4 a4, b4, b-1aba [15] [16, 9] dc4 a8, a4b2, b-1aba [23] [16,10] C4 x C2 x C2 a4,b2,c2 [19] [24, 3] (2, 3, 3) a3, abab-1a-1b-1 [15] [32, 5] a2, b8, c2, bcb-1ac [32, 6] r7a-i a2 ,b2, c2, d4, bdbad-1, cdcbad-1 [32, 7] r7a,2 a8, b2, c2, aba3b, aca-1bc [32, 9] rsai a2, b8, c2, bcbac [32,11] rse a4, b4, c2, bcba-1c [32,17] r2k ai6,b2,aba7b [32,19] r$a2 ai6,b2,abagb [32, 28] r4bi a2,b2,c4,d2, bdbad, (cd)2 [32, 34] r4a2 a4, b4, c2, (ac)2, (bc)2 [32, 42] r3b a8, b2, c2, (ac)2,bcba4c [32,46] C2 x C2 x D4 ~ r^ai a2,b2,c2,d2, (ab)2, (cd)4 [17] [32, 49] rzai a4, b2a2,c2a2, d2 a2, abab-1, cdcd-1 [48, 29] GL(2, 3) a2, b3, c3, (bc)4, (ab)2, (ac)2, [b, c](bc)2 [48, 31] C4 x A4 a4, [+ (1 4)(3 2), (1 2 3)] [18] [48, 33] SL(2, 3) X C2 a2, b3, c3, (bc)4, abac, [b,c]2(bc)2 [48, 50] (C2 xC2 xC2 xC2) XC3 a2,b3,c3, (cb)2, (ab-i)3, c-ib~iabca, cbab-ic-ia [64,128] ri5ai a2,b2, c2, e2, f 2,d2f, [a, b]fd~i, [a, c]e, [a, d]f, [b, d]f [64,134] r26ai a2,b2, c2, e2, f 2,d2f, [a, b]fd-i, [a c]e, [a, d]f, [b, d]f, [b, e]f, [c d]f [64,138] r25ai a2, b2, c2, d2, e2, f2, [a, b]d, [a, c]e, [b, e]f, [c, d]f [64,190] rigai a2,b2, c2, f 2,d2fe-i,e2f, [a, b]e-1d-1, [a, c]f, [a, d]fe-1, [a, e]f, [b,d]fe-1, [b, e]f [96, 70] ((C2 x C2 x C2 x C2) X C3 ) X C2 a2,b6, (bab-1 a)2, (b-2a)3 [96,187] (C2 x S4) X C2 a4, b12, c2, (ab)2, (cb)2, (ac)2, cb-1ab-1a-2 [96,193] GL(2, 3) X C2 a2, b8, c3, (bc)2, (ac)2, (ab)2, b4cb4c-1 [96, 227] ((C2 x C2 x C2 x C2) X C3 ) X C2 a2, b3, c2, d2, e2, f2, (ba)2, cada, cbdb-1, daca, dbdcb-1, eafa, ebfeb-1, faea, fbeb-1 [192, 955] (((C2 x C2 x C2 x C2) X C3) X C2) X C2 a4, b6, c2, (ab)2, (cb)2, (ac)2, (ab-1)4, cb-1ab2a-1b3a-1 Table 6: Groups of symmetric crosscap number 11. GAP G Relations [+ Generators] Ref. [18, 5] Ce x C3 ae,b3 [19] [27, 3] (3, 3 | 3, 3) a3, b3, (6a)3, (b-1a)3 [15] [36,13] C2 x ((C3 x C3) X C2) a2, b3, c3, d2, (6a)2, (ca)2 [54, 5] (2, 3, 6; 3) a3, be, (ab)2, (ba-1b)3 [54, 8] ((C3 x C3) X C3) X C2 a2, b3, c2, (b-1a)2, (ca)3, (b-1c)2(bc)2, (ab-1c)2bac [108,15] ((C3 x C3) X C3) X C4 a2, (b-2a)3, b-1ab4ab-3, b-1abab-2 abab-1a [108,17] G3,e,e a2, b2, c2, (ab)2, (ac)3, (bc)e, (abc)e [216, 87] (((C3 x C3) X C3) a4, be, c2, (ab)2, (cb)2, (ac)2, x C4 ) x C2 c(b-1a)3a Table 7: Groups of symmetric crosscap number 12. GAP G Relations [+ Generators] Reference [20,1] DC5 a10, a5b2, b-1aba [23] [40, 5] C4 x D5 a4, b2, c2, (bc)5 [16] [40, 8] (C10 x C2) X C2 a10, b2, (aba)2, (a-1b)2(ab)2 [40,10] C5 x D4 a5, b2, c2, (bc)4 [16] [40,12] C2 x (5,4, 2) a5, b4, bab-1a3 [80, 39] D5 x D4 a2, b2, c2, d2, (ab)5, (cd)4 [17] [240,189] C2 x S5 a2, [+ (1 2 34 5), (1 2)] Table 8: Groups of symmetric crosscap number 13. GAP G Relations [+ Generators] Reference [42, 3] C7 x S3 ar, [+ (1 2 3), (1 2)] [16] [42, 4] C3 x Dr a3, b2, c2,(bc)r [16] [52, 3] C13 X C4 a4, b13, baba-1 [60, 9] C5 x A4 a5, [+(1 2 3), (1 4)(2 3)] [18] [84, 8] D3 x Dr a2, b2, c2, d2, (ab)3, (cd)r [17] [120, 38] (C5 x A4) X C2 a4, b15, c2, (ab)2, (cb)2, (ac)2, cb-1ab-1a2 Table 9: Groups of symmetric crosscap number 14. GAP G Relations [+ Generators] Ref. [16,12] C2 x Q a4, a2b2, c2, b-1aba [23] [24,4] C2 x DC3 a3, b4, c2, bab-1 a [18] [24, 7] DCe a12,aeb2,b-1aba [23] [24,15] Ce x C2 x C2 ae,b2,c2 [19] [32,48] T2& a4, b2, c2, d2,bcba2c [36,11] C3 x A4 a3, [+ (1 2)(3 4), (1 2 3)] [18] [36,12] Ce x D3 ae,b2,c2, (bc)3 [16] [48, 6] C24 X C2 a24, b2, baba13 [48,14] (C12 x C2) X C2 a3, b4, c4, (bc)2, (b-1c)2, c-1aca [48, 21] C3 x (4,4 | 2, 2) a3, b4, c4, (bc)2, (b-1c)2 [48, 24] C3 x QA4 a8, b2, c3, baba3 [48, 37] (C12 x C2) X C2 a3, b2, c2, d2, dcbcdb, dcbdbc, bdcdbc, (ba)2, (da)2 [48,43] C2 x ((Ce x C2) x C2) a4, be, c2, (ab)2, (a-1b)2 [48,49] C2 x C2 x A4 a2, b2, [+ (1 2)(3 4), (1 2 3)] [48, 51] D2 x De a2, b2, c2, d2,(ab)2, (cd)e [17] [72, 42] C3 x S4 a3, [+(1 2), (1 2 3 4)] [72, 43] (C3 x A4) X C2 a3, b2, c3, d2, (da)2, (dc)2, [+ b = (1 2)(3 4), c =(12 3)] [72, 44] A4 x S3 [+ (1 2)(3 4), (1 2 3), (5 6 7), (5 6)] [72,46] D3 x De a2, b2, c2, d2,(ab)3, (cd)e [17] [96, 89] (D2 x De) x C2 a2, b2, c2, d2, e2, (ab)2, (cd)e, eabea, ecdec [96,115] (C2 x D12) X C2 a2, b2, c2, d2, (bc)12, dcbdc [96, 226] C2 x C2 x S4 a2, b2, [+ (1 2 3 4), (1 2)] [144,183] S3 x S4 [+(1 2 3), (1 2), (4 5 6 7), (4 5)] [180,19] A5 x C3 a3, [+(1 3 24 5), (2 4 3), (2 4)(1 3)] [360,121] A5 x D3 a3, b10, c2, (ab)2, (cb)2, (ac)2, b-2ab3a-1b-4a-1c Table 10: Groups of symmetric crosscap number 15. GAP G Relations [+ Generators] Ref. [24,1] (-2, 2, 3) a8, b8, (a3b)3, a2be, a2(b-1a-1)3, b2(b-1a-1 )3 [15] [39,1] C13 X C3 a3,b13,bab10a-1 [48,15] (C3 x D4) X C2 a2, b8, c3, (ab)2, (ac)2,b-1cbc [78,1] (C13 X C3) X C2 a2, b3, c13, (ca)2,cbc10b-1 [1092, 25] PSL(2,13) a3, b7, c2, (ab)2, (cb)2, (ac)2, b-1(ab-2)ea-1c [21] Table 11: Groups of symmetric crosscap number 16. GAP G Relations [+ Generators] Reference [28,1] dc7 a14, a7b2, b-1aba [23] [56,4] C4 x D7 a4, b2, c2, (bc)7 [16] [56, 7] (C14 x C2) X C2 a2, b14, (bab)2, (b-1a)2(ba)2 [56, 9] C7 x D4 a7, b2, c2, (bc)4 [16] [72,16] C2 x ((C2 x C2) X Cg) ag, b2, c2, d2, bacba-1, caba-1 [112, 31] D7 x D4 a2, b2, c2, d2, (ab)7, (cd)4 [17] [144,109] (C2 x ((C2 x C2) a4, b18, c2, (ab)2, (cb)2, (ac)2, X Cg)) X C2 cb-1ab-1a-2 Table 12: Groups of symmetric crosscap number 17. GAP G Relations [+ Generators] Ref. [25, 2] C5 x C5 a5, b5 [14] [27, 2] Cg x C3 ag,b3 [14] [27,4] Cg X C3 a3, bg, bab5a-1 [15] [36, 6] C3 x DC3 a12, b3, baba-1 [18] [50, 3] C5 x D5 a5, b2, c2, (bc)5 [16] [50,4] (C5 x C5) X C2 a2, b5, c5, (ba)2, (ca)2 [54, 3] C3 x Dg a3, b2, c2, (bc)g [16] [54,4] Cg x D3 ag,b2,c2, (bc)3 [16] [54, 6] (Cg X C3) X C2 a2, bg, c3, (ba)2,cb7c-1b-1 [54, 7] (Cg x C3) X C2 a2, b3, cg, (ba)2, (ca)2 [68, 3] C17 X C4 a4, b17, bab4a-1 [72, 23] (Ce x D3) X C2 ae, b2, c2, d2, (bc)3, bdcbd, dada3 [72, 39] (C3 x C3) X C8 a8, b3, c3, baca-1, cab-1a-1 [100,12] (C5 x C5) X C4 a4, b5, c5, bab3a-1, cac3a-1 [100,13] D5 x D5 a2, b2, c2, d2, (ab)5, (cd)5 [17] [108,16] D3 x Dg a2, b2, c2, d2, (ab)3, (cd)g [17] [200, 43] (D5 x D5) X C2 a4, b10, c2, (ab)2, (cb)2, (ac)2, [ab, ba], cb-1(ab-3)2a-2 [360,118] Ae [+(1 4 2 3 5), (3 5 4), (1 2 4 3)(5 6)] [720, 764] Ae x C2 a3, b8, c2, (ab)2, (cb)2, (ac)2, cb-1ab3ab-2a-1ba-1b-3a-1 5 Groups with symmetric crosscap number 12 k + 3 Firstly, the strong symmetric genus is the minimum genus of any Riemann surface on which G acts, preserving orientation. For this parameter, there is a group of every strong symmetric genus, [25]. The symmetric genus is the smallest non-negative integer g such that the group G acts faithfully on a closed orientable surface of genus g (not necessarily preserving orientation). For this parameter, the spectrum includes every non-negative integer g ^ 8 or 14 (mod 18), and moreover, if a gap occurs at some g = 8 or 14 (mod 18), then the prime-power factorization of g — 1 includes some factor pe = 5 (mod 6), [11]. In the study of the spectrum of the symmetric crosscap number, the groups with symmetric crosscap number of the form 12k + 3 are very interesting. It is known that for all n = 12k + 3, there is a finite group with symmetric crosscap number n, see [6]. Conversely, for some values n = 12k + 3, it is not known whether there exists a group with symmetric crosscap number n. So that, we can enunciate some theorems whereby we find infinite families of groups whose symmetric crosscap number is of the type 12k + 3. The symmetric crosscap numbers obtained in Theorems 5.1 to 5.5, although of 12k + 3 form, were already obtained for other groups, as we can see in the proofs. In the case of Theorem 5.6, also these numbers n were already covered, since the group C7(12fc+7) x C3, in the terms of the statement, has symmetric crosscap number 84k + 51, see [6]. But they are important because they give more examples of groups of this type of n, helping us to see how these groups act. Theorem 5.1. Let n = 12k + 3 be such that n — 2 has all its prime factors congruent to 1 (mod 3). Then Ci2fc+i x C3 and (Cm+i x C3) x C2 have symmetric crosscap number n. Proof. Firstly we have that C12k+1 x C3 has a presentation given by generators a, b such that a3 = b12k+1 = (ab)3 = 1. Now let A be an NEC group with signature (1; —; [3,3]; {—}), whose reduced area is 3. We can define an epimorphism 0: A ^ G given by 0(x1 ) = a-1, 0(x2) = ab, 0(d1) = b6k We have that the images of x1 and x1x2 are the generators a-1 and b respectively, and both are preserving-orientation elements, then we have that it is a group that acts on a non-orientable surface. Besides, the NEC group area is minimal ([6]), and so the symmetric crosscap number of C12k+1 x C3 is n. Now we have (C12k+1 x C3) x C2 that has a presentation given by generators a, b, c and relations a3 = b12fc+1 = c2 = (ab)3 = 1, ca = ac and bc = cb-1. Now let A be an NEC group with signature (0; +; [2,3]; {(—)}), whose reduced area is 6. Therefore, if we define an epimorphism from this NEC group, (C12k+1 x C3) x C2 will have symmetric crosscap number less or equal to n. We can define an epimorphism 0: A ^ G given by 0(x1) = cb, 0(x2) = b-1a-1, 0(e1) = ac, 0(c1,o) = c We have that the element c1,0, the element e1c1,0 and the element c1,0x1 have as images the generators c, a and b respectively. Besides the element (e1c1,0)3 has as image the identity element and it is orientation-reversing, so we have just proved that the group acts on a non-orientable surface. Because of this epimorphism we can say that (C12k+1 x C3) x C2 has symmetric crosscap number at most n. But since it contains C12k+1 x C3, that has symmetric crosscap number n, a((C12fc+1 x C3) x C2) = n. □ Theorem 5.2. Let n = 12k + 3 be such that n — 2 = m2 is a square. Then: (i) (3,3 | 3, m) has symmetric crosscap number n. (ii) There are two groups with algebraic structure (3,3 | 3, m)xC2, namely (2,3,2m; 3) and (2, 3, 6; m), that have symmetric crosscap number n. Proof. Firstly we have that the group (3, 3 | 3, m) of order 3m2 has a presentation given by generators a, b and relations a3 = b3 = (ab)3 = (a-1b)m = 1. From [15], we know that this group has symmetric crosscap number m2 + 2. Now we have two groups with algebraic structure (3,3 | 3, m) x C2: (i) The first one, that is the group (2,3,2m; 3) in the notation of [12], of order 6m2, has a presentation given by generators a, b, c and relations a3 = b3 = c2 = (ab)3 = (a-1b)m = 1, ca = a2c and cb = b2c. Take an NEC group A with signature (0; +; [2]; {(3,3)}), that has reduced area 6. We define an epimorphism 0: A ^ G given by 0(x1) = c, 0(e1) = c, 0(c1,o) = ac, 0(^,1) = cb, 0(c1,2) = a-1 c We have that 0(x1) = c, 0(c1,0x1) = a and 0(x1c1,1) = b, and the element (e1c1,0)3 has as image the identity element and it is orientation-reversing. Thereby we have proved that the group acts on a non-orientable surface. Thereupon we have that this group has symmetric crosscap number at most m2 + 2, but as it contains (3, 3 | 3, m) that has that symmetric crosscap number n, then we have proved that ^ and |B| > ^. This result implies the lower bound s(G) > ^^. In particular, in graphs with perfect matchings the saturation number cannot be smaller than one quarter of the number of vertices, s(G) > n. A polyomino system consists of a cycle C in the infinite square lattice together with all squares inside C. A polyomino graph is the underlying graph of a polyomino system. A benzenoid system consists of a cycle C in the regular infinite hexagonal lattice together with all hexagons inside C. A benzenoid graph is the underlying graph of a benzenoid system. Let G be a benzenoid graph or a polyomino graph. The vertices lying on the outer face of G are called external; other vertices, if any, are called internal. Graph G without internal vertices is called catacondensed. If no inner face in a catacondensed graph G is adjacent to more than two other inner faces, we say that graph G is unbranched or that it is a chain. In each chain G there are exactly two inner faces adjacent to one other inner face; those two inner faces are called terminal, while any other inner faces are called interior. The number of inner faces in chain G is called its length. An interior inner face is called straight if the two edges it shares with other inner faces are parallel, i.e. opposite to each other. If the shared edges are not parallel, the inner face is called kinky. If all interior inner faces of a chain G are straight, the chain is called linear. There is also another terminology, calling straight inner faces linear, and kinky inner faces angular. By introducing abbreviations L and A, respectively, for linear and angular inner faces, each chain can be represented as a word over the alphabet {L, A}, with the restriction that the first and the last letter are always L. Such a word is called the LA-sequence of the chain. A fullerene F is a 3-connected 3-regular plane graph such that every face is bounded by either a pentagon or a hexagon. By Euler's formula, it follows that the number of pentagonal faces of a fullerene is exactly 12. The Cartesian product G □ H of graphs G and H is the graph with the vertex set V(G) x V(H) and (a, x)(b, y) e E(G □ H) whenever ab e E(G) and x = y, or, if a = b and xy e E(H). 3 Polyomino chains and grid graphs In this section we prove some results regarding the saturation number of polyomino chains and rectangular grids. We start with the linear chain Ln, where n denotes the number of squares. Such chain can be obtained as Cartesian product of the path Pn of length n and K2. Here Pn is the path on n edges so that Ln = Pn □ K2. Alternatively, Ln = Pn □ Pi. We draw Ln so that the edges of both copies of Pn are horizontal, see Figure 1. Figure 1: Linear polyomino chain L6 = P6 □ K2. We start by quoting two facts about the saturation number and the structure of smallest matchings in paths. Proposition 3.1 ([6]). Let Pn be a path of length n. Then s(Pn) = [n]. More precisely, r n, 31 n s(Pn)= r , 3 | (n - 1) [ n±l, 3 | (n - 2). Proposition 3.2. Let Pn be a path of length n. Then Pn has a smallest maximal matching that leaves at least one of the end-vertices unsaturated. Proof. Let n be divisible by 3. We form groups of three consecutive edges and construct a matching M by taking the middle edge of each group. M is obviously a smallest maximal matching and leaves unsaturated both end-vertices of Pn. If n = 3k +1, again consider groups of 3 consecutive edges, take the middle edge in each group and add the sole edge that does not belong to any group. Again the constructed matching is a smallest maximal matching. Finally, when n = 3k + 2, construct a matching in the same way by taking the middle edge from each of k groups of three consecutive edges and adding the edge saturating the rightmost vertex. □ Next we show that we can construct a smallest maximal matching in Ln without using vertical edges. This result will enable us to reduce the problem of finding the saturation number of Ln to known results about s(Pn ). Proposition 3.3. Let M be a maximal matching in Ln containing k > 0 vertical edges. Then there is another maximal matching M' in Ln containing k' < k vertical edges such that |M'| < |M|. Proof. We label the vertices in the upper copy of Pn with w0, u1,..., un, from left to right, and vertices in the lower copy with v0, v1,..., vn, in the same direction. There are n +1 vertical edges, each of the form «.¿vj for some 0 < i < n. See Figure 2. >„ -, V. , Figure 2: Linear polyomino chain Ln. Let M be a maximal matching in Ln with k > 0 vertical edges and let the leftmost vertical edge in M be the edge um vm. Obviously, m cannot be equal to 1. We consider first the case m = 0. If uivi is also in M, we construct a matching M' as M' = M - {w0v0, u1v1} U {u0u1, v0v1}. Obviously, M' is a maximal matching of the same cardinality as M containing k - 2 vertical edges. Let now both neighbors u1 and v1 of end-vertices of u0v0 be saturated by horizontal edges. Hence, both m1m2 and v1v2 are in M. Then at least one of u3 and v3 must be saturated by an edge of M. Let u3 be saturated. Then we can construct a matching M' as M' = M — {u0v0, u1 u2} U {m0m1}. Again, M' is a maximal matching, |M= |M| — 1 and k' = k — 1. The case of saturated v3 follows by symmetry. The last case to consider for m = 0 is the one in which only one of u1, v1 is saturated by a, necessarily horizontal, edge of M. Let it be u1. Hence, m1m2 g M and v1v2 G M. Then v3 must be saturated and M' = M — {u0v0} U {v0v1} is a maximal matching of the same cardinality as M but with one vertical edge less. Hence, the claim holds if the leftmost vertical edge in M is u0v0. This case is depicted in Figure 3. 2 M "2 M' Figure 3: The case when m = 0 and only one of u1,v1 is saturated by M. Let now the leftmost vertical edge in M be u2v2. Then both m0m1 and v0v1 must be in M, and at least one of vertices u3 and v3 must be saturated by an edge of M. Let it be u3. Then the matching M' constructed as M' = M - {u2v2, v0vi} U {viv2} will be a maximal matching with smaller cardinality and with one vertical edge less than M. Similar constructions apply when the leftmost vertical edge of M is near the right end of the chain. The simplest is the case m = n - 1, when also the rightmost edge unvn must be in M. Then by switching the edges on the rightmost square one readily obtains a maximal matching of the same size as M but without vertical edges. The case m = n - 2 forces both horizontal edges Mn-1Mn and vn-1vn to be in M. Then at least one of u„_3 and vn_3 must be saturated. Let it be vn_3. Then the matching M' = M - {«n-2vn-2,«n-1Mn} U {«„-2Mn-1j is a maximal matching of smaller size than M without vertical edges. Remains the case when unvn is the leftmost (and hence the only) vertical edge in M. If only one of Mn-1, vn-1 is saturated, let us say Mn-1, it suffices to switch unvn and vn-1vn to obtain a maximal matching M' of the same size without vertical edges. If both wn-1, vn-1 are saturated, they must be saturated by horizontal edges «n_2«n-1 and vn-2vn-1, respectively. Also, at least one of m„_3 and vn_3 must be saturated. Let it be vn_3. Then M' = M - {unvn, v„_2vn-1} U {vn-1vn} is a maximal matching of smaller size than M but without vertical edges. Now we can look at the remaining cases in a unified manner. So, let umvm, 3 < m < n - 3, be the leftmost vertical edge in a maximal matching M. If um+1vm+1 is also in M, we construct M' by switching the edges on the square um, um+1, vm+1, vm, obtaining a maximal matching of the same size but with two vertical edges less. Hence, we can suppose that «m+1vm+1 ^ M. If both wm-1 and um+1 are unsaturated, then both vm-1 and vm+1 must be saturated, necessarily by horizontal edges vm-2vm-1 and vm+1vm+2, respectively. Further, both um_2 and um+2 must be saturated, again by horizontal edges. The situation is shown in Figure 4. Vm-2 Vm_! Vm Vm+1 Vm+2 M v m—2 Vm V„ M' Figure 4: The case when wm-1 and wm+1 are both unsaturated. We construct M' as M' = M - {«mvm} U {wmwm+1}. Obviously, M' is a maximal matching of the same size as M and with one vertical edge less. The situation in which both vm-1 and vm+1 are unsaturated follows by symmetry. It remains to consider the case when at least one of «m-1, wm+1 and at least one of vm-1, vm+1 are saturated. We construct a new matching M'' by keeping the part of M to the left of «mvm, shifting all edges of M that were right of «mvm one place to the left (hence, mimi+1 goes to ui-1ui, to vl_1v; and to ui-1vi-1 for m < l < n) and moving umvm to unvn. Obviously, M'' is a maximal matching of the same size as M and with the same number of vertical edges, but with the leftmost vertical edge at some place l > m. Let us look at the situation on the right-hand side of Ln. If un_1vn_1 is in M'', then M' with the desired properties can be obtained by switching edges on the rightmost square of Ln. If un_1vn_1 is not in M'', then also un_2vn_2 cannot be in M, and M' can be constructed in exactly the same manner as when unvn is the only vertical edge in M. Hence, no matter where in M the leftmost vertical edge appears, we can always construct a maximal matching of the same or smaller size with strictly smaller number of vertical edges. □ Corollary 3.4. There is a maximal matching in Ln of cardinality s(Ln) without vertical edges. Corollary 3.5. 2s(P„) < s(L„) < 2s(P„) + 1. Proof. We know that there is a smallest maximal matching M in Ln (i.e., of the size equal to s(Ln)) without vertical edges. Hence all edges of M are horizontal, and each edge belongs to one of two copies of Pn in Ln. If the cardinality of M is smaller than 2s(Pn), then at least one of two copies of Pn will contain two adjacent unsaturated vertices. This proves the left inequality. To prove the right inequality, let us take a smallest maximal matching Mu in the upper copy of Pn. If Mu saturates exactly one end-vertex of Pn, let us take it so that it saturates u0. Let Mv be a smallest maximal matching in the lower copy of Pn obtained by taking the edges corresponding to the edges of Mu and shifting them one place to the right. Then Mv saturates the vertices in the lower copy of Pn adjacent to the vertices of the upper copy of Pn left unsaturated by Mu. Hence M = Mu U Mv is a maximal matching in Ln of size 2s(Pn ). It remains to consider the case when all smallest maximal matchings in Pn leave both end-vertices unsaturated. In that case, take two smallest maximal matchings Mu and Mv in upper and lower copy of Pn, respectively, and shift Mv one place to the right so that it saturates the neighbors of the vertices left unsaturated by Mu. That leaves unsaturated both end-vertices of v0vi. By adding that edge to the maximal matching constructed from Mu and shifted Mv we obtain a maximal matching of size 2s(Pn) + 1. □ From this we can get the exact expression for the saturation number of the linear poly-omino chain. Theorem 3.6. Let Ln be the linear polyomino chain. Then '2r + 1, 3 | n s(L„) = <( , 3 | (n - 1) ^^, 3 | (n - 2). Proof. If 3 | (n — 1) or 3 | (n — 2) there is a smallest maximal matching M for Pn such that M saturates exactly one end-vertex of Pn. Therefore, it follows from the proof of Corollary 3.5 that s(Ln) = 2s(Pn) and we are done. If 3 | n, the smallest maximal matching of Pn is uniquely defined and it leaves both end-vertices unsaturated. Hence, in this case we obtain s(Ln) > 2s(Pn) and therefore, s(Ln) = 2s(Pn) + 1. Examples of smallest maximal matchings in Ln for all classes of divisibility of the chain length by 3 are given in the Figure 5. □ The above approach can be successfully applied also to obtain non-trivial upper bounds on the saturation number of grid graphs that arise as Cartesian products of two (or more) paths. By taking smallest maximal matchings in all horizontal (or in all vertical) copies of paths in Pm □ Pn, shifting them and adjusting by adding an edge where necessary, and using symmetry, we can obtain following upper bound on s(Pm □ Pn). (b) (c) Figure 5: Linear polyomino chains with maximal matchings. Proposition 3.7. s(Pm □ Pn) < min{(m + 1)[s(Pn) + 1], (n + 1)[s(Pm) + 1]}. This upper bound can be improved a bit by exploiting particular relationships between parities and remainders modulo 3 of m and n. See Figure 6 for an example. We believe, however, that our upper bounds capture the asymptotic behavior of the saturation number of rectangular grids. Figure 6: Graph P8 □ P3 with a maximal matching. Now we go back to polyomino chains. In the following theorem we give the exact closed formulas for the saturation number of polyomino chains where all internal squares are kinky. Theorem 3.8. Let Sk be a polyomino chain with k squares such that all internal squares are kinky. Then k KSfc) = 2 + 1. Proof. We consider two cases. 1. Let k be even. Since the number of vertices in Sk is 2k+2, a perfect matching (which always exists) has k +1 edges. Using Lemma 2.1 we obtain that s(Sk) > . Since k is even, we obtain s(Sk) > [|] +1. To show the upper bound, we construct a maximal matching M from Figure 7. Obviously, |M| = f + 1 = [f] + 1. Hence, s(Sk) = [f] + 1. 2. If k is odd, let M' be a maximal matching from Figure 8. Obviously, |M= f++1 + 1 and therefore, s(Sk) < + 1 = [f] +1. Now suppose that there is a maximal matching N for Sk such that | N | < . It is easy to see that at least one of edges ei, e2, and e3 must be in N. Consider the following cases. Figure 7: Polyomino chain Sk (k even) with maximal matching M. J V /2 /3 e3 e2 Figure 8: Polyomino chain Sk (k odd) with maximal matching M. (a) If e1 G N, then also e3 G N or f3 G N. Therefore, for the graph Sk-3 (see Figure 8) it must hold s(Sk-3) < f-3, which is a contradiction with Case 1. (b) If e2 G N, then for the graph Sk-1 (see Figure 8) it must hold s(Sk-1) < , which is a contradiction with Case 1. (c) If e3 e N, then also one of the edges e1, f1, f2 must be in N. If e1 e N or f1 G N, then for the graph Sk-3 (see Figure 8) it must hold s(Sk-3) < , which is a contradiction with Case 1. Therefore, suppose that f2 G N. But in this case we can use similar reasoning and either obtain a contradiction with the Case 1 or eventually obtain a matching M', which is a contradiction since |M'| > |N|. Since we obtain a contradiction in every case, it follow s that every maximal matching of Sk has at least ff1 + 1 edges. Since + 1 = lf ] +1 it follows s(Sk ) > lf ] +1 and we are done. □ 4 Hexagonal animals In this section we prove some results regarding the saturation number of benzenoid chains and coronenes. 4.1 Benzenoid chains A benzenoid chain of length h will be denoted by Bh. If all interior hexagons of a benzenoid chain are straight, the chain is called a polyacene and denoted by Ah. Saturation number of benzenoid chains has been already studied in a recent paper coau-thored by one of the present authors [5]. We quote without proof some basic results established there. Proposition 4.1 ([5]). Let Bh be a benzenoid chain with h hexagons. Then s(Bh) > h +1. Proposition 4.2 ([5]). For any h it holds s(Bh) + 1 < s(Bh+1) < s(Bh) + 2. Proposition 4.3 ([5]). s(Bh) = h +1 if and only if Bh = Ah. Let Bh, 1 denote a chain of length h = k + m in which hexagon hk is kinky and all other hexagons are straight. An example is shown in Figure 9. Furthermore, let Bh k denote a benzenoid chain of length h with exactly k kinky hexagons. Figure 9: A chain with one kinky hexagon. Proposition 4.4 ([5]). For any h it holds s(Bh,i) = h + 2. Hence one kinky hexagon means one more edge in the smallest maximal matching. The following claim was stated in [5] as Proposition 5. Proposition 4.5 ([5]). Let Bh,k be a benzenoid chain of length h with k kinky hexagons such that no two kinky hexagons are adjacent. Then s(Bh,k) = h + k +1. However, we show in Proposition 4.7, Proposition 4.8, and Proposition 4.9 that the above proposition provides only an upper bound for the saturation number, which is evident from the following proposition. Proposition 4.6. Let Bh k be a benzenoid chain of length h with k kinky hexagons. Then s(Bh,k) < h + k + 1. Proof. Let M be a matching of Bh,k obtained by taking all edges shared by two hexagons, one additional edge in each terminal hexagon and all edges connecting vertices of degree two in kinky hexagons. See Figure 10 for an example. It is easy to see that M is a maximal matching and |M | = h + k + 1. Therefore, we are done. □ Figure 10: Maximal matching M. However, in the same graph shown in Figure 10 we can construct a smaller maximal matching by simply taking all vertical edges. Hence h + k + 1 is only an upper bound on s(Bh,k) and it can be improved in particular cases. Let Bh be a chain of length h. A straight segment in Bh is any sequence of consecutive straight hexagons. Equivalently, it is any sub-word made of consecutive L's in the LA sequence of Bh. The number of consecutive straight hexagons is the length of the straight segment. Figure 11: A benzenoid chain B8 with two straight segments (labelled with bold edges), one of length 2 and one of length 1. In the following we consider the saturation number of benzenoid chains where all straight segments are of length one and no two kinky hexagons are adjacent. It turns out we have to distinguish between three cases. In all of them the upper bound from Proposition 4.6 is improved. Proposition 4.7. Let B2k+1,k be a benzenoid chain such that all straight segments are of length one and no two kinky hexagons are adjacent. Then s(B2fc+i,fc) < 4(10k + 9 - (-1)k). Proof. We build B2k+1,k from left to right by adding blocks of 4 consecutive hexagons at a time. Each block has the form LALL and it is added on the rightmost hexagon of the already constructed chain so that it becomes a kinky hexagon in the new chain. To show an upper bound for the saturation number, we construct a maximal matching M of B2fc+1,fc. For the first four hexagons we need six edges in a maximal matching such that the edge connecting the first and the second block of hexagons is in the matching. See Figure 12. Figure 12: Two possibilities for a maximal matching M of the first block. Afterwards, we can always add four hexagons at a price of five new edges in a maximal matching. Let l be the number of blocks in B2k+i,k. We consider two cases. • If k is odd, then 4 | (2k - 2) and l = ^k-2 = ^r1 and we have three additional hexagons in a benzenoid chain. For these three hexagons, we need 4 additional edges in a maximal matching. We obtain |M| = 6 + 5(l - 1) + 4 = 10 + 5 • = . • If k is even, then l = = | and we have one additional hexagon in a benzenoid chain. For this hexagon, we need 1 additional edge in a maximal matching. Therefore, we get |M| = 6 + 5(l - 1) + 1 = 7 + 5 • ^ = . Combining both cases, we obtain |M| = 1 (10k + 9 - (-1)k). □ Proposition 4.8. Let B2k+2,k, k G N, be a benzenoid chain such that all straight segments are of length one and no two kinky hexagons are adjacent. Then s(B2k+2,k) < 4(10k +13 - (-1)k). Proof. We build B2k+2,k from left to right by adding blocks of 4 consecutive hexagons at a time. Each block has the form LALL or each block has the form LLAL (before adding) and it is added on the rightmost hexagon of the already constructed chain. Because of the symmetry, we can assume that each block has the form LALL (otherwise we can start from right to left). The new block is added on the last hexagon such that it becomes a kinky hexagon. To show an upper bound for the saturation number, we construct a maximal matching M of B2k+2,k. For the first four hexagons we need six edges in a maximal matching such that the edge connecting the first and the second block of hexagons is in the matching. Afterwards, we can always add four hexagons at a price of five new edges in a maximal matching. Let l be the number of blocks in B2k+2,k. We consider two cases. • If k is odd, then 4 | (2k + 2) and l = = ^. Weobtain |M| = 6 + 5(l - 1) = 6 + 5 • ^ = . • If k is even, then l = (2k+42)-2 = | and we have two additional hexagons in a benzenoid chain. For these two hexagons, we need 2 additional edges in a maximal matching. Therefore, we get |M| = 6 + 5(l - 1) + 2 = 8 + 5 • k-2 = . Combining both cases, we obtain |M| = 4 (10k + 13 - (-1)k). □ Proposition 4.9. Let B2k+3,k, k G N, be a benzenoid chain such that all straight segments are of length one and no two kinky hexagons are adjacent. Then s(B2k+3,k) < 1 (10k + 19 + (-l)k). Proof. We build B2k+3 k from left to right by adding blocks of 4 consecutive hexagons at a time. Each block has the form LLAL (before adding) and it is added on the rightmost hexagon of the already constructed chain. The new block is added such that the first hexagon of this block becomes a kinky hexagon. To show an upper bound for the saturation number, we construct a maximal matching M of B2k+3 k. For the first four hexagons we need six edges in a maximal matching such that the edge connecting the first and the second block of hexagons is in the matching. Afterwards, we can always add four hexagons at a price of five new edges in a maximal matching (such that the edge connecting that block with the next block is in the matching). Let l be the number of blocks in B2k+3 k. We consider two cases. • If k is odd, then 4 | ((2k + 3) - 1) and we have to add one additional hexagon (for this hexagon we need one additional edge). Hence, l = = . We obtain |M| = 6 + 5(l - 1) + 1 = 7 + 5 • k-1 = ^kt9. • If k is even, then 4 | ((2k + 3) - 3) and we have to add three additional hexagons (for this three hexagons we need four additional edges in a maximal matching). Hence, l = = §. Therefore, we get |M| = 6 + 5(l - 1) + 4 = 10 + 5 • ^ = ^kt-10. Combining both cases, we obtain |M| = 1 (10k + 19 + (-1)k). □ 4.2 Coronenes In this section we prove bounds for the saturation number of coronenes. These highly symmetric benzenoid systems have long been attracting the attention of both theoretical and experimental chemists. They are suggested as markers for vehicle emissions, since they are produced by incomplete combustion of organic matter. Coronene Hi is just a single hexagon, and Hk is obtained from Hk-i by adding a ring of hexagons around it. See Figure 13 for an example of coronene H4. Proposition 4.10. Let Hk be a coronene. Then 2k2, 3 | (k - 1) 3k2 < s(Hk) <{ 2k2 + f, 3 | k 2k2 + 2k+2, 3 | (k - 2). Proof. Obviously, every coronene has a perfect matching. Since the number of vertices in Hk is 6k2, it follows by Lemma 2.1 that s(Hk) > §k2. For the upper bound, we will consider just the case when 3 | (k -1), since the proofs for other two cases are almost the same. To prove this case, we construct a maximal matching M for Hk. In the matching we put all the vertical edges lying in the center layer of the coronene Hk. Since there are 2k - 1 hexagons in the center layer, we obtain 2k edges in the matching M. See Figure 13. Next, we continue at the top half of the coronene with alternating non-vertical and vertical edges such that two layers of edges are needed Figure 13: A coronene H4. for every three layers of hexagons. Furthermore, for every non-vertical layer of edges we need one additional vertical edge. Let x be the number of edges in M in the top half of the coronene. Then x = (2k - 1) + (2k - 3) +(2k - 4) +(2k - 6) + ••• + (k + 3) + (k + 1) = = 2(k ~ 1) • (2k) - (1 + 3 + 4 + 6 + • • • + (k - 3) + (k - 1)) = 4k2 - 4k (k2 - k ( ,, = -3--[-J- - (2 + 5+ ••• + (k - 2))J = _ 4k2 - 4k ( k2 - k k2 - k = 3 V 2 6 j = = k2 - k. Finally, we obtain |M| = 2k + 2x = 2k + 2(k2 - k) = 2k2. □ In the next proposition we improve the lower bound for any k > 7. Proposition 4.11. Let Hk be a coronene where k > 1. Then s(Hk) > 2k2 - 3k - 1. Proof. For any Hk, k > 2, one can construct a disk-shaped fullerene by taking another copy of Hk and connecting the borders in the following way. We insert 6k edges between vertices of degree 2 such that end-vertices lie in different copies of Hk. Obviously, this can be done in such a way that the resulting graph F is planar with only pentagonal and hexagonal faces. Since F is also 3-regular, it is a fullerene with 12k2 vertices. Let M' be a maximal matching in each copy of Hk. Then this matching can be extended to a maximal matching M of a graph F by adding at most 6k edges between two copies of Hk. Therefore, |M| < 2|M'| + 6k. From Theorem 4.1 in [2] it follows |M| > J-^f^-2 = 4k2 - 2. Therefore, we obtain 2|M+ 6k > 4k2 - 2. Finally, |M> 2k2 - 3k - 1. □ Concluding remarks In the paper we have established some bounds and also exact values for the saturation number of certain families of lattice animals. However, there are still many open problems regarding the exact values for the saturation number of different families of graphs. References [1] V. Andova, T. Doslic, M. Krnc, B. Luzar and R. Skrekovski, On the diameter and some related invariants of fullerene graphs, MATCH Commun. Math. Comput. Chem. 68 (2012), 109-130, http://match.pmf.kg.ac.rs/electronic_versions/Match68/n1/ match68n1_10 9-130.pdf. [2] V. Andova, F. Kardoss and R. Sskrekovski, Sandwiching saturation number of fullerene graphs, MATCH Commun. Math. Comput. Chem. 73 (2015), 501-518, http://match.pmf.kg. ac.rs/electronic_versions/Match73/n2/match73n2_501-518.pdf. [3] A. Behmaram, T. Doslic and S. Friedland, Matchings in m-generalized fullerene graphs, Ars Math. Contemp. 11 (2016), 301-313, doi:10.26493/1855-3974.882.539. [4] T. Doslic, Saturation number of fullerene graphs, J. Math. Chem. 43 (2008), 647-657, doi: 10.1007/s10910-006-9217-3. [5] T. Doslic and I. Zubac, Saturation number of benzenoid graphs, MATCH Commun. Math. Comput. Chem. 73 (2015), 491-500, http://match.pmf.kg.ac.rs/electronic_ versions/Match73/n2/match73n2_4 91-500.pdf. [6] T. Doslic and I. Zubac, Counting maximal matchings in linear polymers, Ars Math. Contemp. 11 (2016), 255-276, doi:10.26493/1855-3974.851.167. [7] N. Tratnik and P. Zigert Pletersek, Saturation number of nanotubes, Ars Math. Contemp. 12 (2017), 337-350, doi:10.26493/1855-3974.1056.ae9. [8] M. Zito, Small maximal matchings in random graphs, Theor. Comput. Sci. 297 (2003), 487-507, doi:10.1016/s0304-3975(02)00653-9. [9] M. A. A. Zito, Randomised Techniques in Combinatorial Algorithmics, Ph.D. thesis, University of Warwick, Warwick, 1999, http://www.dcs.warwick.ac.uk/report/pdfs/ cs-rr-369.pdf. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 205-223 https://doi.org/10.26493/1855-3974.1419.3e9 (Also available at http://amc-journal.eu) Enumerating regular graph coverings whose covering transformation groups are Z2-extensions of a cyclic group* Jian-Bing Liu f Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R. China Jaeun Lee Mathematics, Yeungnam University, Kyongsan, 38541 Korea Jin Ho Kwak Mathematics, POSTECH, Pohang, 37673 Korea Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R. China Received 9 June 2017, accepted 12 October 2017, published online 20 June 2018 Several types of the isomorphism classes of graph coverings have been enumerated by many authors. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by the second and third authors. For regular coverings of a graph, their isomorphism classes were enumerated when the covering transformation group is a finite abelian or dihedral group. In this paper, we enumerate the isomorphism classes of graph coverings when the covering transformation group is a z2-extension of a cyclic group, including generalized quaternion and semi-dihedral groups. Keywords: Graphs, regular coverings, voltage assignments, enumeration, Möbius functions (on a lattice), group extensions. Math. Subj. Class.: 05C30, 20F28, 20K27 *The authors are grateful to anonymous referees for their valuable comments. The authors also would like to thank Young-Soo Kwon for illuminating discussions and remarks. This work was partially supported by the National Natural Science Foundation of China (11671030). t Corresponding author E-mail addresses: jl0068@mix.wvu.edu (Jian-Bing Liu), julee@ynu.ac.kr (Jaeun Lee), jinkwak@postech.ac.kr (Jin Ho Kwak) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/3.0/ 1 Introduction Throughout this paper, all graphs and groups are assumed to be finite. Let G be a connected simple graph with vertex set V(G) and edge set E(G). The neighborhood of a vertex v € V(G), denoted by N(v), is the set of vertices adjacent to v. We use |X| for the cardinality of a set X. The number ft = |E(G)| - |V(G)| + 1 is equal to the number of independent cycles in G and it is referred to as the Betti number of G. Two graphs G and H are isomorphic if there exists a one-to-one correspondence between their vertex sets which preserves adjacency, and such a correspondence is called an isomorphism between G and H. An automorphism of a graph G is an isomorphism of G onto itself. Thus, an automorphism of G is a permutation of the vertex set V(G) which preserves adjacency. Obviously, the automorphisms of G form a permutation group, Aut(G), under composition, which acts on the vertex set V (G). A graph G is called a covering of G with projection p: G ^ G if there is a surjection p: V(G) ^ V(G) such that p|N: N(V) ^ N(v) is a bijection for any vertex v € V(G) and 5 € p-1 (v). Also, we sometimes say that the projection p: G ^ G is a covering, and an n-fold covering if p is n-to-one. A covering p: G ^ G is said tobe regular (simply, A-covering) if there is a subgroup A of the automorphism group Aut(G) of G acting freely on G so that the graph G is isomorphic to the quotient graph G/A, say by h, and the quotient map G ^ G/A is the composition h o p of p and h. The fiber of an edge or a vertex is its preimage under p. Two coverings^: Gj ^ G, i = 1, 2, are isomorphic if there exists a graph isomorphism $: G1 ^ G2 such that p2 o $ = p1, that is, the diagram Gi ---► G2 G commutes. Such a $ is called a covering isomorphism. A covering transformation is just a covering automorphism. Every edge of a graph G gives rise to a pair of oppositely directed edges. By e-1 = vu, we mean the reverse edge to a directed edge e = uv. We denote the set of directed edges of G by D(G). Let A be a finite group. An ordinary voltage assignment (or, A-voltage assignment) of G is a function ^: D(G) ^ A with the property that ^(e-1) = ^(e)-1 for each e € D (G). The values of ^ are called voltages, and A is called the voltage group. The ordinary derived graph G x $ A derived from an ordinary voltage assignment ^: D(G) ^ A has as its vertex set V(G) x A, and as its edge set E(G) x A, so that an edge (e, g) of G x$ A joins a vertex (u, g) to (v,^(e)g) for e = uv € D(G) and g € A. In the (ordinary) derived graph G x$ A, a vertex (u, g) is denoted by ug and an edge (e, g) is denoted by eg. The first coordinate projection p$: G x$ A ^ G commutes with the left multiplication action of the ^(e) and the right multiplication action of A on the fibers, which is free and transitive, so that p$ is a regular |A|-fold covering, called simply an A-covering. Moreover, if the covering graph G x $ A is connected, then the group A becomes the covering transformation group of the A-covering. For a group A, let C1 (G; A) denote the set of A-voltage assignments ^ of G. Choose a spanning tree T of G, and let C1 (G; A) = € C 1(G; A) : ^(uv) is the identity for each uv € D(T)}. Gross and Tucker [4] showed that every A-covering G of a graph G can be derived from an A-voltage assignment ^ in C^ (G; A), say it T-reduced. From now on, let T denote a fixed spanning tree of a graph G, and we consider only an A-voltage assignment ^ in C^(G; A). The enumeration problem of coverings became subject of investigation by many authors starting from the classical paper by Hurwitz published more then 100 years ago. In particular, enumeration of graph coverings became possible after the paper by Hall ([6]) published in 1949. In 1988, Hofmeister [8] counted double covers of graphs. Liskovets enumerated connected non-isomorphic coverings of the graph with a given Betti number, see [19, 20]. The number of connected and disconnected coverings were determined by Kwak and Lee in [15]. Later, Kwak, Lee and A. D. Mednykh counted cyclic and dihedral coverings over surfaces and graphs with prescribed topological characteristics, see [16,17]. Following notations in [14], let IsoR(G; n) denote the number of the isomorphism classes of regular (connected or disconnected) n-fold coverings of G, and use IsocR (G; n) for their connected ones. Similarly, let Iso(G; A) denote the number of the isomorphism classes of (connected or disconnected) A-coverings of G, and use Isoc(G; A) for their connected ones. By the properties of regularity of coverings, one can see that the number of the isomorphism classes of (connected or disconnected) n-fold regular coverings of a graph G is the sum of numbers of the isomorphism classes of connected d-fold regular coverings of G, where d runs over all divisors of n: IsoR(G; n) = ^ IsocR(G; d). d\n Moreover, the number of the isomorphism classes of connected n-fold regular coverings of G is the sum of the numbers of the isomorphism classes of connected A-coverings of G, where A runs over all non-isomorphic groups of order n: IsocR(G; n) = ^ Isoc(G; A). A Consequently, it just needs to determine the numbers Isoc(G; A) for every finite group A. Hong, Kwak and Lee [9] obtained an algebraic characterization of two isomorphic graph regular coverings given as follows. Lemma 1.1. Let ^ G C1 (G; A) and ^ G C1 (G; B) be any two ordinary voltage assignments in G. If their derived (regular) coverings p 4. Then A is isomorphic to one of following six groups. (1) (the cyclic group) z2n = (b | a2"-1 = 1,b2 = a), (2) (the non-cyclic abelian group) Z2n-i x z2 = (a,b | a?n = 1, b2 = 1, b-lab = a), (3) (the dihedral group) d2n = (a,b | a2"-1 = 1, b2 = 1, b-lab = a-1), (4) (the generalized quaternion group) q2n = (a,b | a2"-1 = 1, b2 = a2"-2 ,b-1 ab = a-1), (5) (the ordinary metacyclic group) m2" = (a,b | a2"-1 = 1, b2 = 1, b-1ab = a1+2"-2), (6) (the semidihedral group) sd2n = (a,b | a2"-1 = 1, b2 = 1, b-1 ab = a-1+2"-2). All the six groups are not isomorphic one another. Proof. Since (1) and (2) are trivial cases, we assume that A is not abelian. By Lemma 2.1, A has the following presentation: A = (a,b | a2"-1 = 1, b2 = a*, b-1ab = ar), where t and r satisfy r2 = 1 (mod 2n-1), t(r - 1) = 0 (mod 2n-1). By A non-abelian, one has r = —1 or ±1 + 2n-2 (mod 2n-1), the latter two cases can happen only when n > 4. If r = —1 or — 1 + 2n-2 (mod 2n-1), then 2n-1 | 2t and hence 2n-2 | t, it follows that t = 0 or 2n-2 (mod 2n-1). Now we consider the three cases separately. (i) r = — 1 (mod 2n-1). In this case we get the dihedral group (3) and the generalized quaternion group (4) depending on t = 0 or 2n-2 (mod 2n-1), respectively. These two groups are not isomorphic. Note that the following cases (ii) and (iii) happen only when n > 4. So, when n = 3 we have only the above two groups. (ii) r = — 1 + 2n-2 (mod 2n-1). In this case t = 0 (mod 2n-2). Thus b2 = 1 or a2" .If b2 = a2" , letting b1 = ba, then 1.2/1. \2 7,2/7-1 IN 7.2 -1+2"-2 2"-2 2"-2 -i b2 = (ba)2 = b2(b 1 ab)a = b2a 1+2 a = a2 a2 =1. Thus we get the group (6). (iii) r = 1 + 2n-2 (mod 2n-1). In this case, one has t • 2n-2 = 0 (mod 2n-1) which implies that t is even. Let t = 2s. Since n > 4, there is a j satisfying j(1 + 2n-3) + s = 0 (mod 2n-2). Let b1 = baj. Then b2 = b2(b-1aj b)aj = b2aj(2+2"-2) = a2(j(1+2"-3)+s) = 1. Now the generators a, b1 satisfy the relations in the group (5), with b instead of b1. Finally, we shall show that the mentioned four non-abelian groups are not isomorphic, and we assume that n > 4. It is easy to see that in these four cases the derived group A' = ([a, bj). We calculate the commutator [a, b] and get {a-2 for the groups (3) and (4), a2" for the group (5), a-2+2" 2 for the group (6). So, one has |A'| =2 for (5), and |A'| = 2n-2 for the others. It follows that the group (5) is not isomorphic to any one of the rest. To prove the rest three groups are not isomorphic, we calculate the square of the elements of the form ba® outside (a). We have {1 for the group (3), a2"-2 for the group (4), a®2" 2 for the group (6). This shows that the subgroup of order 2n-1 in A is unique, and outside this subgroup (a), all elements are of order 2 in the group (3), order 4 in the group (4), and some are of order 2 and the others are of order 4 in the group (6). Therefore, all the four groups are not isomorphic to one another. □ Let A be a z2-extension of a cyclic group zn, where n = p^0Pii • • • P?s is the prime decomposition with p0 = 2. First, we consider the case that n is odd, that is, a0 = 0. Theorem 2.4. Let A be a Z2-extension of a cyclic group zn = zp^i x • • • x zpa with n odd. Then, A has a presentation A = (ai, .. ., as, b | aPi = b2 = 1, [aj, aj] = 1, b-1ajb = a^ for all i, j), where r2 = 1 (mod pa) for all i. There are 2s non-isomorphic such extended groups. Proof. By Lemma 2.2, A is split. Since A is a metacyclic group, by Lemma 2.1, A has the presentation A = (a, b | an = b2 = 1, b-1ab = ar), with r2 = 1 (mod n). The action of b on each element of zn by conjugacy is an automorphism of zn of order at most 2. Since Aut(zn) = Aut(zpai) x • • • x Aut(zpj=), the b-conjugation on zn corresponds to an s-tuple (r1,..., rs) with r = ±1 (mod ) for i € {1,..., s}. Thus the s-tuple (r1,..., rs) has 2s choices and A is presented by A = (a1,...,as,b | ap = b = 1, [ai,aj] = 1,b ajb = a^ for all i,j). To finish the proof, it suffices to show that different s-tuples (r1,..., rs) give non-isomor-phic groups. It is easy to see that zpa is a subgroup of the center of A if and only if r = 1. Hence the groups with different s-tuples (r 1,..., rs) have different center of A. Therefore, there are 2s non-isomorphic z2-extensions of zn. □ Next we consider the case of even n. Let A be a z2-extension of a cyclic group zn = zpao x zpai x • • • x zpj= with p0 = 2. We deal with three cases a0 = 1,2 or a0 > 3 in the next theorem. First we determine the Sylow 2-subgroup S0 of A which is a z2-extension of z2ao = (a0). This has been done by Theorem 2.3. Namely, S0 = (a0, b0 | a0 0 = 1, b§ = a0°, b-1 a0b0 = a0°), where t0 = 0,1 or 2a-1, r0 = ±1 or ±1 + 2a°-1 depending on the types of S0 in Theorem 2.3. Next, take b = b0. Thus each Sylow 2-subgroup and each element of order at most 2 in Aut(zp«i) x • • • x Aut(zp?s) gives a unique z2-extension of zn. Theorem 2.5. Let A be a Z2-extension of a cyclic group zn = zpa° x zpai x • • • x zpa with p0 = 2. (1) If a0 = 1, then A has the following presentations (i) A = (a0, a1, .. ., as, b | aPi = b2 = 1, [aj, aj] = 1, b 1ajb = a^ for all i, j) (S0 = z2 x z2). (ii) A = (a0, a1, ..., as, b | aPi = 1, b2 = a0, [aj, aj] = 1, b-1ajb = a^ for all i, j), (S0 = z4). There are 2s+1 non-isomorphic groups. (2) If a0 = 2, then A has the following presentations (i) A = (ao, ai, ... ,as,b | aPi = b2 = 1, [ai ,aj ] = 1, b 1ajb = ai for all i, j), (S0 = z4 x z2 or d8). (ii) A = (ao, ai,. .., as, b | aPi = 1, b2 = ao, [aj, aj] = 1, b-1ajb = a^for all i,j), (So = z8). (iii) A = (ao, ai,. .., as, b | aPi = 1, b2 = a0, [aj, aj] = 1, b-1ajb = a^for all i, j), (So = There are 2s+1 non-isomorphic groups. (3) If ao > 3, then A has the following presentations (i) A = (ao, ai, ... ,as,b | aPi = b2 = 1, [aj, aj ] = 1, b 1ajb = a^ for all i, j), (So = z2ao X z2, d2a0 + 1 , sd2a0 + 1 , or m2a0 + 1 ). (ii) A = (ao, ai,. .., as, b | aPi = 1, b2 = ao, [aj, aj] = 1, b-1ajb = arii for all i,j), (So = z2ao + 1 ). ai a — 1 (iii) A = (ao, ai ,...,as,b | ap = 1, b2 = ao 0 , [aj,aj- ] = 1,b-1ajb = ar^i for all i,j), (So = q2"0+1). There are 6 • 2s non-isomorphic groups. For each extension group A appeared so far, the number Isoc(G; A) shall be determined in the next section. 3 In cases of Z2-extensions of a cyclic p-group For each group A in the classification of z2 -extensions of a cyclic p-group listed in the previous section, we aim to determine the number Isoc(G; A) in this section. However, for an abelian or a dihedral group A, it has already been done in [14]. Hence, we need to do it only for each group A listed in the last three cases of Theorem 2.3. For a z2-extension A of a finite group H, we call an element x normal type if x e H and quotient type otherwise. Note that H is normal in A, and a product of any two normal type elements is normal type. For any two quotient type elements ab, a'b, their product is aba'b = ab2b-1a'b, and hence a product of any two quotient type elements is normal type. A word in {x1,..., xs} is any expression of the form yj1 • • • yj,fc where y1,... ,yk e {x1,... ,xs} and i1,... ,ik e {1, -1}, denoted by w(x1,..., xs). The number k is known as the length of the word. When writing words, it is common to use exponential notation as an abbreviation. Lemma 3.1. Let A be a Z2-extension of a finite group H. For a subset I of S = {1, ...,P}, let Qz (A; ft) = {(x1,..., xp) e 0(A; ft) : xj is quotient type for exactly indices i e I}. Then |Q(A;ft)| = (2p - 1)|Q{i}(A; ft)|. Proof. Recall that Q(A; ft) = {(x i,..., x p) e Ap : (xi,...,xp) = A}. For each tuple (xi,..., x^) g O(A; P), at least one of entries xj should be quotient type to generate the whole group A. Then, Q(A; P) = y Ox (A; P), disjoint union, 0=ICS and |^(A; P)| = £ (A; P)|. 0=ICS For any non-empty subset I of S, choose an index j0 G I and define a map ^: (A; P) ^ n{j0}(A; P) by replacing all quotient type entries xj for i G I by xj0xj except xj0. Then one can see that ^ is well-defined and bijective. It follows (A; P) | = (2^ - 1)|O{j0}(A; P)|. One can assume that j0 = 1 for convenience. □ Lemma 3.2. Let A be a z2-extension of a finite group H. If each xj is normal type except xi, then (x1,..., x^} = A if and only if (xi, x2,...,x^, x-1x2x1,..., x-1x^ x1) = H. Proof. Assume (x2, x2,..., x^, x-1 x2x1,... x-1x^x1) = H and each xj is normal type except x1. Then (x1,..., x^} = (x1, x2, x2,..., x^, x-1x2x1,... x-1x^x1) = (x1, H) = A. Now assume that (x1,..., x^} = A. For any g G A, g can be expressed by a word w(x1,..., x^). For odd k, xjxf = x1 • (x-1xjx1) • (x1)(k-1)/2 and for even k, xjxk = xj • (x^/2). Rewrite g, one has g = w(x1,..., x^) = x1w(x2, x2,..., x^, x-1x2x1,..., x-1x^x1), i = 0,1. It follows that g is normal type if and only if i = 0. Therefore, ( x2,x2,...,x^ , x- x2x1, ...,x-1x^ x1} = H. □ Corollary 3.3. Let A be a Z2-extension of a cyclic group Zn. If each xj is a normal type element except x1, then (x1,..., x^} = A if and only if (x2, x2,..., x^} = zn. We determine |O(A; P) | and | Aut(A) | for each group A listed in the last three cases of Theorem 2.3 in the following. Lemma 3.4. Let A be a Z2-extension of a cyclic group z2n-i and let A be non-abelian. Then |^(A; P)| = 2("-2)^+1(2^ - 1)(2^-1 - 1). Proof. By Lemma 3.1, it just needs to determine ^{1}(A;P). By Corollary 3.3, (x2, x2,..., xs} = z2n-i if and only if (x1,..., xs) G ^{1}(A; P). By the last three cases of Theorem 2.5, one can assume b2 = a4 with t = 0 or 2n-2 for a generator a of z2n-i. Note that x1 is quotient type, say x1 = baj. Then x2 = b2 • b-1ajb • aj = b2aj(1+r) = at+j(1+r) with r G {-1, ±1 + 2n-2}. Suppose x1 generates z2n-i. Then t + i(1 + r) = 1 (mod 2). But it is impossible by checking case by case. So (x2,... ,xs} = z2n-i. By |O(z2n-i; P - 1)| = 2(n-2)(^-1)(2^-1 - 1), which was shown by Kwak et al. in [14], it follows |^(A;P)| = (2^- 1)2n-1 |^(z2n-i;p-1)| = 2("-2)^+1(2^-1)(2^-1 -1). □ Lemma 3.5. For n > 4, (1) |Aut(z2n)| =2n-1, (2) |Aut(z2n-i x z2)| = 2n, .2n-3 2n-3 (3) |Aut(d2n )| = 2' (4) |Aut(q2n )| = 2' (5) |Aut(m2n )| = 2n, (6) |Aut(sd2n )| = 22n-4. Proof. Since the first three cases have been shown in [14], we only need to show the last three cases. To do this separately, let A be a z2 -extension of a cyclic group z2n-1 and let an automorphism a G Aut(A) be of the form a ^ aibk ,b ^ a? b£ with 0 < i, j < 2n-1 - 1 and 0 < k,i < 1. (4) Since the identity (aib)2 = bb-1albalb = b2 gives the orders o(aib) = 4 and o(a) = 2n-1 = 4 for n > 4, the image a(a4) should be of the form a1 with (i, 2n-1) = 1. The surjectivity of a implies that the choices of a(b) are a? b with j = 0,..., 2n-1 -1. Moreover, all of such possible choices a(a) and a(b) satisfy the defining relations of q2n. Hence |Aut(q2n)| = 22n-3 by counting the choices of a(a) and a(b), that is, the choices of i, j, k, i. (5) If k = 0, then a(a) = a1 for some i with (i, 2n-1) = 1. If k = 1, then a(a) = alb for some i with (i, 2n-1) = 1, because the order preserving condition says o(a®b) = o(a) = 2n-1, and (aib)m = bmai(1+ •+rm-1) for all m > 1, where r = 1 + 2n-2. Next, we determine the possible values of a(b). If i = 0, then j should be 2n-2. In this case, all possible values a(a) and a(b) do not satisfy the defining relations of m2n. Thus it should be i = 1. Now the order condition o(akb) = o(b) = 2 implies j = 2n-2 or 0. Consequently, a has four different forms. (i) a ^ ai, b ^ b, (ii) a ^ ai, b ^ a2" b, (iii) a ^ aib, b ^ b, (iv) a ^ aib,b ^ a2" b. In these four cases, a(a) and a(b) satisfy the defining relations of m2n. Therefore, the four different cases give | Aut(m2n) | = 2n. (6) Since (aib)2 = ai2" 2, one gets o(aib) = 2 for even i and o(aib) = 4 for odd i. Hence a should be of the form a ^ ai, b ^ a?b with (i, 2n-1) = 1 and j even. Moreover, all such possible values a(a) and a(b) satisfy the defining relations of sd2n .So |Aut(sd2n )| = 22n-4. □ As a special case, |Aut(q8)| = 24 which is not included in the above lemma. From Theorem 1.2 and Lemmas 3.4 and 3.5, one can get the following theorem. Theorem 3.6. For a Z2-extension of A a cyclic group z2n-i for n > 2, (2(^-1)(n-1)(2^ - 1) if A is z2" , 2(^-2)(n-2) + (n-3)(2^ - 1)(2^-1 - 1) if A is z2"-1 X z2, 2(^-2)(n-2)(2^ - 1)(2^-1 - 1) if A is d2" for n > 3, Isoc(G; A) = <( 2(^-2)(2^ - 1)(2^-1 - 1)/3 if A is 4, 2(^-1)(n-2)-1(2^ - 1)(2^-1 - 1) if A is m2n for n > 4, ^2(^-2)(n-2)+1(2^ - 1)(2^-1 - 1) if A is sd2" for n > 4, where the first three cases were shown in [14]. By using the Mobius function, Isoc(G; A) can also be determined. For example, for a generalized quaternion group q2n, a proper subgroup S of q2n is isomorphic to z2m or q2m, where z2m = (a2" m 1} and = (a2" m, a®6) for m € {1,..., n - 1} and i € {0,..., 2n-m - 1}. From the subgroups lattice of q2n, see Figure 1, one has 1 ifS = q2n , -1 if S = Z2»-i, Q^n-! or , .......... . q(°) )= , ' 2 if S = z2n-2 0 otherwise. Q2n Z q40) Q« Q42) ' ■ ■ ■ Qi2"-2-2) Qi2"-2-1) 1 Figure 1: The subgroup lattice of Q2n It follows from Theorem 1.3 Isoc(G; Q2n) = > 3 l (23^-3 - 3 • 22^-3 + 2^-2) which coincides with the formula given in Theorem 3.6. if n = 3, if n > 3, If A = m2n, then every proper subgroup S of m2n is isomorphic to zm or m2' (i) for m G {2,..., n — 1} and i G {0,1}, where Z2m = (c and m. (1) (1) l>, m2m = ( (a2 „n-2 b). If m = 1, then S is isomorphic to Z2 (0) (an-2> or z2 ; = (an 2b). Now from the subgroups lattice of m2n illustrated in Figure 2 and |Aut(m2n 2n, one can have Isoc(G;M2n) = — (2n^ — 3 • 2(n-1)^ + 2(n-2)^+1), 2 a a Zo Z2 M. M 2n-2 M 2"-2 Z4 M (0) M (1) Figure 2: The subgroup lattice of M2n. which coincides exactly with the result in Theorem 3.6. Also, by using the Mobius function, one can show that Isoc(G; SD2n) = - 3 • 2(n-1^ + 2(n-2^+1). 22n 4 For some small ft and n, the numbers Isoc(G; A) are tabulated in Table 1. Table 1: The number Isoc for small ft and n. (ft, n) Wj2n z^n-1 X z2 d2n Isoc q2n m2n sd2n A (2, 3) 12 3 3 1 0 0 22 (2, 4) 24 6 3 6 24 6 69 (2, 4) 48 12 3 6 48 6 123 (3, 3) 112 42 42 56 0 0 252 (3, 4) 448 168 84 168 672 168 1708 (3, 5) 1792 672 168 336 2688 336 5992 (4, 3) 960 420 420 560 0 0 2360 (4, 4) 7680 3360 1680 3360 13440 3360 32880 (4, 5) 61440 26880 6720 13440 107520 13440 229440 2 n n-1 1 4 In cases of Z2-extensions of any cyclic groups In this section we determine Isoc(G; A) for a z2-extension A of a cyclic group zn (of any order n, not necessarily to be a p-group). Again, let A be a z2-extension of a cyclic group zn = zpao x zpai x • • • x zp?= and let n = p^0Pi1 • • • P?s be the prime decomposition with p0 = 2. Let the b-conjugation on zn correspond to an (s + 1)-tuple (r0, ri,..., rs), where r0 G {±1, ±1 + 2ao-1} and r = ±1 for i G {1,..., s} with —1 in exactly t entries ¿4,..., ¿t. Let n = 2ao n1n2 with n1 = nj=1 P^3. Then A is isomorphic to Bx zn2 where B is a z2-extension of zni since any element of zn2 commutes with each element of A. Since (|B|, |zni |) = 1, one has Isoc(G; B x zni) = Isoc(G; B) • Isoc(G; zni), as shown in [14]. Because Isoc(G; zni) has already been determined, we just need to determine Isoc(G; B). Lemma 4.1. Let B be a Z2-extension of a cyclic group Zn = zp^o x • • • x zpa with p0 = 2 and s > 1, and let n = 2ao m. Let the b-conjugation on zn correspond to an (s + 1)-tuple (r0, r1,..., rs), where r0 G {±1, ±1 + 2ao-1} and all other ri's are —1. Then MB' P)l= — 1)m2ao^lq(zm; P — 1)1 if 2ao+1 I o(b), 1 ( ; P)I \(2p — 1)m2ao |^(z2^om; P — 1)| otherwise. Proof. By Theorem 2.5, one can assume b2 = a0 with t G {0,1, 2ao-1}. By Lemma 3.1, it just needs to determine |Q{1}(B; P)|. Take (x1,..., xp) G Q{1}(A; P). Note that x1 is a quotient type element and other Xj s are all normal type. Since zn = zpao x zpai x • • • x Z„«s , po pi ps any element of zn can be presented gjhj with gj G zp^o and h G f]f=1 zpa. So x1 can be presented by g1h1b and other xj's can be preseioted by gjhj. By Corollary 3.3, (x1,..., xp} = B if and only if (x1, x2,..., xp} = zn. By x2 = (g1 h1b)2 = b2g1+r°, one has (b2g1+ro ,g2h2,... ,gp hp} = zn = zp«o x^ • •xz^s. Recall that b2 = a0 G zp«o withp0 = 2, then (b2g1+ro, g2,..., gp} = zp«o and (h2,..., hp} = zp«i x • • • x zpj=. So (b2g1+ro, g2, . . . , gp) G n(z2^o; P), (h2,...,hp) G ^(zp^i x •••ix zp?s; p — 1). To count the choice of (x1,..., xp), equivalently to count the number of (g1,..., gp) and (h1,..., hp). When computing x2 = b2g1+ro, h1 can be any element of p=1 zpa, and it follows h1 has m choices by m = f]p=1 p"i. The number of choices of (h2,..., hp) is equal to |0(zm; p — 1)|. Hence number of choices of (h1,..., hp) is m|^(zm; p — 1)|. Now we determine the number of choices of (g1,..., gp) in the following. Assume that 2ao+1 | o(b) and it follows t = 1. Then the Sylow 2-subgroup of B is z2ao+i. By Theorem 2.5, (b2} = (ao} = z2«o and ro = 1. Since g1 G z2 ao, one has (b2g1+ro} = (b2g2} = (b2} = z2ao. By (b2g1+ro,g2,...,gp} = (b2} = z2^o, (g1,.. .,gp) has 2aop choices. So |fy1}(B; p)| = 2ao p m|^(zm; p — 1)|, and it follows |fi(B; P)| = (2p — 1)m2aop|fi(zm; P — 1)|. If 2ao+1 does not divide o(b), then, by Theorem 2.5, b2 = a0 with t G {0, 2ao-1}. If t = 0, then b2 = 1 and r0 G {±1, ±1 + 2ao-1}. It follows b2g1+ro = 1, g2, g2ao_i or g2+2ao~i. So g1+ro can not be the generator of z2ao .If t = 2ao-1, then r0 = —1. Then b2g 1+ro = a2ao , again, b2g1+ro can not generate z2ao. So (b2g1+ro,g2,...,gp} = z2ao if and only if (g2,..., gp} = z2ao. It follows (g2,..., gp) G ^(z2ao; P — 1) and g1 is any element in z2«o. Then (g^ ..., gg) has 2ao |n(z2«o; P —1)| choices. So |Q(B; P)| = (2g — 1)m2ao |n(z2a,m; P — 1)|. □ Lemma 4.2. Let B be a Z2-extension of a cyclic group zn = zp°=o x zpai x • • • x zpa with p0 = 2 and s > 1, and let n = 2ao m. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, ri,..., rs), where r0 G {±1, ±1 + 2ao-1} and all other r's are —1. (1) If r0 = 1, then |Aut(B)| ^2aom^(n) f 2ao+1 | o(b), l2my>(n) otherwise. (2) If r0 = —1, then |Aut(B)| = 2ao my(n). (3) If r0 = 1 + 2ao-i, then |Aut(B)| = 2my(n). (4) If r0 = —1 + 2ao-i, then |Aut(B)| = 2a°-1m^(n). Proof. Again, one can assume that b2 = a0 with t G {0,1,2ao-1}. For an automorphism a of B, a(ak) should be of the form a^ with (ik,pak) = 1 for k G {1,..., s} since a is order-preserving. Suppose that a(a0) is quotient type, then b-1akb = ak since a(a0) commutes with a(ak) for each k. So r = 1 for each i G {1,..., s}, which is a contradiction. Then a(a0) is normal type, say a(a0) = a0o with (i0,2ao) = 1. Assume a(b) = aUoa^1 • • • aUUsb. We need to count the number of choices of u0,..., ug. By computing, (a^a?1 • • • a"= b)2 = b2auo(1+ro) • • • auf(1+r^) = b2auo(1+ro). Note that o(a(b)) = o(b) and o(b) is even. By hypothesis, r1 = • • • = rg = —1, and it follows (a(b))2 = b2aUo(1+ro). Then u can be any element of zfor i G {1,..., s}. Now it needs to determine the number of choices of u0. * (1) If r0 = 1 and o(b) = 2ao+1, then (a(b))2 = b2a0Mo. By Theorem 2.5, b2 = a0 in this case. Then o(b2a2wo) = o(b2) = 2ao, and it follows u0 has 2ao choices. Hence |Aut(B)| = 2aomy(n). If r0 = 1 and o(b) = 2, then u0 can be 0 or 2ao-1. So |Aut(B)| = 2m^(n). (2) If r0 = —1, then (a(b))2 = b2 and o(b) is 2 or 4, by Theorem 2.5. So u0 can be any element of z2ao and has 2ao choices. It follows |Aut(B)| = 2aom^(n). (3) If r0 = 1 + 2«o-1, then o(b) = 2. So (a(b))2 = b^2^^ = a^2^^ = 1. If follows u0(2 + 2ao-1) = 0 (mod 2ao). Then u0 has 2 choices: 0 or 2ao-1. Hence |Aut(B)| = 2my(n). (4) If r0 = —1 + 2ao-1,then o(b) = 2. So (a(b))2 = ago2ao-1 = 1. Then u02ao-1 = 0 (mod 2ao), and it follows u0 has 2aR0-1 choices. Hence |Aut(B)| = 2ao-1m^(n). □ The next lemma follows from Theorem 1.2 and Lemmas 4.1 and 4.2. Lemma 4.3. Let B be a Z2-extension of a cyclic group zn = zp^o x zp^1 x • • • x z and let n = paop^1 • • • be the prime decomposition with p0 = 2. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, r1,..., rs), where r0 G {±1, ±1 + 2ao-1} and all other r,'sare — 1. (1) If ro = 1, Isoc(G; B) = i (2f - 1)2aof-ao ^ p(ai-1)(f-1)(pf-1 - 1) if 2ao+1 | 0(b), i=i i (2f - i)2°o-1 n p(ai-1)(f-1)(pf-1 -1) otherwise. i=0 (2) If ro = -1, then Isoc(G; B) = — (2f - 1) n (*i-1)(p-1)( f-1 (pf-1 - 1). (3) If ro = 1 + 2ao-1, then Isoc(G; B) = ^2«o-1(2f - 1) np(ai-1)(f-1)(pf-1 - 1). (4) If r0 = -1 + 2ao-1, then Isoc(G; B) tp(n) 2(2 f -1) n p (ai-1)(f-1), f-1 (Pf-1 - 1). i=o Now one can get main theorem of this section. Theorem 4.4. Let A be a Z2-extension of a cyclic group Zn = Zp°=o x Zpai x • • • x Zpa and let n = pap^1 • • • be the prime decomposition with p0 = 2. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, r1,..., rs), where r0 G {±1, ±1 + 2ao-1} and ri = ±1 for i G {1,..., s} with -1 in exactly t entries £1,..., £t. Let J = {^1,..., £t}, K = {1,..., s} - J and N 1 ^(n) (2 f -1) n p(ai-1)(f-1)(pf-1 -1) n p(fi-1)f (pf -1). ieJ ie/c Then Isoc(G; A) = TN, where 2«o f-ao 2(ao-1)f (2f-1 - 1) T = { 2(ao-1)( f-1)(2f-1 - 1) 2(ao-1)f (2f-1 - 1) if ro = 1 and 2a°+1 | o(b), if ro = 1 and 2ao+1 \ o(b), if ro = -1, if ro = 1 + 2ao-1, 2(ao-1)( f-1)-1(2f-1 - 1) if ro = 1 + 2ao-1. Example 4.5. Let A be a z2-extension of a cyclic group z126o = z4 x z32 x z5 x z7 = (ao) x (a1) x (a2) x (a3). By Theorem 2.5, the b-conjugation on Zn corresponds to a4-tuple (ro, r1, r2, r3), where ri = ±1 for i G {0,1, 2, 3}. Take (ro, r1, r2, r3) = (1, -1, -1,1) and P = 3 as an example. One has Isoc(G; A) = Isoc(G; B) Isoc(G; z7), where B is a z2-extension of z18o. ByLemmas4.1 and4.2, |0(B;3)| = (23 - 1)|Q1(B;3)| = 34836480 and |Aut(B)| = 4320. It follows that Isoc(G; B) = 8064. By Isoc(G; z7) = 57, one gets Isoc(G; A) = 459648. 1 5 In cases of Z2-extensions of an abelian group Naturally, we are interested in extending the counting problem of the previous two sections to the case of a z2-extension of an abelian group. To do this, we need to classify z2-extensions of an arbitrary abelian group, but we can not give a complete answer so far, see Section 6. So we just count two special cases, generalized dihedral groups or generalized dicyclic groups. 5.1 With generalized dihedral groups Let H be an abelian group. A generalized dihedral group Dih(H), as a z2-extension of H, is defined with relations b2 = 1, b-1ab = a-1, for all a € H. It is a semidirect product of H and z2, with z2 acting on H by inverting elements. When H is cyclic, Dih(H) is just a dihedral group. Lemma 5.1. H is a characteristic subgroup of Dih(H). Proof. Take an automorphism a € Aut(Dih(H)). Note that the order of a quotient type element is 2. For any element a of odd order in Dih(H), a(a) should be normal type since a is order-preserving. For an element a0 of even order, suppose that a(a0) is a quotient type element. Since a0 commutes with a as an element of odd order, a(a0) commutes a(a). Then b commutes a, which is a contradiction. Then a(a) € H for any a € H. Hence H is a characteristic subgroup of Dih(H). □ Now, |Aut(Dih(H))| = |H| • |Aut(H)|. By Lemmas 3.1 and 3.2, one can show that |fi(Dih(H);P)| = (2^ - 1)|fi{1}(H;P - 1)| = (2^ - 1)|H||fi(H; P - 1)|. Each abelian group can be decomposed into direct product of abelian p-group, namely, H = Hpi x • • • x Hps withpjprime. Then |fi(Dih(H);P)| = (2^-1)|H||fi(HPi; P-1)| • • • |fi(Hp.; P-1)|. Itjustneeds to determine Isoc(G; Dih(Hp)) for aprime integer p. Since |Q(Hp; P - 1)| is determined in [14], one gets Theorem 5.2. For a generalized dihedral group Dih(Hp) and Hp = m1zpsi x • • • x m^Zpse with m1,..., m^ and s1,..., s£ are positive integers satisfying s£ < • • • < s1; one can obtain Isoc(G; Dih(Hp)) = (2^ - 1)pfllj=1 p-, ( ; ( p)) ( )p n-=1 n==1 p=j-h+1 -1, where m = m1 + • • • + m^ and f (P - 1, mj, sj) = (P - 1 - m) ^ mj(sj - 1) + ^ mj I ^ mj(sj - Sj - 1) \j=1 J j=1 \j=j+1 5.2 With generalized dicyclic groups A generalized dicyclic group Dic(H), as another z2-extension of an abelian group H, is defined with relations b2 = c, b-1ab = a-1, where c is an involution of H and a is an arbitrary element of H. Similarly, one can have the coming lemma. Lemma 5.3. H is a characteristic group of Dic(H). Hence |Aut(Dic(H))| = |H| • |Aut(H)|. Theorem 5.4. For a generalized dicyclic group Dic(Hp) and Hp = m1 zp»i x^ • •xm^zp»« with m1,...,mf and s1,..., s£ are positive integers satisfying s£ < • • • < s1; one can obtain Isoc(G; Dih(Hp)) = 2(2^ — 1)pf iU=1 p-, ( ; ( p)) ( )r n-=1 n==1 r=j-h+1 — 1, where m = m1 + • • • + m^ and e-1 f (^ — 1, m®, s®) = (^ — 1 — m) ^m®(sj — 1) + ^m® I ^ mj(s® — Sj — 1) \i=1 / ®=1 \j=j+1 6 Further remarks In this paper, we enumerate the regular coverings of a graph whose covering transformation groups are z2-extensions of a cyclic group. However, we could not give a complete answer of this problem if A is a z2-extension of any abelian group H. However, we cannot answer the same enumeration problem when the cyclic group is replaced by an abelian group, even by an elementary abelian p-group. In fact the difficulty for authors is how to determine all involutions of Aut(H). The counting problem has studied by many researchers, for example, in [21], it gave a generating function for the number of involutions of GL(n,p) which is isomorphic to automorphism group of Zp x • • • x zp. For more results, see [2], [5], [3] and so on. But it is still hard for us to determine the specific form of each involution of GL(n,p). For further possible problems unsolved in this paper, we list in the following. (1) Isoc(G; A) if A is a z2-extension of any abelian group. (2) Isoc(G; A) if A is a zp-extension of any cyclic group. (3) Isoc(G; A) if A is any metacyclic group. References [1] J. N. S. Bidwell and M. J. Curran, Automorphisms of finite abelian groups, Math. Proc. R. Ir. Acad. 110A (2010), 57-71, doi:10.3318/pria.2010.110.1.57. [2] D. Z. Djokovic, Product of two involutions, Arch. Math. 18 (1967), 582-584, doi:10.1007/ bf01898863. [3] J. Fulman, R. Guralnick and D. Stanton, Asymptotics of the number of involutions in finite classical groups, J. 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Isaacs, Finite Group Theory, volume 92 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 2008, doi:10.1090/gsm/092. [11] D. L. Johnson, Presentations of Groups, volume 15 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2nd edition, 1997, doi:10.1017/ cbo9781139168410. [12] G. A. Jones, Enumeration of homomorphisms and surface-coverings, Quart. J. Math. Oxford 46 (1995), 485-507, doi:10.1093/qmath/46.4.485. [13] G. A. Jones, Counting subgroups of non-Euclidean crystallographic groups, Math. Scand. 84 (1999), 23-39, doi:10.7146/math.scand.a-13930. [14] J. H. Kwak, J.-H. Chun and J. Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11 (1998), 273-285, doi:10.1137/ s0895480196304428. [15] J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Theory 23 (1996), 105-109, doi:10.1002/(sici)1097-0118(199606)22:2(105::aid-jgt2)3.0.co;2-r. [16] J. H. Kwak and J. Lee, Distribution of branched Dp-coverings of surfaces, Discrete Math. 183 (1998), 193-212, doi:10.1016/s0012-365x(97)00030-7. [17] J. H. Kwak, J. Lee and A. Mednykh, Coverings, enumeration and hurwitz problems, in: J. Koolen, J. H. Kwak and M.-Y. Xu (eds.), Applications of Group Theory to Combinatorics, CRC Press, London, pp. 71-107, 2008, selected papers from the Com2MaC Conference on Applications of Group Theory to Combinatorics, Pohang, Korea, 9-12 July 2007. [18] J. H. Kwak and M. Y. Xu, Finite Group Theory for Combinatorists, unpublished. [19] V. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Appl. Math. 52 (1998), 91-120, doi:10.1023/a:1005950823566. [20] V. A. Liskovets, On the enumeration of subgroups of a free group, Dokl. Akad. Nauk BSSR 15 (1971), 6-9. [21] K. E. Morrison, Integer sequences and matrices over finite fields, J. Integer Seq. 9 (2006), Article 06.2.1, https://cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/ morrison37.html. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 225-266 https://doi.org/10.26493/1855-3974.1115.90f (Also available at http://amc-journal.eu) Enumeration of hypermaps of a given genus* Alain Giorgettif FEMTO-ST Institute, Univ. Bourgogne Franche-Comté, CNRS 16 route de Gray, 25030 Besancon cedex, France Timothy R. S. Walsh Department of Computer Science, University of Quebec in Montreal (UQAM) P. O. Box 8888, Station A, Montreal, Quebec, Canada, HC3-3P8 Received 23 May 2016, accepted 17 October 2017, published online 20 June 2018 This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root dart are distinguished. We give functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the vertices containing the distinguished darts. We solve these equations to get parametric expressions of the generating functions of rooted hypermaps of low genus. We also count unrooted hypermaps of given genus by number of darts, vertices, hyperedges and faces. Keywords: Enumeration, surface, genus, rooted hypermap, unrooted hypermap. Math. Subj. Class.: 05C30, 05A15 1 Introduction A (combinatorial) hypermap is a triple (D,R,L) where D is a finite set of darts and R and L are permutations on D such that the group (R, L) generated by R and L acts transitively on D. A (combinatorial ordinary) map is a hypermap (D, R, L) whose permutation L is a fixed-point-free involution on D. For a hypermap (resp. map) the orbits of R, L and * The authors wish to thank Alexander Mednykh, Roman Nedela and the referees for helpful suggestions to improve the presentation of this article. ^For this work Alain Giorgetti was supported by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-03). E-mail addresses: alain.giorgetti@femto-st.fr (Alain Giorgetti), walsh.timothy@uqam.ca (Timothy R. S. Walsh) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ RL (L followed by R) are respectively called vertices, hyperedges (resp. edges) and faces. The degree of a vertex, edge, hyperedge or face is the number of darts it contains. The equivalence of combinatorial maps and topological maps having been established in [14], we use the word "map" to mean "combinatorial map" throughout this paper. The genus g of a map is given by the Euler-Poincare formula [7] v - e + f = 2(1 - g), (1.1) where v is the number of vertices, e is the number of edges and f is the number of faces. The genus of a hypermap with t darts, v vertices, e hyperedges and f faces was defined in [13] by the formula v + e + f = t + 2(1 - g). (1.2) An isomorphism between two maps or hypermaps (D, R, L) and (D', R', L') is a bi-jection from D onto D' that takes R into R' and L into L'; it corresponds to an orientation-preserving homeomorphism between two topological maps. A sensed hypermap (resp. map) is an isomorphism class of hypermaps (resp. maps). We admit the existence of a unique hypermap (resp. map) with an empty set of darts D, called the empty hypermap (resp. map). For both of these objects v = f =1 and g = e = 0. A rooted hypermap (resp. map) is either the empty hypermap (resp. map) or a tuple (D, x, R, L) where (D, R, L) is a non-empty combinatorial hypermap (resp. map) and x G D is a distinguished dart, called the root. The enumeration of maps and hypermaps has several non-trivial applications. One such application is based on the correspondence between hypermaps and algebraic curves established by the Belyi theorem [16]. For instance, the formula for the number of plane trees was used by A. Zvonkin in the computer generation of Shabat polynomials of bounded degree [16]. Another area where the map enumeration plays an important role is theoretical physics, in particular in 2-dimensional gravitation models. Roughly speaking, map enumeration is used to compute matrix integrals determining the properties of gravitational fields (see for instance the works of B. Eynard [9]). Some hypermaps have been shown to be related to contextuality in quantum physics [21]. Also, A. Mednykh and R. Nedela have applied the enumeration of rooted (resp. unrooted) hypermaps to the enumeration of subgroups (resp. conjugacy classes of subgroups) of the triangle group with three generators x, y, z and the relation xyz = 1 [20]. We enumerate rooted hypermaps of a given genus by number of darts, vertices, hyper-edges and faces. To do so we consider more general families of rooted hypermaps and bipartite maps, in which other vertices or darts than the root dart are distinguished. We also use the genus-preserving bijection between hypermaps and 2-vertex-coloured bipartite maps presented in [23]. But since bipartite maps have all their faces of even degree and we're using the degrees of the vertices as parameters, we must instead study the face-vertex dual of a 2-coloured bipartite map, that is, a map whose faces are coloured in two colours (white and black) so that no two faces that share an edge have the same colour. All these maps are Eulerian - that is, all their vertices are of even degree - but not all Eule-rian maps are 2-face-colourable. For example, the map on the torus with one vertex, one face and two edges is Eulerian because its only vertex is of degree 4, but its face cannot be coloured because it shares both edges with itself. Therefore we call the maps we are studying face-bipartite. A sequenced (rooted) map is a rooted map with some vertices other than the root vertex (the vertex that contains the root) distinguished from each other and from all the other vertices. The labels that distinguish these vertices can be taken to be 1, 2,..., k, where k is the number of distinguished vertices. A sequenced (rooted) hypermap is defined similarly. We state (in Section 4) a bijective decomposition for the set H(g, t, f, e, n, D) of sequenced orientable hypermaps of genus g with t darts, f faces and e hyperedges, with the root vertex of degree n and with the sequence of degrees of the distinguished vertices equal to D = (di, d2,..., d|D|), where dj is the degree of the distinguished vertex with label i. We obtain a bijective decomposition of the set F(g, e, w, b, n, D) of sequenced orientable face-bipartite maps of genus g with e edges, w white faces, b black faces, with the root face of degree 2n and with the sequence of half-degrees of the distinguished vertices equal to D. Then we apply face-vertex duality to obtain a bijective decomposition of the corresponding set of 2-coloured bipartite maps with distinguished faces. Next we use the bijection in [23] to obtain a bijective decomposition for hypermaps with distinguished faces, and finally we again apply face-vertex duality to obtain a bijective decomposition of H(g, t, f, e, n, D). A mutirooted hypermap is a hypermap in which a non-empty sequence of darts with pairwise distinct initial vertices is distinguished. We relate multirooted hypermaps to se-quenced hypermaps and thus obtain a recurrence for the number of multirooted hypermaps and functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the initial vertices of the distinguished darts. The paper is organized as follows. Section 2 fixes some notations, recalls a known decomposition for sequenced rooted maps and describes the bijection between hypermaps and bipartite maps presented in [23]. Sections 3 and 4 respectively enumerate sequenced face-bipartite maps and sequenced rooted hypermaps of a given genus. In Section 5 we consider multirooted hypermaps and we give equations for the generating functions that count these objects. In Section 6 we give functional equations relating the generating functions for rooted hypermaps with that for multirooted hypermaps. Then we show how to solve these equations. In Section 7 we obtain parametric expressions for the generating functions that count rooted hypermaps with a given small positive genus. Section 8 presents enumeration algorithms for sensed unrooted hypermaps counted by number of darts, vertices and hyperedges. Appendix A (resp. B) contains a table for numbers of rooted (resp. unrooted) hypermaps of genus g with d darts, v vertices and e hyperedges for d < 14. 2 Background 2.1 Notations We first introduce the notations and conventions we use throughout the paper. Let D and D' be two lists of integers. The inclusion D' C D means that D' is a sublist of D. In this case D - D' is the complementary sublist of D' in D. For instance, the sublists of D = [1,1,2] are the empty list [], [1] (twice), [2], [1,1], [1,2] (twice) and D itself. Their complementary sublists in the same order are D, [1, 2] (twice), [1,1], [2], [1] (twice) and []. We denote by D.D' the concatenation of the lists D and D'. If i is an integer and D is a list of integers, then i.D is a shortcut for [i].D. For 1 < j < |D| we denote by dj the j-th element of the list D of length |D| and by D - {dj} the list obtained from D by removing its j-th element dj. Let p be a positive integer. The abbreviation D1..p denotes the list [d1,..., dp]. The abbreviation Vd p denotes v-1 ... vpp. The sign + (resp. denotes (resp. generalized) disjoint set union in the following decompositions and (resp. generalized) arithmetic sum in the following equations. By con- vention, a disjoint set union (resp. sum) over an empty domain is equal to the empty set (resp. zero). For any logical formula p the notation Av means the singleton set containing only the empty hypermap or map (depending on the context) and the empty set if p is false. The notation 5V means 1 if p is true and 0 if p is false. 2.2 Bijective decomposition of the set of sequenced maps In 1962 W. T. Tutte [22] presented a bijective decomposition of a planar map with all the vertices distinguished and a root in every vertex. In 1972 T. R. Walsh and A. B. Lehman [27] generalized this decomposition to maps of higher genus and used it to count rooted maps of a given genus by number of vertices and faces. In 1987 D. Arques [3] used this latter decomposition to find a closed-form formula for the number of rooted maps of genus 1 by number of vertices and faces. In 1991 E. A. Bender and E. A. Canfield [4] presented a more efficient decomposition that roots only a single vertex and distinguishes only as many other vertices as necessary and used it to obtain explicit formulas for counting rooted maps of genus 2 and 3. In 1998 the first author [11] modified this decomposition and used it to obtain a bijective decomposition satisfied by the set M(g, e, f, n, D) of sequenced orientable maps of genus g with e edges and f faces, with the root vertex of degree n and with D the list of degrees of the distinguished vertices was obtained in [11]. Since this bijective decomposition contains an error, we present the correct bijective decomposition here, and we derive it to make the derivation more accessible than the contents of a Ph. D. thesis. Theorem 2.1. The set M(g, e, f, n, D) of sequenced orientable maps of genus g with e edges and f faces, with the root vertex of degree n and with the list D of degrees of the distinguished vertices is defined by the bijective decomposition M(g, e, f, n, D) = E M(gi,ei,fi,ni,Di) x M(g2, e2, f2,n2, D - Di) gi + 92 = g ei + e2 = e — i fi + f2 = f ni + n2 = n — 2 Di C D n—3 + E M(g - 1, e - 1, f,n - 2 - p,p.D) x {1,...,p} (2.1) p=i p=2e—2 + £ M(g,e - 1, f,p, D) p=n—i |D| + £M(g,e - 1, f, dj + n - 2,D - {dj}) + A(gie,f,n,D)=(0,0,i,0,[]). j=i Proof. If a map m has at least one edge, we reduce by 1 the number of edges by the face-vertex dual of deleting the root edge. There are two cases of this operation, depending upon whether the root edge is a loop or a link, and each of these cases breaks down into two sub-cases. Case 1: The root edge is a loop. We delete the root edge and split the root vertex into two parts, si and s2. If r is the root, then si consists of the darts R(r), R2(r),..., R—i(L(r)) and s2 consists of the darts R(L(r)), R2(L(r)), ..., R x(r). This case breaks down into two cases, depending upon whether or not this operation disconnects the map. Case 1a: This operation disconnects the map into two maps, m1 containing s1 and m2 containing s2 .If m1 has at least 1 edge, its root is r1 = R(r), and if m2 has at least 1 edge, its root is r2 = R(L(r)). Let g1, e1, f1, n1, D1 and g2, e2, f2, n2, D2 be the parameters of the maps m1 and m2, respectively, corresponding to g, e, f, n, D. This operation reduces by 1 the total number of edges; so e1 + e2 = e - 1. It leaves unchanged the total number of faces because r and L(r) simply get deleted from the cycle(s) of RL (L followed by R) containing them; so f1 + f2 = f. It increases by 1 the total number of vertices; so from Formula (1.1), which relates the genus of a map to the number of its vertices, faces and edges, it can easily be deduced that g1 + g2 = g. It decreases by 2 the total number of darts in s1 and s2 since r and L(r), which belonged to the root vertex, get eliminated; so n1 + n2 = n - 2. Finally, D1 can be any sublist of D and D2 is just the complementary sublist, denoted by D - D1. This operation is uniquely reversible; so the set of ordered pairs of sequenced maps obtained in this case is E M(g1,e1,f1 ,n1,D1) x M(g2, e2, f2, n2, D - D1), (2.2) gi + 32 = g ei + e2 = e — 1 fl + f2 = f n i + n 2 = n — 2 Di C D where E means the union of disjoint sets. Case 1b: This operation does not disconnect the map, but instead turns it into a new map m' with e - 1 edges and f faces and, since the number of vertices increases by 1, the genus of m' is g - 1, so that this case only occurs when g > 1. Neither s1 nor s2 can be of degree 0 (otherwise the map would be disconnected); so we can choose for m' the root r1 = R(r) belonging to s1. Let p be the degree of s2. Since the sum of the degrees of s1 and s2 is n - 2, the degree of s1, the root vertex, is n - 2 - p. We distinguish the vertex s2 so that this operation can be reversed, and we put its degree p at the beginning of the list D, turning it into p.D. Now this operation is reversible in p distinct ways, since any of the p darts of s2 can be chosen to be R(L(r)) when we merge the vertices s1 and s2 and replace the deleted root edge. Now p can be any integer from 1 up to n - 3 (so that n - 2 - p > 1). For both p and n - 2 - p to be at least 1, n must be at least 4. The set of sequenced maps obtained in this case is n—3 E M(g - 1,e - 1, f, n - 2 - p,p.D) x {1,...,p}. (2.3) p=1 Case 2: The root edge is a link. We contract the root edge, merging its two incident vertices s1 containing the root r and s2 containing L(r) into a single vertex s with root R(r). This operation decreases by 1 the number of edges and doesn't change the number of faces, since r and L(r) simply get deleted from the cycle(s) containing them. Since the number of vertices is decreased by 1, the genus remains the same. This case breaks down into two sub-cases, depending upon whether or not s2 is one of the distinguished vertices. Case 2a: The vertex s2 is not one of the distinguished vertices. Let p be the degree of the new vertex s. Then p = n - 2 + the degree of s2, and since the degree of s2 must be at least 1, we have p > n - 1. Also, the new map has 2e - 2 darts; so p < 2e - 2. This operation is uniquely reversible for each value of p; so the set of maps so obtained is p=2e-2 E M(g,e - 1, f,p, D). (2.4) p=n-1 Case 2b: The vertex s2 is one of the distinguished vertices. It can be any one of the |D| distinguished vertices. If it is the jth distinguished vertex, then its degree is dj. Then since it gets merged with s1 into the new root vertex, dj gets dropped from D. Finally, the degree of s is dj + n - 2. This operation too is uniquely reversible; so the set of maps so obtained is |D| E M(g, e - 1, f, dj + n - 2, D - {dj}). (2.5) j=1 Finally, suppose that m has no edges. It is of genus 0, has 1 face, its one vertex is of degree 0 and its list D is empty because it has no distinguished vertices; so it constitutes the singleton A(s,e,/,n,D) = (0,0,1,0,[]). (2.6) Then M (g, e, f, n, D) is the disjoint union of the sets given by (2.2) - (2.6). □ 2.3 Bipartite maps and hypermaps To motivate the transformation of (2.2)-(2.6) into the corresponding equations for sequenced hypermaps we briefly describe the bijection in [23] that takes a hypermap h into a 2-coloured bipartite map m = I(h), its incidence map. The bijection I takes the darts, vertices and hyperedges of h into the edges, white vertices and black vertices of m. A root (distinguished dart) of h corresponds to a distinguished edge of m; to make it correspond to a root of m we impose the condition that a root of m belongs to a white vertex. The permutation R in h corresponds to R in m acting on a dart in a white vertex and the permutation L in h corresponds to R in m acting on a dart in a black vertex. The permutation L in m doesn't correspond to any permutation in h; rather, since it takes a dart belonging to a vertex of one colour into a dart belonging to a vertex of the opposite colour, it toggles R in m between R and L in h. A face (cycle of RL) in h corresponds to a face in m with twice the degree. To see this, we follow one application of RL in h starting with a dart d, which corresponds to an edge in m but we make it correspond to the dart d' in that edge that also belongs to a white vertex. Then the L in h takes d' first into L(d'), which belongs to a black vertex, and then into RL(d') and the following R in h takes RL(d') first into LRL(d'), which belongs to a white vertex, and then into RLRL(d'). Since the genus of a hypermap with t darts, v vertices, e hyperedges and f faces is defined by (1.2), m has the same genus as h. Since the root of an incidence map of a rooted hypermap must belong to a white vertex, we impose the condition on a rooted 2-face-coloured face-bipartite map that the root belong to a white face and we transform (2.2)-(2.6) into the corresponding bijective decomposition for these maps. 3 Sequenced face-bipartite maps Let F(g, e, w, b, n, D) be the set of sequenced orientable face-bipartite maps of genus g with e edges, w white faces, b black faces, with the root face of degree 2n and with the list of half-degrees of the distinguished vertices equal to D. For any dart d we denote by f (d) the face containing d and we note that the face f (R(d)) = f (L(d)) must have the opposite colour from f (d) because those two faces share the edge {d, L(d)}. Theorem 3.1. The set F(g, e, w, b, n, D) satisfies the bijective decomposition F(g, e, w, b, n, D) = ^ F(gi,ei,wi,bi,ni,Di) x F(g2, e2, w2, b2, n2, D - D1) g i + 92 = g ei + e2 = e — i wi + b2 = b W2 + bi = w n i + n2 = n — i Di C D n—2 + F(g — 1, e — 1, b, w, n — 1 — p,p.D) x {1,... ,p} (3.1) p=i p=e— i + 53 F(g, e — 1, b, w,p, D) p= n |D| + ^F(g, e — 1, b, w, dj + n — 1, D — {dj}) + A(gie,w,b,n,D)=(0,0,i,0,0,[]). j=i Proof. Case 1: The root edge is a loop. By definition, f (r), where r is the root of the map m, is white, so that since ri = R(r), f (ri) must be black. But when the loop is removed and the vertex s containing r is split, ri becomes a root; so f (ri) must change colour and so must all the faces of the new map m' (in case 1b) or the map mi containing ri (in case 1a). In case 1a, the other map m2 has r2 = RL(r) as a root and f (r2) is white; so its faces stay the same colour. This implies that in case 1a wi + b2 = b and w2 + bi = w, whereas in case 1b w and b switch in going from m to m'. In case 1a, we have, as for general maps, gi + g2 = g, ei + e2 = e — 1 and Di is any subset of D, but instead of ni + n2 = n — 2 we have ni + n2 = n — 1 because the degrees satisfy the equation 2ni + 2n2 = 2n — 2. The analogue of formula (2.2) is thus T: F(gi, ei, wi, bi, ni, Di) xF(g2, e2, w2, b2, n2, D — Di). (3.2) 9i + 92 = 9 ei + e2 = e — i wi + b2 = b W2 + bi = w ni + n2 = n — i Di C D In case 1b, the reduced map m' is still of genus g — 1 and has e — 1 edges, but the degree of s2 is now 2p instead of p and the degree of the new root vertex si is 2(n — 1 — p); so the parameter n — 2 — p in (2.3) changes to n — 1 — p. Also, 1 < 2p < 2n — 3, but since 2p is even, we have 1 < p < n — 2 instead of 1 < p < n — 3, and the condition that n > 4 changes to n > 3. The analogue of formula (2.3) is thus n-2 F(g — 1, e — 1, b, w, n — 1 — p,p.D) x {1,... ,p}. (3.3) p=i Case 2: The root edge is a link. Since the new root R(r) belongs to a black face, all the faces change colour; so b and w switch. In case 2a, we have 2n — 1 < 2p < 2e — 2, but since 2p is even, we now have n < p < e — 1; so the analogue of (2.4) is p=e-1 ^ F(g, e — 1,b,w,p,D). (3.4) p=n In case 2b, the degree of the new root vertex is 2dj + 2n — 2; so the analogue of (2.5) is |D| ^F(g, e — 1,b, w,dj + n — 1, D — {dj}). (3.5) j=i Finally, the map with no edges has one white face and no black ones; so the analogue of (2.6) is ^(g,e,w,6,n,D) = (0,0,1,0,0,[]). (3.6) □ After deriving this bijective decomposition, we became aware of the article [8], which presents a similar bijective decomposition but for multi-rooted face-bipartite maps, which are like sequenced face-bipartite maps except that every distinguished vertex has a root. However, we present our derivation here for several reasons: it makes our article self-contained, we obtained it independently of [8] and our main purpose is to count hypermaps rather than face-bipartite maps. Now [8] does present a construction that converts a hy-permap into a face-bipartite map. However, that construction is not proved and it is far more complicated than the one in [23], which is not cited in [8]. We also recently became aware of the article [6], which generalizes the results of [15] by computing the generating functions for edge-labelled bipartite maps on an orientable surface of genus g with an unbounded number of faces and including the degrees of these faces as parameters. 4 Sequenced rooted hypermaps Theorem 3.1 holds for rooted 2-coloured bipartite maps with distinguished faces, where e is the number of edges, w is the number of white vertices, b is the number of black vertices, n is half the degree of the root face and D is the list of half-degrees of the distinguished faces. By the bijection described in Section 2.3, it also holds for rooted hypermaps with distinguished faces, where e is the number of darts, w is the number of vertices, b is the number of hyperedges, n is the degree of the root face and D is the list of degrees of the distinguished faces. By duality, the theorem also holds for sequenced hypermaps, where e is the number of darts, w is the number of faces, b is the number of hyperedges, n is the degree of the root vertex and D is the list of degrees of the distinguished vertices. To make the letters correspond to the objects they represent, we change F to H, e to t, w to f and b to e. We thus obtain the following results. Theorem 4.1 (Bijective decomposition for sequenced hypermaps). Let H(g, t, f, e, n, D) be the set of sequenced orientable hypermaps of genus g with t darts, f faces and e hy-peredges, with the root vertex of degree n and with the list of degrees of the distinguished vertices equal to D = (d\,d2,... ,d\D\), where di is the degree of the distinguished vertex with label i. The set H(g, t, f, e, n, D) satisfies the bijective decomposition H(g,t,f,e,n,D) = Y H(gi,ti,fi,ei,ni,Di) x H(g2,t2, f2,e2,n2, D - Di) g i + 92 = g ti + t2 = t — i fl + e2 = e f2 + ei = f ni + n 2 = n — 1 D1 C D n—2 + Y H(g - l,t - l,e,f,n - 1 - p,p.D) x{l,...,p] (4.1) p=i p=t—i + Y H(g,t - 1, e, f,p, D) p= n |D| + Y H(g,t - 1 e f,dj + n - 1,D - {dj}) + A(g,t,f,e,n,D) = (0,0,l,0,0,[]). j=i Corollary 4.2 (Recurrence between numbers of sequenced hypermaps). Let H(g, t, f, e, n, D) be the number of rooted sequenced hypermaps of genus g with t darts, f faces and e hyperedges such that the root vertex is of degree n and D is the list of degrees of the distinguished vertices. Then H(0,0,1,0,0, []) = 1 and if t > 1, then H(g,t,f, e, n, D) = Y H(gi,ti,fi,ei,ni,Di) H(g2,t2, f2,e2,n2, D - Di) 91 + 92 = 9 ti + t2 = t — 1 fl + e2 = e f2 + ei = f n i + n2 = n — 1 Di C D n—2 + ¿n>3^g>i Y pH (g - 1,t - 1, e, f, n - 1 - p,p.D) (4.2) p=i p=t—i + Y H(g,t - 1, e, f,p, D) p= n |D| + Y H(g, t - 1,e, f, dj + n - 1,D - {dj}). j=i 5 Multirooted hypermaps For p > 1 a p-rooted hypermap is a hypermap in which a sequence of p darts with pairwise distinct initial vertices is distinguished. A multirooted hypermap is a p-rooted hypermap for some p > 1. This section addresses the enumeration of multirooted hypermaps. Theorem 5.1 (Recurrence between numbers of multirooted hypermaps). Let Hm (g,t,f,e, D) be the number of multirooted hypermaps of genus g with t darts, f faces and e hyper-edges such that D is the list of degrees of the distinguished vertices. Then Hm(0,0,1,0, [ ]) = 1 and if t > 1, then Hm(g,t, f, e, n.D) = Hm(gi,t1 ,fi,ei,ni.Di) Hm(g2 ,h, f2,e2,n2.(D - Di)) g i + g 2 = g ti + t2 = t — i fl + e2 = e f2 + ei = f ni + n2 = n — i Di C D n—2 + Sn>sSg>^ Hm(g - 1,t - 1,e, f, (n - 1 - p).p.D) (5.1) p=i p=t—i + ^ Hm(g,t - 1, e, f,p.D) p= n |D| + ^ dj Hm(g,t - 1, e, f, (dj + n - 1).(D - {dj})). j=i Proof. A multirooted hypermap is similar to a sequenced rooted hypermap except that for each distinguished non-root vertex a dart starting from it is distinguished. If the degree of the j th distinguished vertex is dj, then there are dj ways of distinguishing a dart of this vertex. It follows that for each sequenced rooted hypermap, there are njD}idj multirooted hypermaps. Let Hm(g, t, f, e, D) be the number of multirooted hypermaps of genus g with t darts, f faces and e hyperedges such that such that D is the list of degrees of the initial vertex of the distinguished darts. Then Hm(g, t, f, e, n.D) = H(g, t, f, e, n, D) ngdj. (5.2) Solving (5.2) for H(g, t, f, e, n, D) and substituting into (4.2) proves the theorem. □ For p > 1 let Hg (vi, .. . ,vp,x,y,u,z) = ^ Hm(g,t,f,e,Di..p)vDipp xf yeuv zf (5.3) t > 0, f > i, e > 0 di > i, . . . , dp > i = t + 2(i - g) - e - f be the generating function that counts multirooted hypermaps of genus g with p distinguished darts if g > 0, and 0 otherwise. For 1 < i < p, the exponent di of the variable vi in this series is the degree of the initial vertex of the i-th distinguished dart. The exponent f of the variable x is the number of faces, the exponent e of the variable y is the number of hyperedges, the exponent t of the variable z is the number of darts and the exponent v of the variable u is the number of vertices (v is computable from the other parameters by Formula (1.2)). Corollary 5.2 (Functional equations for multirooted hypermaps). For g > 0 and p > 1 the generating functions Hg of multirooted hypermaps of genus g are defined by the following functional equations: Hg (vi,W,x,y,u,z) = g yviz xu *—' '—' j ^ ^ Hj (vi,X,y,x,u,z)Hg-j (vi,W — X,x,y,u,z) j=0 XCW V\z + — Hg-i(vi,vi, W, y, x, u, z) (5.4) V\u,z vi — 1 j=p + --7 (Hg(vi,W,y,x,u,z) — Hg(1,W,y,x,u, z)) d ( Hg (vj ,W — {vj },y,x,u,z) — Hg (vi,W — {vj },y,x,u,z) Ed I ±±g(Uj ,=_2dj (v'--vt — vi + xuSg=oSp=i, where W = v2,... ,vp. Proof. By summation according to (5.3) of the recurrence between numbers of multirooted hypermaps from Theorem 5.1. □ By vertex-hyperedge duality, we have Hg(vvi, W,y,x,u, z) = Hg(vi,W,x,y,u, z) + Sg=oSp=i(yu — xu) (5.5) and thus another functional equation without x, y swaps is: Hg (vi,W,x,y,u,z) = v g / —[(Hj(vi,X,x,y,u,z) + Sj=oSx=o(yu — xu)) xu j=0 XCW V Hg-j(vi, W — X, x, y, u, z) v z + —Hg-1 (v i,v i,W,x,y,u,z) (5.6) v uz H--7 (Hg(vi,W,x,y,u,z) — Hg(1,W,x,y,u, z)) v — 1 j=P a +v i uzY;vj ¿w j=2 + xuSg=oSp= i. d f Hg (vj ,W — {vj },x,y,u,z) — Hg (v i,W — {vj },x,y,u,z) dvj vj vj — vi The former equation is given here for maximal generality. However, a consequence of the genus formula (1.2) is that three variables among the four variables x, y, u and z are sufficient. In the remainder of the paper we consider the generating functions Hg (v i,W,x,y,u) = Hg (v i,W,x,y,u, 1) with one fewer variable. They are defined by the following functional equations: Hg (vi,W,x,y,u) = g -(Hj(vi,X,x,y,u) + 6j,oS\x\,o(yu - xu)) Hg-j(vi,W - X,x,y,u) j=0 XCW vi + — Hg-i(vi,vi,W,x,y,u) (5.7) + _vru (Hg (vi,w,x,y,u) - Hg (l,W,x,y,u)) vi - 1 + j=p ( Hg(vj, W - {vj},x,y,u) - Hg(vi,W - {vj},x,y,u) + ViU vj dvj vj - vi + xuSg=oSp=i. For g,p = 0,1, after grouping in the left-hand side the terms containing Hg (vi, W, x, y, u) in (5.7), one gets A(vi,x,y,u) -Hg (vi,W,x,y,u) = vi g x(l - vi) ^^ ^^ Hj(vi,X,x,y,u)Hg-j(vi,W - X,x,y,u) j=0 xc w (j,x) = (0, []) (j, X) = (g, W) l — vi +--u— Hg-i(vi,vi, W, x, y, u) + uHg(l, W, x, y, u) + uTg (vi,W,x,y,u) (5.8) A(v, x, y, u) = vu + (l - v)(1 - yv + xv - 2vHo(v, x, y, u)/u) (5.9) Tg (vi,W,x,y,u) = with and j=p g ( v / (1 - vi) E vj j vj \h9 (v,W - {v},x y,u) .x j=2 j - Hg (vi, W-{vj },x,y,u)U . (5.10) 6 Rooted hypermap generating functions Let hg (v, e, f ) be the number of rooted genus-g hypermaps with v vertices, e hyperedges and f faces. Let Hg (x,y,u) = hg (v,e,f )xv yeuf (6.1) v,e,f >1 be the ordinary generating function for counting rooted hypermaps on the orientable surface of genus g > 0, where the exponent of variable x is the number of vertices, the exponent of variable y is the number of hyperedges, and the exponent of variable u is the number of faces. Rooted hypermaps being 1-rooted hypermaps, Hg (x,y,u) = Hg (1,x,y,u), (6.2) where Hg (v\,... ,vp,x, y, u) is the generating function counting p-rooted genus-g hypermaps defined in Section 5 for p > 1. We first recall in Section 6.1 a known parametric expression of the generating function that counts rooted planar hypermaps. Then we explain in Section 6.2 how to solve the functional equation of the generating functions Hg (x, y, u) that count rooted hypermaps with a given positive genus g. 6.1 Rooted planar hypermaps The following proposition is a reformulation of [1, Theorem 3], with the correspondence s = x, f = u and a = y for variables, A = p, p = q and v = r for parameters, and Ho = sf (1 + J) for generating functions. Proposition 6.1 ([1]). The ordinary generating function H0(x,y,u) that counts rooted planar hypermaps by number of vertices (exponent of x), hyperedges (exponent of y) and faces (exponent of u) is the unique solution of the following parametric system: H0(x,y,u) = 1 + pqr(1 — p — q — r) (6.3) with = p(1 - q - r) = q(1 — p — r) (6.4) ky = r(1 — p — q). Proof. The generating function H0(v, x, y, u) that counts rooted planar hypermaps (genus 0) by number of vertices (exponent of x), hyperedges (exponent of y), faces (exponent of u) and degree of the root vertex (exponent of v) satisfies the functional equation yv Ho(v,x,y,u) = —(Ho(v, x,y,u) + yu — xu) Ho(v,x,y,u) xu vu +--- (Ho(v, x, y, u) — Ho(1, x, y, u)) + xu (6.5) v—1 obtained by instantiation of (5.7) with g = 0, p =1 and v\ = v. This equation can be solved by the quadratic method [10, page 515]. The idea is to define auxiliary functions A(v, x, y, u) and B(v, x, y, u) by (5.9) and B (v,x,y,u) = A(v,x,y,u)2 (6.6) and look for a function V(x, y, u) such that A(V (x,y,u),x,y,u) = 0, (6.7) implying that B(V(x, y, u), x, y,u) = 0 and dvB(v, x, y, v,)\v=V(XyVuU) = 0. We get from (6.5), (5.9) and (6.6) that B(v, x, y, u) = 1 — 2yv — 2xv — 2v3y — 2v3x — 2v2u + v4y2 — 2v3y2 + y2v2 + v4 x2 o O O O O O O O O O A — 2v x + x v + v u + 4v yx — 2yv x — 2yv u + 2v yu — 2v yx — 2v3xu + 2xv2u + 4v2x + 4v2y + 2vu + 4v3H0(1, x, y, u) — 4v2H0(1,x,y,u) — 2v + v2. (6.8) The constraints B(V(x, y, u), x, y, u) = 0 and B(v, x, y, u)|v=V(x,y,u) = 0 respectively are 1 - 2yV - 2xV - 2V3y - 2V3 x - 2V2u + V4y2 - 2V3y2 + y2V2 + V4x2 - 2V3x2 + x2V2 + V2u2 + 4V3yx - 2yV2x - 2yV2u + 2V3yux - 2V4y - 2V3xu + 2xV2u + 4V2x + 4V2y + 2Vu + 4V3H0(1,x,y,u) - 4V2H0(1,x,y,u) - 2V + V2 = 0 (6.9) and -2 + 8yV + 8xV + 4V3y2 - 6y2V2 + 4V3x2 - 6x2V2 - 6V2x - 6V2y - 4Vu + 2y2 V + 2x2V + 2Vu2 - 4yVu + 4xVu - 4yVx + 12yV2x + 6yV2u - 8V3yx - 6xV2u + 2V - 2x - 2y + 2u = 0. (6.10) It can be checked that both equations are satisfied by V =1/(1 - q) with x, u, y and H0(1, x, y, u) related to p, q and r by (6.4) and (6.3). (6.11) □ 6.2 Rooted hypermaps with positive genus The following additional notations are used in this section. Let p be a positive integer. Let Hj[ni,..., np] denote the partial derivative of the function Hj(vi,..., vp, x, y, u) with respect to the variables v1,..., vp to the respective orders n1,..., np, computed at v1 = ... = vp = V. The abbreviation [p] denotes the list [2,..., p] if p > 2 and the empty list [] if p =1. The abbreviation N[p] denotes the list [n2,..., np]. For any sublist X C [p] of [p], [p] - X denotes the sublist of the elements of [p] that are not in X, NX denotes the list of those n in N[p] such that i is in X and Nj denotes the list [n2,..., nj_1, nj+1,..., np]. 6.2.1 Equation for rooted hypermaps and recurrence relations The special case of Formula (5.8) for g > 1, p =1 and v1 = V is the following formula: uHg (1, x, y, u) = I '-1 \ (V - 1) I x£Hj(V, x, y,u)Hg-j(V,x,y, u) + Hg_1(V, V, x,y,u)/u I i.e. uHg(1, x, y, u) = (V - 1) |x ]_1 Hj[0]Hg_j[0] + Hg_1[0, 0]/u| . (6.12) In order to derive from (6.12) a value for Hg(1, x, y, u), we are looking for a value for Hj[0], Hg_j[0] and Hg-1[0,0]. More generally, we will derive from the following proposition a closed form for the expressions Hg [n1,..., np]. Proposition 6.2. For g > 0, p > 1 and ni,..., np > 0 the function Hg [ni,..., np] is defined by (ni + 1)A[1] V H g [ni,N[p]] i+j+k=n i + 1 V ,J 7 i>0, k 1, M(v) = 1 — v and L(v) = v(1 — v). Proof. Equation (6.13) is obtained from Equation (5.8) as follows: 1. Partial derivation of (5.8) with respect to the variables vi; v2,..., vp to the respective orders ni + 1, n2, ..., np. 2. Evaluation of this differential equation at vi = • • • = vp = V. The function Hg [ni + 1,..., np] is multiplied by A[0] in the resulting equation, and A[0] is known to be zero (6.7). The functions Tg [...] are replaced by expressions with the functions Fg [...] thanks to Lemma 6.3 below. 3. In the left-hand side of the resulting equation, isolation of the single term involving the function Hg[ni,..., np]. By inspection one can check that the right-hand side of (6.13) depends only on some functions Hg[k, n2,..., np] with k < ni, some functions Hg [ni,..., np,] with p' < p and some functions Hj [...] for j < g. Thus, (6.13) in a recursive definition of the family of functions Hg [ni,..., np] for g > 0, p > 1 and ni,..., np > 0. □ The following lemma relates the partial derivatives of Tg at v = V with the ones of Fg. Lemma 6.3. For p > 2 and g, ni,..., np > 0, Tg [ni + 1,N[p] = E (ni + 1)!nj ! (njFg [ni + nj + 2, Nj] ^ (ni + n + 2)! V j g[ i + j + ' j] + i) x j=2 ' ) + S++3Fg [ni + nj + 3, Nj]). (6.15) Proof. We can easily prove that d f(vj - vi)Hs(vi, [p] - {vj},x, y,u) dvj vj - vi Then, Tg (v1,..., vp, x, y, u) equals (6.16) j=2 (vj - vi) 1 (vj(1 - vi)Hg(vj, [p] - {vj},x,y,u) - vi(1 - vi)Hg (vi, [p] -{vj },x, y,u)) . (6.17) It also holds that ni + i dni dvni+i vj (vj - vi)Hg (vj, [P] - {vj},x,y,u) vj - vi so that d rt^+fi (v1,..., vp, x, y, u) equals (6.18) j=r d " 1 +2 - ( (vj - vi) i (vj (1 - vj )Hg (vj , [p] - {vj }, x, u) - vi(1 - vi)Hg (vi, [p] -{vj },x,y,u) (6.19) i.e. j=P j=2 dvni + i dvj d"1+2 ((vj, [p] -{vj},x,y,u) - (vi, [p] -{vj},x,y,u) vj - vi Formula (6.15) is a consequence of d ni+n2 f ^(x1) — ^(x2) \ dx^1dx^2 V X1 — X2 / ni!n2! xi=x2 = a (ni + n2 + 1)! The formula Fs [n,N] = £ (n)L[k]Hfl [/, N] (6.20) +"2+i)(a). (6.21) □ (6.22) is an easy consequence of (6.14). Thus the right-hand side of (6.13) only depends on some functions [k,..., np] with k < n1, some functions [n1,..., n^,] with p' < p, some functions Hj[...] for j < g and some functions A[i]. A relation between A[i] and some functions H0[j] is established in Section 6.2.2. 6.2.2 Case g = 0 and p =1 The function A[i] can be related to some functions H0[j] as follows: With M(v) = 1 — v and L(v) = v(1 — v), Equation (5.9) is A(v, x, y, u) = vu + M (v) + L(v)(-y + x - 2xHo(v, x, y, u)). (6.23) 0 0 Its instantiation at v = V gives 1 - q Ho[0] = ,-—. (6.24) 1 — q — r For k > 1, the k-th partial derivative of (6.23) in v is d k Qk Qk ^ A(v,x,y,u) = dVk (vu) + dVk M (v) dk + dvk [L(v)(— y + x — 2xHo(v,x,y,u))] (6.25) and its instantiation in v = V is d k A[k] = dv^ (vu)|v=v + M [k] + ^ (k)L[i] (dj(—y+x—2xHo(v,x,y,u))iv=^. (6.26) Solving (6.26) for k = 1 gives (1 — q)2 (A[1] + 1 — p — q — r) Ho[1] = ^-^ 2 (1 -^-". (6.27) 2pq(1 — q — r) For k > 2, one gets i+j=k since M [k] = 0, i.e. A[k] = —2x £ (k)i[i]Ho[j] i+j=k \ / A[k] = — 2x ^L[0]Fo[k] + kL[1]Ho[k — 1] + k(k — 1) L[2]Ho[k — 2] ) (6.28) since L[k] = 0 if k > 3. 7 Explicit formulas for small genera This section proposes explicit parametric expressions for the generating functions that count rooted hypermaps of small positive genus. In Section 7.1 we count by number of vertices, hyperedges and faces; the number of darts can be obtained from these parameters by Formula (1.2). In Section 7.2 we count by number of darts alone. 7.1 Rooted hypermap series enumerated with three parameters For g = 1,..., 5 we have computed an explicit expression of Hg (x, y, u) parameterized by p, q and r, with x = p(1 — q — r), u = q(1 — p — r) and y = r(1 — p — q), by application of formulas in Section 6. For g > 3, the expressions are too large to be included in the present text, but a Maple file with all the generating functions up to genus 5 is available from the first author on request. A parametric expression of Hi (x, y, u) is p q r (1 — p) (1 — q) (1 — r) Hi(x,y,u) = (7.1) [(1 — p — q — r)2 — 4pqr]2 This expression can be derived from [2, Theorem 3], with the correspondence s = x, f = u, and a = y between variables and the correspondence Hi(x, y, u) = xuKi(1, x, y, u) between generating functions. A parametric expression of H2 (x, y, u) is where H2(x,y,u) = p q r (1 — p) (1 — q) (1 — r) P2(p, q,r) [(1 — p — q — r)2 — 4pqr]7 (7.2) P2(p, q, r) = 76p6q2r2 — 8p4q4r2 — 8p4q2r4 + 76p2q6r2 — 8p2q4r4 + 76p2q2r6 + 40p7qr — 76p6q2r — 76p6qr2 — 112p5q3 r — 228p5q2r2 — 112p5qr3 + 8p4q4r + 16p4q3r2 + 16p4q2r3 + 8p4qr4 — 112p3 q5r + 16p3q4r2 + 40p3q3r3 + 16p3q2r4 — 112p3qr5 — 76p2q6r — 228p2q5r2 + 16p2q4r3 + 16p2q3r4 — 228p2q2r5 — 76p2qr6 + 40pq7r — 76pq6r2 — 112pq5r3 + 8pq4r4 — 112pq3r5 — 76pq2 r6 + 40pqr7 + p8 — 20p7q — 20p7r — 35p6q2 — 64p6qr — 35p6r2 + 56p5q3 + 396p5q2r + 396p5qr2 44 + 56p5r3 + 140p4q4 + 264p4q3r + 393p4q2r2 + 264p4qr3 + 140p4r + 56p3q5 + 264p3q4r — 92p3q3r2 — 92p3q2r3 + 264p3qr4 + 56p3r5 — 35p2q6 + 396p2 q5r + 393p2q4r2 — 92p2q3r3 + 393p2q2r4 + 396p2qr5 — 35p2r6 — 20pq7 — 64pq6r + 396pq5r2 + 264pq4r3 + 264pq3r4 + 396pq2r5 — 64pqr6 — 20pr7 + q8 — 20q7r — 35q6 r2 + 56q5r3 + 140q4r4 + 56q3r5 — 35q2r6 — 20qr7 + r8 + 6p7 + 105p6q + 105p6r + 21p5q2 — 116p5qr + 21p5r2 — 420p4q3 — 821p4q2 r — 821p4qr2 34 — 420p4r3 — 420p3q4 — 648p3q3r — 316p3q2 r2 — 648p3 qr3 — 420p3r + 21p2q5 — 821p2 q4r — 316p2q3r2 — 316p2q2r3 — 821p2qr4 + 21p2r5 + 105pq6 — 116pq5r — 821pq4r2 — 648pq3r3 — 821pq2r4 — 116pqr5 25 + 105pr6 + 6q' + 105q6r + 21q5r2 — 420q4r3 — 420q3r4 + 21q2r + 105qr6 + 6r7 — 49p6 — 189p5q — 189p5r + 315p4q2 + 479p4qr + 315p4r2 + 910p3q3 + 1162p3q2r + 1162p3qr2 + 910p3r3 + 315p2q4 + 1162p2q3r + 720p2q2r2 + 1162p2qr3 + 315p2r4 — 189pq5 + 479pq4r + 1162pq3r2 + 1162pq2r3 + 479pqr4 — 189pr5 — 49q6 — 189q5r + 315q4r2 + 910q3r3 + 315q2r4 — 189qr5 — 49r6 + 112p5 + 70p4q + 70p4r — 770p3q2 — 876p3qr — 770p3r2 — 770p2q3 — 1380p2q2r — 1380p2qr2 — 770p2r3 + 70pq4 — 876pq3r — 1380pq2r2 — 876pqr + 70pr4 + 112q5 + 70q4r — 770q3r2 — 770q2r3 + 70qr4 + 112r5 - 105p4 + 210p3q + 210p3r + 735p2q2 + 1034p2 qr + 735p2r2 + 210pq3 + 1034pq2r + 1034pqr2 + 210pr3 - 105q4 + 210q3r + 735q2r2 + 210qr3 - 105r4 + 14p3 - 315p2q - 315p2r - 315pq2 - 672pqr - 315pr2 + 14q3 - 315q2r - 315qr2 + 14r3 + 49p2 + 175pq + 175pr + 49q2 + 175qr + 49r2 - 36p - 36q - 36r + 8. Remark: For g = 0, the formula H0(x, y,u) = pqr(1 - p - q - r) (7.3) can be derived from [1], with the correspondence s = x, f = u, and a = y between variables and the correspondence H0(x, y, u) = xuK0(1,x,y,u) between generating functions. 7.2 Rooted hypermap series enumerated by number of darts Let Hg (z) be the ordinary generating function of rooted hypermaps on the orientable surface of genus g > 0, where the exponent of variable z is the number d of darts. 7.2.1 Generating functions For g from 0 to 6, a parametric expression of Hg (z), where z = t(1 - 2t) and t = 0 when z = 0, is Ho(z) t3 (1 - 3 t ) z2 , (7.4) Hi(z) t 3 (1 - t) (1 - 4 t)2 , (7.5) H2(z) 4 z2 t3 (51 t4 - 77 t3 + 48 t2 - 15 t + 2) (1 - t)5 (1 - 4 t)7 , (7.6) H3(z) 4 z4 t3 P3(z) = (1 - t)9 (1 - 4 t)12 , (7.7) H4(z) 4 z6 t3 P4(z) (1 - t )13 (1 - 4 t )17 , (7.8) H5(z) 4 z8 t3 P5(z) (1 - t)17 (1 - 4 t)22 , (7.9) H6(z) 4 z10 t3P6(z) = (1 - t)21 (1 - 4 T)27 , (7.10) with P3 (z) = 28496 t9 - 36888 t8 - 13164 t7 + 61676 t6 - 61872 t5 + 35172 t4 - 13168 t3 + 3360 t2 - 552 t + 45, P4(z) = 32375616 t 14 + 15509760 t13 - 243313744 t12 + 442844592 t 11 - 389268768 t 10 + 170357328 t9 + 1281984 t8 - 53553072 t7 + 39814032 t6 - 17597520 t5 + 5541192 t4 - 1320920 t3 + 239697 t2 - 30456 t + 2016, P5(z) = 61742404608 t 19 + 239043447552 t 18 - 1163002515456 t 17 + 1403096348736 t 16 + 338393916800 t 15 - 2962590413376 t 14 + 4243997599488 t 13 - 3552865706240 t 12 + 2000782619136 t 11 - 761565230016 t10 + 165542511744 t9 + 7568059872 t8 - 23295865824 t7 + 11016156244 t6 - 3336459144 t5 + 761835465 t4 - 141393220 t3 + 21738240 t2 - 2490480 t + 151200 P6 (z) = 178054771302400 t24 + 1584534210564096 t23 - 4933663711730688 t22 and 6 - 2073822560019456 t21 + 28025505345377280 t20 - 55010184951564288 t 19 + 54283457920223232 t 18 - 22997164994372352 t 17 - 13439214645718272 t 16 + 31734000656779264 t 15 - 29719458122609664 t 14 + 18704646148809216 t 13 - 8736443315384448 t 12 + 3098312828500416 t 11 - 813298324826016 t 10 + 138473163256176 t9 - 4043551301232 t8 - 6580517850696 t7 + 2630924485729 t6 - 626336383104 t5 + 112079088144 t4 - 17314508592 t3 + 2485496880 t2 - 284717376 t + 17107200. We have also computed the generating functions for 7 < g < 11. Their expressions are too large to be included in the present text, but a Maple file is available from the first author on request. A. Mednykh and R. Nedela used our formulas (7.4) to (7.7) to find explicit formulas for the number of rooted hypermaps for genus g = 0,1, 2 and 3 [19]. 7.3 Other parameterization In a private communication to the second author, P. Zograf suggests the parameterization t z =(TT27. ai1) After adding the condition that t = 0 when z = 0, it corresponds to 1 - 4z - V1 - 8z , t =-^-. (7.12) 8z These two parameterizations are equivalent. The one can be transformed into the other by means of the following substitutions: t (7.13) 1 + 2t and 4 = 1-2T. (7-14) By means of these substitutions, the following parametric expressions in t can be obtained from the parametric expressions (7.4)-(7.10) for Hg (t) in t: Ho(z) Hi(z) H2(z) Hs(z) = t (1 - t), (1+ t)(1 - 2t)2 ' 4 t5 (1 + 2t) (t4 - t3 + 6 t2 +1 + 2) (1+1)5(1 - 2t)7 , 4 t7 (1 + 2t) (1 +t)-9(1 - 2t)-12 (80 t9 - 120 t8 + 1500 t7 + 1036 t6 + 3768 t5 + 2820 t4 + 2288 t3 + 1008 t2 + 258 t + 45), H4(z) = 4 t9 (1 + 2t) (1 +t)-13(1 - 2t) 7 (16768 t14 - 33536 t ,13 + 653776 t12 + 786480 t11 4358016 t10 + 6151056 t9 + 10059552 t8 Hs(z) H6(z) For 0 Moreover. + 10217040 t7 + 8418240 t6 + 5227024 t5 + 2365888 t4 + 800128 t3 + 181665 t2 + 25992 t + 2016), = 4 t11 (1 + 2t) (1 + t)-17(1 - 2t)-22 ( 67 3 2800 t19 - 16832000 t18 + 450011520 t17 + 773106240 t16 + 5764983552 t15 + 11910647232 t14 + 29130502912 t13 + 46090300928 t12 + 63452543616 t11 + 68713116608 t10 + 60654218080 t9 + 43591208976 t8 + 25142796864 t7 + 11637842232 t6 + 4232899206 t5 + 1181820745 t4 + 245635580 t3 + 35501760 t2 + 3255120 t + 151200), = 4 t13 (1 + 2t) (1 + t)-21(1 - 2t)-27 (4424052736 t24 - 13272158208 t23 + 452750478336 t22 + 1012254206976 t21 + 9488911137792 t20 + 25803592571904 t19 + 83891900050944 t18 + 180120643165440 t17 + 346626234587904 t16 + 535272874975232 t15 + 701152993531392 t14 + 771688966862592 t13 + 716686355273472 t12 + 563018634260736 t11 + 372549313187520 t10 + 207088794784752 t9 + 96021082581732 t8 + 36765061031004 t7 + 11475757049569 t6 + 2863185376896 t5 + 556090776432 t4 + 80913152016 t3 + 8274846384 t2 + 536428224 t + 17107200). < g < 3, these expressions correspond to Fg (t) in Zograf's communication. they reveal an extra factorization by 4(1 + 2t) for g > 2. 8 Efficient enumeration of rooted and sensed unrooted hypermaps by number of darts, vertices and hyperedges We recall that a sensed map or hypermap is an equivalence class of (unrooted) maps or hypermaps under orientation-preserving isomorphism. Before enumerating sensed hypermaps we first need to enumerate rooted hypermaps. We use an efficient method of counting rooted hypermaps by number of darts, faces, ver- 3 t tices and hyperedges or, equivalently [23], 2-coloured bipartite maps rooted at a white vertex by number of edges, faces, white vertices and black vertices, presented by Kazarian and Zograf [15], and then count sensed 2-coloured bipartite maps and hypermaps with the same parameters using the same method we used [26, 12] to count sensed maps by number of edges, faces and vertices. The recurrence (formula (11) in [15]), with f changed to H, is as follows. Define Hg,d to be a homogeneous polynomial in the three variables t, u, and v. The coefficient of tf ubvw in Hg,d is the number of 2-coloured bipartite maps of genus g with d edges, f faces, b black vertices and w white vertices rooted at a white vertex or, equivalently, the number of rooted hypermaps of genus g with d darts, f faces, b hyperedges and w vertices. Then H0,1 = tuv and (d +1)Hg,d = (2d - 1)(t + u + v)Hg,d—1 + (d - 2) (2(tu + tv + uv) - (t2 + u2 + v2)) Hg,d—2 (8.1) g d—3 + (d - 1)2(d - 2)Hg—1,d—2 + EE(4 + 6J')(d - 2 - j)Hi,jHg—i}d—2—j. i=0j=1 In [26] we collaborated with Mednykh to enumerate rooted and sensed maps. Med-nykh enumerated maps of genus up to 11 by number of edges alone, while we enumerated maps of genus up to 10 by number of edges and vertices. The method we used to enumerate rooted maps is presented in [25]. The method we used to enumerate sensed maps is based on Liskovets' refinement [17] of the method Mednykh and Nedela used to enumerate sensed map of genus up to 3 by number of edges [18]. Later we used a more efficient method of enumerating rooted maps, presented in [5], to enumerate rooted and sensed maps of genus up to 50 [12]. To describe here the modifications we made to pass from maps to 2-coloured bipartite maps we need to briefly discuss a few of the concepts described in more detail in [26]. All the automorphisms of a map on an orientable surface are periodic. If the period is L > 1, then the automorphism divides the map into L isomorphic copies of a smaller map, called the quotient map. Most of the cells (vertices, edges and faces) are in orbits of length L under the automorphism; those that aren't are called branch points. For example, if a map is drawn on the surface of a sphere which undergoes a rotation through 360/L degrees, the two cells through which the axis of rotation pass are fixed; so they are each in an orbit of length 1 for any L. For maps of higher genus, not all the branch points are on orbits of length 1. For example, if a torus is represented as a square with opposite edges identified in pairs, and is rotated by 90 degrees (period 4), then the centre of the square is a branch point of orbit length 1 and so is the point represented by all four corners of the square, but the middle of the sides of the square are two branch points of orbit length 2: the point represented by the middle of both vertical sides of the square is taken by the rotation onto the point represented by the middle of both horizontal sides, and vice versa; so it takes two rotations to take either of these points back onto itself. Also, if the middle of an edge is a branch point, then the quotient map contains half of that edge - a dangling semi-edge. An automorphism of a map M of genus G is characterized by the following parameters: the period L, the genus g of its quotient map and the number of branch points of each orbit length. If each orbit length is replaced by its branch index (L divided by the orbit length), we obtain what is called an orbifold signature in [18]. In [18] a method is presented for determining which orbifold signatures could characterize an automorphism of a map of genus G (a G-admissible orbifold) and how many such automorphisms could be characterized by that orbifold signature; a variant of that method is presented in [17], and this is the one we use except that we deal with orbit lengths instead of branch indices. The method used in [18] to enumerate sensed maps of genus G with E edges by number of edges can be roughly described as follows. For each G-admissible orbifold O, let g be the genus of the quotient map, L be the period and q, be the number of branch points with branch index i. Then the number vo (d) of rooted maps with d darts that could serve as a quotient map for an automorphism with that signature once the branch points are pasted onto the map in all possible ways is given by VO (d) = ± tt((d-*)/2+2-2gk ((d - .)/2), (8.2) V V V92-s,q3,...,qL J where Ng (n) is the number of rooted maps of genus g with n edges (0 if n is not an integer). Here s is the number of dangling semi-edges in the quotient map m, all of which must be in orbits of length L/2 so that they represent normal edges in the original map M. The binomial coefficient is the number of ways of inserting dangling semi-edges into the rooted map multiplied by d/(d - s) because there are d ways to root the map once the dangling edges have been inserted and only d- s ways to root it without the dangling edges. The multinomial coefficient is the number of ways to distribute the branch points with the various branch indices among the non-edges of the quotient map; the number at the top of the multinomial coefficient is the number of non-edges and is given by the Euler-Poincare formula (1.1). Then the number of sensed maps of genus G with E edges is 2E ^^Epi0(n1(O),Zi) vo(2E/L), (8.3) LIE O where O runs over all the G-admissible orbifolds with period L and Epi0(n1(O), ZL) is the number of automorphisms that have the orbifold signature of O. In [26] we distributed the branch points that aren't on dangling semi-edges among the vertices and faces separately. The quotient map of a bipartite map can't contain any dangling semi-edges; otherwise the lifted map would have an edge joining two vertices of the same colour. Here we distribute the branch points among the white vertices, black vertices and faces, and, like in [26], we don't use a formula like (8.3); instead we compute the contribution of each orbifold signature to the number of sensed 2-coloured bipartite maps whose number of white vertices, black vertices, faces and edges are allowed to vary within a user-defined upper bound on the number of edges. Suppose that the quotient map is of genus g and has w white vertices, b black vertices and f faces. Then the number e of edges can be calculated from the formula f - e + w + b = 2(1 - g) (8.4) and the number d of darts is 2e. Suppose also that among the branch points of orbit length i, wj are on a white vertex, b, are on a black vertex and f are in a face. We denote by wL, bL and fL the number of white vertices, black vertices and faces, respectively, that do not contain a branch point. The original map will have W white vertices, B black vertices and F faces, where L L L W = ^ iwi; B = ^ ibi and F = ^ if,, j=1 i=1 i=1 (8.5) and the total number E of edges is equal to Le = F + W + B - 2(1 - g). The binomial coefficient in (8.2) disappears because the quotient map can't contain any dangling semi-edges. The multinomial coefficient must be replaced by the number of ways to distribute the branch points among the white vertices, black vertices and faces. Then (8.2) becomes vo (d, w, b, f) = ( w )(b hb b)(, /, Vg (d,w ,b,f), (8.6) \Wi,W2,...,wLJ \bi,b2,...,bLJ \Ji,J2,...,JlJ where d is the number of edges in the quotient maps on both sides of the formula (or the number of darts in the corresponding hypermaps) and Ng (d, w, b, f) is the number of 2-coloured bipartite maps with d edges with w white vertices, b black vertices and f faces, rooted at a white vertex. For this number to be positive, the sum of all the wi cannot exceed w with a similar bound on the sum of all the bi and the sum of all the fi; so w, b and f each starts at its respective sum and increases by 1 until the number E of edges in the original map exceeds a user-defined maximum. With each increase of w, b or f, one of the multinomial coefficients in (8.6) gets updated using a single multiplication and division. The product of these three multinomial coefficients must be computed for all sets of nonnegative integers such that for each i, wi + bi + fi is equal to the total number of branch points of orbit length i. Once (8.6) is multiplied by the number of automorphisms with the current orbifold signature, we get the contribution of that signature and the numbers wi, bi and fi to E times the number of sensed 2-coloured bipartite maps of genus G with E edges, F faces, B black vertices and W white vertices. This contribution is added to the appropriate element of an array, initially 0, and when all the contributions have been tallied, for each E, F, W and B the corresponding array element is divided by E (not 2E because the root must be incident to a white vertex) to give the number of sensed 2-coloured bipartite maps of genus G with E edges, F faces, B black vertices and W white vertices or, equivalently, the number of sensed hypermaps of genus G with E darts, F faces, B hyperedges and W vertices. This enumeration was done with a program written in C++ using CLN to treat big integers. It enumerated rooted and sensed hypermaps of genus up to 24 with up to 50 darts as fast as it could display the numbers on the screen. The numbers coincide with those obtained by generating the hypermaps [24]. The source code is available from the second author on request. References [1] D. Arques, Relations fonctionnelles et denombrement des hypercartes planaires pointees, in: G. Labelle and P. Leroux (eds.), Combinatoire ¿numirative, Springer, Berlin, volume 1234 of Lecture Notes in Mathematics, pp. 5-26, 1986, doi:10.1007/bfb0072505. [2] D. Arques, Hypercartes pointees sur le tore: Decompositions et denombrements, J. Comb. Theory Ser. B 43 (1987), 275-286, doi:10.1016/0095-8956(87)90003-7. [3] D. Arques, Relations fonctionnelles et denombrement des cartes pointees sur le tore, J. Comb. Theory Ser. B 43 (1987), 253-274, doi:10.1016/0095-8956(87)90002-5. [4] E. A. Bender and E. R. Canfield, The number of rooted maps on an orientable surface, J. Comb. Theory Ser. B 53 (1991), 293-299, doi:10.1016/0095-8956(91)90079-y. [5] S. R. Carrell and G. Chapuy, Simple recurrence formulas to count maps on orientable surfaces, J. Comb. Theory Ser. A 133 (2015), 58-75, doi:10.1016/j.jcta.2015.01.005. [6] G. Chapuy and W. Fang, Generating functions of bipartite maps on orientable surfaces, Electron. J. Comb. 23 (2016), #P3.31, http://www.combinatorics.org/ojs/index. php/eljc/article/view/v2 3i3p31. [7] H. S. M. Coxeter, Regular Polytopes, Dover, New York, 3rd edition, 1973. [8] P. Dunin-Barkowski, N. Orantin, A. Popolitov and S. Shadrin, Combinatorics of loop equations for branched covers of sphere, Int. Math. Res. Notices (2017), rnx047, doi:10.1093/imrn/ rnx047. [9] B. Eynard, Formal matrix integrals and combinatorics of maps, in: J. Hamad (ed.), Random Matrices, Random Processes and Integrable Systems, Springer, New York, pp. 415-442, 2011, doi:10.1007/978-1-4419-9514-8_6. [10] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, New York, NY, USA, 1st edition, 2009, http://ac.cs.princeton.edu/home/. [11] A. Giorgetti, Combinatoire bijective et ¿num^rative des cartes pointees sur une surface, Ph.D. thesis, Universite de Marne-la-Vallee, Institut Gaspard Monge, 1998. [12] A. Giorgetti and T. R. S. Walsh, Constructing large tables of numbers of maps by orientable genus, 2014, arXiv:1405.0 615 [math.CO] . [13] A. Jacques, Sur le genre d'une paire de substitutions, C. R. Acad. Sci. Paris Ser. A 267 (1968), 625-627. [14] 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. [15] M. Kazarian and P. Zograf, Virasoro constraints and topological recursion for Grothendieck's dessin counting, Lett. Math. Phys. 105 (2015), 1057-1084, doi:10.1007/s11005-015-0771-0. [16] S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and Their Applications, volume 141 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004, doi:10.1007/ 978-3-540-38361-1, see especially chapter 2, "Dessins d'Enfants", pp. 79-153. [17] V. A. Liskovets, A multivariate arithmetic function of combinatorial and topological significance, Integers 10 (2010), 155-177, doi:10.1515/integ.2010.012. [18] A. Mednykh and R. Nedela, Enumeration of unrooted maps of a given genus, J. Comb. Theory Ser. B 96 (2006), 706-729, doi:10.1016/j.jctb.2006.01.005. [19] A. Mednykh and R. Nedela, Counting hypermaps by Egorychev's method, Anal. Math. Phys. 6 (2016), 301-314, doi:10.1007/s13324-015-0119-z. [20] A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci. 226 (2017), 635-654, doi:10.1007/s10958-017-3555-5. [21] M. Planat, A. Giorgetti, F. Holweck and M. Saniga, Quantum contextual finite geometries from dessins d'enfants, Int. J. Geom. Methods Mod. Phys. 12 (2015), 1550067, doi:10.1142/ s021988781550067x. [22] W. T. Tutte, A census of slicings, Canad. J. Math. 14 (1962), 708-722, doi:10.4153/ cjm-1962-061-1. [23] T. R. S. Walsh, Hypermaps versus bipartite maps, J. Comb. Theory Ser. B 18 (1975), 155-163, doi:10.1016/0095- 8956(75)90042- 8. [24] T. R. S. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps, J. Integer Seq. 18 (2015), Article 15.4.3, https://cs.uwaterloo.ca/journals/JIS/ VOL18/Walsh/walsh3.html. [25] T. R. S. Walsh and A. Giorgetti, Efficient enumeration of rooted maps of a given orientable genus by number of faces and vertices, Ars Math. Contemp. 7 (2014), 263-280, doi:10.26493/ 1855-3974.190.0ef. [26] T. R. S. Walsh, A. Giorgetti and A. Mednykh, Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices, Discrete Math. 312 (2012), 2660-2671, doi: 10.1016/j.disc.2011.11.027. [27] T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus I, J. Comb. Theory Ser. B 13 (1972), 192-218, doi:10.1016/0095-8956(75)90050-7. A First numbers of rooted hypermaps The following sections show the numbers h of rooted hypermaps of genus g with d darts, v vertices, e edges and d - v - e + 2(1 - g) faces, for g < 6 and d < 14. A.1 Genus 0 d v e f h 6 1 5 2 15 8 2 5 3 2436 1 1 1 1 1 6 2 4 2 135 8 3 4 3 7500 6 3 3 2 262 8 4 3 3 7500 1 sum 1 6 4 2 2 135 8 5 2 3 2436 6 5 1 2 15 8 6 1 3 196 2 1 1 2 1 6 1 6 1 1 8 1 7 2 28 2 1 2 1 1 6 2 5 1 15 8 2 6 2 518 2 2 1 1 1 6 3 4 1 50 8 3 5 2 2436 6 4 3 1 50 8 4 4 2 3985 2 sum 3 6 5 2 1 15 8 5 3 2 2436 6 6 1 1 1 8 6 2 2 518 3 1 1 3 1 8 7 1 2 28 3 1 2 2 3 6 sum 1584 8 1 8 1 1 3 2 1 2 3 8 2 7 1 28 3 1 3 1 1 7 1 1 7 1 8 3 6 1 196 3 2 2 1 3 7 1 2 6 21 8 4 5 1 490 3 3 1 1 1 7 2 1 6 21 8 5 4 1 490 7 1 3 5 105 8 6 3 1 196 3 sum 12 7 2 2 5 280 8 7 2 1 28 7 3 1 5 105 8 8 1 1 1 4 1 1 4 1 7 1 4 4 175 4 1 2 3 6 7 2 3 4 889 8 sum 54912 4 2 1 3 6 7 3 2 4 889 4 1 3 2 6 7 4 1 4 175 9 1 1 9 1 4 2 2 2 17 7 1 5 3 105 9 1 2 8 36 4 3 1 2 6 7 2 4 3 889 9 2 1 8 36 4 1 4 1 1 7 3 3 3 1694 9 1 3 7 336 4 2 3 1 6 7 4 2 3 889 9 2 2 7 882 4 3 2 1 6 7 5 1 3 105 9 3 1 7 336 4 4 1 1 1 7 1 6 2 21 9 1 4 6 1176 7 2 5 2 280 9 2 3 6 5754 4 sum 56 7 3 4 2 889 9 3 2 6 5754 7 4 3 2 889 9 4 1 6 1176 5 1 1 5 1 7 5 2 2 280 9 1 5 5 1764 5 1 2 4 10 7 6 1 2 21 9 2 4 5 13941 5 2 1 4 10 7 1 7 1 1 9 3 3 5 26004 5 1 3 3 20 7 2 6 1 21 9 4 2 5 13941 5 2 2 3 55 7 3 5 1 105 9 5 1 5 1764 5 3 1 3 20 7 4 4 1 175 9 1 6 4 1176 5 1 4 2 10 7 5 3 1 105 9 2 5 4 13941 5 2 3 2 55 7 6 2 1 21 9 3 4 4 42015 5 3 2 2 55 7 7 1 1 1 9 4 3 4 42015 5 4 1 2 10 9 5 2 4 13941 5 1 5 1 1 7 sum 9152 9 6 1 4 1176 5 2 4 1 10 9 1 7 3 336 5 3 3 1 20 8 1 1 8 1 9 2 6 3 5754 5 4 2 1 10 8 1 2 7 28 9 3 5 3 26004 5 5 1 1 1 8 2 1 7 28 9 4 4 3 42015 8 1 3 6 196 9 5 3 3 26004 5 sum 288 8 2 2 6 518 9 6 2 3 5754 8 3 1 6 196 9 7 1 3 336 6 1 1 6 1 8 1 4 5 490 9 1 8 2 36 6 1 2 5 15 8 2 3 5 2436 9 2 7 2 882 6 2 1 5 15 8 3 2 5 2436 9 3 6 2 5754 6 1 3 4 50 8 4 1 5 490 9 4 5 2 13941 6 2 2 4 135 8 1 5 4 490 9 5 4 2 13941 6 3 1 4 50 8 2 4 4 3985 9 6 3 2 5754 6 1 4 3 50 8 3 3 4 7500 9 7 2 2 882 6 2 3 3 262 8 4 2 4 3985 9 8 1 2 36 6 3 2 3 262 8 5 1 4 490 9 1 9 1 1 6 4 1 3 50 8 1 6 3 196 9 2 8 1 36 1 1 1 1 1 1 1 m 0 9 9 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 m 1 0 336 11 2 1 10 55 12 1 3 10 1210 1176 11 1 3 9 825 12 2 2 10 3135 1764 11 2 2 9 2145 12 3 1 10 1210 1176 11 3 1 9 825 12 1 4 9 9075 336 11 1 4 8 4950 12 2 3 9 43098 36 11 2 3 8 23694 12 3 2 9 43098 1 11 3 2 8 23694 12 4 1 9 9075 11 4 1 8 4950 12 1 5 8 32670 339456 11 1 5 7 13860 12 2 4 8 245223 11 2 4 7 105435 12 3 3 8 449988 1 11 3 3 7 194304 12 4 2 8 245223 45 11 4 2 7 105435 12 5 1 8 32670 45 11 5 1 7 13860 12 1 6 7 60984 540 11 1 6 6 19404 12 2 5 7 666996 1410 11 2 5 6 216601 12 3 4 7 1936308 540 11 3 4 6 634865 12 4 3 7 1936308 2520 11 4 3 6 634865 12 5 2 7 666996 12180 11 5 2 6 216601 12 6 1 7 60984 12180 11 6 1 6 19404 12 1 7 6 60984 2520 11 1 7 5 13860 12 2 6 6 925190 5292 11 2 6 5 216601 12 3 5 6 3915576 40935 11 3 5 5 931854 12 4 4 6 6195560 75840 11 4 4 5 1482250 12 5 3 6 3915576 40935 11 5 3 5 931854 12 6 2 6 925190 5292 11 6 2 5 216601 12 7 1 6 60984 5292 11 7 1 5 13860 12 1 8 5 32670 60626 11 1 8 4 4950 12 2 7 5 666996 179860 11 2 7 4 105435 12 3 6 5 3915576 179860 11 3 6 4 634865 12 4 5 5 9032898 60626 11 4 5 4 1482250 12 5 4 5 9032898 5292 11 5 4 4 1482250 12 6 3 5 3915576 2520 11 6 3 4 634865 12 7 2 5 666996 40935 11 7 2 4 105435 12 8 1 5 32670 179860 11 8 1 4 4950 12 1 9 4 9075 288025 11 1 9 3 825 12 2 8 4 245223 179860 11 2 8 3 23694 12 3 7 4 1936308 40935 11 3 7 3 194304 12 4 6 4 6195560 2520 11 4 6 3 634865 12 5 5 4 9032898 540 11 5 5 3 931854 12 6 4 4 6195560 12180 11 6 4 3 634865 12 7 3 4 1936308 75840 11 7 3 3 194304 12 8 2 4 245223 179860 11 8 2 3 23694 12 9 1 4 9075 179860 11 9 1 3 825 12 1 10 3 1210 75840 11 1 10 2 55 12 2 9 3 43098 12180 11 2 9 2 2145 12 3 8 3 449988 540 11 3 8 2 23694 12 4 7 3 1936308 45 11 4 7 2 105435 12 5 6 3 3915576 1410 11 5 6 2 216601 12 6 5 3 3915576 12180 11 6 5 2 216601 12 7 4 3 1936308 40935 11 7 4 2 105435 12 8 3 3 449988 60626 11 8 3 2 23694 12 9 2 3 43098 40935 11 9 2 2 2145 12 10 1 3 1210 12180 11 10 1 2 55 12 1 11 2 66 1410 11 1 11 1 1 12 2 10 2 3135 45 11 2 10 1 55 12 3 9 2 43098 1 11 3 9 1 825 12 4 8 2 245223 45 11 4 8 1 4950 12 5 7 2 666996 540 11 5 7 1 13860 12 6 6 2 925190 2520 11 6 6 1 19404 12 7 5 2 666996 5292 11 7 5 1 13860 12 8 4 2 245223 5292 11 8 4 1 4950 12 9 3 2 43098 2520 11 9 3 1 825 12 10 2 2 3135 540 11 10 2 1 55 12 11 1 2 66 45 11 11 1 1 1 12 1 12 1 1 1 12 2 11 1 66 11 sum 13891584 12 3 10 1 1210 2149888 12 4 9 1 9075 12 1 1 12 1 12 5 8 1 32670 1 12 1 2 11 66 12 6 7 1 60984 55 12 2 1 11 66 12 7 6 1 60984 12 8 5 1 32670 13 8 4 3 5264545 14 3 7 6 80231508 12 9 4 1 9075 13 9 3 3 960960 14 4 6 6 249321114 12 10 3 1 1210 13 10 2 3 74217 14 5 5 6 360078558 12 11 2 1 66 13 11 1 3 1716 14 6 4 6 249321114 12 12 1 1 1 13 1 12 2 78 14 7 3 6 80231508 13 2 11 2 4433 14 8 2 6 10701873 12 sum 91287552 13 3 10 2 74217 14 9 1 6 429429 13 4 9 2 525525 14 1 10 5 143143 13 1 1 13 1 13 5 8 2 1827683 14 2 9 5 4557553 13 1 2 12 78 13 6 7 2 3356522 14 3 8 5 44221632 13 2 1 12 78 13 7 6 2 3356522 14 4 7 5 181925268 13 1 3 11 1716 13 8 5 2 1827683 14 5 6 5 360078558 13 2 2 11 4433 13 9 4 2 525525 14 6 5 5 360078558 13 3 1 11 1716 13 10 3 2 74217 14 7 4 5 181925268 13 1 4 10 15730 13 11 2 2 4433 14 8 3 5 44221632 13 2 3 10 74217 13 12 1 2 78 14 9 2 5 4557553 13 3 2 10 74217 13 1 13 1 1 14 10 1 5 143143 13 4 1 10 15730 13 2 12 1 78 14 1 11 4 26026 13 1 5 9 70785 13 3 11 1 1716 14 2 10 4 1053052 13 2 4 9 525525 13 4 10 1 15730 14 3 9 4 13043030 13 3 3 9 960960 13 5 9 1 70785 14 4 8 4 69432090 13 4 2 9 525525 13 6 8 1 169884 14 5 7 4 181925268 13 5 1 9 70785 13 7 7 1 226512 14 6 6 4 249321114 13 1 6 8 169884 13 8 6 1 169884 14 7 5 4 181925268 13 2 5 8 1827683 13 9 5 1 70785 14 8 4 4 69432090 13 3 4 8 5264545 13 10 4 1 15730 14 9 3 4 13043030 13 4 3 8 5264545 13 11 3 1 1716 14 10 2 4 1053052 13 5 2 8 1827683 13 12 2 1 78 14 11 1 4 26026 13 6 1 8 169884 13 13 1 1 1 14 1 12 3 2366 13 1 7 7 226512 14 2 11 3 122122 13 2 6 7 3356522 13 sum 608583680 14 3 10 3 1919918 13 3 5 7 14019928 14 4 9 3 13043030 13 4 4 7 22089600 14 1 1 14 1 14 5 8 3 44221632 13 5 3 7 14019928 14 1 2 13 91 14 6 7 3 80231508 13 6 2 7 3356522 14 2 1 13 91 14 7 6 3 80231508 13 7 1 7 226512 14 1 3 12 2366 14 8 5 3 44221632 13 1 8 6 169884 14 2 2 12 6097 14 9 4 3 13043030 13 2 7 6 3356522 14 3 1 12 2366 14 10 3 3 1919918 13 3 6 6 19315114 14 1 4 11 26026 14 11 2 3 122122 13 4 5 6 44136820 14 2 3 11 122122 14 12 1 3 2366 13 5 4 6 44136820 14 3 2 11 122122 14 1 13 2 91 13 6 3 6 19315114 14 4 1 11 26026 14 2 12 2 6097 13 7 2 6 3356522 14 1 5 10 143143 14 3 11 2 122122 13 8 1 6 169884 14 2 4 10 1053052 14 4 10 2 1053052 13 1 9 5 70785 14 3 3 10 1919918 14 5 9 2 4557553 13 2 8 5 1827683 14 4 2 10 1053052 14 6 8 2 10701873 13 3 7 5 14019928 14 5 1 10 143143 14 7 7 2 14168988 13 4 6 5 44136820 14 1 6 9 429429 14 8 6 2 10701873 13 5 5 5 64013222 14 2 5 9 4557553 14 9 5 2 4557553 13 6 4 5 44136820 14 3 4 9 13043030 14 10 4 2 1053052 13 7 3 5 14019928 14 4 3 9 13043030 14 11 3 2 122122 13 8 2 5 1827683 14 5 2 9 4557553 14 12 2 2 6097 13 9 1 5 70785 14 6 1 9 429429 14 13 1 2 91 13 1 10 4 15730 14 1 7 8 736164 14 1 14 1 1 13 2 9 4 525525 14 2 6 8 10701873 14 2 13 1 91 13 3 8 4 5264545 14 3 5 8 44221632 14 3 12 1 2366 13 4 7 4 22089600 14 4 4 8 69432090 14 4 11 1 26026 13 5 6 4 44136820 14 5 3 8 44221632 14 5 10 1 143143 13 6 5 4 44136820 14 6 2 8 10701873 14 6 9 1 429429 13 7 4 4 22089600 14 7 1 8 736164 14 7 8 1 736164 13 8 3 4 5264545 14 1 8 7 736164 14 8 7 1 736164 13 9 2 4 525525 14 2 7 7 14168988 14 9 6 1 429429 13 10 1 4 15730 14 3 6 7 80231508 14 10 5 1 143143 13 1 11 3 1716 14 4 5 7 181925268 14 11 4 1 26026 13 2 10 3 74217 14 5 4 7 181925268 14 12 3 1 2366 13 3 9 3 960960 14 6 3 7 80231508 14 13 2 1 91 13 4 8 3 5264545 14 7 2 7 14168988 14 14 1 1 1 13 5 7 3 14019928 14 8 1 7 736164 13 6 6 3 19315114 14 1 9 6 429429 14 sum 4107939840 13 7 5 3 14019928 14 2 8 6 10701873 A.2 Genus 1 d v e f h a 2 S l l470 l0 l a l 330 3 l l l l a 3 4 l 44l0 l0 2 7 l 6930 a 4 3 l 44l0 l0 3 6 l 4lSa0 3 sum l a S 2 l l470 l0 4 S l 97020 a 6 l l l26 l0 S 4 l 97020 4 l l 2 S l0 6 3 l 4lSa0 4 l 2 l S a sum l3l307 l0 7 2 l 6930 4 2 l l S 9 l l 7 2l0 l0 a l l 330 4 sum lS 9 9 l 2 2 l 6 6 3360 3360 l0 sum 97l3a3S S l l 3 lS 9 l 3 S l4700 ll l l 9 49S S l 2 2 40 9 2 2 S 3703S ll l 2 a l3200 S 2 l 2 40 9 3 l S l4700 ll 2 l a l3200 S l 3 l lS 9 l 4 4 23S20 ll l 3 7 l039S0 S 2 2 l 40 9 2 3 4 l0a2aS ll 2 2 7 2S90l7 S 3 l l lS 9 3 2 4 l0a2aS ll 3 l 7 l039S0 9 4 l 4 23S20 ll l 4 6 332640 S sum l6S 9 l S 3 l4700 ll 2 3 6 l493S2S 9 2 4 3 l0a2aS ll 3 2 6 l493S2S 6 l l 4 3S 9 3 3 3 l97a96 ll 4 l 6 332640 6 l 2 3 l7S 9 4 2 3 l0a2aS ll l S S 4aSl00 6 2 l 3 l7S 9 S l 3 l4700 ll 2 4 S 3420a3S 6 l 3 2 l7S 9 l 6 2 3360 ll 3 3 S 6l6S47a 6 2 2 2 4S6 9 2 S 2 3703S ll 4 2 S 3420a3S 6 3 l 2 l7S 9 3 4 2 l0a2aS ll S l S 4aSl00 6 l 4 l 3S 9 4 3 2 l0a2aS ll l 6 4 332640 6 2 3 l l7S 9 S 2 2 3703S ll 2 S 4 3420a3S 6 3 2 l l7S 9 6 l 2 3360 ll 3 4 4 96a4433 6 4 l l 3S 9 l 7 l 2l0 ll 4 3 4 96a4433 9 2 6 l 3360 ll S 2 4 3420a3S 6 sum l6ll 9 3 S l l4700 ll 6 l 4 332640 9 4 4 l 23S20 ll l 7 3 l039S0 7 l l S 70 9 S 3 l l4700 ll 2 6 3 l493S2S 7 l 2 4 S60 9 6 2 l 3360 ll 3 S 3 6l6S47a 7 2 l 4 S60 9 7 l l 2l0 ll 4 4 3 96a4433 7 l 3 3 l0S0 ll S 3 3 6l6S47a 7 2 2 3 269S 9 sum ll3a26l ll 6 2 3 l493S2S 7 3 l 3 l0S0 ll 7 l 3 l039S0 7 l 4 2 S60 l0 l l a 330 ll l a 2 l3200 7 2 3 2 269S l0 l 2 7 6930 ll 2 7 2 2S90l7 7 3 2 2 269S l0 2 l 7 6930 ll 3 6 2 l493S2S 7 4 l 2 S60 l0 l 3 6 4lSa0 ll 4 S 2 3420a3S 7 l S l 70 l0 2 2 6 l04llS ll S 4 2 3420a3S 7 2 4 l S60 l0 3 l 6 4lSa0 ll 6 3 2 l493S2S 7 3 3 l l0S0 l0 l 4 S 97020 ll 7 2 2 2S90l7 7 4 2 l S60 l0 2 3 S 440440 ll a l 2 l3200 7 S l l 70 l0 3 2 S 440440 ll l 9 l 49S l0 4 l S 97020 ll 2 a l l3200 7 sum l4a0S l0 l S 4 97020 ll 3 7 l l039S0 l0 2 4 4 6972S0 ll 4 6 l 332640 a l l 6 l26 l0 3 3 4 l2643l0 ll S S l 4aSl00 a l 2 S l470 l0 4 2 4 6972S0 ll 6 4 l 332640 a 2 l S l470 l0 S l 4 97020 ll 7 3 l l039S0 a l 3 4 44l0 l0 l 6 3 4lSa0 ll a 2 l l3200 a 2 2 4 lll99 l0 2 S 3 440440 ll 9 l l 49S a 3 l 4 44l0 l0 3 4 3 l2643l0 a l 4 3 44l0 l0 4 3 3 l2643l0 ll sum al96a469 a 2 3 3 206a4 l0 S 2 3 440440 a 3 2 3 206a4 l0 6 l 3 4lSa0 l2 l l l0 7lS a 4 l 3 44l0 l0 l 7 2 6930 l2 l 2 9 23S9S a l S 2 l470 l0 2 6 2 l04llS l2 2 l 9 23S9S a 2 4 2 lll99 l0 3 S 2 440440 l2 l 3 a 23S9S0 a 3 3 2 206a4 l0 4 4 2 6972S0 l2 2 2 a SaSSaS a 4 2 2 lll99 l0 S 3 2 440440 l2 3 l a 23S9S0 a S l 2 l470 l0 6 2 2 l04llS l2 l 4 7 990990 a l 6 l l26 l0 7 l 2 6930 l2 2 3 7 44l0l20 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 2 7 4410120 13 3 4 6 260619268 14 1 6 7 38648610 1 7 990990 13 4 3 6 260619268 14 2 5 7 375707570 5 6 1981980 13 5 2 6 93880696 14 3 4 7 1035514340 4 6 13768300 13 6 1 6 9513504 14 4 3 7 1035514340 3 6 24695580 13 1 7 5 6936930 14 5 2 7 375707570 2 6 13768300 13 2 6 5 93880696 14 6 1 7 38648610 1 6 1981980 13 3 5 5 374805834 14 1 7 6 38648610 6 5 1981980 13 4 4 5 582408775 14 2 6 6 512104880 5 5 19920390 13 5 3 5 374805834 14 3 5 6 2020140430 4 5 55785870 13 6 2 5 93880696 14 4 4 6 3126887407 3 5 55785870 13 7 1 5 6936930 14 5 3 6 2020140430 2 5 19920390 13 1 8 4 2642640 14 6 2 6 512104880 1 5 1981980 13 2 7 4 47604648 14 7 1 6 38648610 7 4 990990 13 3 6 4 260619268 14 1 8 5 21471450 6 4 13768300 13 4 5 4 582408775 14 2 7 5 375707570 5 4 55785870 13 5 4 4 582408775 14 3 6 5 2020140430 4 4 87100531 13 6 3 4 260619268 14 4 5 5 4475516612 3 4 55785870 13 7 2 4 47604648 14 5 4 5 4475516612 2 4 13768300 13 8 1 4 2642640 14 6 3 5 2020140430 1 4 990990 13 1 9 3 495495 14 7 2 5 375707570 8 3 235950 13 2 8 3 11674663 14 8 1 5 21471450 7 3 4410120 13 3 7 3 85050784 14 1 9 4 6441435 6 3 24695580 13 4 6 3 260619268 14 2 8 4 145864355 5 3 55785870 13 5 5 3 374805834 14 3 7 4 1035514340 4 3 55785870 13 6 4 3 260619268 14 4 6 4 3126887407 3 3 24695580 13 7 3 3 85050784 14 5 5 4 4475516612 2 3 4410120 13 8 2 3 11674663 14 6 4 4 3126887407 1 3 235950 13 9 1 3 495495 14 7 3 4 1035514340 9 2 23595 13 1 10 2 40040 14 8 2 4 145864355 8 2 585585 13 2 9 2 1225653 14 9 1 4 6441435 7 2 4410120 13 3 8 2 11674663 14 1 10 3 975975 6 2 13768300 13 4 7 2 47604648 14 2 9 3 28283255 5 2 19920390 13 5 6 2 93880696 14 3 8 3 259750218 4 2 13768300 13 6 5 2 93880696 14 4 7 3 1035514340 3 2 4410120 13 7 4 2 47604648 14 5 6 3 2020140430 2 2 585585 13 8 3 2 11674663 14 6 5 3 2020140430 1 2 23595 13 9 2 2 1225653 14 7 4 3 1035514340 10 1 715 13 10 1 2 40040 14 8 3 3 259750218 9 1 23595 13 1 11 1 1001 14 9 2 3 28283255 8 1 235950 13 2 10 1 40040 14 10 1 3 975975 7 1 990990 13 3 9 1 495495 14 1 11 2 65065 6 1 1981980 13 4 8 1 2642640 14 2 10 2 2407405 5 1 1981980 13 5 7 1 6936930 14 3 9 2 28283255 4 1 990990 13 6 6 1 9513504 14 4 8 2 145864355 3 1 235950 13 7 5 1 6936930 14 5 7 2 375707570 2 1 23595 13 8 4 1 2642640 14 6 6 2 512104880 1 1 715 13 9 3 1 495495 14 7 5 2 375707570 13 10 2 1 40040 14 8 4 2 145864355 sum 685888171 13 11 1 1 1001 14 9 3 2 28283255 14 10 2 2 2407405 1 11 1001 13 sum 5702382933 14 11 1 2 65065 2 10 40040 14 1 12 1 1365 1 10 40040 14 1 1 12 1365 14 2 11 1 65065 3 9 495495 14 1 2 11 65065 14 3 10 1 975975 2 9 1225653 14 2 1 11 65065 14 4 9 1 6441435 1 9 495495 14 1 3 10 975975 14 5 8 1 21471450 4 8 2642640 14 2 2 10 2407405 14 6 7 1 38648610 3 8 11674663 14 3 1 10 975975 14 7 6 1 38648610 2 8 11674663 14 1 4 9 6441435 14 8 5 1 21471450 1 8 2642640 14 2 3 9 28283255 14 9 4 1 6441435 5 7 6936930 14 3 2 9 28283255 14 10 3 1 975975 4 7 47604648 14 4 1 9 6441435 14 11 2 1 65065 3 7 85050784 14 1 5 8 21471450 14 12 1 1 1365 2 7 47604648 14 2 4 8 145864355 1 7 6936930 14 3 3 8 259750218 14 sum 4716867857 6 6 9513504 14 4 2 8 145864355 5 6 93880696 14 5 1 8 21471450 A.3 Genus 2 d v e f h l0 2 S l l670l3 l2 l a l aaa03 S l l l a l0 3 4 l 47l240 l2 2 7 l lSaSSa4 l0 4 3 l 47l240 l2 3 6 l 8 6S4 64 6 S sum a l0 S 2 l l670l3 l2 4 S l l932430S l0 6 l l l640l l2 S 4 l l932430S 6 l l 2 a4 l2 6 3 l 8 6S4 64 6 6 l 2 l a4 l0 sum l3S4S2l6 l2 7 2 l lSaSSa4 6 2 l l a4 ll l l 7 39963 l2 a l l aaa03 6 sum 2S2 ll ll l 2 2 l 6 6 SS00ll SS00ll l2 sum la0S0l094a 7 l l 3 469 ll l 3 S 222l06S l3 l l 9 la3la3 7 l 2 2 1183 ll 2 2 S S4090l9 l3 l 2 a 4ll4ll0 7 2 l 2 1183 ll 3 l S 222l06S l3 2 l a 4ll4ll0 7 l 3 l 469 ll l 4 4 346S000 l3 l 3 7 29l3Sl06 7 2 2 l 1183 ll 2 3 4 lS0l4a46 l3 2 2 7 70367479 7 3 l l 469 ll 3 2 4 lS0l4a46 l3 3 l 7 29l3Sl06 ll 4 l 4 346S000 l3 l 4 6 a7933a46 7 sum 4 9S6 ll l S 3 222l06S l3 2 3 6 374l27663 ll 2 4 3 lS0l4a46 l3 3 2 6 374l27663 a l l 4 la69 ll 3 3 3 267l74a2 l3 4 l 6 a7933a46 a l 2 3 aS26 ll 4 2 3 lS0l4a46 l3 l S S l2SaSS730 a 2 l 3 aS26 ll S l 3 222l06S l3 2 4 S a24962S02 a l 3 2 aS26 ll l 6 2 SS00ll l3 3 3 S l4S34l4a46 a 2 2 2 2l229 ll 2 S 2 S4090l9 l3 4 2 S a24962S02 a 3 l 2 aS26 ll 3 4 2 lS0l4a46 l3 S l S l2SaSS730 a l 4 l la69 ll 4 3 2 lS0l4a46 l3 l 6 4 a7933a46 a 2 3 l aS26 ll S 2 2 S4090l9 l3 2 S 4 a24962S02 a 3 2 l aS26 ll 6 l 2 SS00ll l3 3 4 4 22392a0420 a 4 l l la69 ll l 7 l 39963 l3 4 3 4 22392a0420 ll 2 6 l SS00ll l3 S 2 4 a24962S02 a sum 77992 ll 3 S l 222l06S l3 6 l 4 a7933a46 ll 4 4 l 346S000 l3 l 7 3 29l3Sl06 9 l l S S9aS ll S 3 l 222l06S l3 2 6 3 374l27663 9 l 2 4 42Saa ll 6 2 l SS00ll l3 3 S 3 l4S34l4a46 9 2 l 4 42Saa ll 7 l l 39963 l3 4 4 3 22392a0420 9 l 3 3 7702a l3 S 3 3 l4S34l4a46 9 2 2 3 la9999 ll sum l60l74960 l3 6 2 3 374l27663 9 3 l 3 7702a l3 7 l 3 29l3Sl06 9 l 4 2 42Saa l2 l l a aaa03 l3 l a 2 4ll4ll0 9 2 3 2 la9999 l2 l 2 7 lSaSSa4 l3 2 7 2 70367479 9 3 2 2 la9999 l2 2 l 7 lSaSSa4 l3 3 6 2 374l27663 9 4 l 2 42Saa l2 l 3 6 a6S4646 l3 4 S 2 a24962S02 9 l S l S9aS l2 2 2 6 209al337 l3 S 4 2 a24962S02 9 2 4 l 42Saa l2 3 l 6 a6S4646 l3 6 3 2 374l27663 9 3 3 l 7702a l2 l 4 S l932430S l3 7 2 2 70367479 9 4 2 l 42Saa l2 2 3 S a2a97296 l3 a l 2 4ll4ll0 9 S l l S9aS l2 3 2 S a2a97296 l3 l 9 l la3la3 l2 4 l S l932430S l3 2 a l 4ll4ll0 9 sum l074S64 l2 l S 4 l932430S l3 3 7 l 29l3Sl06 l2 2 4 4 l2a420004 l3 4 6 l a7933a46 l0 l l 6 l640l l2 3 3 4 2272S6Sl0 l3 S S l l2SaSS730 l0 l 2 S 167013 l2 4 2 4 l2a420004 l3 6 4 l a7933a46 l0 2 l S 167013 l2 S l 4 l932430S l3 7 3 l 29l3Sl06 l0 l 3 4 47l240 l2 l 6 3 a6S4646 l3 a 2 l 4ll4ll0 l0 2 2 4 llS409S l2 2 S 3 a2a97296 l3 9 l l la3la3 l0 3 l 4 47l240 l2 3 4 3 2272S6Sl0 l0 l 4 3 47l240 l2 4 3 3 2272S6Sl0 l3 sum l9Saa944336 l0 2 3 3 206a070 l2 S 2 3 a2a97296 l0 3 2 3 206a070 l2 6 l 3 a6S4646 l4 l l l0 3SS3SS l0 4 l 3 47l240 l2 l 7 2 lSaSSa4 l4 l 2 9 979a7a9 l0 l S 2 167013 l2 2 6 2 209al337 l4 2 l 9 979a7a9 l0 2 4 2 llS409S l2 3 S 2 a2a97296 l4 l 3 a a729l204 l0 3 3 2 206a070 l2 4 4 2 l2a420004 l4 2 2 a 2l0l64227 l0 4 2 2 llS409S l2 S 3 2 a2a97296 l4 3 l a a729l204 l0 S l 2 167013 l2 6 2 2 209al337 l4 l 4 7 34la2S4a4 l0 l 6 l l640l l2 7 l 2 lSaSSa4 l4 2 3 7 l4444326l2 14 3 2 7 1444432612 14 5 3 4 16427471172 14 7 3 2 1444432612 14 4 1 7 341825484 14 6 2 4 4286172247 14 8 2 2 210164227 14 1 5 6 661320660 14 7 1 4 341825484 14 9 1 2 9798789 14 2 4 6 4286172247 14 1 8 3 87291204 14 1 10 1 355355 14 3 3 6 7523770016 14 2 7 3 1444432612 14 2 9 1 9798789 14 4 2 6 4286172247 14 3 6 3 7523770016 14 3 8 1 87291204 14 5 1 6 661320660 14 4 5 3 16427471172 14 4 7 1 341825484 14 1 6 5 661320660 14 5 4 3 16427471172 14 5 6 1 661320660 14 2 5 5 6100939726 14 6 3 3 7523770016 14 6 5 1 661320660 14 3 4 5 16427471172 14 7 2 3 1444432612 14 7 4 1 341825484 14 4 3 5 16427471172 14 8 1 3 87291204 14 8 3 1 87291204 14 5 2 5 6100939726 14 1 9 2 9798789 14 9 2 1 9798789 14 6 1 5 661320660 14 2 8 2 210164227 14 10 1 1 355355 14 1 7 4 341825484 14 3 7 2 1444432612 14 2 6 4 4286172247 14 4 6 2 4286172247 14 sum 206254571236 14 3 5 4 16427471172 14 5 5 2 6100939726 14 4 4 4 25199010256 14 6 4 2 4286172247 A.4 Genus 3 d v e f h 13 1 7 1 8691683 7 1 1 1 180 11 sum 112868844 13 2 6 1 108452916 13 3 5 1 414918075 7 sum 180 12 1 1 6 2641925 13 4 4 1 636184120 12 1 2 5 24656775 13 5 3 1 414918075 8 1 1 2 3044 12 2 1 5 24656775 13 6 2 1 108452916 8 1 2 1 3044 12 1 3 4 66805310 13 7 1 1 8691683 8 2 1 1 3044 12 2 2 4 159762815 12 3 1 4 66805310 13 sum 28540603884 8 sum 9132 12 1 4 3 66805310 12 2 3 3 280514670 14 1 1 8 25537655 9 1 1 3 26060 12 3 2 3 280514670 14 1 2 7 409732895 9 1 2 2 63600 12 4 1 3 66805310 14 2 1 7 409732895 9 2 1 2 63600 12 1 5 2 24656775 14 1 3 6 2096068975 9 1 3 1 26060 12 2 4 2 159762815 14 2 2 6 4973691275 9 2 2 1 63600 12 3 3 2 280514670 14 3 1 6 2096068975 9 3 1 1 26060 12 4 2 2 159762815 14 1 4 5 4538348815 12 5 1 2 24656775 14 2 3 5 18733893115 9 sum 268980 12 1 6 1 2641925 14 3 2 5 18733893115 12 2 5 1 24656775 14 4 1 5 4538348815 10 1 1 4 152900 12 3 4 1 66805310 14 1 5 4 4538348815 10 1 2 3 659340 12 4 3 1 66805310 14 2 4 4 28579309570 10 2 1 3 659340 12 5 2 1 24656775 14 3 3 4 49719495672 10 1 3 2 659340 12 6 1 1 2641925 14 4 2 4 28579309570 10 2 2 2 1595480 14 5 1 4 4538348815 10 3 1 2 659340 12 sum 1877530740 14 1 6 3 2096068975 10 1 4 1 152900 14 2 5 3 18733893115 10 2 3 1 659340 13 1 1 7 8691683 14 3 4 3 49719495672 10 3 2 1 659340 13 1 2 6 108452916 14 4 3 3 49719495672 10 4 1 1 152900 13 2 1 6 108452916 14 5 2 3 18733893115 13 1 3 5 414918075 14 6 1 3 2096068975 10 sum 6010220 13 2 2 5 988043771 14 1 7 2 409732895 13 3 1 5 414918075 14 2 6 2 4973691275 11 1 1 5 696905 13 1 4 4 636184120 14 3 5 2 18733893115 11 1 2 4 4606910 13 2 3 4 2646424729 14 4 4 2 28579309570 11 2 1 4 4606910 13 3 2 4 2646424729 14 5 3 2 18733893115 11 1 3 3 8141100 13 4 1 4 636184120 14 6 2 2 4973691275 11 2 2 3 19571123 13 1 5 3 414918075 14 7 1 2 409732895 11 3 1 3 8141100 13 2 4 3 2646424729 14 1 8 1 25537655 11 1 4 2 4606910 13 3 3 3 4623070842 14 2 7 1 409732895 11 2 3 2 19571123 13 4 2 3 2646424729 14 3 6 1 2096068975 11 3 2 2 19571123 13 5 1 3 414918075 14 4 5 1 4538348815 11 4 1 2 4606910 13 1 6 2 108452916 14 5 4 1 4538348815 11 1 5 1 696905 13 2 5 2 988043771 14 6 3 1 2096068975 11 2 4 1 4606910 13 3 4 2 2646424729 14 7 2 1 409732895 11 3 3 1 8141100 13 4 3 2 2646424729 14 8 1 1 25537655 11 4 2 1 4606910 13 5 2 2 988043771 11 5 1 1 696905 13 6 1 2 108452916 14 sum 404562365316 A.5 Genus 4 d v e f h 12 3 1 2 75220860 9 1 1 1 8064 12 1 4 1 18128396 14 1 1 6 539651112 12 2 3 1 75220860 14 1 2 5 4736419688 9 sum 8064 12 3 2 1 75220860 14 2 1 5 4736419688 12 4 1 1 18128396 14 1 3 4 12465308856 10 1 1 2 193248 14 2 2 4 29310854804 10 1 2 1 193248 12 sum 684173164 14 3 1 4 12465308856 10 2 1 1 193248 14 1 4 3 12465308856 13 1 1 5 109425316 14 2 3 3 50713072144 10 sum 579744 13 1 2 4 687238552 14 3 2 3 50713072144 13 2 1 4 687238552 14 4 1 3 12465308856 11 1 1 3 2286636 13 1 3 3 1194737544 14 1 5 2 4736419688 11 1 2 2 5458464 13 2 2 3 2820651496 14 2 4 2 29310854804 11 2 1 2 5458464 13 3 1 3 1194737544 14 3 3 2 50713072144 11 1 3 1 2286636 13 1 4 2 687238552 14 4 2 2 29310854804 11 2 2 1 5458464 13 2 3 2 2820651496 14 5 1 2 4736419688 11 3 1 1 2286636 13 3 2 2 2820651496 14 1 6 1 539651112 13 4 1 2 687238552 14 2 5 1 4736419688 11 sum 23235300 13 1 5 1 109425316 14 3 4 1 12465308856 13 2 4 1 687238552 14 4 3 1 12465308856 12 1 1 4 18128396 13 3 3 1 1194737544 14 5 2 1 4736419688 12 1 2 3 75220860 13 4 2 1 687238552 14 6 1 1 539651112 12 2 1 3 75220860 13 5 1 1 109425316 12 1 3 2 75220860 14 sum 344901105444 12 2 2 2 178462816 13 sum 16497874380 A.6 Genus 5 d v e f h 13 1 1 3 292271616 14 2 1 3 11947069680 11 1 1 1 604800 13 1 2 2 686597184 14 1 3 2 11947069680 13 2 1 2 686597184 14 2 2 2 27934773440 11 sum 604800 13 1 3 1 292271616 14 3 1 2 11947069680 13 2 2 1 686597184 14 1 4 1 2961802480 12 1 1 2 19056960 13 3 1 1 292271616 14 2 3 1 11947069680 12 1 2 1 19056960 14 3 2 1 11947069680 12 2 1 1 19056960 13 sum 2936606400 14 4 1 1 2961802480 12 sum 57170880 14 1 1 4 2961802480 14 sum 108502598960 14 1 2 3 11947069680 A.7 Genus 6 d v e f h 13 1 1 1 68428800 14 1 1 2 2699672832 14 sum 8099018496 14 1 2 1 2699672832 13 sum 68428800 14 2 1 1 2699672832 These tables extend to 14 darts the part of Appendix B of [24] about rooted hypermaps. B First numbers of unrooted hypermaps The following sections show the numbers H of unrooted hypermaps of genus g with d darts, v vertices, e edges and d - v - e + 2(1 - g) faces, for g < 6 and d < 14. B.1 Genus 0 d v e f H 6 1 5 2 3 8 2 5 3 309 1 1 1 1 1 6 2 4 2 24 8 3 4 3 946 6 3 3 2 46 8 4 3 3 946 1 sum 1 6 4 2 2 24 8 5 2 3 309 6 5 1 2 3 8 6 1 3 26 2 1 1 2 1 6 1 6 1 1 8 1 7 2 4 2 1 2 1 1 6 2 5 1 3 8 2 6 2 67 2 2 1 1 1 6 3 4 1 10 8 3 5 2 309 6 4 3 1 10 8 4 4 2 505 2 sum 3 6 5 2 1 3 8 5 3 2 309 6 6 1 1 1 8 6 2 2 67 3 1 1 3 1 8 7 1 2 4 3 1 2 2 1 6 sum 291 8 1 8 1 1 3 2 1 2 1 8 2 7 1 4 3 1 3 1 1 7 1 1 7 1 8 3 6 1 26 3 2 2 1 1 7 1 2 6 3 8 4 5 1 64 3 3 1 1 1 7 2 1 6 3 8 5 4 1 64 7 1 3 5 15 8 6 3 1 26 3 sum 6 7 2 2 5 40 8 7 2 1 4 7 3 1 5 15 8 8 1 1 1 4 1 1 4 1 7 1 4 4 25 4 1 2 3 2 7 2 3 4 127 8 sum 6975 4 2 1 3 2 7 3 2 4 127 4 1 3 2 2 7 4 1 4 25 9 1 1 9 1 4 2 2 2 5 7 1 5 3 15 9 1 2 8 4 4 3 1 2 2 7 2 4 3 127 9 2 1 8 4 4 1 4 1 1 7 3 3 3 242 9 1 3 7 38 4 2 3 1 2 7 4 2 3 127 9 2 2 7 98 4 3 2 1 2 7 5 1 3 15 9 3 1 7 38 4 4 1 1 1 7 1 6 2 3 9 1 4 6 132 7 2 5 2 40 9 2 3 6 640 4 sum 20 7 3 4 2 127 9 3 2 6 640 7 4 3 2 127 9 4 1 6 132 5 1 1 5 1 7 5 2 2 40 9 1 5 5 196 5 1 2 4 2 7 6 1 2 3 9 2 4 5 1549 5 2 1 4 2 7 1 7 1 1 9 3 3 5 2890 5 1 3 3 4 7 2 6 1 3 9 4 2 5 1549 5 2 2 3 11 7 3 5 1 15 9 5 1 5 196 5 3 1 3 4 7 4 4 1 25 9 1 6 4 132 5 1 4 2 2 7 5 3 1 15 9 2 5 4 1549 5 2 3 2 11 7 6 2 1 3 9 3 4 4 4671 5 3 2 2 11 7 7 1 1 1 9 4 3 4 4671 5 4 1 2 2 9 5 2 4 1549 5 1 5 1 1 7 sum 1310 9 6 1 4 132 5 2 4 1 2 9 1 7 3 38 5 3 3 1 4 8 1 1 8 1 9 2 6 3 640 5 4 2 1 2 8 1 2 7 4 9 3 5 3 2890 5 5 1 1 1 8 2 1 7 4 9 4 4 3 4671 8 1 3 6 26 9 5 3 3 2890 5 sum 60 8 2 2 6 67 9 6 2 3 640 8 3 1 6 26 9 7 1 3 38 6 1 1 6 1 8 1 4 5 64 9 1 8 2 4 6 1 2 5 3 8 2 3 5 309 9 2 7 2 98 6 2 1 5 3 8 3 2 5 309 9 3 6 2 640 6 1 3 4 10 8 4 1 5 64 9 4 5 2 1549 6 2 2 4 24 8 1 5 4 64 9 5 4 2 1549 6 3 1 4 10 8 2 4 4 505 9 6 3 2 640 6 1 4 3 10 8 3 3 4 946 9 7 2 2 98 6 2 3 3 46 8 4 2 4 505 9 8 1 2 4 6 3 2 3 46 8 5 1 4 64 9 1 9 1 1 6 4 1 3 10 8 1 6 3 26 9 2 8 1 4 1 1 1 1 1 1 1 m 0 9 9 8 8 8 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 m 1 0 38 11 2 1 10 5 12 1 3 10 104 132 11 1 3 9 75 12 2 2 10 265 196 11 2 2 9 195 12 3 1 10 104 132 11 3 1 9 75 12 1 4 9 765 38 11 1 4 8 450 12 2 3 9 3605 4 11 2 3 8 2154 12 3 2 9 3605 1 11 3 2 8 2154 12 4 1 9 765 11 4 1 8 450 12 1 5 8 2736 37746 11 1 5 7 1260 12 2 4 8 20472 11 2 4 7 9585 12 3 3 8 37545 1 11 3 3 7 17664 12 4 2 8 20472 5 11 4 2 7 9585 12 5 1 8 2736 5 11 5 1 7 1260 12 1 6 7 5102 56 11 1 6 6 1764 12 2 5 7 55633 144 11 2 5 6 19691 12 3 4 7 161455 56 11 3 4 6 57715 12 4 3 7 161455 256 11 4 3 6 57715 12 5 2 7 55633 1226 11 5 2 6 19691 12 6 1 7 5102 1226 11 6 1 6 1764 12 1 7 6 5102 256 11 1 7 5 1260 12 2 6 6 77174 536 11 2 6 5 19691 12 3 5 6 326432 4111 11 3 5 5 84714 12 4 4 6 516507 7606 11 4 4 5 134750 12 5 3 6 326432 4111 11 5 3 5 84714 12 6 2 6 77174 536 11 6 2 5 19691 12 7 1 6 5102 536 11 7 1 5 1260 12 1 8 5 2736 6081 11 1 8 4 450 12 2 7 5 55633 18019 11 2 7 4 9585 12 3 6 5 326432 18019 11 3 6 4 57715 12 4 5 5 752940 6081 11 4 5 4 134750 12 5 4 5 752940 536 11 5 4 4 134750 12 6 3 5 326432 256 11 6 3 4 57715 12 7 2 5 55633 4111 11 7 2 4 9585 12 8 1 5 2736 18019 11 8 1 4 450 12 1 9 4 765 28852 11 1 9 3 75 12 2 8 4 20472 18019 11 2 8 3 2154 12 3 7 4 161455 4111 11 3 7 3 17664 12 4 6 4 516507 256 11 4 6 3 57715 12 5 5 4 752940 56 11 5 5 3 84714 12 6 4 4 516507 1226 11 6 4 3 57715 12 7 3 4 161455 7606 11 7 3 3 17664 12 8 2 4 20472 18019 11 8 2 3 2154 12 9 1 4 765 18019 11 9 1 3 75 12 1 10 3 104 7606 11 1 10 2 5 12 2 9 3 3605 1226 11 2 9 2 195 12 3 8 3 37545 56 11 3 8 2 2154 12 4 7 3 161455 5 11 4 7 2 9585 12 5 6 3 326432 144 11 5 6 2 19691 12 6 5 3 326432 1226 11 6 5 2 19691 12 7 4 3 161455 4111 11 7 4 2 9585 12 8 3 3 37545 6081 11 8 3 2 2154 12 9 2 3 3605 4111 11 9 2 2 195 12 10 1 3 104 1226 11 10 1 2 5 12 1 11 2 6 144 11 1 11 1 1 12 2 10 2 265 5 11 2 10 1 5 12 3 9 2 3605 1 11 3 9 1 75 12 4 8 2 20472 5 11 4 8 1 450 12 5 7 2 55633 56 11 5 7 1 1260 12 6 6 2 77174 256 11 6 6 1 1764 12 7 5 2 55633 536 11 7 5 1 1260 12 8 4 2 20472 536 11 8 4 1 450 12 9 3 2 3605 256 11 9 3 1 75 12 10 2 2 265 56 11 10 2 1 5 12 11 1 2 6 5 11 11 1 1 1 12 1 12 1 1 1 12 2 11 1 6 11 sum 1262874 12 3 10 1 104 215602 12 4 9 1 765 12 1 1 12 1 12 5 8 1 2736 1 12 1 2 11 6 12 6 7 1 5102 5 12 2 1 11 6 12 7 6 1 5102 12 8 5 1 2736 13 8 4 3 404965 14 3 7 6 5731330 12 9 4 1 765 13 9 3 3 73920 14 4 6 6 17809776 12 10 3 1 104 13 10 2 3 5709 14 5 5 6 25720986 12 11 2 1 6 13 11 1 3 132 14 6 4 6 17809776 12 12 1 1 1 13 1 12 2 6 14 7 3 6 5731330 13 2 11 2 341 14 8 2 6 764633 12 sum 7611156 13 3 10 2 5709 14 9 1 6 30711 13 4 9 2 40425 14 1 10 5 10247 13 1 1 13 1 13 5 8 2 140591 14 2 9 5 325652 13 1 2 12 6 13 6 7 2 258194 14 3 8 5 3159069 13 2 1 12 6 13 7 6 2 258194 14 4 7 5 12995424 13 1 3 11 132 13 8 5 2 140591 14 5 6 5 25720986 13 2 2 11 341 13 9 4 2 40425 14 6 5 5 25720986 13 3 1 11 132 13 10 3 2 5709 14 7 4 5 12995424 13 1 4 10 1210 13 11 2 2 341 14 8 3 5 3159069 13 2 3 10 5709 13 12 1 2 6 14 9 2 5 325652 13 3 2 10 5709 13 1 13 1 1 14 10 1 5 10247 13 4 1 10 1210 13 2 12 1 6 14 1 11 4 1868 13 1 5 9 5445 13 3 11 1 132 14 2 10 4 75283 13 2 4 9 40425 13 4 10 1 1210 14 3 9 4 931845 13 3 3 9 73920 13 5 9 1 5445 14 4 8 4 4960016 13 4 2 9 40425 13 6 8 1 13068 14 5 7 4 12995424 13 5 1 9 5445 13 7 7 1 17424 14 6 6 4 17809776 13 1 6 8 13068 13 8 6 1 13068 14 7 5 4 12995424 13 2 5 8 140591 13 9 5 1 5445 14 8 4 4 4960016 13 3 4 8 404965 13 10 4 1 1210 14 9 3 4 931845 13 4 3 8 404965 13 11 3 1 132 14 10 2 4 75283 13 5 2 8 140591 13 12 2 1 6 14 11 1 4 1868 13 6 1 8 13068 13 13 1 1 1 14 1 12 3 172 13 1 7 7 17424 14 2 11 3 8741 13 2 6 7 258194 13 sum 46814132 14 3 10 3 137217 13 3 5 7 1078456 14 4 9 3 931845 13 4 4 7 1699200 14 1 1 14 1 14 5 8 3 3159069 13 5 3 7 1078456 14 1 2 13 7 14 6 7 3 5731330 13 6 2 7 258194 14 2 1 13 7 14 7 6 3 5731330 13 7 1 7 17424 14 1 3 12 172 14 8 5 3 3159069 13 1 8 6 13068 14 2 2 12 440 14 9 4 3 931845 13 2 7 6 258194 14 3 1 12 172 14 10 3 3 137217 13 3 6 6 1485778 14 1 4 11 1868 14 11 2 3 8741 13 4 5 6 3395140 14 2 3 11 8741 14 12 1 3 172 13 5 4 6 3395140 14 3 2 11 8741 14 1 13 2 7 13 6 3 6 1485778 14 4 1 11 1868 14 2 12 2 440 13 7 2 6 258194 14 1 5 10 10247 14 3 11 2 8741 13 8 1 6 13068 14 2 4 10 75283 14 4 10 2 75283 13 1 9 5 5445 14 3 3 10 137217 14 5 9 2 325652 13 2 8 5 140591 14 4 2 10 75283 14 6 8 2 764633 13 3 7 5 1078456 14 5 1 10 10247 14 7 7 2 1012271 13 4 6 5 3395140 14 1 6 9 30711 14 8 6 2 764633 13 5 5 5 4924094 14 2 5 9 325652 14 9 5 2 325652 13 6 4 5 3395140 14 3 4 9 931845 14 10 4 2 75283 13 7 3 5 1078456 14 4 3 9 931845 14 11 3 2 8741 13 8 2 5 140591 14 5 2 9 325652 14 12 2 2 440 13 9 1 5 5445 14 6 1 9 30711 14 13 1 2 7 13 1 10 4 1210 14 1 7 8 52634 14 1 14 1 1 13 2 9 4 40425 14 2 6 8 764633 14 2 13 1 7 13 3 8 4 404965 14 3 5 8 3159069 14 3 12 1 172 13 4 7 4 1699200 14 4 4 8 4960016 14 4 11 1 1868 13 5 6 4 3395140 14 5 3 8 3159069 14 5 10 1 10247 13 6 5 4 3395140 14 6 2 8 764633 14 6 9 1 30711 13 7 4 4 1699200 14 7 1 8 52634 14 7 8 1 52634 13 8 3 4 404965 14 1 8 7 52634 14 8 7 1 52634 13 9 2 4 40425 14 2 7 7 1012271 14 9 6 1 30711 13 10 1 4 1210 14 3 6 7 5731330 14 10 5 1 10247 13 1 11 3 132 14 4 5 7 12995424 14 11 4 1 1868 13 2 10 3 5709 14 5 4 7 12995424 14 12 3 1 172 13 3 9 3 73920 14 6 3 7 5731330 14 13 2 1 7 13 4 8 3 404965 14 7 2 7 1012271 14 14 1 1 1 13 5 7 3 1078456 14 8 1 7 52634 13 6 6 3 1485778 14 1 9 6 30711 14 sum 293447817 13 7 5 3 1078456 14 2 8 6 764633 1 f 1 m 2 1 1 m 3 2 2 1 1 1 m 4 3 3 2 2 2 1 1 1 1 m 5 4 4 3 3 3 2 2 2 2 1 1 1 1 1 m 6 5 5 4 4 4 3 3 3 3 2 2 2 2 2 1 7 31 31 31 78 31 7 31 31 7 285 10 80 80 150 385 150 80 385 385 80 10 80 150 80 10 2115 17 187 187 557 1409 557 557 2597 2597 557 187 1409 2597 1409 187 17 8 2 5 1 187 10 1 8 1 34 8 3 4 1 557 10 2 7 1 698 8 4 3 1 557 10 3 6 1 4172 8 5 2 1 187 10 4 5 1 9724 8 6 1 1 17 10 5 4 1 9724 10 6 3 1 4172 8 sum 16533 10 7 2 1 698 10 8 1 1 34 9 1 1 7 24 9 1 2 6 374 10 sum 972441 9 2 1 6 374 9 1 3 5 1634 11 1 1 9 45 9 2 2 5 4115 11 1 2 8 1200 9 3 1 5 1634 11 2 1 8 1200 9 1 4 4 2616 11 1 3 7 9450 9 2 3 4 12033 11 2 2 7 23547 9 3 2 4 12033 11 3 1 7 9450 9 4 1 4 2616 11 1 4 6 30240 9 1 5 3 1634 11 2 3 6 135775 9 2 4 3 12033 11 3 2 6 135775 9 3 3 3 21990 11 4 1 6 30240 9 4 2 3 12033 11 1 5 5 44100 9 5 1 3 1634 11 2 4 5 310985 9 1 6 2 374 11 3 3 5 560498 9 2 5 2 4115 11 4 2 5 310985 9 3 4 2 12033 11 5 1 5 44100 9 4 3 2 12033 11 1 6 4 30240 9 5 2 2 4115 11 2 5 4 310985 9 6 1 2 374 11 3 4 4 880403 9 1 7 1 24 11 4 3 4 880403 9 2 6 1 374 11 5 2 4 310985 9 3 5 1 1634 11 6 1 4 30240 9 4 4 1 2616 11 1 7 3 9450 9 5 3 1 1634 11 2 6 3 135775 9 6 2 1 374 11 3 5 3 560498 9 7 1 1 24 11 4 4 3 880403 11 5 3 3 560498 9 sum 126501 11 6 2 3 135775 11 7 1 3 9450 10 1 1 8 34 11 1 8 2 1200 10 1 2 7 698 11 2 7 2 23547 10 2 1 7 698 11 3 6 2 135775 10 1 3 6 4172 11 4 5 2 310985 10 2 2 6 10434 11 5 4 2 310985 10 3 1 6 4172 11 6 3 2 135775 10 1 4 5 9724 11 7 2 2 23547 10 2 3 5 44091 11 8 1 2 1200 10 3 2 5 44091 11 1 9 1 45 10 4 1 5 9724 11 2 8 1 1200 10 1 5 4 9724 11 3 7 1 9450 10 2 4 4 69790 11 4 6 1 30240 10 3 3 4 126519 11 5 5 1 44100 10 4 2 4 69790 11 6 4 1 30240 10 5 1 4 9724 11 7 3 1 9450 10 1 6 3 4172 11 8 2 1 1200 10 2 5 3 44091 11 9 1 1 45 10 3 4 3 126519 10 4 3 3 126519 11 sum 7451679 10 5 2 3 44091 10 6 1 3 4172 12 1 1 10 62 10 1 7 2 698 12 1 2 9 1976 10 2 6 2 10434 12 2 1 9 1976 10 3 5 2 44091 12 1 3 8 19694 10 4 4 2 69790 12 2 2 8 48846 10 5 3 2 44091 12 3 1 8 19694 10 6 2 2 10434 12 1 4 7 82652 10 7 1 2 698 12 2 3 7 367645 o 33 12 3 2 7 367645 13 3 4 6 20047636 14 1 6 7 2760990 12 4 1 7 82652 13 4 3 6 20047636 14 2 5 7 26837442 12 1 5 6 165262 13 5 2 6 7221592 14 3 4 7 73967488 12 2 4 6 1147628 13 6 1 6 731808 14 4 3 7 73967488 12 3 3 6 2058329 13 1 7 5 533610 14 5 2 7 26837442 12 4 2 6 1147628 13 2 6 5 7221592 14 6 1 7 2760990 12 5 1 6 165262 13 3 5 5 28831218 14 1 7 6 2760990 12 1 6 5 165262 13 4 4 5 44800675 14 2 6 6 36580432 12 2 5 5 1660331 13 5 3 5 28831218 14 3 5 6 144298902 12 3 4 5 4649379 13 6 2 5 7221592 14 4 4 6 223353280 12 4 3 5 4649379 13 7 1 5 533610 14 5 3 6 144298902 12 5 2 5 1660331 13 1 8 4 203280 14 6 2 6 36580432 12 6 1 5 165262 13 2 7 4 3661896 14 7 1 6 2760990 12 1 7 4 82652 13 3 6 4 20047636 14 1 8 5 1533950 12 2 6 4 1147628 13 4 5 4 44800675 14 2 7 5 26837442 12 3 5 4 4649379 13 5 4 4 44800675 14 3 6 5 144298902 12 4 4 4 7259140 13 6 3 4 20047636 14 4 5 5 319684549 12 5 3 4 4649379 13 7 2 4 3661896 14 5 4 5 319684549 12 6 2 4 1147628 13 8 1 4 203280 14 6 3 5 144298902 12 7 1 4 82652 13 1 9 3 38115 14 7 2 5 26837442 12 1 8 3 19694 13 2 8 3 898051 14 8 1 5 1533950 12 2 7 3 367645 13 3 7 3 6542368 14 1 9 4 460245 12 3 6 3 2058329 13 4 6 3 20047636 14 2 8 4 10419653 12 4 5 3 4649379 13 5 5 3 28831218 14 3 7 4 73967488 12 5 4 3 4649379 13 6 4 3 20047636 14 4 6 4 223353280 12 6 3 3 2058329 13 7 3 3 6542368 14 5 5 4 319684549 12 7 2 3 367645 13 8 2 3 898051 14 6 4 4 223353280 12 8 1 3 19694 13 9 1 3 38115 14 7 3 4 73967488 12 1 9 2 1976 13 1 10 2 3080 14 8 2 4 10419653 12 2 8 2 48846 13 2 9 2 94281 14 9 1 4 460245 12 3 7 2 367645 13 3 8 2 898051 14 1 10 3 69765 12 4 6 2 1147628 13 4 7 2 3661896 14 2 9 3 2020530 12 5 5 2 1660331 13 5 6 2 7221592 14 3 8 3 18554641 12 6 4 2 1147628 13 6 5 2 7221592 14 4 7 3 73967488 12 7 3 2 367645 13 7 4 2 3661896 14 5 6 3 144298902 12 8 2 2 48846 13 8 3 2 898051 14 6 5 3 144298902 12 9 1 2 1976 13 9 2 2 94281 14 7 4 3 73967488 12 1 10 1 62 13 10 1 2 3080 14 8 3 3 18554641 12 2 9 1 1976 13 1 11 1 77 14 9 2 3 2020530 12 3 8 1 19694 13 2 10 1 3080 14 10 1 3 69765 12 4 7 1 82652 13 3 9 1 38115 14 1 11 2 4659 12 5 6 1 165262 13 4 8 1 203280 14 2 10 2 172040 12 6 5 1 165262 13 5 7 1 533610 14 3 9 2 2020530 12 7 4 1 82652 13 6 6 1 731808 14 4 8 2 10419653 12 8 3 1 19694 13 7 5 1 533610 14 5 7 2 26837442 12 9 2 1 1976 13 8 4 1 203280 14 6 6 2 36580432 12 10 1 1 62 13 9 3 1 38115 14 7 5 2 26837442 13 10 2 1 3080 14 8 4 2 10419653 12 sum 57167260 13 11 1 1 77 14 9 3 2 2020530 14 10 2 2 172040 13 1 1 11 77 13 sum 438644841 14 11 1 2 4659 13 1 2 10 3080 14 1 12 1 99 13 2 1 10 3080 14 1 1 12 99 14 2 11 1 4659 13 1 3 9 38115 14 1 2 11 4659 14 3 10 1 69765 13 2 2 9 94281 14 2 1 11 4659 14 4 9 1 460245 13 3 1 9 38115 14 1 3 10 69765 14 5 8 1 1533950 13 1 4 8 203280 14 2 2 10 172040 14 6 7 1 2760990 13 2 3 8 898051 14 3 1 10 69765 14 7 6 1 2760990 13 3 2 8 898051 14 1 4 9 460245 14 8 5 1 1533950 13 4 1 8 203280 14 2 3 9 2020530 14 9 4 1 460245 13 1 5 7 533610 14 3 2 9 2020530 14 10 3 1 69765 13 2 4 7 3661896 14 4 1 9 460245 14 11 2 1 4659 13 3 3 7 6542368 14 1 5 8 1533950 14 12 1 1 99 13 4 2 7 3661896 14 2 4 8 10419653 13 5 1 7 533610 14 3 3 8 18554641 14 sum 3369276867 13 1 6 6 731808 14 4 2 8 10419653 13 2 5 6 7221592 14 5 1 8 1533950 B.3 Genus 2 d v e f H 10 2 S 1 1672S 12 1 a 1 7417 S 1 1 1 4 10 3 4 1 47164 12 2 7 1 132202 10 4 3 1 47164 12 3 6 1 7213a2 S sum 4 10 S 2 1 1672S 12 4 S 1 1610617 10 6 1 1 1649 12 S 4 1 1610617 6 1 1 2 16 12 6 3 1 7213a2 6 1 2 1 16 10 sum 13SS400 12 7 2 1 132202 6 2 1 1 16 11 1 1 7 3633 12 a 1 1 7417 6 sum 4a 11 11 1 2 2 1 6 6 S0001 S0001 12 sum 1S0429a19 7 1 1 3 67 11 1 3 S 20191S 13 1 1 9 14091 7 1 2 2 169 11 2 2 S 491729 13 1 2 a 316470 7 2 1 2 169 11 3 1 S 20191S 13 2 1 a 316470 7 1 3 1 67 11 1 4 4 31S000 13 1 3 7 2241162 7 2 2 1 169 11 2 3 4 1364986 13 2 2 7 S412aa3 7 3 1 1 67 11 3 2 4 1364986 13 3 1 7 2241162 11 4 1 4 31S000 13 1 4 6 6764142 7 sum 70a 11 1 S 3 20191S 13 2 3 6 2a7790S1 11 2 4 3 1364986 13 3 2 6 2a7790S1 a 1 1 4 237 11 3 3 3 242aa62 13 4 1 6 6764142 a 1 2 3 1072 11 4 2 3 1364986 13 1 S S 96a1210 a 2 1 3 1072 11 S 1 3 20191S 13 2 4 S 634Sa6S4 a 1 3 2 1072 11 1 6 2 S0001 13 3 3 S 111a01142 a 2 2 2 2664 11 2 S 2 491729 13 4 2 S 634Sa6S4 a 3 1 2 1072 11 3 4 2 1364986 13 S 1 S 96a1210 a 1 4 1 237 11 4 3 2 1364986 13 1 6 4 6764142 a 2 3 1 1072 11 S 2 2 491729 13 2 S 4 634Sa6S4 a 3 2 1 1072 11 6 1 2 S0001 13 3 4 4 1722S2340 a 4 1 1 237 11 1 7 1 3633 13 4 3 4 1722S2340 11 2 6 1 S0001 13 S 2 4 634Sa6S4 a sum 9a07 11 3 S 1 20191S 13 6 1 4 6764142 11 4 4 1 31S000 13 1 7 3 2241162 9 1 1 S 667 11 S 3 1 20191S 13 2 6 3 2a7790S1 9 1 2 4 4736 11 6 2 1 S0001 13 3 S 3 111a01142 9 2 1 4 4736 11 7 1 1 3633 13 4 4 3 1722S2340 9 1 3 3 aS60 13 S 3 3 111a01142 9 2 2 3 21113 11 sum 14S61360 13 6 2 3 2a7790S1 9 3 1 3 aS60 13 7 1 3 2241162 9 1 4 2 4736 12 1 1 a 7417 13 1 a 2 316470 9 2 3 2 21113 12 1 2 7 132202 13 2 7 2 S412aa3 9 3 2 2 21113 12 2 1 7 132202 13 3 6 2 2a7790S1 9 4 1 2 4736 12 1 3 6 7213a2 13 4 S 2 634Sa6S4 9 1 S 1 667 12 2 2 6 174a723 13 S 4 2 634Sa6S4 9 2 4 1 4736 12 3 1 6 7213a2 13 6 3 2 2a7790S1 9 3 3 1 aS60 12 1 4 S 1610617 13 7 2 2 S412aa3 9 4 2 1 4736 12 2 3 S 690a644 13 a 1 2 316470 9 S 1 1 667 12 3 2 S 690a644 13 1 9 1 14091 12 4 1 S 1610617 13 2 a 1 316470 9 sum 119436 12 1 S 4 1610617 13 3 7 1 2241162 12 2 4 4 10702449 13 4 6 1 6764142 10 1 1 6 1649 12 3 3 4 1a93a994 13 S S 1 96a1210 10 1 2 S 1672S 12 4 2 4 10702449 13 6 4 1 6764142 10 2 1 S 1672S 12 S 1 4 1610617 13 7 3 1 2241162 10 1 3 4 47164 12 1 6 3 7213a2 13 a 2 1 316470 10 2 2 4 11S47a 12 2 S 3 690a644 13 9 1 1 14091 10 3 1 4 47164 12 3 4 3 1a93a994 10 1 4 3 47164 12 4 3 3 1a93a994 13 sum 1S06a41a72 10 2 3 3 20 68 9S 12 S 2 3 690a644 10 3 2 3 20 68 9S 12 6 1 3 7213a2 14 1 1 10 2S40S 10 4 1 3 47164 12 1 7 2 132202 14 1 2 9 70004S 10 1 S 2 1672S 12 2 6 2 174a723 14 2 1 9 70004S 10 2 4 2 11S47a 12 3 S 2 690a644 14 1 3 a 623SS26 10 3 3 2 206895 12 4 4 2 10702449 14 2 2 a 1S012496 10 4 2 2 11S47a 12 S 3 2 690a644 14 3 1 a 623SS26 10 S 1 2 1672S 12 6 2 2 174a723 14 1 4 7 24417030 10 1 6 1 1649 12 7 1 2 132202 14 2 3 7 10317S7aS 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 B. d 7 7 8 8 8 8 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 3 2 7 103175785 14 5 3 4 1173398706 14 7 3 2 103175785 4 1 7 24417030 14 6 2 4 306159286 14 8 2 2 15012496 1 5 6 47238510 14 7 1 4 24417030 14 9 1 2 700045 2 4 6 306159286 14 1 8 3 6235526 14 1 10 1 25405 3 3 6 537417269 14 2 7 3 103175785 14 2 9 1 700045 4 2 6 306159286 14 3 6 3 537417269 14 3 8 1 6235526 5 1 6 47238510 14 4 5 3 1173398706 14 4 7 1 24417030 1 6 5 47238510 14 5 4 3 1173398706 14 5 6 1 47238510 2 5 5 435785878 14 6 3 3 537417269 14 6 5 1 47238510 3 4 5 1173398706 14 7 2 3 103175785 14 7 4 1 24417030 4 3 5 1173398706 14 8 1 3 6235526 14 8 3 1 6235526 5 2 5 435785878 14 1 9 2 700045 14 9 2 1 700045 6 1 5 47238510 14 2 8 2 15012496 14 10 1 1 25405 1 7 4 24417030 14 3 7 2 103175785 2 6 4 306159286 14 4 6 2 306159286 14 sum 147326131 3 5 4 1173398706 14 5 5 2 435785878 4 4 4 1799940644 14 6 4 2 306159286 Genus 3 v e f H 13 1 7 1 668591 1 1 1 30 11 sum 10260804 13 2 6 1 8342532 13 3 5 1 31916775 sum 30 12 1 1 6 220244 13 4 4 1 48937240 12 1 2 5 2054974 13 5 3 1 31916775 1 1 2 385 12 2 1 5 2054974 13 6 2 1 8342532 1 2 1 385 12 1 3 4 5567550 13 7 1 1 668591 2 1 1 385 12 2 2 4 13314231 12 3 1 4 5567550 13 sum 2195431068 sum 1155 12 1 4 3 5567550 12 2 3 3 23377106 14 1 1 8 1824323 1 1 3 2900 12 3 2 3 23377106 14 1 2 7 29267487 1 2 2 7070 12 4 1 3 5567550 14 2 1 7 29267487 2 1 2 7070 12 1 5 2 2054974 14 1 3 6 149721473 1 3 1 2900 12 2 4 2 13314231 14 2 2 6 355267058 2 2 1 7070 12 3 3 2 23377106 14 3 1 6 149721473 3 1 1 2900 12 4 2 2 13314231 14 1 4 5 324171185 12 5 1 2 2054974 14 2 3 5 1338142324 sum 29910 12 1 6 1 220244 14 3 2 5 1338142324 12 2 5 1 2054974 14 4 1 5 324171185 1 1 4 15308 12 3 4 1 5567550 14 1 5 4 324171185 1 2 3 65972 12 4 3 1 5567550 14 2 4 4 2041388556 2 1 3 65972 12 5 2 1 2054974 14 3 3 4 3551405485 1 3 2 65972 12 6 1 1 220244 14 4 2 4 2041388556 2 2 2 159608 14 5 1 4 324171185 3 1 2 65972 12 sum 156469887 14 1 6 3 149721473 1 4 1 15308 14 2 5 3 1338142324 2 3 1 65972 13 1 1 7 668591 14 3 4 3 3551405485 3 2 1 65972 13 1 2 6 8342532 14 4 3 3 3551405485 4 1 1 15308 13 2 1 6 8342532 14 5 2 3 1338142324 13 1 3 5 31916775 14 6 1 3 149721473 sum 601364 13 2 2 5 76003367 14 1 7 2 29267487 13 3 1 5 31916775 14 2 6 2 355267058 1 1 5 63355 13 1 4 4 48937240 14 3 5 2 1338142324 1 2 4 418810 13 2 3 4 203571133 14 4 4 2 2041388556 2 1 4 418810 13 3 2 4 203571133 14 5 3 2 1338142324 1 3 3 740100 13 4 1 4 48937240 14 6 2 2 355267058 2 2 3 1779193 13 1 5 3 31916775 14 7 1 2 29267487 3 1 3 740100 13 2 4 3 203571133 14 1 8 1 1824323 1 4 2 418810 13 3 3 3 355620834 14 2 7 1 29267487 2 3 2 1779193 13 4 2 3 203571133 14 3 6 1 149721473 3 2 2 1779193 13 5 1 3 31916775 14 4 5 1 324171185 4 1 2 418810 13 1 6 2 8342532 14 5 4 1 324171185 1 5 1 63355 13 2 5 2 76003367 14 6 3 1 149721473 2 4 1 418810 13 3 4 2 203571133 14 7 2 1 29267487 3 3 1 740100 13 4 3 2 203571133 14 8 1 1 1824323 4 2 1 418810 13 5 2 2 76003367 5 1 1 63355 13 6 1 2 8342532 14 sum 28897471080 B.5 Genus 4 d v e f H 12 3 1 2 6268712 9 1 1 1 900 12 1 4 1 1510846 14 1 1 6 38547144 12 2 3 1 6268712 14 1 2 5 338317960 9 sum 900 12 3 2 1 6268712 14 2 1 5 338317960 12 4 1 1 1510846 14 1 3 4 890383128 10 1 1 2 19344 14 2 2 4 2093639428 10 1 2 1 19344 12 sum 57017238 14 3 1 4 890383128 10 2 1 1 19344 14 1 4 3 890383128 13 1 1 5 8417332 14 2 3 3 3622371084 10 sum 58032 13 1 2 4 52864504 14 3 2 3 3622371084 13 2 1 4 52864504 14 4 1 3 890383128 11 1 1 3 207876 13 1 3 3 91902888 14 1 5 2 338317960 11 1 2 2 496224 13 2 2 3 216973192 14 2 4 2 2093639428 11 2 1 2 496224 13 3 1 3 91902888 14 3 3 2 3622371084 11 1 3 1 207876 13 1 4 2 52864504 14 4 2 2 2093639428 11 2 2 1 496224 13 2 3 2 216973192 14 5 1 2 338317960 11 3 1 1 207876 13 3 2 2 216973192 14 1 6 1 38547144 13 4 1 2 52864504 14 2 5 1 338317960 11 sum 2112300 13 1 5 1 8417332 14 3 4 1 890383128 13 2 4 1 52864504 14 4 3 1 890383128 12 1 1 4 1510846 13 3 3 1 91902888 14 5 2 1 338317960 12 1 2 3 6268712 13 4 2 1 52864504 14 6 1 1 38547144 12 2 1 3 6268712 13 5 1 1 8417332 12 1 3 2 6268712 14 sum 24635879496 12 2 2 2 14872428 13 sum 1269067260 B.6 Genus 5 d v e f H 13 1 1 3 22482432 14 2 1 3 853365360 11 1 1 1 54990 13 1 2 2 52815168 14 1 3 2 853365360 13 2 1 2 52815168 14 2 2 2 1995345826 11 sum 54990 13 1 3 1 22482432 14 3 1 2 853365360 13 2 2 1 52815168 14 1 4 1 211558928 12 1 1 2 1588218 13 3 1 1 22482432 14 2 3 1 853365360 12 1 2 1 1588218 14 3 2 1 853365360 12 2 1 1 1588218 13 sum 225892800 14 4 1 1 211558928 12 sum 4764654 14 1 1 4 211558928 14 sum 7750214770 14 1 2 3 853365360 B.7 Genus 6 d v e f H 13 1 11 5263764 14 1 1 2 192834612 14 sum 578503836 14 1 2 1 192834612 13 sum 5263764 14 2 1 1 192834612 ARS MATHEMATICA CONTEMPORANEA Author Guidelines Before submission Papers should be written in English, prepared in ETEX, and must be submitted as a PDF file. 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By mail: Ars Mathematica Contemporanea UP FAMNIT Glagoljaska 8 SI-6000 Koper Slovenia By fax: +386 5 611 75 71 By e-mail: info@famnit.upr.si ARS MATHEMATICA CONTEMPORANEA Groups and Graphs, Designs and Dynamics Yichang, China, August 12 - 25, 2019 http://math.sjtu.edu.cn/conference/G2D2 China Three Gorges University is organizing the International Conference and PhD-Master's Summer School on "Groups and Graphs, Designs and Dynamics" (G2D2). All scientific activities will take place in the Three Gorges Mathematical Research Center at China Three Gorges University, Yichang, China during August 12 - 25, 2019. The summer school part of G2D2 consists of 4 short courses and 4 colloquium talks; its conference part consists of about 20 invited talks and some contributed talks. G2D2 is concerned with all aspects of mathematics, especially those relating to simple structures and simple processes. We will bring together experts and students to exchange ideas and to enrich their mathematical horizons. Four short courses and four colloquium talks will let participants see order and simplicity from possibly new perspectives and share insights with experts. We will also schedule invited talks (45 minutes) and contributed talks (25 minutes) with topics ranging from coding theory, design theory, ergodic theory, graph theory, group theory, matrix theory, optimization theory, and quantum information theory to symbolic dynamics. Selected papers based on talks in G2D2 will be published in a special issue of The Art of Discrete and Applied Mathematics. The guest editors of the special issue: • Alexander Ivanov, Imperial College London, UK • Elena Konstantinova, Sobolev Institute of Mathematics, Novosibirsk State University, Russia • Jack Koolen, University of Science and Technology of China, China • Yaokun Wu, Shanghai Jiao Tong University, China Confirmed short courses: • Rosemary A. Bailey & Peter Cameron, University of St Andrews, UK, Laplacian Eigenvalues and Optimality • Mike Boyle & Scott Schmieding, University of Maryland and Northwestern University, USA, Symbolic Dynamics and the Stable Algebra of Matrices • Tullio Ceccherini-Silberstein, Universita del Sannio, Italy, Topics in Representation Theory • Nobuaki Obata, Tohoku University, Japan, Spectral Analysis of Growing Graphs -A Quantum Probability Point of View Steering committee: • Eiichi Bannai, China Three Gorges University, China • Sergey Goryainov, Shanghai Jiao Tong University, China • Alexander Ivanov, Imperial College London, UK • Vladislav Kabanov, Krasovskii Institute of Mathematics and Mechanics, Russia • Elena Konstantinova, Sobolev Institute of Mathematics, Novosibirsk State University, Russia • Jiping Zhang, Peking University, China ARS MATHEMATICA CONTEMPORANEA Vladimir Batagelj is 7G Vladimir Batagelj, known as Vlado to his friends, is one of the most prolific Slovenian scientists. Although he has a PhD in mathematics and has worked most of his active life at the Department of Mathematics of the University of Ljubljana where he is now Professor Emeritus, he has very broad research interests and a passion for teaching. Over ten years before obtaining his PhD, Vlado published a solo paper on quadratic hash method in the distinguished journal Communications of the ACM. When I was a graduate student of computer science at Penn State, one of our textbooks cited his paper. I was very proud to tell my fellow graduate students that Vlado and I attended the same courses as undergraduates. Actually, during compulsory military service we shared a room in barracks in Zagreb for a year. Vlado's scientific work has been cited over 11000 times in Google Scholar; over 5400 times since 2013. His most cited work, with over 3300 citations, is his book Exploratory Social Network Analysis with Pajek, written together with W. de Nooy and A. Mrvar. The book was also translated into Chinese and Japanese. The revised and expanded 3rd edition of this successful textbook was published by Cambridge University Press this year. Pajek is a highly successful, freely available software package for large networks analysis, authored by Vlado and his former PhD student Andrej Mrvar and used widely in social sciences. Vlado is one of the pioneers of discrete mathematics and theoretical computer science in Slovenia, who chartered his academic course on his own and works on problems that he finds interesting. Nevertheless, he understands the vital need for a nation of 2M to receive fresh knowledge in its own language. Vlado is the author of over 20 textbooks in Slovenian, covering a wide range of topics, from TEX to Combinatorics and Discrete Mathematics. Vladimir Batagelj Exploratory Social Network Analysis with Pajek, 3rd Edition Vlado, I wish you a very happy birthday and many more productive years! Tomo (Tomaž Pisanski) ARS MATHEMATICA CONTEMPORANEA Branko Grunbaum, Geometer Branko Grunbaum passed away on September 14, 2018, just a few weeks short of his 89th birthday. Dr. Griinbaum was an early contributor to this journal, and was a major influence on a lot of the people who have been involved with it over the years. He contributed an article [6] that appeared in its second issue that remains one of the most cited papers to have appeared here, helping raise the profile of AMC, and his most recent contribution is currently available online and will appear in 2019 [1]. Branko was a prodigious author. Over the course of his career he published over 250 articles and several books. Probably the most influential was his book Convex Polytopes [2], which first appeared in 1967. This was an indispensable reference for mathematicians working in the theory of convex polytopes, linear programming, and related combinatorial problems in geometry for at least the next two decades. Its value came not only from the thoroughness of his treatment, but the care and skill he applied in presenting some of the latest ideas and techniques in the study of convex polytopes, and the wealth of material he had collected from sometimes obscure references and then presented in an approachable and clear style. It introduced the world to Micha Perles' theory of Gale diagrams and included Branko's easy to follow proof of Steinitz's Theorem on convex polyhedra. The text also included numerous open problems and spurred much subsequent activity. The text was so esteemed as a reference that a second addition was assembled and prepared by Voker Kaibel, Victor Klee and Gunter Ziegler, and released in 2003 [5]. In the new edition the original text was presented in its entirety and supplemented with commentaries at the end of each section, these provide insight into more recent developments and discuss the status of open problems discussed in the original text. There was a long period prior to the publication of the second edition when copies were incredibly hard to obtain, and I was once told it was the most stolen book in mathematics as a consequence. His volume Tilings and Patterns, with Geoffrey C. Shephard [9], was also very influential from the moment of its publication in 1987. It is filled with beautiful diagrams and interesting mathematical results and it inspired many researchers. It also functioned very well as a coffee table book! Branko and Zdenka Griinbaum at the author's wedding in 2002. To give you a sense of the scope of the reach of his work, these two texts alone have over 1000 citations on MathSciNet, involving 1209 distinct authors! In addition to his work on convex polytopes and tilings, he also inspired many mathematicians to take up the study of configurations and arrangements of points and lines. His 1972 monograph Arrangements and Spreads was based on a series of keynote lectures he gave at the Conference on Convexity and Combinatorial Geometry at the University of Oklahoma [10]. To help set the stage he began his lectures by reading part of McElligot's Pool by Dr. Seuss [11], a book he had read to his sons when they were small, enjoining his audience to join him in the unexpected adventures that awaited in this subject area if only they would use their imaginations. His fascination with arrangements and configurations continued into his retirement, resulting in the publication of the graduate text Configurations of Points and Lines in 2009 [7]. This text is an essential reference on the subject, covering the key developments in the study of configurations since their introduction in 1876 and presenting many open problems that have inspired a new generation of mathematicians to take up their investigation. At least seventeen different articles in this journal alone have listed it as a reference. Branko was also well known for having an encyclopedic knowledge about the state of the field for a wide variety of topics in discrete geometry. His office had rows of cases filled with note cards with bibliographic information and notes on the many articles he had read over the years. People were always writing him to ask what might be known about questions they were interested in, and he often had excellent references to point them to (this was especially true before MathSciNet became popular). He was equally well known for disseminating open problems in geometry, a testimony to which are the 58 articles he wrote for Geombinatorics, a journal devoted to the discussion of open problems in combinatorial and discrete geometry. Branko also had a talent for spotting mistakes. Many of his articles contain corrections to the literature, and he often used finding such mistakes as a springboard for reopening and exploring old questions from a new perspective. Probably the most famous mistake he ever caught was in the logo of the Mathematical Association of America. From the period from 1971-1985, the official logo of the MAA — a drawing of a supposedly regular icosahedron — was drawn in such a way that it could not have been the product of any geometric projection onto the plane, a point Branko made in the pages of Mathematics Magazine [4]. The MAA, much to its credit, immediately revised its logo and started using one with greater respect for geometry. Unfortunately, Branko caught them using the bad one again a few years later, but a follow up letter from him on the question seems to have permanently resolved the issue. A recurring theme in Branko's writing was the importance of teaching geometry, and not as some highly refined and abstract activity, but through teaching the study of geometry as an area of applied mathematics. Tied closely to this was his concern that as mathematicians we have a responsibility to communicate our ideas and our proofs in a manner that not only achieves the desired result — such as proving a theorem — but doing so in a way that preserves the inherent beauty of the objects under investigation and provides genuine insight into what motivates their study [3]. He was deeply concerned by the approach of the Bourbaki to geometric subjects, and Dieudonne's famous slogan "Euclid must go!" epitomized a movement to treat geometry as a purely formal and abstract subject (so much so that the only diagram in any of the texts on geometry published by the Bourbaki is of a ARS MATHEMATICA CONTEMPORANEA Coxeter-Dynkin diagram). Now that you know something of the mathematician, I'd like to say something about the history of the man. Branko was born on October 2, 1929 in the small city of Osijek, in what was then the Kingdom of Yugoslavia and is now Croatia; Zdenka Bienenstock was born there a year later. Zdenka's family, and all of the family on Branko's father's side were Jewish. When World War II came to Yugoslavia in 1941 it uprooted their lives. Zdenka survived the war hidden in a Catholic convent, but the rest of her entire extended family was killed, many in Auschwitz. Branko's mother was Catholic, and his family survived the war by moving to live with his maternal grandmother, benefitting from protections given to families in mixed marriages in Croatia. Branko and Zdenka met after the war while high school students, and soon fell in love. Branko was admitted to the university in Zagreb, but quickly realized that he might not be able to demonstrate sufficient ardor for Marxism-Leninism and could be potentially denied a degree or future employment. This, combined with his father's experience of having been forced to "donate" his share in a successful business to the local government soured him on the idea of staying in Yugoslavia. In 1948, the Communist regime arranged for Jews wishing to emigrate from Yugoslavia to register for transport to Israel. When it turned out that a ship really did arrive and it was announced that there would be a second opportunity to emigrate the following year, Branko convinced his family and Zdenka to seize the opportunity, arriving in Haifa, Israel in July 1949. As was the case for many immigrants to Israel at the time, conditions were very difficult, but both Branko and Zdenka were determined to resume their studies. In the fall of 1950, Branko quit a job in Tel Aviv to go to Jerusalem to study mathematics. In 1954 he received his M.Sc., and he and Zdenka married on June 30, 1954. In the fall of 1955 Branko was called to active duty in the Israeli Air Force, where he worked in the Operations Research unit; meanwhile Zdenka earned her M.Sc. in Chemistry. Their first son, Rami, was born in 1956. Branko completed his Ph.D. in 1957 and in 1958 he was discharged from the military. Soon afterward he was awarded a scholarship to the Institute for Advanced Study in Princeton, NJ, where he and his family spent two years. In the fall of 1960 he obtained a visiting appointment at the University of Washington in Seattle, where their second son Daniel was born in November. While they were planning their return to Israel where Branko had accepted a position as a lecturer at Hebrew University, they learned his marriage to Zdenka was annulled because he was not legally Jewish according to Orthodox interpretation (his mother having not been a Jew), so he and Zdenka remarried at the City Hall in Seattle before moving to Jerusalem. Within three years Branko had been promoted to Associate Professor. He spent the summer of 1963 in Seattle as a visitor at the University of Washington, and he spent a sabbatical in 1965-66 at Michigan State University as a visiting professor. The story in the news that another Israeli immigrant from a mixed marriage had her passport and citizenship revoked in 1966 for reasons similar to those used to annul Branko and Zdenka's marriage resulted in them deciding not to return to Israel, even though this meant Zdenka would be unable to complete her Ph.D. in Chemistry. Branko joined the faculty at the University of Washington as a full professor in 1966. He retired in 2001, but continued to teach and work with graduate students as an emeritus professor (Leah Berman and I were his last two doctoral students at the University of Washington, completing our degrees in 2002). I would like to close by saying a bit about what it was like being his student. I moved to Seattle in 1997 at the suggestion of Marjorie Senechal. I was immediately welcomed into a vibrant and generous community of discrete geometers. At the core of this community were Vic Klee and Branko Griinbaum, both of whom provided me with much valuable advice and guidance during my time there, and both of whom were unfailingly kind to me. As a mathematics graduate student, a visit to Branko's office was like a visit to the candy store. His office was filled with models he had built over the years to help him think through geometric problems. They covered shelves and hung from the ceiling tiles on bits of string (which I'm sure caused the fire marshal fits of apoplexy), and they were colorful and intriguing. It seemed like every time I went into his office I noticed something new, and he was always happy to explain the math behind the model and pull a copy of a preprint from his filing cabinet of the paper that had provided the need for the model in the first place. Questions I brought to Branko were often answered by him pulling a model from a shelf to illustrate a point, and would lead us into a discussion of other questions the model helped to illuminate. One of my most prized possessions is a model he built, and I was immensely proud when he asked me to contribute a copy of a model I had built for my own research into his collection. The garage at his house was equally a treasure trove of mathematical models, and interesting examples often made their way from his home to lecture halls at the university. In this way I learned the importance of visualization and model building as tools to gain deeper insight into geometric questions, and as an important step in verifying my understanding of mathematical ideas (if I couldn't build it, I clearly didn't understand it). When I began preparing my first paper for publication, I got invaluable advice from Branko about what I should be trying to achieve in my writing. He believed strongly that an article should be written as an invitation to engage in a conversation with the author(s). This means making sure it has the necessary background, trying to make the writing as clear and engaging as possible, and asking thought provoking questions. Because a paper is a conversation between the author and the reader, even solo authored papers should use "we". He encouraged the inclusion of conjectures, because they excited the reader to a challenge. He enjoyed and fostered collaboration at every turn. The slight exception to this was co-authoring papers with his students (especially while they were his students), because he wanted to make sure readers gave us the credit instead of assuming the significant contributions were his. This was a little frustrating to me because I wanted to lower my Erdos number and get my Grimbaum number down to 1, so I've always been a bit envious of my wife Eeah Berman, who was a student of his at the same time, who co-authored a paper with him a few years after we graduated. After completing my Ph.D., Branko continued to be a significant presence and influence on my life and career. His signature graces our marriage contract, and he and Zdenka welcomed us into their home when we would visit Seattle. He and Zdenka always made us feel welcome and cherished, and they doted on our children. I was always a little in awe of how much in love they seemed, even after 61 years of marriage. Zdenka sadly passed away in her sleep on December 28, 2015. Having been his student has been a constant source of open doors for me. Anywhere I go in the world of discrete geometry, I am always greeted with warmth and delight when someone learns I was his student. One of my favorite interactions along these lines was when I first met Janos Pach while he was at MSRI; when he learned I was Branko's student he said that Branko "always had great taste in problems." That always struck me as very high praise indeed. It was also clear that many mathematicians I met not only respected him for his achievements and contributions to the field, but also treasured his company, hospitality and generosity — that they held in high esteem not just the mathematician but the man. There is an old Jewish tradition that no one is truly gone as long as their memory and name survive. I will treasure his memory and the influence he has had on the course of my life for the remainder of my days. May his name be a blessing to you as well. Gordon Williams * Department of Mathematics and Statistics, University of Alaska Fairbanks E-mail address: giwilliams@alaska.edu References [1] N. Basic, J. Groselj, B. Griinbaum and T. Pisanski, Splittable and unsplittable graphs and configurations, ArsMath. Contemp. 16 (2019), 1-17, doi:10.26493/1855-3974.1467.04b. [2] B. Grunbaum, Convex Polytopes, volume 16 of Pure and Applied Mathematics, John Wiley & Sons, 1967. [3] B. Grunbaum, Shouldn't we teach GEOMETRY?, The Two-Year College Mathematics Journal 12 (1981), 232-238, doi:10.2307/3027069. [4] B. Grunbaum, Geometry strikes again, Mathematics Magazine 58 (1985), 12-17, doi:10.2307/ 2690229. [5] B. Grunbaum, Convex Polytopes, volume 221 of Graduate Texts in Mathematics, SpringerVerlag, New York, 2nd edition, 2003, doi:10.1007/978-1-4613-0019-9, prepared and with a preface by Volker Kaibel, Victor Klee and Gunter M. Ziegler. [6] B. Grunbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Math. Contemp. 2 (2009), 1-25, doi:10.26493/1855-3974.88.e12. [7] B. Grunbaum, Configurations of Points and Lines, volume 103 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009, doi:10.1090/gsm/103. [8] B. Grunbaum, Branko and Zdenka — Our Story, February 2017, private communicaton. [9] B. Grunbaum and G. C. Shephard, Tilings and Patterns, A Series of Books in the Mathematical Sciences, W. H. Freeman and Company, New York, 1989. [10] D. C. Kay (ed.), Proceedings of the Conference on Convexity and Combinatorial Geometry, Department of Mathematics, University of Oklahoma, Norman, Oklahoma, 1971, held at the University of Oklahoma, Norman, Oklahoma, 21 - 25 June 1971. [11] D. Seuss, McElligot's Pool, Random House, 1947. ^I would like to thank Rami Grunbaum for providing a copy of a transcript of a speech Branko gave in 2017 about his life with Zdenka that was used in the preparation of this memoriam, and for confirming certain dates and details. The author takes full responsibility for any errors that remain. I also would like to thank Dr. Marilyn Breen who provided important background on the Conference on Convexity and Combinatorial Geometry at the University of Oklahoma. Printed in Slovenia by IMEJTISKARNE