ARS MATHEMATICA CONTEMPORANEA Volume 7, Number 1, Spring/Summer 2014, Pages 1-262 Covered by: Mathematical Reviews 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 publication is partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications. ARS MATHEMATICA CONTEMPORANEA Bled'11 - Part 2 This is the second of two special issues of Ars Mathematica Contemporanea dedicated to the 7th Slovenian Graph Theory Conference (Bled'11). The conference was held (by tradition) at Lake Bled, Slovenia, and took place from June 19 to June 25, 2011. In total, 9 keynote lectures and 213 contributed talks (within 16 Minisymposia and a general session) were given. The first special issue (published as the first issue of Volume 6 of Ars Mathematica Contemporanea) contained 15 research papers. The current issue contains another 18 papers, on some more of the high quality research presented at the conference, and accepted for publication after a thorough refereeing process. These papers, together with the 15 from the first special issue, present the readers of Ars Mathematica Contemporanea with a selection of 33 fascinating and valuable contributions, covering a wide range of aspects of graph theory. A small number of submissions for these special issues have not yet completed the peer review process. Those that succeed will be published in forthcoming regular issues of Ars Mathematica Contemporanea. In a sense, this special issue represents a formal conclusion to the Bled'11 conference. We are already looking forward to the next Bled conference, to be held in June 2015. Klavdija Kutnar and Primož Šparl Guest Editors Contents Compression ratio of Wiener index in 2-d rectangular and polygonal lattices Jelena Sedlar, Damir Vukičević, Franco Cataldo, Ottorino Ori, Ante Graovac................................. 1 On the packing chromatic number of square and hexagonal lattice Danilo Korže, Aleksander Vesel....................... 13 Product irregularity strength of certain graphs Marcin Anholcer...............................23 Polytopes associated to dihedral groups Barbara Baumeister, Christian Haase, Benjamin Nill, Andreas Paffenholz . 31 A Class of semisymmetric graphs Li Wang, Shaofei Du, Xuewen Li.......................41 Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian Ebrahim Ghaderpour, Dave Witte Morris................... 55 Poly-antimatroid polyhedra Yulia Kempner, Vadim Levit.........................73 The L2 (11) -subalgebra of the Monster algebra Sophie Decelle................................ 83 On fat Hoffman graphs with smallest eigenvalue at least —3 Hye Jin Jang, Jack Koolen, Akihiro Munemasa, Tetsuji Taniguchi .....105 Convex cycle bases Marc Hellmuth, Josef Leydold, Peter F. Stadler...............123 From spanning forests to edge subsets Martin Trinks.................................141 Revised and edge revised Szeged indices of graphs Morteza Faghani, Ali Reza Ashrafi......................153 Petersen-colorings and some families of snarks Jonas Hägglund, Eckhard Steffen.......................161 Constructions for large spatial point-line (nk) congurations Gäbor Gévay.................................175 Distinguishing graphs with infinite motion and nonlinear growth Johannes Cuno, Wilfried Imrich, Florian Lehner...............201 Isomorphic tetravalent cyclic Haar graphs Hiroki Koike, Istvän Koväcs.........................215 Small cycles in the Pancake graph Elena Konstantinova, Alexey Medvedev...................237 Fat Hoffman graphs with smallest eigenvalue at least — 1 — t Akihiro Munemasa, Yoshio Sano, Tetsuji Taniguchi.............247 Volume 7, Number 1, Spring/Summer 2014, Pages 1-262 ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 1-12 Compression ratio of Wiener index in 2-d rectangular and polygonal lattices Jelena Sedlar * Faculty of civil engeneering, architecture and geodesy Matice hrvatske 15, HR-21000 Split, Croatia * Damir VukiCeviC Faculty of Science, University of Split, Nikole Tesle 12, HR-21000 Split, Croatia Franco Cataldo , Ottorino Ori Actinium Chemical Research, Via Casilina 1626/A, 00133 Rome, Italy Ante Graovac Faculty of Science, University of Split, Nikole Tesle 12, HR-21000 Split, Croatia IMC, University of Dubrovnik, Branitelja Dubrovnika 29, HR-20000 Dubrovnik, Croatia Received 20 October 2011, accepted 31 October 2012, published online 7 January 2013 In this paper, we establish leading coefficient of Wiener index for open and closed 2-dimensional rectangular lattices, for various open and closed polygonal lattices, and for open and closed multidimensional cubes. These results enable us to establish compression ratio of Wiener index when number of rows and columns in the lattice tends to infinity. Keywords: Graph theory, 2D rectangular and polygonal lattices, Wiener index, Compression ratio. Math. Subj. Class.: 05C12, 92E10 * corresponding author E-mail addresses: jsedlar@gradst.hr (Jelena Sedlar), vukicevi@pmfst.hr (Damir Vukicevic), franco.cataldo@fastwebnet.it (Franco Cataldo), ottorino.ori@alice.it (Ottorino Ori), graovac@irb.hr (Ante Graovac) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 1 Introduction Topological indices are very important in chemistry, since they can be used for modeling and prediction of many chemical properties. One of the most famous and the most researched indices is Wiener index. Topological indices are invariants defined on graphs representing various chemical compounds. For example, one such compound is graphene which is represented with hexagonal lattice graph. Also, nanotubes and nanotori received much attention recently, and they are represented with a graph which is rectangle shaped lattice with opposite sides identified. Recent theoretical investigations point out that the minimization of distance-based graph invariants, namely the Wiener index W [7] and the topological efficiency index p recently introduced [3], provides the fast determination of the subsets of isomers with relative structural stability of a given chemical structure. This method has been applied to important classes of carbon hexagonal systems like fullerenes, graphene with nanocones and graphene. This elegant computational topological approach quickly sieves the most stable C66 cages among 4478 distinct isomers as reported in [8]. Moreover, the same method gives the correct numbers of NMR resonance peaks and relative intensities. The Wiener index has been computed for monodimensional infinite lattices to describe conductibilty features of conjugated polymers [1]. Present article reports about a relevant property - the compression factor [3] - of the topological invariants computed on infinite lattices. For different topological indices, there is quite much recent interest [4-6] in the ratio of the value of the index on open and closed lattices (i.e. nanotubes and nanotori). In this article, we investigate compression ratio of 2-dimensional and multidimensional rectangular lattices, and various 2-dimensional polygonal lattices. The present paper is organized as follows. In the second section named 'Preliminaries', we introduce some basic notions and notation that will be used throughout the paper. In the third section, we establish the leading coefficient in Wiener index for open and closed 2-dimensional rectangular lattices, which leads us to compression ratio in asymptotic case. In the fourth section, we use the results from the second section to derive the same kind of result for hexagonal and similar lattices. In the fifth section, we establish the limit of compression ratio for d-dimensional rectangular cube. Finally, in the last section named 'Conclusion', we summarize the main results of this paper. 2 Preliminaries In this paper, we consider only simple connected graphs. We will use the following notation: G for graph, V(G) or just V for its set of vertices, E(G) or just E for its set of edges. With N we will denote number of vertices in a graph. For two vertices u,v e V, we define distance d(u, v) of u and v as the length of shortest path connecting u and v. Given the notion of distance, Wiener index of a graph G is defined [9] as W(G) = E d(u, v). Here, the pair of vertices u, v is unordered. In some literature, the summation goes over all ordered pairs (u, v) of vertices and then the sum needs to be multiplied by a half. Now, let us introduce some special kinds of graphs that will be of interest to us in this paper, namely open and closed lattice graphs. First, let Rn,k be rectangular lattice containing 2n rows and 2kn columns of squares. Lattices R3,1 and R3,2 are shown in Figure 1. Figure 1: Lattices R3,1 and R3,2. Let us denote vertices of Rn,k with integer coordinates (i, j) for i = 0,..., 2nk and j = 0,..., 2n as if the lattice was placed in first quadrant of Cartesian coordinate system. Therefore V(Rn,k) = {vi,j : i = 0,..., 2nk and j = 0,..., 2n} . Open lattice is a graph ORn,k obtained from Rn,k by deleting vertices v0,j and vi,0. Closed lattice is a graph CRn,k obtained from Rn,k by identifying vertices v0,j and v2nk,j, and also vertices vi 0 and vi 2n. Therefore, open and closed lattice graphs have the same number of vertices which is |V (O Rn,k)\ = |V (C Rn,fc)| =4n2k. Further, for a fixed integer k let us consider polygons Pk with 4k + 2 vertices. Let Ln,k be rectangle shaped lattice consisting of 2n rows of polygons Pk, with rows containing n and n - 1 polygons alternatively, such that two neighboring polygons from one row share exactly two vertices, while two neighboring polygons from different rows share exactly k + 1 vertices. For k =1, lattice Ln,k is actually hexagonal lattice with 2n rows and n columns of hexagons. Note that lattice Ln,k can be considered as subgraph of rectangular lattice Rn,k assigning to polygons the size of 2k square cells. Therefore, we can use the same vertex notation in Ln,k as in Rn,k. Lattices L3<1 and L3,2 for k = 1 and k = 2 are shown in Figure 2. Again, open and closed lattices O Ln,k and C Ln,k can be obtained as for rectangular lattice. Finally, a d-dimensional rectangular cube Rn is defined in a following manner: set of vertices V(Rn) is defined with V (Rdn) = {(xi,...,xd): xi g{0,1,..., n}} , and two vertices (x1,..., xd) and (y1,... ,yd) are connected with an edge if and only if there is a coordinate j such that xi = yi for every i G {1,..., d} \ {j} , and for j-th coordinate holds |yj - xj | = 1. Open lattice O Rn is a graph obtained from Rn by deleting all vertices that have at least one zero coordinate. Closed lattice C Rn is a graph obtained from Rn by identifying every pair of vertices (x1,..., xd) and (y1,..., yd) such that xi = yi for every i G {1,..., d} \ {j} and for j-th coordinate holds xj = 0 and yj = n. Obviously, the number of vertices in O Rn and C Rn is the same and it equals nd. Now, if OG is an open lattice graph and CG its closed version, compression ratio of G is defined [3] as W (C G) W (O G)' Figure 2: Lattices L3,1 and L3,2. Obviously, Wiener index depends on the size of the lattice i.e. on n and k, and therefore compression ratio depends on them too. Our goal is to establish the limit of compression ratio for Rn,k, Ln,k and Rn when n tends to infinity for a fixed k. 3 Compression ratio of Rn,k For k = 1 (i.e. for square shaped rectangular lattices), the result cr(Rn,k ) = 3 was already obtained in reference [4]. The result f for bidimensional square is the same as the result for monodimensional lattices (polymer chains) obtained in [3]. We will here derive the same result for k > 1, i.e. for some rectangle shaped lattices. Let k be fixed integer. We have following theorems. Theorem 3.1. For the lattice graph ORn,k, Wiener index W(ORn k ) is a polynomial in n of degree 5 whose leading coefficient equals f k2 (1 + k). Proof. For vertices vi,j and vpq of ORn,k holds d (vi,j ,vp,q) = 1 p - i\ + |q - j\ . Obviously, 2 kn 2n 2 kn 2n 2 kn 2n 2n W (O Rn,k ) = EEEE d(vi,j , vp,q ) - EEE d(vi,j ,vi,q ). i= 1 j = 1 p=i q=1 i=1 j = 1 q=j The second (subtracted) sum does not influence leading term in n, therefore we can neglect it. To avoid absolute value, we can rewrite the first sum as 2kn 2n 2kn 2n 2kn 2n 2kn W(ORn,k) « 2 E E E E d(vi,j, vp,q) - E E E d(vi,j, vp,j). i=1 j=1 p=i q=j i=1 j=1 p=i Again, the second (subtracted) sum does not influence leading term in n, therefore we can neglect it. Now we have 2kn 2n 2kn 2n w (O Rn,k ) « 2 EEEE (p - i + q - j ) i=1 j=1 p=i q=j and the result then follows by easy calculation. □ Theorem 3.2. For the lattice graph CRn,k, Wiener index W(CRn,k) is a polynomial in n of degree 5 whose leading coefficient equals 4k2 (1 + k). Proof. Obviously, all vertices in C Rn,k have the same sums of distances to all other vertices. Therefore, to obtain W(CRn,k) it is enough to calculate distances from one vertex (vi,i is easiest for calculation) to all other vertices. Since CRn,k is a torus, to do that we will calculate the sum of distances from v1,1 to vp,q where 1 < p < kn and 1 < q < n and multiply it by 4. Now, the obtained number should be multiplied by number 2n • 2kn of vertices in the lattice, and then divided by 2 since each distance was counted twice. Therefore we have kn n W(CRn,k) « 2 • 2n • 2kn • 4 • ^^ (p - 1 + q - 1), p=1 q=1 and the result now follows by direct calculation. □ Corollary 3.3. Holds 3 lim cr(Rn k) = 7. n—^^o 4 4 Compression ratio of Ln,k This section is devoted to the exact determination of the compression factors for the 2-dimensional polygonal lattice Ln,k. We start from a numerical example devoted to the case of the graphene lattice. Figure 3 shows the rectangular L31 portion of this hexagonal system. Figure 3: View of L31 hexagonal lattice with bold vertices common to neighboring polygons along one row, whereas the dotted ones are shared by two neighboring polygons from two different rows. The numerical determination of invariants W (O G) and W (C G) for this infinite graph is based on the results summarized in Table 1 where, for an increasing number of vertices N, values of both descriptors are listed. The exact polynomial forms are given in Table 1 5 5 for O W and C W with leading terms O W « ^N2 and C W « T^2 respectively, producing N OW = 15 (6N5 - 5N3 - N2 ) C W = 24 (7N 2 - 4N i) 36 3 038 2 232 100 39 666 29 000 196 214 214 156 408 324 753 882 550 152 484 2 057 902 1 501 368 676 4 746 690 3 462 472 900 9 710 998 7 083 000 Table 1: Exact polynomial forms for the Wiener index of the open (O W) and closed (C W) rectangular graphene lattices Lnj1 with N vertices. the value cr(Ln,1) = f§ for the compression factor of rectangular graphene. More details about the numerical determination of various topological descriptors of the graphene rectangular lattices are given in [2]. Now, we want to establish the compression ratio of Ln,k. We will use the fact that Ln,k can be considered as the subgraph of Rn,k. Therefore, distances between vertices in Ln,k for some pairs of vertices are equal as in Rn k, while for some other pairs of vertices are greater than in Rn k. We will establish for which pairs of vertices the distance is greater in Ln k than in Rn,k, and how much greater. We will not establish the exact value of Wiener index for OLn,k or CLn,k as that would be tedious for all the possible cases, but we will neglect some quantities which do not influence the leading coefficient. Theorem 4.1. For the lattice graph O Ln,k, Wiener index W (O Ln k ) is a polynomial in n of degree 5 whose leading coefficient equals 15k2 (7k + 5). Proof. Since Ln,k is a subgraph of Rn,k that means distances in Ln,k are equal or greater than in Rnk. Therefore W (O Ln,k )= W (O Rn,k )+A. If we establish leading coefficient in A, then by combining that result with Theorem 3.1 we obtain desired result. Therefore, we are interesting in establishing A, i.e. for which pairs of vertices the distance in Ln k is greater than in Rn k and how much greater. For a vertex vi j lattice Lnk can be divided into four areas as illustrated with Figure 4. Vertices in areas left and right to vijj have the same distance to vijj in Ln,k as in Rn,k, while the vertices in areas up and down to vijj have the greater distance to vijj in Ln,k than in Rn,k. For easier calculation, we will approximate zig-zag lines that divide Ln k into areas with lines q = qi(p) = j - k"(p - ^ q = q2(p) = j + 1(p - i). as also illustrated in Figure 4, and we denote upper and lower areas with L^ k and left and right areas with L^ k. With such an approximation, we make an error in some vertices near Figure 4: Division of lattice Ln,k into areas for a vertex vitj. the lines, but since number of such vertices is linear in n, that error does not influence the leading term of A and we can neglect it. Now that we established the pairs of vertices for which the distance is greater, we want to establish how much greater. Obviously, the trouble is if we have to go vertically, since some vertical edges are missing now. Therefore, we go vertically as little as we can (the shortest path is illustrated in Figure 4), and for each vertical step the path is k edges longer in Ln,k than in Rn,k. Now, to calculate all these exactly, we have to divide into cases, regarding the position of vjj (since then lines q1 (p) and q2 (p) intersect boundaries in different sides, which influences calculation). The division into cases is illustrated with Figure 5. There are four areas, but upper and lower Figure 5: Division into cases with respect to the position of vj,j in the lattice. are equal up to symmetry, and also left and right. Again, we will approximate division with lines j = ji (i) = n - k-(i - kn), j = j2 (i) = n + k-(i - kn) and denote upper and lower areas with L^k and left and right areas with L^k. This approximation again produces an error in calculating W(Ln k) but not great enough to influence the leading term, therefore we can again neglect it. Before we proceed note that lines we introduced can also be expressed as i = i i (j ), i = i2 (j), p = pi(q) and p = p2(q). Now we distinguish two cases. CASE I: Let vie L^). For an arbitrary vertex vp,q, we are interested in establishing the difference in d(vi,j, vp,q) between Ln,k and Rn,k. The difference is greater than zero only if vp,q e LAk. To calculate the difference Ai that occurs for pairs of vertices in this case, we have to divide L A k into 4 subareas Ai, A2, A3, A4 as illustrated in Figure 6. Now Figure 6: The division of the area LA k into subareas in the case vi, j e L^) we have 2n i 2 ( j ) i q2(p) A(Ai) = 2 E E EE (q2(p) - q) • k, j=n i=ii (j) p=1 q=1 2n i2 ( j ) 2knqi(p) A(A2) = 2 E E EE (qi(p) - q) • k, j=ni=i1(j) p=i q=i 2n i2(j) i 2n A(A3) = 2 £ E E E (q - qi(p)) • k, j=n i=ii (j) p=pi (2n) q=qi (p) 2n i2(j) p2 (2n) 2n A(A4) = 2£ E E E (q - q2(p)) • k. j=n i=ii (j) p=i q=q2(p) Therefore Ai = A(Ai) + A(A2) + A(A3) + A(A4) and by direct calculation we establish that Ai is a polynomial in n of degree 5 with leading coefficient being k3. CASE II: Let vi , j e L^). Again, the difference d(vi , j ,vp , q) is greater than zero only if vp , q e LAk. Again, we divide LA k into four areas Ai,..., A4 as shown in Figure 7. Now, we calculate Figure 7: The division of the area La k into subareas in the case vi}j e L)i 'k. kn ji(i) i q2 (p) A(A1) = 2 £ £ ££ (q2(p) - q) ■ k, i= 1 j=j2 (i) p=i q=i kn ji(i) pi(1) qi(p) A(A2) = 2 £ £ £ £ (qi(p) - q) ■ k, i=1 j=j2 (i) p=i q=1 kn ji (i) i 2n A(A3) = 2 £ £ £ £ (q - qi(p)) ■ k, i=1 j=j2 (i) P=1 q=qi(p) kn ji(i) P2 (2n) 2n A(A4) = 2 £ £ £ £ (q - q2(p)) ■ k. i=1 j=j2(i) p=i q=q2(p) Now A2 = A(A1) + A(A2) + A(As) + A(A4) and by direct calculation we establish that A2 is a polynomial in n of degree 5 with leading coefficient being 25 k3. Therefore, we conclude that A is a polynomial in n of degree 5 with leading coefficient being equal to 1 ( 14k3 + 22k3) = 32k3. 2 V 5 15 ) 15 Now the result follows from this and Theorem 3.1. Theorem 4.2. For the lattice graph CLn,k, Wiener index W(CLn,k ) is a polynomial in n of degree 5 whose leading coefficient equals 4k2 (4k + 3). Proof. Let us introduce the same vertex notation in CLn,k as in CRn,k (we can do that as CLn,k is a subgraph of CRn,k). Since, CLn,k is a subgraph of CRn,k we have W (CLn,k) = W (CRn,k) + A. Therefore, if we establish A, the result will follow from Theorem 3.2. Since all vertices in C Ln k are equivalent in the sense that they have the same distances to all other vertices, it is enough to calculate difference in distances for one vertex, and then multiply it by number of vertices, and divide by two since each difference is thus calculated twice. It is easiest if we calculate for v1;1. Since CLn,k is a torus, we will calculate the difference in distance from v1;1 to vp,q where 1 < p < kn and 1 < q < n and multiply it by 4. In that area difference in distances is greater than 0 only if 1 < p < kn and q1 (p) < q < n where qi(p) = k" • p. Therefore, 1 kn n A « ^ • 2kn • 2n • 4 • E E (q - ^W) . p=1 q=qi(p) By direct calculation, we obtain that A is a polynomial in n of degree 5 with leading coefficient being 4 -k3. 3 Now the result follows from Theorem 3.2. □ Corollary 4.3. Holds 5 (4k + 3) lim cr(Ln,k) = 4 (7k + 5)' Now, we can derive from this result compression ratio for some specific lattices. For example, lattice Ln1 is hexagonal lattice with 2n rows and n columns of hexagons. Therefore, from Corollary 4.3 follows that compression factor for such lattice equals lim cr(Ln i) = 5(4 + 3) = 35 = 0.729 17 v n'u 4(7 + 5) 48 confirming the result numerically derived in [2]. For k = 2, lattice Ln k is a lattice consisting of 10-gons, with 2n rows and n columns of rectangular unit cells with size 2k = 4 squares as in Figure 2. From Corollary 4.3 follows that compression factor for such lattice equals lim cr(Ln 2) = 5(4 ^ 2 + 3) = 55 = 0.723 68. v n'2; 4(7 • 2 + 5) 76 Corollary 4.4. Holds k lim cr(Ln,k) = lim cr(Rn,k) - n,k n,k - n^TO n^TO 4(7k + 5) Invariant limn^TO cr(Ln,k) reaches its maximum value limn^TO cr(Lnj1) = 48, whereas for large k the limit limn^TO cr(Ln,k) constantly decreases toward its lower limit 5. 5 Compression ratio of R In this section, we will establish the limit of compression ratio of d-dimensional rectangular cube Rd when n tends to infinity, and we will show that it does not depend on dimension d. Theorem 5.1. Holds 3 lim cr (Rt) = -. nmro V n' 4 Proof. Let us denote [n] = {1,2,..., n} . Note that lim W 2dR"^ = lim "wr I E E E lvi - Xil nmro n2d+1 nmro n2d+1 I \(xi,...xd)e[u]d (y1,...yd)e[u]d 1 i. Here dG(u,v) denotes the usual shortest path distance between u and v. Then Xi is said to be an i-packing, while such a partition is called a packing k-coloring. The smallest integer k for which there exists a packing k-coloring of G is called the packing chromatic number of G and it is denoted by xP (G). Let G = (V, E) be a graph. A walk is a sequence of vertices v1, v2,..., vk and edges vivi+1, 1 < i < k - 1. A path on n vertices is a walk on n distinct vertices and denoted Pn. A walk is closed if v1 = vn. A closed walk in which all vertices (except the first and the last) are different, is a cycle. The cycle on n vertices is denoted Cn. For u, v e V (G), dG (u, v) or d(u, v) denotes the length of the shortest walk (i.e., the number of edges on the shortest walk) in G from u to v. These definitions extend naturally to directed graphs. A set S C V (G) is independent if xy e E (G) for any pair of vertices x,y e S. Cardinality of a largest independent set S of G is the independence number a(G) of G. This paper studies the packing chromatic number of hexagonal lattice and of some infinite subgraphs of square lattice. Section 2 contains the search for the lower bound on the packing chromatic number in hexagonal lattice. The bound is obtained by a computer program using the dynamic approach for computing graph invariants, as described in the first part of the section. Section 3 discusses the packing chromatic number for some infinite subgraphs of the square lattice. We establish the packing chromatic number for P6 DZ and provide upper bounds on these numbers for PnDZ, where 7 < n < 13. We conclude the paper with the packing chromatic number for C4DZ as well as with some partial results on upper bounds for some infinite subgraphs of Z2DP2 provided in Section 4. The results in our paper were partially obtained by computers, mainly in Windows environment, but some also using Linux Ubuntu operating system. The machines used for computations were also diverse: Intel i7 930 based personal computer, Intel Q9400 based machine and a computer cluster (with up to 24 processor cores). All computations were carried out during six months, starting in the middle of 2010. The development environment and class libraries Lazarus (version of Pascal language) were used to write all necessary programs. 2 Hexagonal lattice The hexagonal lattice H plays a crucial role in many network applications, particularly in frequency assignments, e.g. see [5]. It was proved by Bresar et al. [1] that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. The result was improved by Fiala et al. [3], where the packing 7-coloring of the hexagonal lattice is presented. We show in this section that actual lower bound on the packing chromatic number of the infinite hexagonal lattice is 7 and therefore xP(H) = 7. We now present the algorithms, that have been used to provide the main result. We first describe the concept needed to describe our computer checking. The idea is introduced in [9] in a more general framework, but for our purposes the following description will be sufficient. Figure 1: Graph Hi. Observe first the graph Hi depicted in Fig. 1. We construct Hi for i > 1 as follows. Take the graph which is composed of an isomorphic copy of Hi-i and of an isomorphic copy of Hi. Then add additional four edges that connect vertices u', v', w', and z' of the last added copy of Hi in Hi-i with the vertices u, v, w, and z of the new copy of Hi. As an example see Fig. 2 where H2 is depicted. Figure 2: Graph H2. Obviously, Hi is a subgraph of H for i > 1. We next define a directed graph Dk as follows. The vertices of Dk are all packing k-colorings of Hi. Let u and v be two distinct vertices of Dk. Then uv denotes a k-coloring of H2 such that u and v induce the respective packing k-coloring of the first and the second copy of Hi. Note that uv need not to be a packing k-coloring of H2. uv is an arc in Dk if and only if uv is a packing k-coloring of H2. Lemma 2.1. Let k < 6. Then Hi admits a packing k-coloring if and only if Dk possesses a walk P = vi,v2,..., vi with vj corresponding to the j-th copy of Hi. Proof. Suppose first that Dk possesses a walk P — v\, v2,..., v,. If i — 2, then P is an arc from vi to v2 in Dk and the claim is obvious. Let then i > 2. Suppose the claim holds for P ' — vi, v2,..., vj_i, i.e. Hi—i admits a packing k-coloring. Since P has an arc from vi-i to vi, the corresponding colorings induce a packing k-coloring in a copy of H2 that corresponds to vi—i and vi. In order to see that the assertion holds, note that the distance between a vertex of a copy of Hi that corresponds to vi and a vertex of a copy of Hi that corresponds to vi—2 is at least 7. Suppose now that Hi admits a packing k-coloring. If i = 2, then by definition of Dk a packing k-coloring of H2 induce an arc in Dk. Let then i > 2. Note that Hi is composed of an isomorphic copy of Hi—i, say X, and of an isomorphic copy of Hi, say Y. Y is connected in Hi to an isomorphic copy of Hi, say Z. Suppose the claim holds for Hi—i and let P' — vi, v2,..., vi—i denote a walk in Dk that corresponds to X. Since Hi admits a packing k-coloring, Y induces a packing k-coloring of Hi, say vi. Y and Z together induce a packing k-coloring of H2 and therefore vi—ivi forms an arc in Dk. Then P — vi, v2,..., vi—i, vi is a walk in Dk and the proof is complete. □ Lemma 2.2. Let k < 6. Then H admits a packing k-coloring only if Dk contains a closed directed walk. Proof. Let H for a given k admit a packing k-coloring denoted f. Suppose that Dk is acyclic. Since Hi is finite, there is obviously only a finite number of vertices (packing k-colorings of Hi) in Dk, say nk. Let then d < nk denotes the length of a longest directed path in Dk. Take now a subgraph of H isomorphic to Hd+2. A restriction of f to Hd+2 is obviously a packing k-coloring of Hd+2. From Lemma 2.1 it follows that Hd+2 admits a packing k-coloring if and only if Dk possesses a walk P — vi, v2,..., vd+2 with vj corresponding to a packing k-coloring of the j-th copy of Hi. But since Dk is acyclic, the length of the longest walk in Dk is at most d and we obtain a contradiction. □ Theorem 2.3. xP(H) — 7. Proof. Since it is proved in [1] that the packing chromatic number of the infinite hexagonal lattice is at least 6 and since in [3] a coloring of the hexagonal lattice using 7 colors is presented, we have to show that H does not admit a packing 6-coloring. We first constructed the graph D6 by using a computer program. The graph consists of 26660 vertices with a maximum output degree of 37 (see also the concluding remark). By the depth first search algorithm we next established that D6 is an acyclic graph. From Lemma 2.2 then it follows that the hexagonal lattice cannot admit a packing 6-coloring. This assertion completes the proof. □ An alternative approach to prove Theorem 2.3 is to use a naive brute force search for a large enough subgraph of H. The approach used in the proof Theorem 2.3, however, is potentially much more interesting and utile in order to search for the packing chromatic number in other families of graphs since it uses the packing k-colorings of a relatively small graph. 3 Square lattice Cartesian product of graphs provide a setting which has been widely used in designing large scale computer systems and interconnection networks. The Cartesian product of graphs G and H is the graph QUH with vertex set V(Q) x V(H) and (xi, x2)(yi, y2) G E(QUE) whenever x1y1 G E (Q) and x2 = y2, or x2y2 G E (H ) and x1 = y1. The Cartesian product is commutative and associative, having the trivial graph as a unit, cf. [8]. The subgraph of QUE induced by u x V(H) is isomorphic to H and it is called an H-fiber. It will be convenient to view the square lattice as the Cartesian product of two infinite paths, i.e ZUZ. Goddard et al. [6] determined the packing chromatic number for infinite subgraphs of the square lattice Z2 with up to 5 rows. In the same paper the question of determining the packing chromatic number of the infinite square lattice was posed. The best upper bound 17 was given by Holub and Soukal [7], while the best lower bound 12 was determined by Ekstein et al. [2]. We have considered infinite subgraphs of ZUZ with up to 13 rows. The main results are summarized in the following proposition. Proposition 3.1. (i) Xp(PßUZ) = 10, (ii) Xp(PrUZ) < 11, (iii) Xp(PsUZ) < 12, (iv) Xp(PqUZ) < 13, (v) Xp(PwUZ) < 14, (vi) Xp(PuUZ) < 14, (vii) Xp(P12UZ) < 15. (viii) Xp(P13UZ) < 15. Proof. Note first that if f is a packing k-coloring of PnUC^, k < I, then we can construct from f a packing k-coloring of PnUPm for every m. One can use f to color every Pn-fibre (uj x Pn) of PnUPm in the same way as the Pn-fibre (vj mod£ x Pn) of P„UC^ 1213121312131 10 31418151914121 16121312131715 21317141612131 19151213151814 312131 10 1213121 Figure 3: A packing 10-coloring of PeUC14 1 2 1 3 1 2 1 3 1 2 1 3 1 4 1 5 3 1 6 1 4 1 7 1 5 1 6 1 2 1 7 1 1 8 1 2 1 3 1 2 1 3 1 9 1 3 1 2 4 1 3 1 5 1 10 1 4 1 2 1 5 1 11 1 1 2 1 9 1 2 1 3 1 8 1 3 1 2 1 3 5 1 7 1 3 1 6 1 2 1 7 1 4 1 6 1 1 3 1 2 1 4 1 5 1 3 1 2 1 3 1 2 Figure 4: A packing 11-coloring of PtUC16 1 2 1 3 1 2 1 3 1 2 1 3 1 4 3 1 5 1 4 1 10 1 11 1 5 1 2 1 1 8 1 2 1 3 1 2 1 3 1 6 1 9 2 1 3 1 6 1 5 1 4 1 7 1 3 1 1 4 1 7 1 2 1 3 1 2 1 12 1 5 3 1 2 1 3 1 9 1 8 1 3 1 2 1 1 11 1 5 1 4 1 2 1 5 1 4 1 10 2 1 3 1 2 1 3 1 6 1 2 1 3 1 Figure 5: A packing 12-coloring of PglUC^ 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 4 1 5 1 6 1 4 1 5 1 7 1 4 1 5 1 6 1 7 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 12 1 8 1 7 1 10 1 11 1 6 1 9 1 13 1 4 1 5 1 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 6 1 4 1 9 1 5 1 4 1 8 1 5 1 7 1 10 1 11 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 7 1 5 1 13 1 6 1 7 1 12 1 4 1 6 1 5 1 4 1 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 Figure 6: A packing 13-coloring of P9DC20 In order to obtained the upper bounds, we therefore first tried to find a packing k-coloring of P„DCm for every n of the interest with k (and m) being as small as possible. The obtained colorings for n G {6,7,8,9,11,13} are depicted in Figs. 3 - 8, while a packing 14-coloring of P10DC16 and a packing 15-coloring of P12DC16 can be obtained from the first 10 rows of the packing 14-coloring of P11DC16 depicted in Fig. 7 and the first 12 rows of the packing 15-coloring of P13UC16 depicted in Fig. 8, respectively. In order to provide the lower bound for xp(P6DZ) we applied the backtracking search, e.g. see [10], adapted to packing colorings. Since the procedure did not find a packing 9-coloring in xp(P6OP12), the assertion follows. □ Results in Proposition 3.1 provide general upper bounds for infinite families of Cartesian products of two paths. For some graphs of these families however, better bounds or even the exact numbers can be computed. The results are depicted on the web page presented in the concluding remark. We again applied the backtracking search, which it is guaranteed to find a solution, if one exits, but it is relatively time consuming and therefore not usable for larger graphs. The colorings depicted in Figs. 3 - 8 have something in common: every second vertex in a row (column) is colored by the color 1. We therefore conjecture, that if a packing k-coloring of P„DPm exists, one can always find a packing k-coloring such that the class X1 is distributed as described above. This conjecture is formally stated below. Conjecture 3.2. Let n > 4 and let xP(PmOPn) = k. Then exists a packing k-coloring of PmDP„ with |X11 = a(PmDPn) = \nm 1. 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 4 1 5 1 8 1 4 1 6 1 7 1 5 1 12 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 7 1 9 1 13 1 5 1 10 1 11 1 4 1 6 1 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 5 1 4 1 6 1 7 1 4 1 5 1 8 1 14 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 10 1 11 1 5 1 12 1 9 1 6 1 7 1 4 1 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 6 1 7 1 4 1 8 1 5 1 4 1 13 1 5 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 Figure 7: A packing 14-coloring of P^DC^ 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 4 1 5 1 8 1 14 1 9 1 6 1 7 1 12 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 6 1 7 1 4 1 15 1 5 1 4 1 10 1 5 1 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 11 1 9 1 5 1 6 1 7 1 8 1 13 1 4 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 5 1 4 1 10 1 12 1 4 1 5 1 6 1 7 1 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 8 1 6 1 7 1 5 1 11 1 9 1 4 1 14 1 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 4 1 5 1 13 1 4 1 6 1 7 1 5 1 15 1 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 Figure 8: A packing 15-coloring of P13nC16 If the conjecture holds, the vertices of the class Xi can be fixed and therefore the backtracking is capable to provide results for much larger graph. In order to provide lower bounds for graphs of moderate siže we therefore applied backtracking with no additional constraints, while for larger graphs the vertices of the class Xi were fixed. The results of these computations are summarized in Table 1. The results in the table are of two types: exact values and upper bounds. Some of the upper bounds are exact values of xP if the Conjecture 3.2 holds. If a value k in the table is exact, that means that a packing k-coloring for the graph of interest is found and that the backtracking procedure confirmed that a packing (k - 1)-coloring does not exist. An upper bound k means that a packing k-coloring for the graph of interest exists, but we could not prove that a packing (k - 1)-coloring does not exist. On the other hand, if an upper bound k is marked with asterisk, the backtracking proved that a packing (k - 1)-coloring with the vertices of the class X1 fixed as stated in the conjecture does not exist. m\n 6 7 8 9 10 11 12 13 14-15 16-24 25-27 28-41 > 41 6 8 9 9 9 9 9 10 10 10 10 10 10 10 7 9 9 9 10 10 10 10 10 < 11* < 11* < 11* < 11* 8 9 10 10 10 < 11* < 11* < 11* < 11* < 11* < 12 < 12 9 10 < 11* < 11* < 11* < 11* < 11* < 12* < 12* < 12* < 13 10 < 11* < 11* < 11* < 12* < 12* < 12* < 14 < 14 < 14 11 < 11* < 12* < 12* <14 < 14 < 14 < 14 < 14 12 < 12* < 12* <15 < 15 < 15 < 15 < 15 13 < 13 < 15 < 15 < 15 < 15 < 15 Table 1: Packing chromatic numbers and bounds for PmDPn. 4 Subgraphs of ZDZDP2 It is known that XP(Z3) = to [4]. Moreover even the packing chromatic number of ZDZDP2 is unbounded [3]. On the other hand, it was proved that xp(GDZ) < to for any finite graph G [3]. Hence, it is worthy to study the packing chromatic number of some infinite subgraphs of ZDZDP2. In particular we considered C4DZ, C6DZ, C8DZ, C10DZ, C12DZ, and P2DP3DZ. We were able to obtain exact results for the packing chromatic number of C4DZ, while for the other families some partial results and bounds were found. Proposition 4.1. (i) Xp(C4DZ) = 9, (ii) Xp(CeDZ) < 13, (iii) Xp(C8DZ) < 15, (iv) Xp (C10DZ) < 22, (v) Xp(Ci2DZ) < 17, (vi) Xp(P2DP3DZ) < 18. Proof. The upper bounds follow from the packing 9-coloring of C4DC16 and from the packing 15-coloring of C8DC24 depicted in Fig 9 and Fig 10, respectively. The packing 13-coloring of C6DC48, the packing 22-coloring of C10DC48, the packing 17-coloring of C12DC48 and the packing 18-coloring of P2DP3DC48 can be obtained from the authors or from the web page presented in the concluding remark. 1416151814161519 2131213121312131 1517141915171418 3121312131213121 Figure 9: A packing 9-coloring of C4 DC16 The lower bound for C4DP10 is obtained by using the backtracking procedure which confirms that 8-coloring of C4DP10 does not exist. □ Note that a coloring of G which provides the upper bound in Proposition 4.1 has the vertices of the class X1 distributed such that the cardinality of X1 equals the independence number of G. We therefore generalize Conjecture 3.2 as follows. Let Xm denote Pm or Cm. 1 13 1 8 1 4 1 5 1 9 1 4 1 5 1 8 1 4 1 5 1 9 1 4 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 1 6 1 7 1 12 1 14 1 6 1 7 1 13 1 15 1 6 1 7 1 10 1 5 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 1 4 1 9 1 5 1 4 1 8 1 5 1 4 1 9 1 5 1 4 1 8 1 11 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 1 5 1 15 1 6 1 7 1 10 1 11 1 6 1 7 1 12 1 14 1 6 1 7 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 2 1 3 1 Figure 10: A packing 15-coloring of C8DC24 Conjecture 4.2. Let n > 4 and let xP(XmDPiDPn) = k. Then there exists a packing k-coloring of XmDP^DPn such that |Xi| = o.(XmDPiDPn). Analogous as in Section 3 we therefore applied backtracking with no additional constraints for graphs of moderate size, while for larger graphs the vertices of the class X1 were fixed. The packing colorings of the graphs of interest can be obtained from the authors or from the web page presented in the concluding remark. The results of these computations are summarized in Table 2, where an upper bound k marked with asterisk means that the backtracking proved that a packing (k - 1)-coloring with the vertices of the class X1 fixed as stated in the conjecture does not exist. m\n 2 3 4 5 6 7 8 9 10 11 12-15 16-18 19-34 > 34 4 5 5 7 7 7 7 8 8 9 9 9 9 9 9 6 5 8 8 8 10 10 11 11 11 12 < 12* < 12* < 13 < 13 8 7 7 9 9 10 10 11 < 12* < 12* < 13* < 13* < 14 < 14 < 15 P2OP3 5 8 8 10 10 11 < 12* < 12* < 14 < 15 < 18 < 18 < 18 < 18 Table 2: Packing chromatic numbers for CmDPn and P2DP3DPn (below). Concluding remark All obtained packing colorings as well as the graph D6 can be obtained from the authors or directly from the web page http://matematika-racunalnistvo.fnm. uni-mb.si/personal/vesel/constructions.aspx. Acknowledgments We would like to thank the referees for their careful reading and helpful suggestions. References [1] B. Bresar, S. Klavzar and D. F. Rall, On the packing chromatic number of Cartesian products, hexagonal lattice, and trees, Discrete Appl. Math. 155 (2007), 2303-2311. [2] J. Ekstein, J. Fiala, P. Holub and B. Lidicky, The packing chromatic number of the square lattice is at least 12, manuscript. [3] J. Fiala, S. Klavžar and B. Lidicky, The packing chromatic number of infinite product graphs, European J. of Combin. 30 (2009), 1101-1113. [4] A. S. Finbow and D. F. Rall On the packing chromatic number of some lattices, Discrete Appl. Math. 158 (2010), 1224-1228. [5] A. Gamst, Some lower bounds for a class of frequency assignment problems, IEEE Trans. Veh. Technol. 35 (1986), 8-14. [6] W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, J. M. Harris and D. F. Rall, Broadcast chromatic numbers of graphs, Ars Combin. 86 (2008), 33-49. [7] P. Holub and R. Soukal, A note on packing chromatic number of the square lattice, Electron. J. Combin. 17 (2010), R#17. [8] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley-Interscience, New York, 2000. [9] S. Klavžar and A. Vesel, Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of (2,1)-colorings and independence numbers, Discrete Appl. Math. 129 (2003), 449-460. [10] A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms, Academic Press, 1978. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 23-29 Product irregularity strength of certain graphs Marcin Anholcer * Poznati University of Economics, Al. Niepodleglosci 10, 61-875 Poznati, Poland Received 31 October 2011, accepted 2 October 2012, published online 7 January 2013 Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w : E (G) ^ {1,2,... ,m} is called product - irregular, if all product degrees pdG(v) = f]e3v w(e) are distinct. The goal is to obtain a product - irregular labeling that minimizes the maximal label. This minimal value is called the product irregularity strength and denoted ps(G). We give the exact values of ps(G) for several families of graphs, as complete bipartite graphs Km,n, where 2 < m < n < (m+2), some families of forests, including complete d-ary trees, and other graphs with 5(G) = 1. Keywords: Product-irregular labeling, product irregularity strength, tree. Math. Subj. Class.: 05C05, 05C15, 05C78 1 Introduction Assume we are given simple undirected graph G = (V (G), E (G)) with neither loops nor isolated edges and with at most one isolated vertex. Let us define integer labelling w : E(G) ^ {1, 2,..., s}. For every vertex v G V (G) we define the product degree as The product irregularity strength ps(G) of G is the smallest value of s that allows some product-irregular labelling. *http://kbo.ue.poznan.pl/anholcer E-mail address: m.anholcer@ue.poznan.pl (Marcin Anholcer) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract (1.1) (where dG(v) denotes the degree of vertex v in G). We call w product-irregular if for every pair of vertices u,v G V (G), u = v pda(u) = pda(v). (1.2) The concept was introduced by M. Anholcer in [4]. As we can see, it is the multiplicative version of the well known irregularity strength introduced by Chartrand et al. in [5] and studied by numerous authors (the best result for general graphs can be found in Kalkowski, Karoriski and Pfender [6], while e.g. trees and forests have been studied e.g. by Aigner and Triesch [1], and Amar and Togni [3]). On the other hand, the problem of founding the product irregularity strength of graph is connected with the product antimagic labellings investigated e.g. by Pikhurko [7]. Indeed, the result from the last publication implies that for sufficiently large graphs ps(G) < |E(G)|. Let nd denote the number of vertices of degree d, where 6(G) < d < A(G). In [4] M. Anholcer showed that ps(G) > max S(G) r —n e 1/r - r + 1 in the case of r-regular graph. Note that these bounds are not tight. Also the bounds on ps(Cn) were given, where Cn is cycle on n vertices. It was proved that if n > 17, then ps(Cn) > 1 — ln 2 1/2- while for every e > 0 there exists n0 such that for n > n0 ps(Cn) < [(1 + e)V2nlnn|. Similarly the upper bounds on the irregularity strength of grids and toroidal grids were proved: k k Ps(Tn i xn2 x-xnk ) < r(1+ e)V2(E V^j)ln(E nj )1' j=1 k j = 1 k ps(Gm nixn2x-^^xnfc ) <[(1 + e)V2(£ Vn)ln^ nj)]. j=1 j=1 In [9] Skowronek-Kaziow showed, that Proposition 1.1. For every n > 3 ps(Kn) = 3. Let us recall that nd denotes the number of vertices of degree d in G, where 6(G) < d < A(G). In this paper we are going to give the exact value of ps(G) for complete bipartite graphs Km,n, where 2 < m < n < , and some families of graphs with 6(G) = 1. The main results are as follows. e n Proposition 1.2. Let m and n be two integers such that 2 < m < n. Then ps(Km,n ) = 3 if and only if n < (m+2). Otherwise ps(Km,n) > 4. Theorem 1.3. Let D > 3 be arbitrary integer. For almost all forests F such that (i) A(F) = D, n2 = 0 and n0 < 1, (ii) if we remove all the pendant edges, then in the resulting forest F', n2 = 0, the product irregularity strength equals to ps(F ) = n\. The proofs of the above results are given in two following sections. 2 Complete bipartite graphs Proof of Proposition 1.2 Let Km,n = (U, V, E), where U = {u1,..., um}, V = [v\,..., vn} and E = {{«, vj}, 1 < i < m, 1 < j < n}. If we used only labels 1 and 2, we would be able to obtain at most n +1 distinct products, 1,2,..., 2n while we have n + m > n + 2 vertices. Thus ps(Kn,n) > 3. On the other hand, assume that we are using only the labels 1, 2 and 3. The number of possible multisets of m elements is equal to (m+2), and it is the maximal number distinct products for the vertices in V. Thus it is impossible to find product-irregular labeling of Km,n if | V| = n > (m+2). Now we are going to prove that labels 1, 2 and 3 are enough if m < n < (m+2). Let us consider the set of all (m+2) multisets of m elements equal to either 1, 2 or 3. Let us denote the elements of jth multiset, where 1 < j < (m+2), with aj, where 1 < i < m. Assume they are arranged in non decreasing order, i.e. in such a way that aj < aj+1 for 1 < j < (m+2) and 1 < i < m - 1. Now we arrange the obtained sequences in decreasing lexicographic order, i.e. in such a way that for every 1 < j1 < j2 < (m+2) there exists io, 1 < i0 < m such that aj = aj if 1 < i < i0 and aj0 > aj0. Now if m < n, then we put j1 j2 j1 j2 w({ui, vj}) = aj, 1 < i < m, 1 < j < n. Observe that the weighted degrees in V are distinct, as the respective multisets are. It is also straightforward to see that the degrees in U are distinct, as the numbers of factors equal to 3 are. Moreover the number of factors different than 1 in the weighted degrees in V are equal at most m, while in U they equal at least m +1. Thus finally the obtained labeling is product-irregular. If m = n, then we label any Kn-1n subgraph of Kn n as above and then put 1 on all the edges incident to the remaining vertex. As in the case m = n - 1 none of the vertices obtains the weighted degree 1, the resulting labeling is product-irregular. □ 3 Graphs with Ö(G) = 1 Proof of Theorem 1.3 Let us consider a forest F. We distinguish two kinds of non pendant vertices. The external vertex is such a vertex that at least one of its neighbours is pendant vertex. The internal vertex has no pendant vertices in the neighbourhood. The product degree of every pendant vertex is equal to the label of the only edge incident to it. Thus ps(F) > n1. So in order to prove the theorem we have to show that there exists a product-irregular labelling of F with n1 labels. The proof consists of two parts. First using the Probabilistic Method (more precisely the Linearity of Expectation, see e.g. [2], pp.13-21) we will prove the existence of partial labeling that distinguishes the product degrees of internal vertices. Then, by labeling the pendant edges we will extend the product-irregular labeling on whole forest F. Let us choose the label for every non-pendant edge uniformly at random from the set of all odd primes p, n1/2 < p < n1. The number of such primes n1/2 equals 1/2 n1 - 2.51012n11/2 n1/2 = n(n1) - n(n1 ) > -]- ' In n1 provided n1 > 17 (see e.g. [8]). Let us enumerate in any way all the m1 pairs of non-adjacent internal vertices and m2 pairs of adjacent internal vertices. As for every forest we have D D n1 > 2 + J2(i - 2)ni it follows that the total number of internal vertices nint satisfies the inequality n1 nint < . Thus n1 m-2 < nint - 1 < "2" and . (nint\ n1 m1 n 2 ) < i-. For every i, 1 < i < m1 + m2, let vi and ui be the vertices forming pair i and let Xi be random variable such that X = I 1, pd(ui) = pd(vi), 1 0, otherwise. We have ui vi E(Xi) = Pr(pd(ui) = pd(vi)) < J ^l^2^ . D!n1/2 3, otherwise. Let X be random variable counting the"bad" pairs. Of course, mi+m2 E(X)= ^ E(Xi) < m1D!n1/2-3 + m2D!n1/2-2 < 1, (3.1) i=1 if only n1 is large enough. This implies that for almost every forest satisfying given conditions, there exists a labeling w* using only the primes lower or equal to n1 that distinguishes all the internal vertices. Moreover, for every internal vertex v we have pd(v) > ni and pd(v) mod 2=1. Let us choose any such labeling. Now we are going to extend w* on the pendant edges in order to obtain complete labeling w. We will do it in two steps. Let next be the number of external vertices. In the first step, for each such vertex we leave two pendant edges incident to it not labeled. If there are other pendant edges (i.e. for at least one external vertex v, d(v) > 3), then we put on them distinct labels from the set {1,,..., n1 - 2next} (in any order). Let pd*(v) be the product of all edges incident with v that have been labeled. In the second step we order the external vertices with non-decreasing value of pd*(v). Then we label two edges incident with ith external vertex (1 < i < next) using labels n1 - 2next + i and n1 - i + 1. Observe that the products of the pairs of labels increase with i. After the second step, the product degrees of external vertices satisfy the conditions: pd(vj) < pd(vj ) if i < j, pd(vi) > n1 and pd(vi) mod 2 = 0. For pendant vertices we have in turn: pd(vi) < n1 and pd(vi) = pd(vj ) if i = j. As there can be at most one vertex v with pd(v) = 0, this finishes the proof. □ Corollaries From the above one can deduce the following two corollaries. Corollary 3.1. Let d > 2 be arbitrary integer. Then for almost all the complete d-ary trees ps(T ) = n1. Proof. We proceed as in the proof of Theorem 1.3. Even if d =2, there is only one vertex of degree 2 (the root) and its product degree is the product of two (not necessarily distinct) primes p1,p2 > %/2. It distinguishes this vertex from all other internal and pendant vertices. And even if n1 G {p1,p2}, it is impossible to obtain the external vertex with same product degree, as the triple 1, p1, p2 cannot appear (pendant edges should be labeled 1 and n1 this time, what would imply n1 mod 2 = 0, contradiction). Thus all the product degrees are distinct. □ One of the most important facts used in the proof of Theorem 1.3 is that n1 > n/2. However, as it may be easily checked, the inequality analogous to (3.1) will be satisfied even for smaller number of pendant vertices. Corollary 3.2. Let D > d > 3 be arbitrary integers. For almost all graphs G such that (i) 0(G) = 1, A(G) = D, (ii) if we remove all the pendant edges, then for the resulting graph G', S(G') = d > 4, (d— 1)/2 (iii) n1 >> n2/(d-1) lnn (i.e. n = o( niln )), (iv) none of the external vertices is adjacent to exactly one pendant vertex, the product irregularity strength equals to ps(G) = ni. Proof. We proceed as in the proof of Theorem 1.3. The difference is that this time we do not distinguish pairs of adjacent and non-adjacent vertices. The inequality (3.1) takes the form: E(X) < (n)D!ni/2-d+1 < 1, and the deterministic part of the proof remains unchanged. □ Two simple observations Finally let us add two simple observations on some special families of trees. Proposition 3.3. Let K1,n be star with n pendant vertices, n > 2. Then ps(K1,n) = max{3,n}. Proof. In the case n = 2, two labels 1 and 2 are not enough (either we use two equal labels and obtain same product degrees of pendant vertices, or we label the edges with 1 and 2 and obtain two product degrees 2). On the other hand, using labels 2 and 3 we produce product-irregular labeling. If n > 3, we need at least n labels to distinguish the product degrees of pendant vertices and this is enough, as the product degree of central vertex equals n! > n. □ Centipede Qn is the graph with V(Qn) = {u1,..., un, v1,..., vn} and E(Qn) = {{Ui, Vi}, 1 < i < n} U {{vi, vi+1}, 1 < i < n - 1}. Proposition 3.4. Let Qn be a centipede, n > 2. Then ps(Qn) = max{3, n}. Proof. If n = 2, two labels are not enough as it would be possible to obtain at most three distinct products 1, 2 and 4 and we have four vertices. If n > 3, we need at least n labels to distinguish all the pendant vertices. So ps(Qn) > max{3, n}. The product-irregular labelings realizing this bound are given as follows: (i) If n = 2, put w({ui,vi}) = i, i = 1, 2 and w({v^v2}) = 3. Then pd(u1) = 1, pd(u2) = 2, pd(vi) = 3 andpd(v2) = 6. (ii) If n = 3, put w({u2,v2}) = 1, w({ui,vi}) = w({vi,v2}) = 2, w({u3,v3}) = w({v2,v3}) = 3. Thenpd(ui) = 2,pd(u2) = 1,pd(u3) = 3,pd(vi) = 4,pd(v2) = 6 and pd(v3) = 9. (iii) If n > 4, put w({ui, vi}) = n — 1, w({un, vn}) = n — 2, w({un_i, vn_i}) = n, w({uj, vj}) = i — 1, 2 < i < n — 2 and w({vj, vj+1}) = n, 1 < i < n — 1. Then product degrees of pendant vertices are distinct numbers from the set {1,2,..., n} and the product degrees of external vertices - distinct numbers from the set {n(n — 2), n(n — 1)} U {in2, 1 < i < n — 3} U {n3}. □ Aknowledgments I would like to thank Professor Tomasz Luczak for inspiring discussions. I also thank two anonymous referees for all the remarks that allowed to improve the quality of the paper. References [1] M. Aigner M and Triesch, Irregular assignments of trees and forests, SIAM J. Discrete Math. 3 (1990), 439-449. [2] N. Alon and J. H. Spencer, The Probabilistic Method, John Wiley & Sons, 2000. [3] D. Amar and O. Togni, Irregularity strength of trees, Discrete Math. 190 (1998), 15-38. [4] M. Anholcer, Product irregularity strength of graphs, Discrete Math. 309 (2009), 6434-6439. [5] G. Chartrand, M. S. Jacobson, J. Lehel, O. R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congressus Numerantium 64 (1988), 187-192. [6] M. Kalkowski, M. Karonski, F. Pfender, A new upper bound for the irregularity strength of graphs, SIAM J. Discrete Math 25 (2011), 1319-1321. [7] O. Pikhurko, Characterization of product anti-magic graphs of large order, Graphs Combinator. 23 (2007), 681-689. [8] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-97. [9] J. Skowronek-Kaziow, Multiplicative vertex-colouring weightings of graphs, Inform. Process. Lett. 112 (2012), 191-194. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 31-39 Polytopes associated to dihedral groups Barbara Baumeister * Universität Bielefeld, Germany Christian Haase f Goethe-Universität Frankfurt, Germany Benjamin Nill * Case Western Reserve University, Cleveland, OH, USA Andreas Paffenholz § Technische Universität Darmstadt, Germany Received 2 December 2011, accepted 30 October 2012, published online 8 January 2013 In this note we investigate the convex hull of those n x n permutation matrices that correspond to symmetries of a regular n-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart h*-vector. Keywords: Permutation polytopes, dihedral groups, lattice polytopes. Math. Subj. Class.: 20B35, 52B12; 05E10, 52B05, 52B20 *The first author likes to thank for the support by the DFG through the SFB 701 "Spectral Structures and Topological Methods in Mathematics". t The second author is supported by DFG Heisenberg (HA 4383/4). * The third author is supported in part by the US National Science Foundation (DMS 1203162). §The last author is supported by the Priority Program 1489 "Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory" of the German Research Council (DFG). E-mail addresses: b.baumeister@math.uni-bielefeld.de (Barbara Baumeister), haase@math.uni-frankfurt.de (Christian Haase), benjamin.nill@case.edu (Benjamin Nill), paffenholz@mathematik.tu-darmstadt.de (Andreas Paffenholz) Abstract 1 Introduction To any finite group G of real n x n permutation matrices we can associate the permutation polytope P (G) given by the convex hull of these matrices in the vector space Rnxn. A well-known example of such a polytope is the Birkhoff polytope Bn, which is defined as the convex hull of all n x n permutation matrices [9, 8]. This polytope appears in various contexts in mathematics from optimization to statistics to enumerative combinatorics. (See, e.g., [24, 20, 21, 2, 1].) It is also a famous example of a Gorenstein polytope (see Section 5). Gorenstein polytopes turn up in connection to mirror symmetry in theoretical physics. Guralnick and Perkinson [15] studied polytopes associated to general subgroups G of the symmetric group and proved results about their dimension, and about the diameter of their vertex-edge graph. A systematic exposition of general permutation polytopes is given in [5]. There, we studied which groups lead to affinely equivalent polytopes, we considered products of groups and polytopes, classified low-dimensional cases, and we formulated several open conjectures. In order to get an intuition about what one can expect in general, it is instructive to consider some special classes of permutation groups. A seemingly very difficult case is when G equals the group of even permutation matrices. Just to exhibit exponentially many facets is already a daunting task, for this see [17]. Even for cyclic G we showed in [6] that these polytopes have a surprisingly complex and not yet fully understood facet structure. In [12] Collins and Perkinson studied polytopes given by Frobenius groups. A special case is the dihedral group Dn for n odd, which was considered in more detail by Steinkamp [22]. Since Dn is the automorphism group of a regular n-gon, the cases where n is even and odd are quite different. The most recent paper on permutation polytopes [11] focused on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, and Frobenius groups. In order to compute the volume of P(Dn), the authors find a Gale dual combinatorial description, which they use to provide an explicit formula for the Ehrhart polynomial of P (Dn). The dihedral group Dn is the automorphism group Aut(Cn) of a cycle Cn, and any permutation matrix M (a) of an element a e Dn commutes with the adjacency matrix A of Cn. So any point in P(Dn) commutes with A, and P (Dn) c {M e Rnxn | M is doubly stochastic and MA = AM} . Here, a matrix is doubly stochastic if all entries are non-negative and each row and column sum is 1. Tinhofer [24] asks, more generally, for a classification of those undirected graphs G where the two sets above are equal, i.e. where the commutation condition MA = AM already suffices to characterize the elements of P(Aut(G)) among all doubly stochastic matrices. The Birkhoff-von Neumann theorem is the special case where A is the unit matrix. Tinhofer shows that this also holds for the adjacency matrices of cycles and trees [24, Theorems 2&3]. In this note, we independently investigate P(Dn) in a more direct and elementary way. We give a complete list of its facet inequalities (Theorem 3.3, Theorem 4.1). As an application, we observe that these lattice polytopes are Gorenstein polytopes, and we get a nice description of the generating function of their Ehrhart polynomials (Theorem 5.3, Corollary 5.4). Acknowledgments: Many results are based upon experiments and computations using the package polymake [14] by Gawrilow and Joswig. We would like to thank the referees for carefully reading and improving the text. 2 Notation and preliminary results Let Sn be the permutation group on n > 3 elements. Every permutation a e Sn can be represented by an n x n matrix Ma with entries Si(j)a. So the entries are in {0,1} and there is exactly one 1 in each row and column. Notice that we apply matrices and permutations 2 from the right. We can view such a matrix as a vector in Rn . For a subgroup G of Sn we define the polytope PG := conv(Mff | a e G). This is a 0/1-polytope, so all matrices are in fact vertices of the polytope. We denote by Dn the subgroup of Sn corresponding to the symmetry group of the regular n-gon, the dihedral group of order 2n. This group is generated by two elements. If n is odd, then these may taken to be the rotation p of the n-gon by 360/n degrees, and the reflection t along a line through one vertex and the midpoint of the opposite edge. If n is even, then the second generator t is instead the reflection along a line through two opposite vertices. Thus p is the permutation (1,2,... ,n) and t the reflection (2, n)(3, n -1) •• • ((n + 1)/2, (n + 3)/2) if n is odd and (2,n)(3,n - 1) •• • (n/2, (n/2) + 2) if n is even. The associated permutation polytope is the convex hull of the corresponding matrices, DPn := conv(Mff | a e Dn). The dihedral group Dn has 2n elements p°,pv, , pn 1,T,Tp,Tp2 ,Tp3, . . . ,Tpn 1. , wn-1 in this order. Let us give a We label the vertices of DPn by v°,..., vn-1, w°, more convenient way to write these matrices. Let I be the n-dimensional identity matrix and R be the n x n matrix that has 0's everywhere except at the n entries (i,j), where 0 < i, j < n — 1 and j = i +1 mod n: R 01 00 00 00 10 01 00 Reading the matrices Ma row by row, we can identify Ma with a (row) vector in Rn . For instance, the 2 x 2 identity matrix would be identified with (10 0 1). Under this identification the vertices of DPn are (in the order given above) the rows of the 2n x n2 matrix R° R1 R° R 1 R2 R n- 1 2 R R-(n-1) Permuting the coordinates (corresponding to a linear automorphism of Rn ) we may write the vertices in the form V III ••• I I R-2 R-4 ... R-2(n-1) (2.1) Clearly, the first 2n coordinates of the vertices linearly determine the remaining coordinates. So we can project onto R2n without changing the combinatorics of the polytope. Hence, we observe that the dimension of DPn is at most 2n. 3 The situation for odd n In this section we completely describe DPn for n odd. As it will turn out, it is useful to introduce a new polytope that will serve as a basic building block for both situations of even n and odd n. Definition 3.1. Let Qn be the polytope defined as the convex hull of the rows of the 2n x n2 matrix W := II I ... I I R1 R2 ... Rn-1 (3.1) While Qn differs from DPn for even n, for odd n the R2k for 0 < k < n — 1 are a permutation of the Rk for 0 < k < n — 1. So we deduce from (2.1) that, for n odd, Qn is up to a permutation of coordinates just the polytope DPn. Proposition 3.2. For odd n, the polytopes DPn and Qn are affinely isomorphic. □ The following theorem examines the structure of Qn for arbitrary n. For n odd, this result is a special case of Theorem 4.4 in [12]. Let us fix some convenient notation. We denote by Ar the r-dimensional simplex. We also use for any two integers s, k, the term [s]k G {0,..., k — 1} to denote the remainder of s upon division by k. The free sum of two polytopes P and P ' of dimensions d and d' is the polytope P e P' := conv({(p, 0) G Rd+d' | p G P} U {(0,p') G Rd+d' | p' G P'}). Theorem 3.3 (Collins&Perkinson [12]). Let n be odd or even. The polytope Qn has dimension 2n — 2 and is a free sum of two copies of An-1. Taking coordinates x0,..., xn2-1 for Rnxn, its affine hull is given by the equations (l + 1)n-1 1= Y xi (aff) i=ln 0 = xkn+[j]n — x(k + 1)n+[j]„ — x(k+1)n+[j + 1]„ + x(k+2)n+[j+1]„ (Aj,k) for 0 < I < n — 1, 0 < j < n — 2, 0 < k < n — 3. An irredundant system of inequalities defining the polytope inside its affine hull is given by the inequalities for 0 < i < n2 — 1. Proof. All the given equations are satisfied by the vertices of Qn. There are n equations of type (aff) and n2 - 3n + 2 equations of type (Aj,k). They are easily seen to be linearly independent, so the dimension of Qn is at most 2n - 2. On the other hand, deleting any row of W leaves us with a linearly (and hence affinely) independent set of row vectors. (Observe that deleting a row leaves us with a column that contains exactly one 1.) Hence, dim(Qn) = 2n - 2 and the given equations define the affine hull of Qn in Rn . Further, we see that every 2n - 1 of the 2n rows of W span the affine hull of Qn. So any facet of Qn has 2n - 2 vertices. Since the inequalities xj > 0 are 0 on exactly 2n - 2 of the rows, they all define facets. In order to prove that Qn is a free sum of simplices we observe that the first n and the last n vertices define (n - 1)-dimensional simplices sitting in transversal subspaces (intersecting in the matrix corresponding to the row vector (1/n,..., 1/n)). Therefore, the combinatorial dual of Qn corresponds to the product of An-1 with itself. In particular, Qn has precisely n2 facets, so the facet description given above is complete. □ 4 The situation for even n Recall that the join P * Q of two polytopes P and Q is the convex hull of P U Q after embedding P and Q in skew affine subspaces. The dimension of P * Q equals dim(P) + dim(Q) + 1. For instance, the join of two intervals is a tetrahedron. Theorem 4.1. Let n be even. The polytope DPn is a join of two copies of Qn/2. In particular, its dimension is 2n - 3. Combined with Theorem 3.3, this result gives a complete description of the facet inequalities and the affine hull equations of DPn for n even. Proof. Permuting the coordinates, we can transform V (see (2.1)) into III ■■■ I III ■■■ I " E>0 E>2 t?4 on—2 t?0 t?2 t?4 K>n —2 R R R • • • R R R R • • • R Clearly, projecting onto the first nr coordinates yields an affine isomorphism of DPn onto n2 the convex hull of the rows of the 2n x matrix I I I ... I R0 R2 R4 ... Rn-2 In the representation given by this matrix let us partition the set of 2n vertices (labelled from 0 to 2n - 1) into two sets: consisting of the n rows with even index and the n rows with odd index. sort rows sort columns Then we permute the nr coordinates in such a way that in the first set of rows (correspond- n2 ing to even vertices) all nonzero entries are in the first half (i.e. in the first ^ columns). Then all nonzero entries in the second set of rows (corresponding to the odd vertices) will 2 be in the second half (i.e. in the last ^ columns). By a permutation of the coordinates within the first half we get that the rows of even vertices yield precisely the vertex set of Qn/2 x {0} (for 0 g R t ). In the same way, the coordinates in the second half can be n2 permuted so that the rows of odd vertices equal the vertices of {0} x Qn/2 (for 0 g R t ). Since 0 is not in the affine hull of Qn/2, we deduce that DPn is a join of two copies of Qn/2. Hence, its dimension equals 2dim(Qn/2) + 1 = 2(n - 2) + 1 = 2n - 3 by Theorem 3.3. □ 5 Lattice properties n2 DPn and Qn are lattice polytopes, i.e. their vertices lie in the lattice Zn of integral vectors. It is readily checked that all above affine isomorphisms respect lattice points. In this section, we will show that these lattice polytopes have especially nice properties which allow us to completely describe their Ehrhart h*-vectors. A d-dimensional lattice polytope P containing 0 in its interior is reflexive, if its polar (or dual) polytope P * := {x g Rd | (x,v) > -1 V v g P} is again a lattice polytope (in the dual lattice). This notion was introduced by Batyrev in [3]. A generalization of this is the class of Gorenstein polytopes. A lattice polytope is a Gorenstein polytope ofcodegree k, if there is a positive integer k and an interior lattice point m in kP such that kP — m is a reflexive polytope. Such polytopes play an important role in the classification of Calabi-Yau manifolds for string theory. See [4] for basic properties. The next proposition tells us that the polytopes Qn belong to this class. The normalized volume of Rn is the volume form which assigns to the standard simplex the volume 1. Proposition 5.1. Let n be odd or even. The polytope Qn is Gorenstein ofcodegree n and normalized volume n. Proof. By Theorem 3.3, the point n (1,1,..., 1) is an interior point of Qn with equal integral distance 1/n to all facets, and m := (1,1,..., 1) is the unique interior lattice point in nQn. Hence nQn - m is a reflexive polytope. By Theorem 3.3, all facets of Qn are simplices of facet width 1, hence they are all unimodular. As we have seen, multiplying by n gives (up to translation) a reflexive polytope with the unique interior lattice point m = (1,1,..., 1). The normalized volume of nQn is the sum of the volumes of n2 pyramids over facets with apex m. But in nQn each facet has normalized volume n2n-3, and the apex has lattice distance 1 from the facet, so each pyramid has normalized volume n2n-3. There are n2 of these pyramids, so the normalized volume of nQn equals n2n-1. Dividing by n2n-2 to get from nQn back to Qn gives the normalized volume n of Qn. □ A polytope P is compressed if every so-called pulling triangulation is regular and unimodular. Equivalently, P is compressed if for any supporting inequality a4x < b with a primitive integral normal a, i.e. with a normal vector whose entries are integers and which is not an integral multiple of some other integer vector, the polytope is contained in the set {x | b - 1 < a/'x < b}. For a more detailed explanation of these terms we refer to [13]. This property has strong implications on the associated toric ideal, see e.g. [23]. The next proposition follows immediately from Theorem 1.1 of [19] and Theorem 3.3. Proposition 5.2. Let n be odd or even. The polytope Qn is compressed. □ The Ehrhart polynomial LP (k) := | kP nZd| ofa d-dimensional lattice polytope counts the number of integral points in integral dilates of P. It is well known that the generating function of LP is given by E Lp (m)tm = m> 0 V ' for some polynomial h* of degree at most d with integral non-negative coefficients, see [7]. Hence, determining the Ehrhart polynomial is equivalent to finding the h*-vector (also called the ^-vector) of coefficients of h* (t). As is well-known, P is Gorenstein if and only if the h*-vector is symmetric. The following theorem shows that in our case this vector has a particularly nice form. Theorem 5.3. Let n be oddor even. The h*-vector of Qn satisfies h* = 1 for 0 < i < n —1 and h/ =0 otherwise. Proof. Since the codegree of Qn is n and its dimension is 2n — 2 by Theorem 3.3, the maximal non-zero entry of the h*-vector has to be hn_i, see [7]. By a theorem of Bruns and Römer [10] we know that the h* -vector of a Gorenstein polytope that has a regular unimod-ular triangulation is symmetric and unimodal. In particular, h* > 1 for i = 0,..., n — 1. Since by Proposition 5.1 the sum of the entries of the h*-vector equals n, the statement follows. □ In particular, if n is odd, the previous result describes the h*-vector of DPn. Finally, let us deal with the even case. Corollary 5.4. Let n be even. The h*-vector of DPn equals (1,2,3,...,n — i,n,n — 1,..., 2,1). In particular, the polytope DPn is Gorenstein of codegree n and normalized volume n2/4. Proof. By the proof of Theorem 4.1, DPn convex hull of the rows of is given up to coordinate permutation as the W 0 ^ 0 W where W is the n x (n)2 matrix whose rows are the vertices of Qn as given in (3.1). The integral linear functional which sums the first f coordinates evaluates to 1 on the first n rows, and to 0 on the second half. Hence, the two copies of Q n (say, P1 x {0} and {0} x P2) have lattice distance 1 in the lattice Zn aff DPn. In other words, there is an affine isomorphism respecting lattice points which maps DPn onto the convex hull of n2 P1 x {0} x {1} and {0} x P2 x {0} in Rtt+1. Therefore, the statement follows from the well-known fact [7, Example 3.32] that in this case the h*-polynomial equals the product of the h* -polynomials of P1 and P2. □ 6 Substructures In [5] the authors discussed which subgroups of a permutation group yield faces of P (G). An obvious class of such subgroups are stabilizers: Take a partition [n] := {1,..., n} = [J /i. Then the polytope of the stabilizer of the subsets / stab(G; (/i)i) := {a G G | a(/i) = /i for all i} < G is a face of P (G). The authors conjecture that there are no other examples. Conjecture 5.8 [5] Let G < Sn. Suppose H < G is a subgroup such that P (H ) ^ P (G) is a face. Then H = stab(G; (/i)i) for a partition [n] = [J /. We have verified the conjecture for G = Sn as well as for cyclic subgroups G < Sn, see Proposition 5.9 of [5]. Meanwhile Jessica Nowack and Daniel Heinrich studied this question for the dihedral groups in their Diploma theses. Proposition 6.1. (Heinrich, Nowack [16, 18]) Conjecture 5.8 holds for G = Dn < Sn for every n. Sketch of the proof. For n odd Heinrich first shows that, if H is the subgroup of all rotations of G, then PH is not a face of PG. The remaining subgroups are precisely the stabilizers of their orbits, see Theorem 7.1.1 of [16]. For n even the main work is to show that the subgroup of all rotations, the subgroup of the squares of the rotations and finally the subgroup generated by the squares of the rotations and by the reflections through two edges are precisely those subgroups H of G for which PH is not a face of PG. Nowack shows that the remaining subgroups are precisely the stabilizers of their orbits, see Section 4.2 of [18]. □ References [1] C. A. 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Nowack, Beweis der Seitenvermutung für Permutationspolytope vom Typ D2n mit geradem n, Diploma thesis, FU Berlin, Mai 2011. [19] H. Ohsugi and T. Hibi, Convex polytopes all of whose reverse lexicographic initial ideals are squarefree, Proc. Am. Math. Soc. 129(9) (2001), 2541-2546. [20] S. Onn, Geometry, complexity, and combinatorics of permutation polytopes, J. Comb. Theory, Ser. A 64(1) (1993), 31-49. [21] I. Pak, Four questions on Birkhoff polytopes, Ann. Comb. 4(1) (2000), 83-90. [22] H. Steinkamp, Convex polytopes of permutation matrices, Bachelor thesis, The Division of Mathematics and Natural Sciences, Reed College, 1999. [23] B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series (vol. 8), American Mathematical Society, Providence, RI, 1996. [24] G. Tinhofer, Graph isomorphism and theorems of Birkhoff type, Computing 36 (1986), 285300. /^creative , ars mathematica ^commons contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 41-54 A class of semisymmetric graphs Li Wang School of Mathematical Sciences, Capital Normal University, Beijing, 100048, PR China and School of Mathematics and Information Sciences, Henan Polytechnic University, Jiaozuo, 454000, P R China Shaofei Du * School of Mathematical Sciences, Capital Normal University, Beijing, 100048, PR China Xuewen Li Department of Mathematics and Information Sciences, Tangshan Normal University, Tangshan, 063000, P R China Received 20 November 2011, accepted 1 October 2012, published online 14 January 2013 Abstract A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. Let p be a prime. In this paper, a class of semisymmetric graphs of order 2p3 are determined. This work is a partial result for our long term goal to classify all semisymmetric graphs of order 2p3. Keywords: Permutation group, primitive group, vertex-transitive graph, semisymmetric graph. Math. Subj. Class.: 05C10, 05C25, 20B25 1 Introduction All graphs considered in this paper are finite, undirected, connected and simple. For a graph X, we use V(X), E(X) and A := Aut(X) to denote its vertex set, edge set and the full automorphism group, respectively. The graph is said to be vertex-transitive and edge-transitive, if A acts transitively on V(X) and E(X), respectively. If X is bipartite with bipartition V(X) = W(X) U U(X), we let A+ be the subgroup of A preserving both * Corresponding author. E-mail addresses: wanglimath@hpu.edu.cn (Li Wang), dushf@mail.cnu.edu.cn (Shaofei Du), lixuewen@sina.com (Xuewen Li) W(X) and U(X). Since X is connected, we have that either |A : A+1 = 2 or A = A+, depending on whether or not there exists an automorphism which interchanges the two parts. For G < A+, the graph X is said to be G-semitransitive if G acts transitively on both W(X) and U(X), and semitransitive if X is A+-semitransitive. We call a graph semisymmetric if it is regular and edge-transitive but not vertex-transitive. It is easy to see that every semisymmetric graph is a semitransitive bipartite graph with two parts of equal size. The first person who studied semisymmetric graphs was Folkman. In 1967 he constructed several infinite families of such graphs and proposed eight open problems see [13]. Afterwards, Bouwer, Titov, Klin, I.V. Ivanov, A.A. Ivanov and others did much work on semisymmetric graphs see [2, 4, 16, 17, 18, 30]. They gave new constructions of such graphs and nearly solved all of Folkman's open problems. In particular, Iofinova and Ivanov [16] in 1985 classified biprimitive semisymmetric cubic graphs using group-theoretical methods. This was the first classification theorem for such graphs. More recently, following some deep results in group theory which depend on the classification of finite simple groups and some methods from graph coverings, some new results on semisymmetric graphs have appeared. For instance, in [11] Du and Xu classified semisymmetric graphs of order 2pq for two different primes p and q. For more papers on semisymmetric graphs see [5, 7, 8, 9, 10, 11, 12, 19, 21, 22, 23, 24, 25, 26, 27, 28, 33]. In [13], Folkman proved that there are no semisymmetric graphs of order 2p and 2p2 where p is a prime. Then we are interesting in determining semisymmetric graphs of order 2p3, where p is prime. Since the smallest semisymmetric graphs have the order 20 [13], we let p > 3. It is proved in [25] that the Gray graph of order 54 is the only cubic semisymmetric graph of order 2p3. To classify all semisymmetric graphs of order 2p3 is still one of attractive and difficult problems. These graphs X are naturally divided into two subclasses: (1) Aut(X ) acts unfaithfully on at least one part; (2) Aut(X) acts faithfully on both parts. Now we are going to concentrate on Subclass (1). To state our main theorem, we first introduce two concepts. Let Y be a connected semitransitive and edge-transitive graph with bipartition V (Y) = W (Y)U U (Y),where W (Y ) = Zp and U (Y) = Zp for an odd prime p. For distinguishing the vertices of W (Y) and U (Y) convenience, the vertices of W (Y) and U (Y) are denoted by (i, j, k, 0) and (y, z, 1), respectively, where i, j, k, y, z e Zp. Now we define a bipartite graph X with bipartition W(X) U U(X), where W(X) = W(Y), U(X) = Zp X U(Y) = {(x, y, z, 1) | x, y, z e Zp}, such that two vertices (i, j, k, 0) e W(X) and (x, y, z, 1) e U(X) are adjacent if {(i, j, k, 0), (y, z, 1)} e E(Y). From now on, we shall say that the graph X is the graph expanded from Y and that the graph Y is the graph contracted from X. Clearly X is edge-transitive and regular. Furthermore, since for any (y, z, 1) e U(Y), the p vertices {(x, y, z, 1) | x e Zp} in U(X) have the same neighborhood, X is semisymmetric, provided there exist no two vertices in W (X ) which have the same neighborhood. Clearly, Aut(X) acts unfaithfully on W(X) and Aut(X)/Sp = Aut(Y). Note that the semisymmetric graphs where two vertices have the same neighbourhood have been studied in several papers see [11, 20, 29, 33], with different definitions, for instance, X is a derived graph from Y, X is a unworthy graph, X is contracted to from Y and so on. Let Y be a connected graph and B an imprimitive system of Aut(Y). Define a graph Z with the vertex set B such that two blocks are adjacent in Z if there exists at least one edge in Y between two blocks. This graph Z is called the block graph of Y. Moreover, if B is the set of orbits of some nontrival normal subgroup N of Aut(Y), then we call Z the block graph induced by N. The following proposition gives a characterization for Subclass (1) given in [31]: Proposition 1.1. Suppose X is a semisymmetric graph of order 2p3, where p is an odd prime, such that Aut(X ) acts unfaithfully on at least one part. Then Aut(X ) must act unfaithfully on one part and faithfully on the other part, and X is the graph expanded from the graph Y with bipartition V(Y) = W(Y)UU(Y), where W(Y) = Zp and U(Y) = Zp;. Moreover, we have that either (1) p = 3, Aut(Y ) = S3 I S3 which acts primitively on W (Y ); or (2) Aut(Y) has blocks of length p2 on W(Y) and of length p on U(Y). Let Y be the block graph of Y. Then either (2.1) the block graph Y is of valency at least 3, and Aut(Y ) is solvable and contains a normal regular subgroup on W(Y); or (2.2) the block graph Y is of valency 2, where Aut(Y) may be solvable or insolv-able. Following Proposition 1.1, in this paper we shall determine the graphs in Case (2.2), while Cases (1) and (2.1) will be determined in our another paper. Before giving the main theorem of this paper, we first define six families of graphs Y. Definition 1.2. We shall define six families of bipartite graphs X with bipartition V(X ) = W(X) U U(X), where W(X) = {(i,j,k, 0) I i,j,k G Zp}, U(X) = {(x,y,z, 1) I x,y,z G Zp}, and edge set E(X) = {{(i,j,k, 0), (x,i + b,k + p—■, 1)}^,j,k,x G Zp,b G S}U {{(i,j,k, 0), (x,j + sb,k + p++1, 1)} I i,j,k,x G Zp,b G S}, p — 1 where s = 0, Zp = (0) for the family of graphs X2(p, r), s =1 for other five families of graphs X^(p, r), and S is given by (1) Graphs Xi(p, r): Let p > 3 and let S be a subgroup of Zp of order r, where (p,r) = (7, 3), (11,5). Moreover, the valency of X1 (p,r) is 2pr and the smallest examples are X1(3,1) and X1(3, 2). (2) Graphs X2(p, r): Let p > 5 and let S be a subgroup of Zp of order r > 2, where (p, r) = (7,3), (11, 5) and 2r | (p — 1). Moreover, the valency of X2(p, r) is 2pr and the smallest example is X2 (5,2). (3) Graphs X3(11,5): Let p =11 and let S = {0, 2,3,4,8} C Zii. Moreover, the valency of X3(11, 5) is 110. (4) Graphs X4(11,6): Let p = 11 and S = {1,5,6, 7, 9,10} C Zn. Moreover, the valency of X4 (11, 6) is 132. (5) Graphs X5(p, r): Choose a point (v) and a hyperplane L in the project space PG(n - 1, q), where q—- = p > 7, and let G = (t) be a Singer subgroup of PGL(n, q). Let S = {l G Zp | (v) G L }, where r = |S| = . Moreover, the valency of X5 (p, r) is 2pq q--1 and the smallest example is X5 (7, 3). (6) Graphs X6(p, r) : Adopting the same notation as in (5), set S = {l G Zp | (v) G L }, where r = qn-1. Moreover, the valency of X6(p, r) is 2pqn-1 and the smallest example is X6(7,4). Remark 1.3. For 1 < i < 6, let Xj(p, r) be as in Definition 1.2. Then (1) For any given y, z G Zp, the p vertices {(x, y, z, 1) | x G Zp} have the same neighborhood. Let Yj(p, r) be the contracted graph from Xj(p, r), obtained by contracting each such p vertices into one vertex while preserving the adjacent relation, that is, W(Yi(p,r)) = W(Xi(p,r)), U(Y(p,r)) = {(y,z, 1) | y, z G Zp}. Then we shall see from the proof of Theorem 1.4 that Aut(Yj(p, r)) = K x D2p, where the subgroup K is the following (i) Y1(p, r) and Y>(p,r): K = Sp if r G {1,p - 1}; K = (Zp x Zr)p if r G {1,p -1}; (ii) Ys(11, 5) and Yt(11,6): K = (PSL(2,11))p; (iii) Y5(p, r) and Y6(p,r): K = (PrL(n,q))p. (2) For any k G Zp, let Wk(Y) = {(i, j,k, 0) G W(Y) | i, j G Zp}, Uz(Y) = {(y, z, 1) | y G Zp}. Then we shall see from the proof of Theorem 1.4 that {Wk (Y ) | k G Zp} and {Uz (Y ) | z G Zp} are orbits of the group K on W (Y ) and U (Y ), respectively. Let Y be the block graph induced by K. Then Y is a cycle of length 2p. Now we give the main theorem of this paper. Theorem 1.4. For an odd prime p, suppose that X is a semisymmetric graph of order 2p3 expanded from a graph Y such that Aut(Y) has the blocks of length p2 on W (Y ) and of length p on U (Y) while the block graph Y is a cycle of length 2p. Then X is isomorphic to one of graphs Xj(p, r) where 1 < i < 6, defined in Definition 1.2. After this introductory section, some preliminary results will be given in Section 2, and the main theorem will be proved in Sections 3. For group-theoretic concepts and notation not defined here the reader is refereed to [6, 15]. 2 Preliminaries First we introduce some notation. By H char G, we mean that H is a characteristic subgroup of G. Given a group G and a subgroup H of G, by Cos(G, H) we denote the set of right cosets of H in G. The action of G on Cos(G, H) is always assumed to be the right multiplication action. For two subgroups N < G and H < G, by N x H we denote the semi-direct product of N by H, where N is normal. For a group G, by Exp (G) we denote the least common multiple of orders of all the elements of G. By H i K, we denote the wreath product of H and K. A group-theoretic construction of semitransitive and semisymmetric graphs were given in [11]. Here we quote one definition and two results. Definition 2.1. Let G be a group, let L and R be subgroups of G and let D be a union of double cosets of R and L in G, namely, D = Ui Rdi L. Define a bipartite graph X = (G, L, R; D) with bipartition V(X) = Cos(G, L) U Cos(G, R) and edge set E(X) = {(Lg, Rdg) | g G G, d G D}. This graph is called the bi-coset graph of G with respect to L, R and D. Proposition 2.2. [11] The graph X = b(G, L, R; D) is a well-defined bipartite graph. Under the right multiplication action of G on V (X ), the graph X is G-semitransitive. The kernel of the action of G on V(X) is CoreG(L) n CoreG (R), the intersection of the cores of the subgroups L and R in G. Furthermore, we have (i) X is G-edge-transitive if and only if D = RdL for some d G G; (ii) the degree of any vertex in Cos(G, L) (resp. Cos(G, R)) is equal to the number of right cosets of R (resp. L) in D (resp. D-1), so X is regular if and only if |L| = |R|; (iii) X is connected if and only if G is generated by elements of D-1D; (vi) X = b(G, La, Rb; D') where D' = Ui Rb (b-1dia)La,for any a,b G G; (v) X = b(G, La ,Ra ; Da ) where a is an isomorphism from G to G (it does not appear in [11] but it is easy to prove.) Proposition 2.3. [11] Suppose Y is a G-semitransitive graph with bipartition V (Y) = U(Y) U W(Y). Take u G U(Y) and w G W(Y). Set D = {g G G | wg G Yi (u)}. Then D is a union of double cosets of Gw and Gu in G, and Y = b(G, Gu, Gw ; D). Proposition 2.4. [32, 11.6, 11.7] Every permutation group of prime degree p is either insolvable and 2-transitive, or isomorphic to Zp x Zs for some s dividing p — 1. Proposition 2.5. [14] The insolvable permutation groups of prime degree p are given as follows, where T denotes be the socle of the group and H denotes a point stabilizer of T : (i) T = Ap and H = Ap-i; (ii) T = PSL(n, q) and H is the stabilizer of a projective point or a hyperplane in PG(n — 1, q), and |T : H| = (qn — l)/(q — 1) = p; (iii) T = PSL(2,11) and H = A5, and T has two conjugacy classes of subgroups isomorphic to A5; (iv) T = Mi1 and H = Mi0; (v) T = M23 and H = M22. Lem ma 2.6. [31] Let G be an imprimitive transitive group of degree p2 with p > 3 and p3 j |G|. Suppose that G has an imprimitive system B with p-blocks and the kernel K. Let P be a Sylow p-subgroup of G and N = P n K. Then (1) Exp (P ) < p2, |Z (P )| = p and P = N (t), where tp G Z (P ); (2) K is solvable, N char K and so N < G, provided either p = 3; or p > 5 and |N| < pp-1. 3 Proof of the main theorem To prove Theorem 1.4, we assume that p is an odd prime and that X is a semisymmetric graph of order 2p3 expanded from the graph Y, where Aut(Y) acts edge transitively on Y and has blocks of length p2 on W(Y) and of length p on U(Y), and the block graph Y is a cycle C2p of length 2p. Let F = Aut(Y) and let B = {Bo,Bi, ■■■ ,Bp-1} and B' = {B'0 ,B[, ■■■ ,B'p_ 1} be blocks system of F on U (Y) and W (Y), respectively. Label E(Y) = {(Bo, B'p+i ), (B'p+i , Bi), • • • , (B, B'0), (B'0, Bp+i ), • • • , (Bp_i, B'p_1 ), (B'p_i , B0)}, 22 2 2 22 so that Y = Cc2p. Set /p — 1 p + 1 ~2 a =(0,1, ■■■ ,p — 1) and t = (0)(1, —1) ■■■ (*—-) G Sp. Then Aut(Y) = (a,t) = by defining (Bj)7 = Bìy and (Bj)Y = BjT for any Y G (a, t). Label the vertices in Bj by j for j G Zp. By considering the imprimitive action of F on U (Y ), we know that F < Sp ? (a, t) = Sp X (a, t), where, for any e =(e(0),e(1), ■■■ ,e(p-1)) G Sp and 7 G (a, t ), we have ,(e;Y) = a ... j = VW iY . In particular, by identifying (1,7 ) with 7 so that aji = , we have that (a, t) can be viewed as a subgroup of F. From now on, for any t G T < Sp and i G Zp, we set i+1 ti = (mT-"-,t, 1, ■■■ , 1) and Ti = (ti | t G T), where Ti acts transitively on Bi and fixes Bj pointwise for all j = i. Moreover, we have j.(e;7) ___ rp _ rp j^a^ _ r> ti = tiY , T0 = = Ti, B 0 = Bi. a Since Kis a transitive group of degree p, following Propositions 2.4 and 2.5 we need to consider the following four cases separately in four subsections: (i) p > 5 and KBi is insolvable; (ii) p > 5 and KB- = Zp x Zr for r = 1; (iii) p > 5 and KBi = Zp. (iv) p = 3. 3.1 KBi is insolvable for p > 5 Lemma 3.1. Suppose that p > 5 and KBi is insolvable. Then Y is isomorphic to one of the following graphs: (i) Yi(p, r), and Aut(Y) = Sp I D2p, where r = 1 or p — 1; (ii) Y3(11,5) and Y4( 11,6), and Aut(Y) = PSL(2,11) I D22; (iii) Y5(p, ^— ) and Ye(p, qn-1), and Aut(Y) = PrL(n, q) I D2p. Proof. Suppose that p > 5 and KBi is insolvable. Then by Lemma 2.6 we know that K = To X T1 x • • • x Tp-1, where T is an insolvable group of degree p and Ti is defined as before. In particular, a Sylow p-subgroup of F is of order pp+1, and so we may assume that F contains a defined as above. Let u g B0 and take an element g0 G Fu \ K. Since g0 fixes B0 setwise and exchanges B'p-i and B'IJ±1_, there exists a d = (d(0), d(1), • • • , d(p-1) ) G Sp suchthat g0 = dr, where 2 2 t is defined as before. Since F/K = D2p, by considering the order of F we get F = KR where R = (a, dr}. Let Ho = (To)u. Then Ku = H0 X T1 X • • • X Tp-1 and Fu = Ku x (dr}. By Kut = Ku, we know that d(0) G N{sp)0 (Ho) and d(i) G N(sp)4 (Ti) for i = 0. Now dr fixes the block B' setwise and exchanges B p— and B p+i. Take w G B'0. Since Tp-i xTp+i fixes u and acts transitively on B', there exists a k G Tp-i xTp+i < Ku 2 2 0 2 2 such that kdr fixes both u and w, where without loss of generality, we denote kd by d again so that dr fixes both u and w. Then Kw = T0 X • • • X T p-3 X L p—i X N p+i x T p+3 x • • • x T„ 1 and Fw = Kw (dr}. 2 2 2 2 L By KWt = Kw, we know that L = N and d(i) G N(Sp)i (Li) for i G { pT1, p+1}. Now the corresponding groups H and L are two maximal subgroups of T of index p. Following Proposition 2.5 we need to consider three cases separately. (1) H and L are conjugate in T. Without loss of generality, let H = L. For any almost simple group T in Sp, its point stabilizers have two orbits in each block Bi with the respective length 1 and p — 1. We may therefore let T = Sp so that H = Sp-1 and F = SpR = Sp(a, dr} = Sp(a, t}. Thus, we may set d = 1. For later use, we set t = (0,1, • • • ,p — 1), E1 = {0} and S2 = Zp. (2) soc(T) = PSL(2,11), and H and L are not conjugate in T. In this case T = PSL(2,11), and T has two nonequivalent representations on the set of right cosets of A5 of cardinality 11. Now F = Tp(a,dr). Since d(i) G Ns11 (Ti) = Ti, we have d G Tp and so F = Tp(a, t). Therefore, we set d = 1. Moreover, T may be considered as the automorphism group of a (11, 5, 2)-design D. Let V = Zu be the point set and let t = (0,1, • • • , 10) be an element of order 11 in T. Then M = {0, 2, 3,4,8} C V is a block (see [1, p.55]) of D. Without loss of generality, we choose L and H to be the stabilizes of the block M and point 0, respectively. Again, for later use, set S3 = M and S4 = Zn \ M. (3) soc(T) = PSL(n, q), and H and L are not conjugate in T. In this case, PSL(n, q) < T < Nsp(PSL(n, q))=PrL(n, q). With the same reason as (1), we let T = NSp (PSL(n, q)) and d =1. Let Si and S2 be the set of points and hyperplanes ofPG(n,q - 1), respectively, where |S1| = |S2| = q—i1 = P. Without loss of generality, we choose L and H to be the stabilizers of a given point (v) and a hyperplane L, respectively. Let G = (t) = Zp be a singer subgroup of PGL(n, q). Let £5 = {l G Zp j (v) G L }, Se = Zp \ Si = {l G Zp j (v) G L }, where IS5I = and |S6| = qn-i. Now for the above three cases (1)-(3), we have Cos(F,Fw) = {FwtUtjp+1 ak | i,j,k G Zp}, Cos(F,F„) = {Futy0az j y, z G Zp}. 2 —2~ Clearly, Fw has two orbits on Bp-1 U Bp+1, that is, 2 2 D, = {FutOaV,Fu faj b G E,}, where l = 1, 3,4, 5,6. For any point Fwtip_1 tjp+1 ak in W(Y), since _ 1 p+1 p+1 _ 1 _ 1 FutOatUtjp+1 ak = Fut0(tL! )i(t^+1 )ja+k = Fu^a+k 22 22 = Fut0+bak+ ^, and similarly, F tbap±iti tj ak = F tj+bak+ p±i Fut0a 2 t p_1t p+1a = Fut0 a 2 , 2 2 its neighbor is Diti-1 tjp+1 ak = {Fut0+bak+, Fut0+bak+ ^ j b G E,}. 2 2 By labeling Fwti-tjp+1 ak by (i, j, k, 0) and Fut0az by (y, z, 1), we get the respective 2 edge set of two graphs Y(l) Ei = {((i, j, k, 0), (y, z, 1))jy = i + b, z = k + p-1 ; and y = j + b, z = k + p+1, i, j, k, y, z G Zp,b G £,}. In cases (2) and (3), we get the graphs Y3(11,5),Y4(11, 6) with the automorphism group PSL(2,11) 1 D2p, and Y5(p, r), Y6(p, r) with the automorphism group PrL(n, q) 1 D p. For case (1), the graph with the edge set E2 is exactly Yi(p,p - 1) for p > 5. As for the graph with the edge set E1, let ^ be a map on W (Y) U U (Y) which fixes W (Y) pointwise and sends (y, z, 1) to (y +1, z, 1). Then ^ is an isomorphism between the present graph and Y1(p, 1). From the proof we know that both Y1(p, 1) and Y1(p,p - 1) have the automorphism group Sp 1 D2p. □ 3.2 KBi Zp x Zr for p > 5 and r = 1 Lemma 3.2. Suppose KBi = Zp x Zr for p > 5 and r = 1. Then Y = Y1(p,r) or Y2(p,r) where p > 5, r = 1, p - 1 and (p, r) = (7,3), (11, 5), where Aut(Y) = (Zp x Zr) \ D2p. Proof. Step 1: Determination of the structure of F. Proof. Suppose KBi = Zp x Zr for r = 1. Let S = (t) x (c) = Zp x Zp-1 < Sp. Then p— 1 we may set T = (t) x (h), where h = c. Let P be a Sylow p-subgroup of F and take d0a e P where do e (to) X (t1) X • • • X (tp-1) = Zp. Then K < Tp and F = K (d0a, dr) for some d e Sp. Moreover, F < F = Tp (d0a, dr) = Tp (a, dr) and (a, dr) / (Tp n (a, dr) ) = D2p. Let w e B0 and (B0, Bp-1 ), (B0,Bp±ì) e E(Y). Let (w,u1) e E(Y) for «1 e Bp-i. Then E = (w,u1)F < (w, u1)F. Since the orbits of Fw and Fw on the block Bp-i in U(Y) are completely the same, we have |(w,u1)F| = 2rp3 = |E|, which implies E = (w, u1)F. Therefore, we may just consider the case F = F = Tp(a, dr ). As in the last Lemma, we choose two vertices u e B0 and w e B'0 which are fixed by dr. Without loss of generality, let H = (h) so that Fu = (H0 XT1 X • • • xTp-1)(dr), Fw = (T0 xT1 x-xHp_i xHp±i x • • • xTp-1)(dr). We then need to determine the element d. Let d = (d(0), d(1), • • • , d(p-1)) e Sp. Since dr normalizes K, Ku and Kw, it follows that d(i) e Nsp(Hi) = (c) for i e {0, p±1}, and d(i) e NSp(Ti) = S = (t)(c) for i e {0, }. Suppose that i e {0, p±±1} and write d(i) = tmcn. Since Ti < Fu and Fw we may re-choose d(i) = cn. Therefore, for any i e Zp, we get d(i) e (c). (1) Since (dr)2 e K, we have drdr = ((d(0))2, d(1)d(p-1), • • • , d(p-1)d(1)) e K, and by taking into account (1) we get (d(0))2,d(1)d(p-1),••• ,d()d(p±i) e H. (2) p—1 P+1 Since Kw fixes only one point ua 2 in Bp— and ua 2 in B p+1 and since dr normalizes 2 2 Kw and exchanges B p— and B p+1 , it follows that dr must exchange these two points. 2 2 Therefore, Fua 1 (dr ) = Fu(d( 1 ),d( "+1 ), ••• ,d(P-1),d(°),d(1), ••• ,d( 3 )}ra = Fud(d(p—1 ),d( ^ ), ••• ,d(1),d(°),d(P-1), ••• ,d( "+1 ) )a = Fu(d(°)d(1 ), d(Dd(p—3), • • • , d(1 )d(°), d()d(P-D, ••• ,d(P-!)d( ^ V ^ p+1 = Fua 2 . Hence, d(0)d(^), d(1)d(3), • • • , d(^)d(0), d(^ )d(P-1), • • • , d(P-1)d(^ ) G H. (3) From (2) and (3) we get d(0),d(1) • • • , d(p-1) G H, or d(0),d(1) • • • , d(p-1) G (cp2—1 > \ H if 2r | (p - 1). (4) Therefore, if 2r f (p - 1) then we set d = 1; if 2r | (p - 1), we set d = (c'm, c'm, • • • c'm) p — 1 where c' = cand m = 0,1. To unify these two cases, in the first case we still write d = (c'm, c'm, • • • c'm) for m = 0. Suppose that 2r | (p - 1). Let F1 = K x (a, t> and F2 = K x (a,dr>, where p— 1 d = (c', • • • , c') with c' = c, noting that c' G T. we may then state the following fact Fact: F1 = F2 Proof: Assume the contrary. Suppose that 7 is an isomorphism from F1 to F2. Since ((t, t, • • • , t)> is characteristic in F1 and F2, we get 7((t,t••• t)) = (tk,tk,tk ••• ,tk) for some k G FP. Assume that y(t) = edT, where e = (e(0), e(1) • • • , e(p-1)) G K. Since t-1(t,t, ••• ,t)T = (t,t, ••• ,t). we have that is which implies y(t-1)7((t,t, ••• ,t))Y (t ) = Y(t,t, ••• ,t), (edT )-1(tk ,tk, ••• ,tk )(edT ) = (tk ,tk, ••• ,tk ), (tk )e(0)c' = tk. Therefore, e(0)c' G (t> and so c' G T, a contradiction. Step 2: Determination of the bicoset graphs. Set D(l) = Fut'a p—1Fw and by Z = Z (p, r, d, l) we denote the corresponding bicoset graph. We consider two cases separately. (1) I = 0. Since FuaP—~ Kw = FuaP—1 and FuaP—1 Kw (dr) = Fuap++1, we have p — 1 - p—1 P+1 - D(0) = Fua — Fw = {Fua—, Fua—}. For any point FwtiE—1 tjp+1 ak in W(Z), since 2 2 _ i p+1 p+1 _ 1 _ 1 Fua ^ t)—1 tjp+1 ak = Fu (tU )i(tU )j ak+ ^ = Futi0t[ak+ ^ = Futi 2 2 2 2 and similarly F a p+1 ti tj ak = F tj ak+ p+21 Fua 2 t p—1 t p+1a = Fut0a 2 : 2 2 its neighbor is N = {Fut0ak+ E2L, FutJ0ak+p+1}. In this case, d(w) = 2. Let p : Fw t)——1 tjp+1 ak ^ Fw f+l tj+1 ak, Futy0 az ^ Futy0 az 2 2 2 2 be the mapping of V(Z(p, r, d, 0)) to V(Y1(p, 1)). Then one may check that p is an isomorphism from Z(p, r, d, 0) to Y1(p, 1). Therefore, Aut(Z(p, r, d, 0)) = Sp I D2p, contrary to our hypothesis KBi = Zp y Z r . (2) I = 0. In Sp 1 D2p, there exists some cl such that the inner automorphism I(cl ) fixes Fu p and Fw and maps D(1) to D(l). Therefore, up to graph isomorphism, we only consider Z (p, r, d, 1). Since p-1 p-1 p-1 Futiia — Kw = Futa ap21 (T0 x T x • • • x H p—1 x Hp,+1 x • • • Tp_1) = Fu(tH )0a —, Futia ^ Kw dr = Fu (tH )oa ^ dr = Fu(tH )o(c'm, c'm, ••• , c'm)ra p+ Fu (tHc'm )oa p+1, it follows that D(1) = {Fu(th')oa^,Fu(th'c'm)oa^ | h' G H}. For any i, j, k G Zp, since Fu(th')oap—21 tip—! tjp+1 ak = Fu(th')ot0ak+ ^ = Fu(th'+i)aak+^ 2 2 and Fu(th'c'm )oa ^ tU tjp+1 ak = Fu(th'c'm )otiak+p+1 = Fu(th' )„ ak+ p+1 2 2 the neighbor of FwtiE—1 tjp+1 ak is 2 2 {Fu(th'+i)oak+^,Fu(th'c'm+j)oak+p2 | h' G H}. Suppose d(w) = 2(p — 1). Then KBi = Zp x Zp-1 and F = K {a, t ). In this case, for any FwtiE=1 tjp+1 ak in W(Y), its neighbor is 2 N = [Fu(th +i)oak+ V ,Fu(th'+j)oak+p+i | h' e H} where H = Zp_i. Clearly, the corresponding graph is isomorphic to YL(p,p — 1), with KBi = Sp, a contradiction. Therefore, 2 < d(w) < 2(p — 1). Let th = tb and b e S where S is a subgroup of Z* of order r. Define a mapping $ from V(Z) to V(Y2(p, r)), for r = 1,p — 1, by Fwtp_it{ak ^ Fwtp_it{ak, and Futv0az ^ Futy0az. Then $ is clearly an isomorphism between the two graphs. Step 3: Determination of isomorphic classes and automorphism groups. Let A = Aut(Z) and KZ be the kernel of A on Z, where Z is a cycle of length 2p. Clearly, A/K = D2p. If (p,r) = (7, 3) and (11,5), then Z is Y>(7, 3) with KB = PrL(3,2) and Yi(11,5) with PSL(2,11), respectively, contradicting our condition. Suppose that (p, r) = (7,3), (11, 5). Since r | (p — 1) and r = 1,p — 1, KBi can not be insolvable and hence an affine group. Therefore, K < KZ = (Zp x Zr)p and then KZ = K. Therefore, A = F. In the case of 2r | (p — 1), let F1 = K x {a, t) and F2 = K x {a, dT) be defined as in Step 1. Let Z1 and Z2 be the corresponding graphs. Suppose that p is an isomorphism from Zi to Z2. Then F2 < {p-1Fip, F2) < Aut(Z2) = F2. Therefore, F2 = p-1Fip = Fi, a contradiction. Therefore, the two graphs are not isomorphic. □ 3.3 K Bi = Zp for p > 5 Lemma 3.3. The case KB = Zp cannot occur. Proof. Suppose that KB = Zp. Then |E(Y)| = 2p3. As above, let w e B'0 and (B'0, Bp-i ), (B0,Bp+i ) e E(Y). Let (wv,ui) e E(Y) for ui e Bp-i. Then E = (w,ui)F. We may consider the group F = Sp x {a,T) > F. From the proof of Lemma 3.1, we may construct two representations of F with respective degree p3 and p2 such that both Kw and (Sp)w fix ui. Then (w, ui)F C (w, ui)F. Since |(w, ui)F| = 2p3 = |(w, ui)F|, we have (w, ui)F = (w, ui)F = E(Y) and so Aut(Y) = F, contrary to our hypothesis KBi = Zp. Therefore, this case cannot occur. □ 3.4 p = 3 Lemma 3.4. If p = 3, then Y = Yl(3, r) for r =1,2. Proof. In this case, take F = S3 I D6 and H = L = Z2. Checking the proof of Lemma 3.1(1), one may find that the arguments in there still hold for p = 3. Therefore, Y = Yl(3, r) for r = 1, 2. □ Acknowledgments: The authors thank the referees for their helpful comments and suggestions. 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ARS MATHEMATICA CONTEMPORANEA 7 (2014) 55-72 Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian Ebrahim Ghaderpour, Dave Witte Morris Department of Mathematics and Computer Science, University of Lethbridge Lethbridge, Alberta, T1K3M4, Canada Received 27 November 2011, accepted 2 July 2012, published online 22 February 2013 Abstract We show that if G is any nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle. Keywords: Cayley graph, hamiltonian cycle, nilpotent group, commutator subgroup. Math. Subj. Class.: 05C25, 05C45 1 Introduction It has been conjectured that every connected Cayley graph has a hamiltonian cycle. See [4, 10, 11, 14, 15, 17] for references to some of the numerous results on this problem that have been proved in the past forty years, including the following theorem that is the culmination of papers by Marusic [12], Durnberger [5, 6], and Keating-Witte [9]: Theorem 1.1 (D.Marusic, E.Durnberger, K.Keating, and D. Witte, 1985). Let G be a nontrivial, finite group. If the commutator subgroup [G, G] of G is cyclic of prime-power order, then every connected Cayley graph on G has a hamiltonian cycle. It is natural to try to prove a generalization that only assumes the commutator subgroup is cyclic, without making any restriction on its order, but that seems to be an extremely difficult problem: at present, it is not even known whether all connected Cayley graphs on dihedral groups have hamiltonian cycles. (See [1, 2] and [15, Cor. 5.2] for the main results that have been proved for dihedral groups.) In this paper, we replace the assumption on the order of [G, G] with the rather strong assumption that G is nilpotent: Theorem 1.2. Let G be a nontrivial, finite group. If G is nilpotent, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle. E-mail addresses: Ebrahim.Ghaderpoor@uleth.ca (Ebrahim Ghaderpour), Dave.Morris@uleth.ca, http://people.uleth.ca/~dave.morris/ (Dave Witte Morris) The proof of this theorem is based on a variant of the method of D. Marusic [12] that established theorem 1.1 (cf. [9, Lem. 3.1]). Remark 1.3. Here are some previous results on the hamiltonicity of the Cayley graph Cay (G; S) when G is nilpotent: 1. Assume G is nilpotent, the commutator subgroup of G is cyclic, and S has only two elements. Then a hamiltonian cycle in Cay(G; S) was found in [9, §6] (see proposition 3.16). The present paper generalizes this by eliminating the restriction on the cardinality of the generating set S. 2. For Cayley graphs on nilpotent groups (without any assumption on the commutator subgroup) it was recently shown that if the valence is at most 4, then there is a hamiltonian path (see [13]). 3. Every nilpotent group is a direct product of p-groups. For p-groups, it is known that every Cayley graph has a hamiltonian cycle ([16], see theorem 3.13). Unfortunately, we do not know how to extend this to direct products. 4. Every abelian group is nilpotent. It is well known (and easy to prove) that Cayley graphs on abelian groups always have hamiltonian cycles. In fact, there is usually a hamiltonian path from any vertex to any other vertex (see [3]). Acknowledgments. We thank Dragan Marusic and Mohammad Reza Salarian for their comments that encouraged this line of research. We also thank an anonymous referee for reading an earlier version of the paper very carefully, and providing many corrections and helpful comments. 2 Assumptions and notation We begin with some standard notation: Notation 2.1. Let G be a group, and let S be a subset of G. • Cay(G; S) denotes the Cayley graph of G with respect to S. Its vertices are the elements of G, and there is an edge joining g to g s for every g G G and s G S. • G' = [G, G] denotes the commutator subgroup of G. • Sr = { sr | s G S } for any r G Z. (Similarly, Gr = { gr | g G G }.) • S±x = S U S-1. • #S is the cardinality of S. Note that if S happens to be a cyclic subgroup of G, then Sr is a subgroup of S. We now fix notation designed specifically for our proof of theorem 1.2: Notation 2.2. • G is a nilpotent, finite group, • N is a cyclic, normal subgroup of G that contains G', • g ^ g is the natural homomorphism from G to G/N = G, • S = {<71; 2, The minimality of S implies that e G S, and that if s G S and |s| > 3, then s-1 G S. Sk = { ai | i < k } for 1 < k < i, Gk = (Sk)N, and • mk = |Gk : Gk-1|. Definition 2.3. • If (si)n=1 is a sequence of elements of S±1 and g G G, we use g(si)"=1 to denote the walk in Cay(G; S) that visits (in order) the vertices g,gS1, gS1S2, .. .,, gS1S2 • • • sn. • If C = g(si)n=1 is any oriented cycle in Cay(G; S), its voltage is f]n=1 si. This is an element of N, and it may be denoted nC. • For S0 C S, we say the walk g(si)n=1 covers S± if it contains an oriented edge labeled s and a (different) oriented edge labeled s-1, for every s g S0. (That is, there exist i, j with i = j, such that si = s and s j = s-1. When | s | = 2, this means si sj s.) • Vk is the set of voltages of oriented hamiltonian cycles in the graph Cay(Gk; Sk ) that cover S^1. Notation 2.4. For kG Z+, we use (s1;..., sn)k to denote the concatenation of k copies of the sequence (s1;..., sn). Abusing notation, we often write sk and s-k for (s)k = (s, s,..., s) and (s-1)k = (s-1, s-1,..., s-1), respectively. Furthermore, we often write ((s1;..., sm), (t1;... ,tn)) to denote the concatenation (s1;..., sm, t1;..., tn). For example, we have ((a3, b)2, c-2)2 = (a, a, a, b, a, a, a, b, c-1, c-1, a, a, a, b, a, a, a, b, c-1, c-1). The following well-known, elementary observation is the foundation of our proof: Lemma 2.5 ("Factor Group Lemma" [17, §2.2]). Suppose • N is a cyclic, normal subgroup of G, • C = g(si)n=1 is a hamiltonian cycle in Cay(G; S), and • the voltage nC generates N. Then (s1,..., sn)|N 1 is a hamiltonian cycle in Cay(G; S). With this in mind, we let N = G', and we would like to find a hamiltonian cycle in Cay(G; S) whose voltage generates N. In almost all cases, we will do this by induction on l = #S, after substantially strengthening the induction hypothesis. Namely, we consider the following assertion (a|) for 2 < k < l and e G {1,2}: there exists hk G N, such that, for every x G N, (Vk n hk (G'k)e)x contains a generator of (a%) a subgroup of N that contains (G'k)e. For e = 2, we also consider the following slightly stronger condition, which we call a\+ : ak holds, and (hk, (G'k)2} contains G'k. (ak+) Lemma 2.6. Let N = G'. If either a] or a2+ holds, then there is a hamiltonian cycle in Cay(G; S) whose voltage generates N. Proof. Note that G' = G' = N. Since V' consists of voltages of hamiltonian cycles in Cay(G; S), it suffices to find an element of V' that generates G'. If we assume a], then the desired conclusion is immediate, by taking x = e in that assertion. Similarly, if we assume a/2+, then taking x = e in a2 tells us that some element 7 of V' n h'(G')2 generates a subgroup of N that contains (G')2. Then, since 7 G (G')2h', and (h', (G')2} contains G', we have N D (7} = (7, (G')2} = (h', (G')2} D G' = N. □ Remark 2.7. 1. If |G'k | is odd, then (G'k)2 = G'k, so we have ak ^ a| ^ a2k+ in this case. Thus, the parameter e is only of interest when |G'| is even. 2. It is not difficult to see that ak ^ ak+, but we do not need this fact. Our proof of a] or a2+ is by induction on k. Here is the outline: I. We prove a base case of the induction: a2 is usually true (see proposition 4.1). II. We prove an induction step: under certain conditions, ak ^ ak+1 and o?k+ ^ a2k+1 (see proposition 5.4). III. We prove a] or a2+ is usually true, by bridging the gap between a2 and either a3 or a|+, and then applying the induction step (see corollary 6.1 and proposition 6.2). 3 Preliminaries 3A Remarks on voltage Remark 3.1. By definition, it is clear that all translates of an oriented cycle C in Cay(G; S) have the same voltage. That is, n(g(si)n=i) = n((si)n=i). Remark 3.2. If | N| is square-free (which is usually the case in this paper), then N is contained in the center of G (because N is the direct product of normal subgroups of prime order, and it is well known that those are all in the center [8, Thm. 4.3.4]). In this situation, the voltage of a cycle in Cay (G; S ) is independent of the starting point that is chosen for its representation. That is, if (tj)"= 1 is a cyclic rotation of (sj)"= so there is some r G {0,1, 2,... ,n} with tj = si+r for all i (where subscripts are read modulo n), then n(ti)n=1 = Sr+1 Sr+2 • • • Sn SlS2 • • • «r = («1 • • • «r )"^n(si)n=j Si • • • Sr = n(sj)n=1, because n(si)n=1 g N c Z(G). The following observation is useful for calculating voltages: Lemma 3.3. If a,b G G, G' C Z (G), and p1; q1,... ,pr, qr G Z, then aPl bqi ap2 bq2 •• • apr bq = api+-+pr bqi+'+qr [a,b]_s, where £ = piqó. Proof. The desired conclusion is easily proved by induction on r, using the fact that, since G' C Z (G), we have [ap, bq ] = [a, b]pq for p, q G Z [7, Lem. 2.2(i)]. □ 3B Facts from group theory Lemma 3.4. If |G'fc| is square-free, then |G'fc/G'fc_1| is a divisor of both |Gk_11 and |Gfc/Gfc_1|. Proof. We may assume k = I, so G = Gk. Let p be a prime factor of |G'/G'fc_1|, let P be the Sylow p-subgroup of G, and let ip : G ^ P be the natural projection. Since |G'| is square-free, it suffices to show that |Gk_ 11 and |Gk/Gk_1| are divisible by p. We may assume |G'| = p and G'k_1 = {e}, by modding out the unique subgroup of index p in G'. Therefore p(Gk_1) is abelian, so it is a proper subgroup of P. Since G' = P' C $(P), this implies p(Gk_1)G' is a proper subgroup of P, so its index is divisible by p. Hence |G/Gk_11 is divisible by p. There must be some t g Sk_1, suchthat [uk,t] is nontrivial. Hence p(t) G Z (G) D G', so p is a divisor of |p(t)|, which is a divisor of |Gk_11. □ The following fact is well known and elementary, but we do not know of a reference in the literature. It relies on our assumption that G' is cyclic. Lemma 3.5. We have ( [s, t] | s, t G S ) = G' if N C Z (G). Proof. Let H = ( [s, t] | s, t G S ). Then H is a normal subgroup of G, because every subgroup of a cyclic, normal subgroup is normal. In G/H, every element of S commutes with all of the other elements of S (and with all of N), so G/H is abelian. Hence G' C H. □ 3C Elementary facts about cyclic groups of square-free order Lemma 3.6. Assume |N | is square-free, and H and K are two subgroups of N. Then: 1. There is a unique subgroup KL of N, such that N = K x K 2. KL is a normal subgroup of G. 3. K C H iff H = N in G = G/K Proof. (1) Since N is cyclic, it has a unique subgroup of any order dividing |Nlet KL be the subgroup of order |N/K |. Since |N | is square-free, we have gcd( |K |K = 1, so N = K X K^ (2) It is well known that every subgroup of a cyclic, normal subgroup is normal (because no other subgroup of N has the same order). (3) We prove only the nontrivial direction. Since H = N, we know that |K | = |N| is a divisor of |H|. So |H| has a subgroup whose order is |K|. Since K is the only subgroup of N with this order, we must have K C H. □ Remark 3.7. When we want to show that some subgroup H of N contains some other subgroup K, Lemma 3.6 often allows us to assume K = N (by modding out Kwhich means we wish to prove H = N. Lemma 3.8. Suppose • Y is a generator of N, • x G N, and • a > max(|N5). Then, for some i with 1 < i < [(a — 1)/2J, we have N2 C (y-2ìx). Proof. Write x = Yh, where 1 < h < |N|, choose r G {1, 2} such that h — r is even, and let . \ r if h G {1,2}, i [(h — r)/2 if h> 2. Then h — 2i G {±r} C {±1, ±2}, so N2 C (Yh-2i) = (y-2ìx). □ Lemma 3.9. If • N is a cyclic group of square-free order, • m > |N|, • k > 2, • T = {y1; ..., Yk} generates N, • h G N, and • Cay(N; T) is not bipartite, then we may choose a sequence (j®)™-1 of elements of {1,2,..., k}, and y2 G {Yj'j1} for each i, such that y®+1 = Yi whenever ji+1 = j®, and (hYiYa ••• Y^-1) = N. (3.10) Proof. Let us assume |N | > 3. (The smaller cases are very easy to address individually.) We begin by finding Yi, Y®,..., G T ±1, suchthat (hYìY2 ' ' ' Ym-1) contains N2 (or N, if appropriate), but without worrying about the requirement that y2+1 = Yi whenever ji+1 = ji. Let Y be a generator of N, and assume h-1Y = e (by replacing y with its inverse, if necessary). Since Cay(N; T) is not bipartite, there is a walk (y2)ì=1 from e to h-1Y, such that r = m — 1 (mod 2). We now show the walk can be chosen to satisfy the additional constraint that r < |N | (so r < m - 1). We know that Cay(N; T) has a hamiltonian cycle C (since N is abelian). Since Cay(N; T) is not bipartite, C must have a chord L of even length. We may assume one endpoint of L is e, since Cay(N; T) is vertex transitive. Let z be the other endpoint of L. Being a hamiltonian cycle, C can be written as the union of two edge-disjoint paths from e to h-1Y. Let P be the one of these paths that contains a subpath of even length from e to z, and let P be obtained from P by replacing this subpath with the edge L. Then P and P are two paths from e to h-1Y. Both have length less than |N |, and their lengths are of opposite parity. Now h-Yi y* • • • Yr* (YiY-1)(m-1-r)/2 = Y generates N, as desired. To complete the proof of the lemma, we modify the above sequence y *, Y*,..., Ym-1 to satisfy the condition that y*+1 = Yi whenever ji+1 = ji. First of all, since N is commutative, we may collect like terms, and thereby write =+= =+= =+= mi m2 mk —ni —"2 -"k Y1Y2 ••• Ym-1 = Y1 1Y2 2 ••• Yk kY1 1Y2 2 ••• Yk k where m1 + ••• + mk + n1 + ••• + nk = m -1. Notice that if mk and n1 are both nonzero, then no occurrence of Yi is immediately followed by y- 1 ; so we have y*+1 = Yi whenever ji+1 = ji, as desired. Therefore, by permuting y1; ..., Yk, we may assume mi = ni = 0 for all i > 1. Also, we may assume m1 and n1 are both nonzero, for otherwise we have Yi = Y* for all i and j. Then, since y1Y-1 = Y2Y-1, we have m1 n1 Y:lY2 ••• Ym-1 = Y1 1Y1 1 m1 1 -(n1-1) 1 : Y1 1 Y2Y1 Y2 . We can assume n1 > m1 (by replacing y1 with its inverse, if necessary). Then > m - 1 > riN i- n 2 2 > 2, so this new representation of the same product satisfies the condition that Yi is never immediately followed by y-1 . This completes the proof. □ Corollary 3.11. Assume |N| is square-free and k > 2 (and e G {1, 2}, as usual). For convenience, let m = mk+1 and a = ak+1. If h G N, then there exists a sequence (si)m=11 of elements of Sk, and s2 Si+1 = si, and G {sf1} for each i, such that s i+1 whenever 1 h ^ [a, s*], (G'k)j contains G k+1 if there exist s, s' G Sk, such that | [a, s] | is even and | [a, s'] | is odd, i=1 (G'k+1r if |G'k+1/G'k | is odd, ,(G'k+1)2 otherwise. Proof. For each s G Sk, let ys = [a, s]. Also, let (3.12) T = { Ys | s G Sk }C G'k+1 C N. =s Let 1 if there exist t, t' g T, such that |t| is even and |t'| is odd, e if |G'fc+1/G'fc| is odd, 2 otherwise. Lemma 3.6 allows us to assume (G'fc+1)e = N, by modding out ((G'fc+1)e)±. We can also assume (G'k ) is trivial, by modding it out. We claim that (T) = N. We have (T, G') D ({ [ak+i,s] | s G Sk },{ [s,t] | s,t g Sk }) = ({ [s,t] | s,t G Sk+i }) = Gk+1 (see Lemma 3.5) = N. Since (Gk) is trivial, this implies |N : (T)| < e. Thus, if the claim is not true, must have |N : (T)| = e = 2. In particular, |N| is even. Since (G'+1)? = N, we conclude that e = 1, so the definition of e (together with the fact that e = 2) implies that T contains an element of even order. So |(T)| is even, which contradicts the fact that |N : (T)| = 2 is even (and |N| is square-free). This completes the proof of the claim. We also claim that Cay(N; T) is not bipartite. We may assume |N| is even (for otherwise the claim is obvious). Since (G'+1 ) = N, this implies ? = 1. However, the claim is also obviously true if there exist t,t' g T, such that |t| is even and |t'| is odd. Hence, we may assume |Gk+1/Gk| is odd, and e = ? = 1. Since (Gk)e is trivial, this implies |G' | is odd, which contradicts the fact that | N| is even. This completes the proof of the claim. The desired conclusion is now immediate from Lemma 3.9 (since ys = [a, s] and (Ys)-1 = [a, s-1]). □ 3D Results from [9] and [16] The following result from [16] allows us to assume G is not a 3-group. (Since we always assume that G' is cyclic, a short proof of the special case we need can be found in [15, Thm. 6.1].) Theorem 3.13 (Witte [16]). If |G| is apower of some primep, then every connected Cayley graph on G has a hamiltonian cycle. The following simple observation usually allows us to assume |N| is square-free. Lemma 3.14 ([9, Lem. 3.2]). Let G = G/$(N), where $(N) is the Frattini subgroup of N [8, §10.4]. Then: 1. |N| is square-free, and 2. if there is a hamiltonian cycle in Cay(G/N; S) whose voltage generates N, then there is a hamiltonian cycle in Cay(G/N; S) whose voltage generates N. Lemma 3.15 (Keating-Witte [9, Case 6.1]). If |G2| is even, then Cay(G2; S2) has a hamiltonian cycle whose voltage is a generator of G'2. Proof. For the reader's convenience, we provide a proof. We may assume |(i | is even (by interchanging 71 and 72 if necessary). For convenience, let n = |7i | and m = m2. Then / m —1 / -(m-2) m-2\(n-2)/2 -(m-1) -(n-1)\ V72 , (71,72 ,71,72 ) ,71,72 ,71 ) is a hamiltonian cycle in Cay(G2; S2). Lemma 3.14 allows us to assume |G2| is square-free, which implies G'2 is in the center of G2 (see remark 3.2). Also, from Lemma 3.4, we know that | 7, 72] | is a divisor of both m and n. Therefore (7172-(m-2)7172m-2)(n-2)/2 = (72 [71,7m-2])(n-2)/2 = 7n-2 [71,72](m-2)(n-2)/2 = 7n-2 [71, 72]2 and [7-(m-1), 7-(n-1)] = [72,71](m-1)(n-1) = [72,71] = [71, 72]-1, so the voltage of this cycle is m-1 ( -(m-2) m-2)(n-2)/2 -(m-1) -(n-1) 72 71 72 71 72 71 72 71 m-1/ n-2 r „.-(m-1) -(n-1) = 72 l71 [71,72] J7172 71 m-1 n-1 -(m-1) -(n-1) r i2 = 72 71 72 71 [71,72] r -(m-1) -(n-1)n r -|2 = [72 ;,7i ] [71, 72] = [71,72], which generates G2 (see Lemma 3.5). □ The following result allows us to assume I > 3. Proposition 3.16 (Keating-Witte [9, §6]). If t = 2 and N = G', then Cay(G; S) has a hamiltonian cycle. Proof. For the reader's convenience, we provide a proof (using the main result of Section 4 below). We may assume |G/G'| is odd, for otherwise a hamiltonian cycle is obtained by combining Lemma 3.15 with the Factor Group Lemma (2.5). We may also assume that |G| is not a power of 3, for otherwise theorem 3.13 applies. This implies it is not the case that |s| = 3 for every s G S. If |G'| is square-free, then proposition 4.1 tells us that a" is true. Also, since |G/G'| is odd, we know |G'| is odd (cf. Lemma 3.4), so «2 implies that «2 is true (see remark 2.7(1)). Therefore, the Factor Group Lemma (2.5) provides a hamiltonian cycle in Cay(G; S) (see Lemma 2.6). Then Lemma 3.14 tells us there is a hamiltonian cycle even without the assumption that |G'| is square-free. □ 4 Base case of the inductive construction Recall that the condition aek is defined in Section 2. Proposition 4.1 (cf. [9, Case 6.2]). Assume |N| is square-free (and t > 2). Then «2 is true unless |G2| = m2 = |77| = |72| = 3. Proof. For convenience, let a = <7i, b = <72, and m = m2, and define r by b™ = ar and 0 < r < |a|. We may assume: • I = 2, so S = S2 = {a, b} and G = G2. • (G')2 is nontrivial. (Otherwise, the condition about generating (G')2 is automatically true, so it suffices to show V2 = 0, which is easy.) • Either |a| is even, or m is odd (by interchanging <1 and <2 if necessary). • |a| = 3 (by interchanging <1 and <2 if necessary: if |7l| = |72| = 3, then m = 3 and, from Lemma 3.4, we also have |G' | =3, which means we are in a case in which the statement of the proposition does not make any claim). • r > |a|/2 (by replacing a with its inverse if necessary). Note that |G'| is a divisor of both |a| and m (see Lemma 3.4). Since (G')2 is nontrivial, this implies that |a| and m both have at least one odd prime divisor. Case 1. Assume m = 3. Since |G'| is a divisor of m, we must have |G'| = 3, so |a| must be divisible by 3. Then, since |a| = 3, we must have |a| > 6. Furthermore, by applying Lemma 3.4 with a and b interchanged, we see that |G/(b}| is also divisible by |G'| = 3, which means that r is divisible by 3. We claim that it suffices to find two elements y^Y2 g V2, such that y1 = y2 and Y1 G y2G'. To see this, note that, for any x g N, there is some i G {1,2}, such that (y%x) has nontrivial projection to G' (with respect to the unique direct-product decomposition N = G' X (G')^). Since |G'| is prime, this implies that the projection is all of G', so Lemma 3.6 tells us that (yìx) contains G'. This establishes a1, which is equivalent to «2 (see remark 2.7(1)). This completes the proof of the claim. Assume, for the moment, that r = 3. Then, since r > |a|/2 and |a| > 6, we must have |a| = 6. Here are two hamiltonian cycles in Cay(G; a, b) that cover S±1: (b-1, a-2, b-4, a-2, b-1, a3, b2, a, b-2) and (b-1, a-2, b-1, a, b-1, a-1, b-2, a-1, b-1, a2, b2, a, b-2) (see Figure 1). By using Lemma 3.3, we see that their voltages are b-1a-2b-4a-2b-1a3b2ab-2 = b-6[a, b]-(-10) = b-6[a, b] and b-1a-2b-1ab-1a-1b-2a-1b-1a2b2ab-2 = b-6[a, b]-(-8) = b-6[a,b]2, respectively. So we may let y1 = b-6 [a, b] and y2 = b-6 [a, b]2. We may now assume r > 6 (since r is divisible by 3). Let I = |{0,1} if r = |a|, {1, 2} if r =a Figure 1: Two hamiltonian cycles in Cay(G; ja, b}) when m = r = 3. Figure 2: A hamiltonian cycle in Cay(G; ja, b}) when m = 3 and r > 6. Then, for i G I, we have 0 < i < |a| — 4, and 4 < r — i < |a| — 1, so the walk Ci = (a\ b-1, a-( 1 a| -r+i-1), b-1, a 1 a| -4, b-1, a-( 1 a 1 -i-4), b-1, ar-i-3, b-2, a, b2, a, b-2) is as pictured in Figure 2. It is a hamiltonian cycle in Cay(G; a, b) that covers S±1. Furthermore, since aib-1a-i = b-1(baib-1 a-i) = b-1[b-1, a-i] = b-1[b,a]i = b-1[a,b]-i, its voltage is of the form h2 [a, b]-2i, where h2 is independent of i. Thus, we may let {71,72} = j h2 [a, b]-2i | iel}. Case 2. Assume m = 3. (Cf. [9, Case 4.3].) Since m and |a| both have at least one odd prime divisor, we must have m > 5 and |a| > 5. Let X = ' (b-(m-2), a, bm-3, a|a|-3, b-1, (a-(|a|-4), b-1, a|a|-4, b^1)(m-3)/2) if |a| is odd, (b-1, (b-(m-3), a, bm-3, a)(|a|/2)-1, b-(m-2)) if |a| is even. For each i with 1 < i < |_(|a| — 1)/2J, we have 1 < i < mi^r — 1, |a| — 3) (since r > |a|/2 and |a| > 5), so we may let Ci = (ai, b-1 ,a-(|a|+i-r-1),X,a-(|a|-i-2),b-1,ar-i-1,b-(m-1)) (see Figures 3 and 4). Then Ci is a hamiltonian cycle in Cay(G; a, b). Figure 3: A hamiltonian cycle Ci in Cay(G; {a, b}) when m = |G/(a)| is odd. Note that both possibilities for X contain oriented edges labelled a, b, and b-1. Furthermore, since |a| — i — 2 > 1, we see that Ci also contains at least one oriented edge labelled a-1. Therefore Ci covers {a, b, a-1, b-1} = S±1. As in Case 1, the voltage nCi of Ci is of the form h2 [a, b]-2i, where h2 is independent of i. Since |a| > |G'| (see Lemma 3.4) and ([a, b]) = G' (see Lemma 3.5), Lemma 3.8 (combined with Lemma 3.6) tells us that for any x e N, we may choose i so that ((nCi)x) contains (G')2. □ 5 The main induction step The induction step of our proof uses the following well-known gluing technique that is illustrated in Figure 5. Definition 5.1. Let • C1 and C2 be two vertex-disjoint oriented cycles in Cay(G; S), • g e G, and • a, s e S. If • C1 contains the oriented edge g(s), and • C2 contains the oriented edge gsa( s -1 ), then we use C1 C2 to denote the oriented cycle obtained from C1 U C2 as in Figure 5, by • removing the oriented edges g(s) and gsa(s-1), and i-I 1-2 a-1 Figure 4: A hamiltonian cycle C® in Cay(G; {a, ò}) when |a| is even. • inserting the oriented edges g(a) and gsa(a-1). This is called the connected sum of C1 and C2. Lemma 5.2. If C1, C2, g, s, and a are as in definition 5.1, and N C Z (G), then n(C1 #a C2) = (nC1)(nC2)[a, s]. Proof. Write C1 = gš^®)^ and C2 = ga(tj)n=1. Then C1 #a C2 = gša(a-1, (si)™^1, a, (jj-1), so (m— 1 \ l n—1 H sj a ( JJtj m n 1 ns® p™1 a i n tj it = a M |si lsm a V®=1 / \j=1 = a-1 (nC1) s—1 a (nC2) s = (nC1) (nC2) a-1s—1as = (nC1)(nC2) [a, s]. j n- 1 (nC® g n c z (G)) □ Corollary 5.3. Assume • 2 < k < and (to eliminate some subscripts) m = mk+1 and a = afc+1, • n1, n2,..., nm are elements of Vk, b b Figure 5: C1 and C2 are merged into a single cycle by replacing the two white edges labelled s and s-1 with the two black edges labelled a and a-1. • s1, s2,..., sm-1 are elements of Sk, and, for each i, a choice s* g {sf1} has been made in such a way that if si+1 = sž, then s*+1 = s*, and • N C Z (G). Then there is a hamiltonian cycle in Cay(Gk+1 ; Sk+1 ) that covers S±+11, and whose voltage is (m \ /m-1 \ n MJ . Proof. For each i, let Ci be an oriented hamiltonian cycle in Cay(Gk; Sk) that covers S±\ and has voltage nj. We inductively construct sequences (g-i)™1 and (xj)m=1 of elements of Gk, as follows. Let g1 = e. Since C1 covers Sf1, we know there is some x1 G Gk, such that ag1C1 contains the oriented edge äxl(sf ). Now, suppose g1, x1, g2, x2,..., gi, xi G Gk are given, such that the connected sum ag1C1 #as» a2g2C2 #?. • • • ajgjCj exists, and contains the oriented edge aixi(s*). Since Ci+1 covers S±\ we know that Ci+1 contains an oriented edge labelled (s* )-1, and a different oriented edge that is labelled s*+1. Therefore, there exist gi+1, xi+1 G Gk, such that ai+1gi+1Ci+1 contains the oriented edges ai+1xis* ((s*)-^ and ai+1xi+1(s*+1). The first of these edges is removed when we form the connected sum (ag1C1 #11 a2g2C2 #as. • • • #1._i ajgiCj) ai+1gj+1Cj+1, but the second edge remains, and will be used to form the next connected sum (unless i + 1 = m). Since each Q is a hamiltonian cycle in Cay(Gk; Sk), the resulting connected sum agiCi a • • • , a" passes through all of the vertices in aGk U a2Gk U • • • U amGk. That is, it passes through every element of Gk+1, so it is a hamiltonian cycle in Cay(Gk+1; Sk+1). Its voltage is calculated by repeated application of Lemma 5.2. To complete the proof, we verify that the hamiltonian cycle covers S±+11. Since each Q covers S±\ the disjoint union ag1C U a2g2C2 U • • • U amgmCm contains (at least) m disjoint pairs of edges labelled s and s-1, for each s g Sk. Each invocation of the connected sum removes only one such pair, and the operation is performed only m -1 times, so at least one of the m pairs must remain, for each s g Sk. Therefore, the hamiltonian cycle covers S^1. Also, the cycle certainly covers a±1, since each invocation of the connected sum inserts a pair of edges labelled a and a-1. Hence, the hamiltonian cycle covers S±1 U{a±1} = S±+1. □ We can now prove the main result of this section. (Recall that the condition is defined in Section 2.) Proposition 5.4. Assume |N | is square-free and |G'k+1/G'k | is odd. Then 1. ak ^ ak+1, and 2. ak+ ^ ak+1 if | [s, t] | is even for all s, t g Sk+1 with s = t. Proof. For convenience, let m = mk+1 and a = ak+1. Let hk g N be as in (a|), and choose an oriented hamiltonian cycle C in Cay(Gk; Sk ) that covers S±\ and has its voltage in hk (G'k )e. There is no harm in assuming that the voltage is precisely hk. Let hk+1 = (hk)m[a,CT1]; 1 Given any x g N, corollary 3.11 provides a sequence (si)"=11 of elements of Sk, and s* g {sf1} for each i, such that s*+1 = s* whenever si+1 = sž, and -1 " (x (hk )" n [a, s*], (Gk yj contains (Gk+1 )e. (5.5) From (ak) we know there exists n g Vk n hk (G'k)% such that, if we let " -1 Y = n (hk)"-1 H [a, s*], i=1 then (xy) contains (G'k)e. Since n = hk (mod (G'k)e), combining this with (5.5) shows that (xy) contains (G'k+1)e. Also, since we are assuming | [a, s*] | is even if e = 2, we have [a, s*] = [a, a1] (mod (G'k+1)e) for all i, so Y G (hk)"[a,a1]"-1(G'k+1)e = h^G'^)6. Furthermore, corollary 5.3 tells us that there is a hamiltonian cycle in Cay(G; S) whose voltage is Y, and this hamiltonian cycle covers Si1 . This establishes «k+1. Now, if e = 2, then our assumptions imply that | hk | and | [a, a 1] | are both even. Since m and m - 1 are of opposite parity, this implies that |hk+1| is even, so (hk, (G'fc+1)2) contains G'fc+1. This establishes ak+1. □ 6 Combining the base case with the induction step Recall that the condition aek is defined in Section 2. Corollary 6.1. Assume |N| is square-free and I > 2. If |G'| is odd, then « is true unless |G'| = |s| = 3 for all s G S. Proof. Assume it is not the case that |G'| = |s| = 3 for all s G S. Then we may assume (by permuting the elements of S) that either |G2| =3 or |a1| = 3. Therefore proposition 4.1 tells us that «2 is true. Also, since |G'| is odd, we have « ^ (see remark 2.7(1)), so «2 is true. Then repeated application of proposition 5.4(1) establishes «1 □ Proposition 6.2. Assume |N| is square-free and I > 3. If |G'| is even, then: 1. «1 is true if there exist s, t G S, such that | [s, t] | is odd and s = t. 2. «2+ is true if | [s, t] | is even for all s, t G S with s = t. Proof. Since |G'| is even, we may assume (by permuting the elements of S) that | [a3, a1] | is even. It suffices to prove a3 or «3+ (as appropriate), for then repeated application of proposition 5.4 establishes the desired conclusion. Thus, we may assume I = 3, so G3 = G. Let m = m.3 and a = <73 = a^. By permuting the elements of S, we may assume that either: odd case: | [<3, <2] | is odd, or even case: | [s, t] | is even for all s, t G S with s = t. Furthermore, in the even case, we may assume that either: even subcase: («T, <2) has even index in G, or odd subcase: (s, t) has odd index in G, for all s, t g S, such that s = t. Since | [a3, a1] | is even, we know |<571 is even (see Lemma 3.4). In particular, we have | 3, for otherwise proposition 3.16 applies. • |G| is not a power of 3, for otherwise theorem 3.13 applies. Let S be a minimal generating set of G, and let N = G'. Note that S is a minimal generating set of G (because G' is contained in the Frattini subgroup $(G) [8, Thm. 10.4.3]). We claim there is a hamiltonian cycle in Cay(G; S) whose voltage generates G'. While proving this, there is no harm in assuming that |G'| is square-free (see Lemma 3.14). Also note that, since |G| is not a power of 3, we cannot have |G'| = |s| = 3 for all s G S. Then, by applying either corollary 6.1 or proposition 6.2 (depending on the parity of |G'|), we obtain either a] or a2+. Each of these yields the desired hamiltonian cycle in Cay(G; S) (see Lemma 2.6). Now that the claim has been verified, the Factor Group Lemma (2.5) provides a hamil-tonian cycle in Cay(G; S). □ References [1] B. Alspach, C. C. Chen and M. 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Levit Ariel University, Israel Received 31 October 2011, accepted 28 October 2012, published online 4 March 2013 Abstract The notion of "antimatroid with repetition" was conceived by Bjorner, Lovasz and Shor in 1991 as an extension of the notion of antimatroid in the framework of non-simple languages. Further they were investigated by the name of "poly-antimatroids" (Nakamura, 2005, Kempner & Levit, 2007), where the set system approach was used. If the underlying set of a poly-antimatroid consists of n elements, then the poly-antimatroid may be represented as a subset of the integer lattice Zn. We concentrate on geometrical properties of two-dimensional (n = 2) poly-antimatroids - poly-antimatroid polygons, and prove that these polygons are parallelogram polyominoes. We also show that each two-dimensional poly-antimatroid is a poset poly-antimatroid, i.e., it is closed under intersection. The convex dimension cdim(S) of a poly-antimatroid S is the minimum number of maximal chains needed to realize S. While the convex dimension of an n-dimensional poly-antimatroid may be arbitrarily large, we prove that the convex dimension of an n-dimensional poset poly-antimatroid is equal to n. Keywords: Antimatroid, polyhedron, convex dimension, lattice animal, polyomino. Math. Subj. Class.: 05B35 1 Preliminaries An antimatroid is an accessible set system closed under union [3]. An algorithmic characterization of antimatroids based on the language definition was introduced in [5]. Another algorithmic characterization of antimatroids that depicted them as set systems was developed in [14]. Dilworth (1940) was the first to study antimatroids, using another axiomatiza-tion based on lattice theory, and they have been frequently rediscovered in other contexts. The most updated survey on the subject may be found in [19]. E-mail addresses: yuliak@hit.ac.il (Yulia Kempner), levitv@ariel.ac.il (Vadim E. Levit) Antimatroids can be viewed as a special case of either greedoids or semimodular lattices, and as a generalization of partial orders and distributive lattices. While classical examples of antimatroids connect them with posets, chordal graphs, convex geometries, etc., game theory gives a framework in which antimatroids are interpreted as permission structures for coalitions [1]. There are also rich connections between antimatroids and cluster analysis [16]. In mathematical psychology, antimatroids are used to describe feasible states of knowledge of a human learner [9]. A poly-antimatroid [21] is a generalization of the notion of the antimatroid to multisets. If the underlying set of a poly-antimatroid consists of n elements, then the n-dimensional poly-antimatroid may be represented as a subset of the integer lattice Zn. In this paper we investigate the correspondence between poly-antimatroids and poly-ominoes. In the digital plane Z2, a polyomino [12] is a finite connected union of unit squares without cut points. If we replace each unit square of a polyomino by a vertex at its center, we obtain an equivalent object named a lattice animal [13]. Further, we use the name polyomino for the two equivalent objects. A polyomino is called column-convex (row-convex) if all its columns (rows) are connected. In other words, each column/row has no holes. A convex polyomino is both row-convex and column-convex. The parallelogram polyominoes [6], sometimes known as staircase polygons [4, 13, 22], are a particular case of this family. In the staircase polygon each element may be reached from the source (lowest left) point by a path made only of north and east unit steps, and similarly this element may be reached from the target (highest right) point by a path made only of south and west unit steps. Hence staircase polygons are defined by a pair of monotone north-east paths that have common ending points. We prove that a representation of a two-dimensional poly-antimatroid on the plane is a staircase polygon. Let E be a finite set. A set system over E is a pair (E, F), where F is a family of sets over E, called feasible sets. We will use X U x for X U {x}, and X - x for X - {x}. Definition 1.1. [18]A finite non-empty set system (E, F) is an antimatroid if (A1) for each non-empty X e F, there exists x G X such that X - x G F (A2) for all X, Y G F, and X £ Y, there exists x G X - Y such that Y U x G F. Any set system satisfying (A1) is called accessible. In addition, we use the following characterization of antimatroids. Proposition 1.2. [18] For an accessible set system (E, F) the following statements are equivalent: (i) (E, F) is an antimatroid (ii) F is closed under union (X, Y gF^ X U Y G F ) An "antimatroid with repetition" was invented by Bjorner, Lovasz and Shor [2]. Further it was investigated by the name of "poly-antimatroid" as a generalization of the notion of the antimatroid for multisets. A multiset A over E is a function fA : E ^ N, where fA(e) is a number of repetitions of an element e in A. A poly-antimatroid is a finite nonempty multiset system (E, S) that satisfies the antimatroid properties (A1) and (A2). So antimatroids may be considered as a particular case of poly-antimatroids. Examples of an antimatroid ({x, y, z}, F) and a poly-antimatroid ({x, y}, S) are illustrated in Figure 1. {x,y {y,y} 0 0 (a) (b) Figure 1: (a) Antimatroid. (b) Poly-antimatroid. Definition 1.3. A multiset system (E, S) satisfies the chain property if for all X,Y e F, and X c Y, there exists a chain X = X0 C Xi C ... C Xk = Y such that X, = Xj_i U xi and X, e S for 0 < i < k. It is easy to see that the chain property follows from (A2), but they are not equivalent. It is clear that each poly-antimatroid satisfies the chain property. Antimatroids have already been investigated within the framework of lattice theory by Dilworth [7]. The feasible sets of an antimatroid ordered by inclusion form a lattice, with lattice operations: X V Y = X U Y, and X A Y is the maximal feasible subset of set X n Y called a basis. Since an antimatroid is closed under union, it has only one basis. A finite lattice L is called join-distributive [3] if for any x e L the interval [x, y] is Boolean, where y is the join of all elements covering x. Such lattices have appeared under several different names, e.g. locally free lattices [18] and upper locally distributive lattices (ULD) [10, 20]. Distributive lattices are exactly those that are both upper and lower locally distributive. Theorem 1.4. [3, 18] A finite lattice is join-distributive if and only if it is isomorphic to the lattice of feasible sets of some antimatroid. It easy to see that feasible sets of a poly-antimatroid ordered by inclusion form a join-distributive lattice as well. Consider a particular case of antimatroids called poset antimatroids [18]. A poset an-timatroid has as its feasible sets the lower sets of a poset (partially ordered set). Poset antimatroids can be characterized as the unique antimatroids, which are closed under intersection [18]. We extend this definition to poly-antimatroids. Definition 1.5. A poly-antimatroid is called a poset poly-antimatroid if it is closed under intersection. Evidently, feasible sets of a poset poly-antimatroid ordered by inclusion form a distributive lattice. 2 Two-dimensional poly-antimatroids and polyominoes In this section we consider a geometric characterization of two-dimensional poly-anti-matroids. Let E = {x, y}. In this case each point A = (xA, yA) in the digital plane Z2 may be considered as a multiset A over E, where xA is a number of repetitions of an element x, and yA is a number of repetitions of an element y in multiset A. Thus a set of points in the digital plane Z2 that satisfies the properties of an antimatroid is a representation of a two-dimensional poly-antimatroid. Definition 2.1. A set of points S in the digital plane Z2 is a poly-antimatroid polygon if (A1) for every point (xA, yA) e S, such that (xA, yA) = (0,0), either (xA - 1, yA) e S or (xA, yA - 1) e S (A2) for all A £ B e S, if xA > xB and yA > yB then either (xB + 1, yB ) e S or (xB ,yB + 1) e S if xa < xb and yA > yB then (xb , yB + 1) e S if xa > xb and yA < yB then (xb + 1, yB ) e S Notice that accessibility implies 0 e S. For example, see a poly-antimatroid polygon in Figure 2. A • • • • • • *B • • • • • • • • • 0 1 2 Figure 2: A poly-antimatroid polygon. We use the following notation [17]. If A 4-neighborhood N4(x, y) is the set of points (x, y) is a point in a digital plane, the N4(x, y) = {(x - 1, y), (x, y - 1), (x +1, y), (x, y + 1)} and 8-neighborhood N8(x, y) is the set of points Ng(x,y) = {(x - 1, y), (x, y - 1), (x + 1,y), (x,y + 1), (x - 1,y - 1), (x - 1, y + 1), (x + 1,y - 1), (x + 1,y + 1)}. Let m be any of the numbers 4 or 8. A sequence A0, Ai,..., An is called an Nm-path if Aj e Nm(Ai-1) for each i = 1, 2, ...n. Any two points A, B e S are said to be Nm-connected in S if there exists an Nm-path A = A0, A1,..., An = B from A to B such that Aj e S for each i = 1,2, ...n. A digital set S is an Nm-connected set if any two points y 1 x P,Q from S are Nm-connected in S. An Nm-connected component of a set S is a maximal subset of S, which is Nm-connected. An Nm-path A = A0, A1,..., An = B from A to B is called a monotone increasing Nm-path if Ai C Ai+1 for all 0 < i < n, i.e., (xAi < XAi+1 ) A (yAi < yA.+1 ) or (xAi < xAi+i ) A (yA. < yAi+i ). The chain property and the fact that the family of feasible sets of a poly-antimatroid is closed under union mean that for each two points A, B: if B C A, then there is a monotone increasing N4-path from B to A, and if A is incomparable with B, then there is a monotone increasing N4-path from both A and B to A U B = (max(xA,xB), max(yA,yB)). In particular, for each A g S there is a monotone decreasing N4-path from A to 0. So, we can conclude that a poly-antimatroid polygon is an N4-connected component in the digital plane Z2. Definition 2.2. A point set S C Z2 is defined to be orthogonally convex if, for every line L that is parallel to the x-axis (y = y*) or to the y-axis (x = x*), the intersection of S with L is empty, a point, or a single interval ([(x1, y*), (x2, y*)] = {(x1, y*), (x1 + It follows immediately from the chain property that every poly-antimatroid polygon S is both connected and orthogonally convex. In what follows we prove that poly-antimatroid polygons are closed not only under union, but under intersection as well. Lemma 2.3. A poly-antimatroid polygon is closed under intersection, i.e., if two points A = (xA, yA) and B = (xB, yB ) belong to a poly-antimatroid polygon S, then the point A H B = (min(xA, x^), min(yA, yB)) G S. Proof. The claim of the lemma is evident for two comparable points. Consider two incomparable points A and B, and assume without loss of generality that xA < xB and yA > yB. Then there is a monotone decreasing N4-path from A to 0, and so there is a point C = (xC, yB ) G S on this path with xC < xA. Hence, the point A H B belongs to S, since it is located on the monotone increasing N4-path from C to B. □ Thus, every poly-antimatroid polygon is a distributive lattice polyhedron [11], since x, y G S ^ min(x, y), max(x, y) G S. Consider the following rectangles: R = {(x, y) G Z2 : xmin < x < xmax A ymin < y < ymax} and |R| > 1 Definition 2.4. The sequence of n rectangles C1, C2,..., Cn is called regular if 1, y*),..., (x2,y*)}). (a) x0 . = y0 ■ =0 and for each 1 < i < n - 1 at least one of the inequality is strong (c) xi+1 < xi A yi+1 < yi for each 1 < i < n - 1 .i+1 ,i+1 and for each 1 < i < n - 1 at least one of the inequality is strong Lemma 2.3 implies that every poly-antimatroid polygon is a union of rectangles built on each pair of incomparable points. The following is even more explicit. Lemma 2.5. Every poly-antimatroid polygon is a regular sequence of rectangles. Proof. Consider the set of rectangles built on pairs of incomparable points and leave only maximal rectangles, i.e., rectangles that are not covered completely by other rectangles. These rectangles forms a regular sequence. Indeed, the property (a) and (c) follows from the definition of a poly-antimatroid polygon (A1). Suppose there are two maximal rectangles R1 and R2 with two incomparable minimal points (x^in, yüiin) and (x2min ,y2min). Then, since poly-antimatroid polygons are closed under union and under intersection, these rectangles are covered by two rectangles. The minimal and maximal points of the first rectangle are (min(x1lnin, x^in), mintóin, ^in)), (min^ax, xLx), min^ax , ymax)) • The minimal and maximal points of the second rectangle are (max(xmin, xmin), max(ymin, ^min)), (max(xmax, Xmax), max(ymax, ^max)) respectively. There are the cases that these two rectangles are identical. See Figure 3. Thus the rectangles Rm and R2 are not maximal and so all minimal points are comparable. The same is true for maximal points as well. □ Rm Rm Rm Rm y y 0 x x Figure 3: Two examples for proof of Lemma 2.5. The following theorem shows that a poly-antimatroid polygon is a parallelogram poly-omino. Theorem 2.6. A set of points S in the digital plane Z2 is a poly-antimatroid polygon if and only if it is an orthogonally convex N4-connected set that is bounded by two monotone increasing N4-paths between (0,0) and the maximum point of the set (xmax, ymax). To prove the "if" part of Theorem 2.6 it remains to give a definition of the boundary. A point A in set S is called an interior point in S if N8(A) G S. A point in S which is not an interior point is called a boundary point. All boundary points of S constitute the boundary of S. We can see a poly-antimatroid polygon with its boundary in Figure 4. Since poly-antimatroid polygons are closed under union and under intersection, there are six types of boundary points that we divide into two sets - lower and upper boundary: Blower = {(x, y) G S : (x + 1,y) GS V (x,y - 1) GS V (x + 1,y - 1) GS} B 10 B18 (xmax ,ymax) B8 u Bo B i B4 x Figure 4: A boundary of a poly-antimatroid polygon. = {(x,y) G S • (x — 1, y) GS V (x,y + 1) GS V (x - 1 ,y + 1) G S} It is possible that Blower n Bupper = 0. For example, the point B10 in Figure 4 belongs to both the lower and upper boundaries. From Lemma 2.5 it follows immediately that lower and upper boundaries are boundaries of regular sequence of rectangles and so a poly-antimatroid polygon is an orthogonally convex N4-connected set bounded by two monotone increasing N4-paths. The following lemma is the "only-if" part of Theorem 2.6. Lemma 2.7. An orthogonally convex N4-connected set S that is bounded by two monotone increasing N4-paths between (0,0) and (xmax, ymax) is a poly-antimatroid polygon. Proof. By Definition 2.1 we have to check the two properties (A1) and (A2): (A1) Let A = (x, y) G S. If A is an interior point in S then (x — 1, y) G S and (x, y — 1) G S. If A is a boundary point, then the previous point on the boundary ((x, y — 1) or (x - 1 , y) ) belongs to S. (A2) Let A ^ B g S. Consider two cases: (i) xa > xb and ya > yb. If B is an interior point in S then (xb + 1, yb) G S and (xb, yb + 1) G S. If B is a boundary point, then the next point on the boundary ((xb, yb + 1) or (xb + 1, yb)) belongs to S. (ii)xa < xb and ya > yb. We have to prove that (xb, yb +1) G S. Suppose the opposite. Then the point B is an upper boundary point. Since ya > yb there exists an upper boundary point (xa ,y) with y > yb that contradicts the monotonicity of the boundary. □ Corollary 2.8. Any poly-antimatroid polygon S may be represented as the union of its boundary: S Blower V Bupper {X ^ Y • X G Blower, Y G Bupper } y 3 Convex dimension Definition 3.1. Convex dimension [18] cdim(S) of any antimatroid S is the minimum number of maximal chains 0 = Xo C Xi C ... C Xk = Xmax with Xi = Xi-1 U xi whose union gives the antimatroid S. The set of maximal chains sufficient to realize a poly-antimatroid is called a convex realizer. The result of Corollary 2.8 shows that the convex dimension of a two-dimensional poly-antimatroid equals two. Note, that the convex dimension of an arbitrary three-dimensional poly-antimatroid may be arbitrarily large [8]. Let S be a set of points: S = {(x, y, z) : (0 < x, y < N) A (0 < z < 1) A (z = 1 ^ x + y > N)}. It is easy to check that S is a three-dimensional poly-antimatroid. Consider N +1 points (x, y, 1) with x + y = N. Since each of these points cannot be represented as a union of any points from S with smaller coordinates, the convex dimension of S is at least N + 1. In the sequel we prove that the convex dimension of an n-dimensional poset poly-antimatroids is at most n. An endpoint of a feasible set X is an element e e X such that X - e is a feasible set too. A feasible set that has only one endpoint is called a path of the antimatroid. It is easy to see that a path is an union-irreducible element of the lattice and each feasible set is the union of its path subsets. The family of ideals in the set of paths partially ordered by set inclusion forms path poset antimatroid. Theorem 3.2. [18] The convex dimension of antimatroid is equal to the width of its path poset. It is easy to check that the theorem holds for poly-antimatroids as well. Denote by dim(S) the order dimension of the lattice of feasible multisets of a poly-antimatroid. The order (or the Dushnik-Miller) dimension of a poset is the smallest number of total orders the intersection of which gives the partial order. Corollary 3.3. [18] dim(S) < cdim(S) Since any n-dimensional poly-antimatroid may be represented as a subset of the integer lattice Zn, its order dimension is at most n [23]. For poset poly-antimatroids their convex dimension should equal to order dimension, since feasible sets of a poset poly-antimatroid form a distributive lattice and order dimension of distributive lattice is equal to the width of its path poset (ideals of the union-irreducible elements)[23]. Eventually, we obtain the following. Proposition 3.4. The convex dimension of the n-dimensional poset poly-antimatroids is at most n. 4 Conclusions It turned out that a two-dimensional case of a poly-antimatroid known in this paper as a poly-antimatroid polygon is equivalent to special cases of polyominoes, lattice animals, and staircase polygons. It seems to be a challenging problem to generalize the path structure of poly-antimatroid polygons to upper dimensions. References [1] E. Algaba, J. M. Bilbao, R. van den Brink and A. Jimenez-Losada, Cooperative Games on Antimatroids, Discrete Math. 282 (2004), 1-15. [2] A. Björner, L. Lovasz and P. R. 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Dimension Theory, Johns Hopkins University Press, Baltimore, 1992. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 83-103 The L2 (11) -subalgebra of the Monster algebra Sophie Decelle * Department of Mathematics, Imperial College, 180 Queen's Gate, London, SW7 2AZ, UK Received 31 October 2011, accepted 25 August 2012, published online 15 March 2013 We study a subalgebra V of the Monster algebra, VM, generated by three Majorana axes ax, ay and az indexed by the 2A-involutions x, y and z of M, the Monster simple group. We use the notation V = {{ax,ay,az}}. We assume that xy is another 2A-involution and that each of xz, yz and xyz has order 5. Thus a subgroup G of M generated by {x, y, z} is a non-trivial quotient of the group G(5'5'5) = {x, y, z | x2, y2, (xy)2, z2, (xz)5, (yz)5, (xyz)5}. It is known that G(5,5,5) is isomorphic to the projective special linear group L2(11) which is simple, so that G is isomorphic to L2(11). It was proved by S. Norton that (up to conjugacy) G is the unique 2A-generated L2(11)-subgroup of M and that K = CM(G) is isomorphic to the Mathieu group M12. For any pair {t, s} of 2A-involutions, the pair of Majorana axes {at, as} generates the dihedral subalgebra {{at, as}} of VM, whose structure has been described in [16]. In particular, the subalgebra {{at, as}} contains the Majorana axis atst by the conjugacy property of dihedral subalgebras. Hence from the structure of its dihedral subalgebras, V coincides with the subalgebra of VM generated by the set of Majorana axes {at 11 G T}, indexed by the 55 elements of the unique conjugacy class T of involutions of G = L2(11). We prove that V is 101-dimensional, linearly spanned by the set { at ■ as | s, t G T }, and with CVm(K) = V © iM, where iM is the identity of VM. Lastly we present a recent result of A. Seress proving that V is equal to the algebra of the unique Majorana representation of L2 (11). Keywords: Majorana representation, Monster group, Conway-Griess-Norton algebra. Math. Subj. Class.: 20C99, 20F05, 20C34, 20D05 1 Main result We let (VM, ■ , ( , )) be the Monster algebra, a commutative non-associative algebra of dimension 196, 884 over R, as described in [2]. As an RM-module, VM = VM © 1M, where VM is the minimal faithful irreducible RM-module of dimension 196,883 and 1M * This article was written under the supervision of Prof. Alexander A. Ivanov. E-mail address: sophie.decelle@imperial.ac.uk (Sophie Decelle) Abstract is the trivial RM-module which is the R-span of the identity iM of the algebra VM. The automorphism group of (VM, • , ( , )) is M the Monster simple group ( [2], [7]). By 2A we denote the conjugacy class of involutions in M with the largest centraliser as in the Atlas [3]. For each 2A involution t of M, the centraliser CM(t) = 2.BM stabilises a 2-subspace W of VM which has two non-trivial idempotents at and iM - at. In [2], J. Conway constructed an M-invariant bijection ^ sending each 2A involution t to the nontrivial idempotent at of W with eigenvalue 1 and multiplicity 1 . We denote by at := ^(t) the image of t. In [8] A. A. Ivanov axiomatises some of the properties of the idempotents at into the definition of a Majorana axis. A Majorana axis a of a real commutative non-associative algebra (V, • , (, )), where • associates with ( , ) in the sense that (u • v, w) = (u, v • w) for all u, v, w e V, is an idempotent of length 1, whose adjoint operator ada is semi-simple on V with spectrum {1,0, , 25}. The eigenspaces of ada are denoted by VM(a), with p an eigenvalue, and satisfy the following conditions. The 1-eigenvectors of ada are exactly the scalar multiples of a. There exists a linear transformation t (a) of V, called a Majorana involution, negating the 25-eigenvectors, fixing the other eigenvectors and preserving both the algebra and inner products. Lastly there exists a linear transformation a (a ) of V+(a) = V5(a) © V0( ) © V1 ) + 22 negating the 22 -eigenvectors, fixing the 0- and 1-eigenvectors, and preserving both products on V++a). From [8], this definition is equivalent to the 'Fusion Rules'. For two eigenvectors u VA(a) and v e VM(a) of a fixed Majorana axis a, the Fusion Rules specify in which part of the spectrum of ada the product u • v lies. Sp 1 0 1 22 1 25 1 1 0 1 22 1 25 0 0 0 1 22 1 25 1 22 1 22 1 22 1, 0 1 25 1 25 1 25 1 25 1 25 1,0, 22 Table 1: Fusion rules Definition 1.1. We denote by ((A)) the subalgebra of VM generated by a set A of Majorana axes. The classification of subalgebras ((at, as)) of VM, where {as, at} is a pair of Majorana axes, was started in [2] and completed in [16]. We call them dihedral subalgebras as the corresponding pair of 2A-involutions {t, s} generates a dihedral subgroup of M. We say the dihedral subalgebra has type C if the product of involutions ts belongs to the conjugacy class C of M. Some subalgebras of VM generated by triples of Majorana axes are described by A. A. Ivanov et al in [11], [12], [13], [10], and [9]. In this paper, we investigate a subalgebra V = {{ax, ay, az}} of VM such that the dihedral subalgebra {{ax, ay}} has type 2 A and each of the dihedral subalgebras {{ax, az}}, {{ay, az}}, and {{axy, az}} has type 5A. The vector axy is the Majorana axis ^(xy) (since a dihedral subalgebra {{as, at}} of type 2A contains the axis ast). Keeping in mind the bijection ^ we might ask whether there exists a subgroup of M generated by a triple of 2A involutions {x, y, z} satisfying the relations: x2 = y2 = z2 = (xy)2 = (xz)5 = (yz)5 = (xyz)5 = 1. A group affording the presentation {x, y, z | x2, y2, (xy)2, z2, (xz)5, (yz)5, (xyz)5} defines the Coxeter group G(5'5,5) and from [4] it is isomorphic to the projective special linear group L2(11). From classical results on Ln(pk), [5], L2(11) is a simple group of order 660 = 22 • 3 • 5 • 11 and it has a single conjugacy class of involutions which we denote by T, and whose size is 55. Proposition 1.2. There exists a monomorphism i : L2(11) ^ M such that i(T) C 2A and i is unique up to conjugacy in M. Proof. In Table 5 of [17] S. Norton gives the list of simple subgroups of M having their elements of order 5 in the M-conjugacy class 5A. For i(T) it is a requirement since if a product of two 2A involutions has order 5 it belongs to the conjugacy class 5A of M [2]. By Norton's list there is only one conjugacy class of groups isomorphic to L2 (11) containing 5A elements and their involutions belong to class 2A. □ Throughout the paper i denotes the monomorphism as in Proposition 1.2, G = L2(11) denotes the image of i, and T denotes the conjugacy class of involutions in G. By the conjugacy property of dihedral subalgebras1, the axis atst is contained in {{at, as}}. Hence from the dihedral subalgebras of V, we can restate our aim to be the study of the subalgebra V of VM generated by the set of 55 Majorana axes {at 1t g T}. We determine the dimension of V and find a spanning set for V. In the next section we prove the following theorem. Theorem. Let V be the subalgebra of VM generated by the set of 55 Majorana axes {at | t g T}, where T is the class of involutions of the unique 2A-generated L2(11)-subgroup G of M. Then (1) dim(V) = 101, (2) V is linearly spanned by the set {at • as 1t, s g T}. (3) If K = Cm(G) then CyM (K ) = V e im . 1Let t and s be two 2A involutions, then tst is an involution conjugate to s. Hence tst is a 2A involution with corresponding Majorana axis atst := ^(tst). In the last section we give some evidence towards the uniqueness of the map ^ : t ^ at, where t G T, within the class of Majorana representations of L2(11) satisfying conditions (2A) and (3A) (the terminology is explained in the last section)2. Lastly we state a recent result of A. Seress proving that V is equal to the algebra of the unique Majorana representation of L2 (11). 2 Some properties of L2 (11) We present some of the standard properties of G = L2( 11) used when calculating inner product values for V. The group G is the automorphism group of the (11,5,2)-biplane, which we denote B (see [19]). B is a 2-symmetric design with 11 points, {p1,... ,pn}, and 11 lines, {l1,..., l11}, such that each line contains 5 points, each point lies on 5 lines, two lines intersect in exactly 2 points, and two points share exactly 2 lines. We call the incidence relation pi g j a flag, which we denote ai,j, and the relation pi G lj an anti-flag, which we denote by wi,j. From [14], the lines of B can be obtained by finding a difference set l1 of size 5, with elements from Z11, such that every integer modulo 11 appears exactly twice as a difference i — j mod 11 for i and j in l1. We have that l1 = {1,3,4, 5,9}, which is the set of non-zero perfect squares in Z11, and all other lines lk can be defined by lk = {1 + k, 3 + k, 4 + k, 5 + k, 9 + k}, where k G Z11 and addition is modulo 11. The incidence matrix N of B is given below with the rows indexed by the points of B, the columns indexed by the lines, and each flag is represented by a '1' and each anti-flag by a '0'. 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 0 1 We can represent G as a permutation group on 11 letters, so that G C Sym(11), by letting G act on the indices of the points or lines such that the incidence structure of B is preserved. The stabiliser G(aij) of a flag ai,j is isomorphic to A4, the stabiliser G(wkj) of an anti-flag wk,l is isomorphic to D10, and the stabiliser of a line (or a point) is isomorphic to A5. We can associate to a flag ai,j a unique subgroup S(ai,j ) = C2 x C2 and to an anti-flag wk,l a unique subgroup S(wk,l) = C5 such that NG(S(ai,j)) = G(ai,j) and NG(S(wk,i)) = G(wkl). It is easy to see that each involution t stabilises 3 flags and to deduce that CG(t) = 2When the first draft of this article was written, the author has learned that Akos Seress has written a GAP program, [6], capable of checking this uniqueness conjecture. D12. Similarly for (h) a subgroup of order 3 we can deduce NG((h)) = D12. There are only one class of involutions and one class of elements of order 3 in G, so we can let d be the G-invariant bijection between subgroups of order 2 and 3 sending each involution t to the unique subgroup of order 3 commuting with t. Furthermore, by [14], G contains one class of subgroups isomorphic to the Frobenius group of order 55, which we denote F55. These are the four conjugacy classes of maximal subgroups of G; two non-conjugate classes of subgroups isomorphic to A5 each of size 11 and each stabilising a point or a line, one class of subgroups isomorphic to D12, and one class of subgroups isomorphic to the Frobenius group F55. 3 The algebra V We start this section by finding an upper bound for dim(V) based on the work of S. Norton ([15], [16], and [17]). We then calculate the Gram matrix of a particular subset of V which provides a lower bound for dim(V). 3.1 S. Norton's observations The upper bound on dim(V) stems from the following inclusion. Lemma 3.1. V C CvM (Cm(G)) Proof. By the definition of a Majorana axis, at is fixed by CM(t) = 2.BM. Therefore Cm(G) = CM((x, y, z)) = P| CM(t) fixes V = ((ax, ay, az)) by M-invariance of the t=x,y,z algebra VM. □ We denote by K the group CM(G). The dimension of the fixed space of K in VM can be obtained by calculating the fusion of the character table of K in that of M (since the character of VM is known [3]). It is equal to the inner product of characters (xVm Ik , 1K )RK, where 1K is the trivial character of K, and xvM iK is the character of VM restricted to K. We thus need to determine the isomorphism type of K and the inclusions of the conjugacy classes of G and K into those of M. We call an A5-subgroup H of M an A5 of type (2A, 3A, 5A) if the elements of order 2, 3 and 5 of H are in the M-conjugacy classes 2A, 3A and 5A respectively. Clearly all A5-subgroups of G are of type (2A, 3A, 5A). Proposition 3.2. (i) For K as above, K = M12. (ii) All A5-subgroups H as above are conjugate in M and CM(H) = A12. (iii) The conjugacy classes of G fuse into those M as follows: Class in G 1a 2a = T 3a 5a 5b 6a 11a 11b Class in M 1A 2A 3A 5A 5A 6A 11A 11A Proof. The result from part (i) can be read from the entry 31 of Table 3 of [15]. Part (ii) is proved in Lemma 4 of [15]. To prove (iii) we carry on from the proof of Proposition 1.1. From Table 5 of [17] we deduce the inclusion 3a C 3A. In the character table of M, given in [3], the information on p-powers3 of elements g G 6A gives g2 G 3A and g3 G 2A, and 6A is the unique conjugacy class of elements of order 6 with those p-powers, hence to avoid a contradiction we must have 6a C 6A. Since M has a unique class 11A of elements of order 11 the classes 11a and 11b are subsets of 11A. □ Proposition 3.3. For the algebra CViI (K) we have dim(CViI (K)) = 102. Within the proof of Proposition 3.3 we determine the fusion of the conjugacy classes of K into those of M. We follow the Atlas's notation, [3], by writing the conjugacy of elements of order N in M: NA, NB,... (etc) in increasing order of the size of the class. Similarly for K we use the notation NAK, NBK, • • • (etc). The character tables used are those of the Atlas [3]. Proof. By part (i) of the previous proposition we have the inclusion of groups H C G, where H = A5 is of type (2A, 3A, 5A), which implies K C Cm (H) = A12. In A12, the elements with cycle decompositions 2218, 2414, and 26 have 8, 4, and no fixed points respectively in the natural action of A12 on 12 points, and so by Lemma 6 in [15] they are mapped to the M-conjugacy classes 2A, 2B and 2A respectively. There is a doubly transitive action of M12 on 12 points with character x1 + x11a where x1 is the trivial character of M12 and x11a is the first irreducible character of degree 11 (as in the Atlas, [3]). This character takes the value 0 for the elements in the class 2AMl2, and the value 4 for the elements in the class 2BMl2, hence 2AMl2 C 2A and 2BMl2 C 2B. The structure class constants4 for any pair of 2A involutions in M give the number of elements in each conjugacy class of M expressible as a product of two 2A involutions. The constants can be calculated directly from the character table. For M the product of two 2A involutions lies in either of the M classes: 1A, 2A, 2B, 3A, 3C, 4A, 4B, 5A or 6A (see [2] or [15]). Similarly for K = M12 we obtain that the product of two 2AK involutions lies in either of the K classes 1Ak , 2Ak , 2Bk , 3Bk , 4Ak , 4Bk , 5Ak or 6Ak .To avoid a contradiction on the monomorphism i we have 5AK C 5A, and 6AK C 6A and 3AK is a subset of either 3A or 3C. The class 6AK has p-powers 3BK, 2AK in K, and the class 6A has p-powers 3A, 2A in M. Hence 3BK C 3A. From lemma 6 of [15] no elements of order 3 in CM(H) = A12 belongs to class 3C of M, hence 3AK belongs to either 3A or 3B. If 3Ak C 3A then 6BK C 6C and if 3BK C 3B then 6BK is in either 6B or 6E according to the relevant p-powers in K and M. We determine the fusion in M of 3AK and 6Bk at the end of the proof. The classes 4AK, 4BK contain products of 2A involutions and their squares lie in class 2BK C 2B hence 4AK, 4BK C 4A as 4A is the unique class of elements of order 4 squaring to 2B. The classes 8AK, 8BK have their squares in classes 4AK, 4BK respectively, and in M the unique conjugacy class of elements of order 8 squaring to 4A is 8B. Hence 8AK, 8BK C 8A. The class 10AK in M12 has p-powers 5Ak C 5A and 2AK C 2A and in M the class 10A is the unique class of elements of 3For a finite group L, the p-power line in the character table of L records for each conjugacy class C of L, and for each prime p dividing the order of the elements of C, to which conjugacy class the pth-power of the elements of C belongs to. 4For a finite group L, the structure class constants give the number of solutions «1,2,3 to equations in the group of the type x1 .x2 = x3, where each xi belongs to a conjugacy class Ci of L. From the table of complex characters of L: = _|L|_ y^ x(xi)x(x2)x(x3) s1,2,3 = |cL(xi)HcL(x2)| x(l) order 10 with such p-powers so that 10AK c 10A. There is a unique class of elements of order 11 in M so 11AK, 11BK c 11 A. If 3AK c 3A then 6BK c 6C and the completed fusion of conjugacy classes of K in those of M gives a value of (xvm , 1K}rk which is not integral, a contradiction. Hence 3AK c 3B and 6BK is in 6B or 6E. To determine which, we look at the fusion of the conjugacy classes of A := CM(H) = Ai2 in M. Apart from the conjugacy classes 6GA, 9Aa, 9Ba and 9CA, the fusion of the classes of A in M is straightforward using the information on p-powers and the fusion of the classes of K in M already obtained. From a calculation of S. Shpectorov in [12] we know that (xvM Ia, 1 a}ra = 26. This can only happen if 6GA c 6B and 9Aa, 9Ba, 9Ca c 9A. In particular elements of order 6 in Ai2 cannot be subsets of 6E, hence neither can the elements of order 6 in K. Hence 6BK c 6B. We have obtained the fusion of K in M Class in K 1Ak 2Ak 2BK 3Ak 3BK 4Ak 4Bk p-powers A A A A A B B Class in M 1A 2A 2B 3B 3A 4A 4A 5Ak 6Ak 6BK 8Ak 8Bk 10Ak 11 Ak A BA AB A B AA A 5A 6B 6A 8B 8B 10A 11A and we can now compute the inner product of real characters of K (XVm Ik, 1K}RK = 102. □ The following useful observation was made by S. Norton (in a more general context). Lemma 3.4. The identity iM of VM cannot be contained in V. Proof. The groups G and K centralise each other in M and G x K is a subgroup of M. The unique conjugacy class of involutions T of G is in class 2A of M, and we have proved that the classes 2AK and 2BK are in 2A and 2B respectively. Claim : there exist elements s G T and t G 2AK such that the element ts of G x K is in class 2B of M. From Proposition 3.2, part (iii), the element s can be taken from a A5-subgroup X of G of type (2A, 3A, 5A). From the proof of Proposition 3.2, the elements of 2AK act fixed-point freely on 12 points, and the centraliser in M of an A5-subgroup of type (2A, 3A, 5A) is isomorphic to Ai2. By Table 4, line 7 of [17], there exists a subgroup Y C Ai2, Y = A5, that acts transitively on 12 points. Let t G Y. Then, by Table 3, line 8 of [17], the involutions in the diagonal subgroups of X x Y are in 2B. In particular ts G 2B and the claim is proved. Since ts G 2B, the axes at and as generate a dihedral algebra of type 2B and at ■ as =0 (see [2] or [11]). For all z G T there is an element g G G such that z = sg, so by invariance of the algebra product (at ■ as )g = 0 = at ■ az since G normalises K. Now, V is generated by the 55 Majorana axes az for z G T and the 0-eigenspace of at is closed under the algebra product, so if the identiy iM were in V we would get the contradiction at ■ iM = 0. □ The identity of a commutative algebra being unique and therefore stable under the automorphism group we have iM € CVm (K). And since iM is not in V we obtain the main result of this subsection. Proposition 3.5. For the algebra V we have dim(V) < 101. □ 3.2 Inner product values for V In this subsection we calculate all inner products on a well-chosen subset of V and compute the rank of the corresponding Gram matrix to bound below the dimension of V. We do so using the information on some subalgebras of V which have already been classified. 3.2.1 Dihedral subalgebras of V The algebra V contains the dihedral subalgebras of types 2 A, 3 A, 5 A, and 6 A (obtained by calculating the relevant structure class constants in the character table of G). From [16], for each type of dihedral algebra, we know the dimension of the algebra, and a basis for which all algebra and inner products are known. We follow the exposition given in [8] which is now accepted as standard in the Majorana Theory and where a different scaling to [16] is used. Table 2 is taken from [8], which notation we explain below. Each dihedral subalgebra corresponds to a dihedral subgroup D of M generated by two 2 A involutions t and s, whose product we denote by p := ts. We denote by a0, a1 and a the Majorana axes at, as and atpi in Table 2. In the subalgebra of type 2A, we have ap = ^(p) which is also a Majorana axis, and in the types 3A and 5A the vectors up and wp are introduced to close the algebra product. They correspond to elements of order 3 or 5 in D respectively. From [2], the 1-dimensional subspace linearly spanned by the vector up or wp is invariant under the normaliser NM ( (p) ) which is isomorphic to 3.F24 or (D10 x F5).2 respectively. Also, in the type 3A the vector itself is stable under NM ( (p) ), so that up = up-1, and in the type 5 A the vector is stabilised up to negation wp = — wp2 = — wp3 = wp4. Any element of order 3 or 5 in G can be expressed as a product of two involutions, and any two involutions correspond to a dihedral subalgebra of V. Hence to study the algebra V we can consider the span of the vectors corresponding to the cyclic subgroups of order 2, 3 and 5 of G. We let G(i) be a set of non-trivial representatives of each cyclic subgroup of order i for i = 2, 3, 5, of size 55, 55 and 66 respectively, where for i = 5 the representatives are taken from the same conjugacy class of G. We use the notation A := {at | t € G(2)}, U := {uh | h € G(3)}, and W := {wf | f € G(5)}, and we let S := A U U U W. 3.2.2 As subalgebras of V The algebra V also contains 22 A5-subalgebras of type (2A, 3A, 5A) as G contains two conjugacy classes of A5-subgroups of type (2A, 3A, 5A), of size 11 each. The structure of a subalgebra VH generated by the Majorana axes indexed by the involutions of an A5-subgroup H of type (2A, 3A, 5A) follows from [12]. We reformulate it so as to present VH as a subalgebra of VM generated by a triple of Majorana axes. Type Basis Products and angles 2A 3A 5A 6A ao, ai, ap a— i, ao, ai, Up a—2, a— i, ao, ai, a2, Wp a—2, a—i, ao, ai, a2, a3 ap3, up2 •*p2 ao ■ ai = 23 (ao + ai — ap), ao ■ ap = |3(ao + ap — ai) (ao, ai) = (ao, ap) = (ai, ap) = ao ■ ai = Tir(2ao + 2ai + a_i) — ^n5? ao ■ Up = 32(2ao — ai — a_i) + |5Up p (ao, ai) = TI, (ao,Up) = 22, (up,Up) = ao ■ ai = 27 (3ao + 3ai — a2 — a_i — a_2) + Wp ao ■ a2 = 57(3ao + 3a2 — ai — a_i — a_2) — Wp ao ■ Wp = 2^2 (ai + a_i — a2 — a_2) + 55 Wp Wp ■ Wp = "2tq-(a_2 + a_i + ao + ai + a2) (ao, ai) = 27, (ao, Wp) = 0, (Wp, Wp) = Irr ■ ai = 56 (ao + ai — a_2 — a_i — a2 — a3 + ap3 ) + ^In5- u( ao ■ a2 = 25(2ao + 2a2 + a_2) — |ttUp2 ao ■ Up2 = |2 (2ao — a2 — a_2) + 25Up2 ao ■ a3 = 2— (ao + a3 — ap3 ), ap3 ■ Up2 = 0, (ap3, Up2 ) = 0 (ao,ai) = 28, (ao,a2) = ^^, (ao,ai) = |3 p U p p Table 2: Dihedral subalgebras Proposition 3.6. Let VH = {{ax, ay, az}} be a subalgebra of VM where the dihedral subalgebra {{ax, ay}} has type 2A and where the dihedral subalgebras {{ax, az}}, {{ay, az}}, and {{axy, az}} have types 5A, 5A and 3A respectively. Then VH has dimension 26 and it is linearly spanned by the products of all pairs of Majorana axes indexed by the involutions of H. □ For explicit formulas for the algebra product in VH or a list of all inner product values, we refer the reader to [12]. In the rest of the paper we will simply refer to an A5-subgroup H to mean an A5-subgroup of type (2A, 3A, 5A). For an A5-subgroup H, we denote by H(2), H(3) and H(5) the sets of non-trivial conjugate representatives of cyclic subgroups of order 2, 3 and 5 and in the corresponding algebra VH we denote by AH, UH and WH the sets of vectors {at 1t g H(2)}, {uh | h g H(3)}, and {wf | f g H(5)}. Let wH be the sum of all vectors in WH. By [12], the set SH := Ah U Uh U Wh is a spanning set of size 31 for VH, and VH is 26-dimensional with a basis Ah U Uh U {wH}. The five independent linear relations on SH, which can be found in [12] or [16], are called the Norton Relations. Proposition 3.7. The Norton Relations In the algebra VH corresponding to an A5-subgroup H of type (2A, 3A, 5A), all vectors wf g Wh satisfy: ( wf = 6 wH + 27 E at - E at I + ^it \t£H<2)(f ) teH<2)(f ) / 32.5 \ E uh - E uh heH(3), heH(3), \0([h,f])=3 o([h,f])=5 / where H5(2)(f ): = {t g H(2) | o(tf) = 5}, H32)(f): = {t g H(2) | o(tf) = 3}. □ We denote by Hi = {Hi,..., Hü} and H2 = {Hi,..., Hi J the two classes of A5-subgroups in G. One class corresponds to the rows of N and the other to the columns, so the intersection between A5's taken from different classes can be read directly from the entries of N. For a given A5-subgroup Hj in G let Wj be the sum of all vectors in WHi. For a vector wf g WHi we rewrite the Norton relation for wf as wf = 6Wj + A Aj(f ) + MUj(f ) (3.1) where the meaning of A, Aj(f ) and Uj(f ) is clear from Proposition 3.7. Corollary 3.8. Let w be the sum of all vectors wf in W C V. Then S' := A U U U {w} is a spanning set of size 111 for S. Proof. Consider an A5-subgroup Hi e Hi. From N, any subgroup Hi, e H2 intersect Hi ina Dio oran A4. If Hi n Hi> = Dio there exists a representative f of H(5) in Hi nH for which the Norton relations give r 6 W1 + A Ai(f ) + pUi(f ) wf = s [ 1Wi + A Ai (f )+ pUi, (f ) and so W^ is in Sp(A U U U {Wx}), the R-linear span of A U U U {W1}. If Hi n Hi, = A4 then the situation can be visualized as the following submatrix of N, where each anti-flag has been replaced by the unique element of G(5) stabilising it, and the rows and columns are indexed with the copy of A5 stabilising the corresponding line or point of B. Hi Hi Hi f 1 wg\ Hj ywf wk J From the Norton relations for g e Hi n Hi,, k e Hi n Hj, and f e H n Hj,, we also get Wi e Sp(A U U U {Wx}). From N, for any subgroup Hi e Hi there are 3 elements of H2 intersecting both Hi and Hi in a Di0, with say H;, being one of them: Hi Hi Hi, ( wf w; ) so the Norton relations for f and l give Wi e Sp(A U U U {Wx}). Hence there exists v e Sp(A U U) such that 22 W1 = E*eH Wi + Eh,eH W+ v, and from N every element of G(5) is contained in exactly one element of Hi and one of H2 so that E Hiem Wi + E Hi, eH2 Wi' =2 w, where w is the sum of all vectors wf in W, and hence W1 = w + v' for some v' e Sp(A U U). □ 4 Inner product values Definition 4.1. For each pair (G(i), G(j)) with i, j e {2,3, 5} we call the inner product values on G(i) X inner products of type (i, j). If we let E(i) be the equivalence class of elements of order i in G belonging to the same cyclic subgroup, then the orbits of G acting by conjugation on E(i) x form a subpartition of the distinct inner products values of type (i, j ) (these orbits were calculated using [1]). We will only explain the inner products values (uk, v;) for which the subgroup (k, l) is isomorphic to F55 or to the whole of G. They arise as the solutions of equations of intersecting subalgebras inside V, or equivalently as particular configurations of subgroups inside G, which can be read from the incidence matrix N or found using a code written in [1]. 4.1 Inner products of type (2, 2) From the dihedral subalgebras of V we know all possible inner product values of any two Majorana axes in V, see [11]. Case o(ts) ((at, as)) (t, s) (at, as) 1 2 3 4 5 1 1A 11 2 2A D4 T? 3 3A De if 5 5A Dig 27 6 6A Di2 28 Table 3: Inner Products of type (2, 2), with t, s e G(2) 4.2 Inner products of type (2, 3) The value for case 5) of the inner product of type (2, 3) was computed using the following lemma. Lemma 4.2. For t e G(2) and h e G(3) such that (t, h) = L2 (11) we have (at^ uh) = 2X5. Proof. We fix an element h e G(3) and we let t e G(2) such that (t, h) = G. Since Ng((h)) = Di2 then (h) is contained in exactly two distinct dihedral groups of order 6. Let Sh be one of the two sets of 3 involutions, Sh := {s, sh, sh2}, such that (S, h) = De, and up to permutation of the set Sh we have (t, s) = De, (t, sh) = Di2, (t, sh2) = Dig. In the 3A-dihedral subalgebra ((as, uh)) we have the equality uh — 33. ash — 25 (2as + 2ash — ash2 ) so taking the inner product with at gives (at , uh) = -33-5 (at , as • ash) - ö5(at , 2as + 2ash - ash2) a s 5 Case o(ht) (t, h> (at, Uh) 1 2 3 4 5 2 De 52 3 A4 -2 5 A5 2.32 6 Ce 0 11 L2 (11) 2A5 Table 4: Inner Products of type (2,3), with t e G(2) and h e G(3) Case o(tf )= o(tf-1) o([t,/ ]) (t,/ > (at,Wf ) 1 2 5 D10 0 2 3 5 A5 72 214 3 5 3 A5 72 214 4 5 5 L2(11) 1 214 5 6 6 L2(11) 3 212 6 11 5 L2(11) 19 214 Table 5: Inner Products of type (2,5), with t e G(2) and / e G(5) and by associativity of the algebra product with the inner product (at, as ■ ash) = (as, at ■ ash). Since (t, sh> = D12, the element p = tsh has order 6 so the algebra product at ■ ash is contained in the dihedral algebra ((at, ash>> of type 6A, and so (as, at ■ ash) = 56(as, at + ash - atp2 - atp3 - atp4 - atps + ap3) + -^n-(as,Wp2), where (s, p2> = A4, so the value of (as , up3) is known to be 9 from [11]. Since all the required inner products are now known, one can compute (at, uh) = . . . □ 4.3 Inner products of type (2, 5) The next lemma justifies the values found in cases 4), 5) and 6). We omit its proof which is similar to the proof of Lemma 4.2. Lemma 4.3. Let t G G(2) and f G G(5) such that (t, f} = L2(11). Then exactly one of the following holds. (i) There exists s G G(2) commuting with t and inverting f, and (at,wf ) = - 2"rr + 26p - 23q, where p = 7(at, af ) + (at, asf2) and q = (ats,asf ). (ii) There exists s G G(2) inverting f and generating with t a dihedral group of order 6, and / \ q2 1 q3 r (at, Wf ) = 234 + 27p - ifr q, where p = (at, 5asf + asf2 + asf3 + asf4) and q = (wst, asf ). (iii) There exists s G G(2) inverting f and generating with t a dihedral group of order 12, and (at, Wf ) = -2f5 + 26p + 27q - |1f r, where p = (asf, ap3 - 2at - atp2 - atp3 - atp4 - atps), q = (at, asf2 + asf3 + asf4) and r = (up2, asf ), for p := ts of order 6 in (t, s} = D12. □ 4.4 Inner products of type (3, 3) In the next lemma part (i) addresses cases 4) and 6) and part (ii) addresses case 5). The lemma assumes all products of type (2,3) are known. Lemma 4.4. Let h, k G G(3) with (h, k} = L2(11). Then exactly one of the following holds. (i) There exists an involution t inverting both h and k, and (uh,Uk) = g2^(5 - 23.32p + 26q), where p =(uh,atk2) = (uk,ath2) and q = (ath, atk2) + (ath, atk) + (ath2, atk) + (ath2, atk2). (ii) There exists an involution t inverting h and generating with k an alternating group A4, and (uh,uk) = 335 (523 - p - 2q - r), where p = (ath, 3utk - 4utk - 4utk2), q = Kh2,uk) and r =(ath,«k). Case {o(hk),o(hk x)} (h, k) (uh,uk) 1 {1, 3} C3 23 5 2 {2, 3} A4 23.17 34.5 3 {5, 5} A5 24 34.5 4 {5, 6} L2(11) 23 3.52 5 {5,11} L2(11) 23.7 33.52 6 {6, 6} L2(11) 25 34.5 Table 6: Inner Products of type (3, 3), with h, k G G(3) 4.5 Inner products of type (3, 5) Part (i) of the next lemma addresses case 3), and part (ii) addresses cases 5) and 6). Lemma 4.5. Let h G G(3) and f G G(5) such that (h, f ) = L2(11). Then exactly one of the following holds. (i) There exists an involution t inverting f and h, and (uh,wf ) = -331 (p - q), where P = 2T2(atf + atf4 - atf2 - atf 3, ath) + 25(wf ,ath) and q = (2at + 2ath + ath2, Wf). (ii) There exists an involution t inverting f and generating with h a subgroup isomorphic to A5, and (uh,Wf ) = 27P + 27q - 24r, where p = (asf + asf4, uh), q = (asf2 + asf3, uh) and r = (asf, ush + ush2 ). □ The inner product value for case 4) of the inner product of type (3, 5) can be found using the Norton relations inside some A5-subalgebras of V, see equation (1). The proof of the following lemma uses similar arguments to the proof of Corollary 3.8. We use the notation of (1). Lemma 4.6. Let f G and h If there exists an element g G G(5) such that Ai := (f, g) and A2 := (h, g) are two non-conjugate A5-subgroups of G, then (uh, Wf ) = 6 {uh ,W2) + 27 (uh ,la) + 22 (v'h,lu), where la = Ai(g) + Ai(f) - A2(g) and lu = Ui(g) + Ui(f ) - U2 (g). Case >(hf ) o(hf-1) (h,f) (uh,wf ) 1 2 5 A5 2 3 5 A5 3 3 6 L2(11) 4 5 5 L2(11) 5 6 11 L2(11) 6 11 11 L2(11) -5.7 29.32 5.7 29.32 -67 29.32.5 -1 28.32.5 7 26.32.5 -7 27.32.5 Table 7: Inner Products of type (3, 5), with h G G(3) and f G G(5) 4.6 Inner products of type (5, 5) In the next lemma, part (i) justifies the values of the inner product of type (5, 5) for the cases 2), 3) and 4), and part (ii) justifies case 6). The proof is similar to that of Corollary 3.8 and the notation is the same as the one used in the previous lemma. Lemma 4.7. Let f, g G G(5) not contained in a common A5-subgroup, with f and g belonging to the pairs {Hi, Hi'} and {Hj, Hj'} respectively, of distinct non-conjugate A5-subgroups of G. Then exactly one of the following holds. (i) Hi n Hj' = D10, or Hj n Hi' = D10, with k an element of order 5 in Hi n Hj/, say, then (wf , w g ) = 62 (W W ) + ^ (la , W ) + 13 ) + {la ,Aj (g)) 32 5 32 5 34 52 + 21F (la , Uj (g)) + 21F (l« ,Aj (g)) + -2Š2T (lu ,Uj (g)) , where la = Aj(k) — Ai' (k) + Ai' (f) and lu = Uj (k) — Ui' (k) + Ui' (f). (iii) Hi n Hj' = Hj n Hi' = A4, and there exist two elements k = l G G(5) such that k belongs to Hi and Hm' and l belongs to Hj and Hm', so that (wf ,wg ) = 62 (Wj , Wj ) + ^^L. (la , Wj ) + 35 (lu , ) + ^^ (la , A (g)) 32 5 32 5 34 52 + 21F(la,Uj(gV + ^^(lu ,Aj(g)) + (lu ,Uj(g)), where la = Aj (l) — Am (l) — Ai (k) + Am (k) + Ai (f ), and lu = Uj(l) — Um'(l) — Ui(k) + Um'(k) + Ui(f). Case {o(/g),o(/g-1)} o([/, g]) (uh) 1 {1, 5} 1 C5 53.7 219 2 {3, 5} 5 A5 7.29 219 3 {5,11} 11 F55 -ii 219 4 {3, 6} 5 LI (11) 3. 151 221 5 {2, 6} 5 LI (11) 157 I20 6 {5,11} 2 Li (11) 59 I20 7 {5, 5} 3 Li (11) -3.41 I20 Table 8: Inner Products of type (5,5), with /, g G G(5) Corollary 4.8. The inner product values between the vector w, and the vectors at G A, uh G U, wf G W and w itself are as follows 32 (i) (at, w) = 2TT; 32 (ii) (uh, w) = - 27.5 ; (iii) (wf, w) = ^^iTr1 ; (iv) (w,w) = . □ 4.7 Dependence relations in the algebra We let Vs' be the R-vector space having the subset S' = A U U U{w} of V as a basis. We turn VS' into a G-module by the natural action of G on S', and we let n be the natural projection n : VS' ^ V. Using [1] we find the rank of the Gram matrix of the set S' and give a description of the kernel of n. We recall the bijection d introduced at the beginning of section 2 between subgroups of order 2 and 3 in G: d :G(2) ^ G(3) ^ since Vt G G(2) 3! h G G(3) where [t, h] = 1. For a fixed involution t G G(2) its normaliser NG(t) = Di2 has the following orbits on G(2) (the action is conjugation): Ol, 03, 03, O6, O2, 03, 04, OÌ2 and Of2, where the subscript indicates the size of the orbit. If we write NG(t) = (p) x (s) then p3 = t, so O1 = {p3}, O3 = {s, sp2, sp4} and O3 = {sp, sp3, sp5} wlog. Further we can describe the orbits as follows: O3 uO3 ={s e G(2)|(s, t) = 22} O6 u O2 ={s e G(2)|(s, t) = D12} O3 U O4 ={s e G(2)|(s, t) = De} O32 U O32 ={s e G(2)|(s, t) = D10}. For (t1, t2) in O3 x O3 or O3 x O3 the subgroup (t1, t2) in G is isomorphic to either 22 or D10. For (t1, t2) in O3 x O3 or O3 x O3 the subgroup (t1, t2) is isomorphic to either De or D12. Proposition 4.9. (i) The rank of the Gram matrix for the set S ' is 101. (ii) The kernel of n is 10-dimensional and consists of 10 linearly independent relations, between the vectors of A U U, taken from a set of 55 G-invariant relations R(t) indexed by the involutions of G. For a fixed involution t in G(2), R(t) defines the following NG(t)-invariant relation: R(t) := ^ ar - Y as + ^^ ( ^ Uh - ^ Uk r£Ti s£T2 hed(Ti) ked(T2) where T1 and T2 can taken to be O1 U O,1 and O| U O^ respectively (or vice versa). □ From the rank of the Gram matrix of the set S' we obtain the following proposition. Proposition 4.10. For the algebra V we have dim(V) > 101. □ The above, together with Proposition 3.5, proves that dim(V) = 101. Hence the set S' spans V, so that {at • as | t, s e T} also spans V. From Lemma 3.4, the identity of VM is not in V. The space CVm (K) is 102-dimensional, containing iM and having V as a subspace. Hence Cvm (K) decomposes as V © iM, and we have proved our main theorem. 5 A Majorana representation of L2 (11) The dihedral and A5-subalgebras of V can be characterised under the axioms of Majorana theory; they are equal to the algebra of the Majorana representations of the dihedral groups D4 of type 2A, De of type 3A, D10 of type 5A, and D12 of type 6A, and of the alternating group A5 of type (2A, 3A, 5A). Majorana theory was introduced by A. A. Ivanov in [8] to axiomatise some of the properties of VM and its Majorana axes. We refer the reader to [8] and [11] for a full description. Definition 5.1. A Majorana representation of a finite group G is a tuple R = (G, T, X, ( , ), •, ^, where T is a union of conjugacy classes of involutions generating G, and X is a commutative non-associative R-algebra endowed with an inner product ( , ) associating with its algebra product • in the sense that (u • v, w) = (u, v • w) for all u, v, w g X and satisfying the Norton Inequality (u • u, v • v) > (u • v, u • v), for all u, v G X. The image of the homomorphism ^ : G ^ GL(X) is an automorphism of (X, • , (, )), and the map ^ is an injection sending each involution t of T to a Majorana axis at of X, as defined in the second paragraph of section 1 (the properties of the spectrum of ada and the Fusion Rules are assumed to hold ), such that ^ and ^ commute in the sense that : ag-itg = (at)v(g) for every g G G. We require that the algebra X be generated by the set of Majorana axes ^(T) and that it must satisfy conditions (2A) and (3A) below. Conditions (2A) and (3A) ensure that when constructing X in the above definition we get the right number of 3A vectors uh from the Majorana axes. (2A) Let t0, ti G T and p := t0t1 such that (a) if p G T and the vectors at0, atl generate a dihedral subalgebra of type 2A then aP = ^(p^ (b) if pj G T for p of order 4 or 6 and the vectors at0 and atl generate a subalgebra of type 4B or 6A, then ^(pj) coincides with the axis api ; (3A) Let to,ti,t2,t3 G T with {to,ti} = {t2,t3} = De. We let pi := toti and p2 := t2t3 both of order 3. If the following two conditions are satisfied: (i) pi = p2 or p-1, and (ii) the dihedral subalgebras generated by {at0, atl} and {at2, at3} have type 3A, then the corresponding 3A-axial vectors uPl and uP2 in the above subalgebras are equal in X. We call dim(X) the dimension of R, and we say that R is based on an embedding of G into M if there exists a monomorphism i : G ^ M with i(T) C 2A and such that R is isomorphic to the subalgebra of VM generated by the Majorana axes corresponding to i(T). Definition 5.2. The shape of a Majorana representation R of G specifies the types of dihedral subalgebras associated with all pairs of involutions on T. Theorem 5.3. A Majorana representation of G = L2(11) must have shape (2A, 3A, 5A, 6A). Proof. Let R be a Majorana representation of G with associated algebra X. The group L2(11) has a single conjugacy class of involutions, 2a, and a single class 3a of elements of order 3. From the structure class constants the product of any 2a involutions is in either of the L2(11) classes 1a, 2a, 3a, 5a, 5b or 6a. Hence X contains dihedral subalgebras of type 5A and 6A since they are the only dihedral subalgebras associated with dihedral groups of order 10 and 12. By the inclusion of the dihedral subalgebras 3A ^ 6A and 2A ^ 6A the classes 3a and 2a are mapped to 3A and 2A under Hence X also contains dihedral subalgebras of type 3A and 2A and we have accounted for all possible dihedral subalgebras in X. □ Let R be a Majorana representation of G with associated algebra X. From the above theorem, R has the same shape as the subalgebra V of VM and the same inner product values for the sets S' and S. Moreover the dihedral and A5-subalgebras of X are equal to their Majorana representations from [11] and [12]. Proposition 5.4. (i) The dihedral subalgebras of type 2A, 3A, 5A and 6A are equal to the unique Majorana representations of D4, D6, Di0, and Di2 of shape 2A, 3A, 5A and 6A respectively. (ii) The A5-subalgebra of type (2A, 3A, 5A) is equal to the unique Majorana representation of shape (2A, 3A, 5 A) of a group A5 of type (2A, 3A, 5A), which has dimension 26. □ We would like to show that the shape of R uniquely determines the algebra product in X so that X = V .In particular it is necessary to find the closure of the algebra generated by S. This can be inspected computationally. We let S2 := {u • v | u, v e S} and S3 := {(u • v) • w, u • (v • w) | u, v, e S} and for any positive integer n the set Sn is defined in a similar way. Already for the set of vectors S U S2 the Majorana axioms yield a very large number of eigenvectors for each Majorana axis and the first computational step is to check whether or not the linear span of S U S2 over R is contained in the closure of X. In fact during the reviewing stage of this paper, the author has learned that A. Seress has proved in [18] that the system of linear equations in S U S2, obtained from the eigenvectors of the axes {at 11 e T}, has a unique solution and that dimR(X) = 101. The result was obtained computationally with an algorithm written with [6]. Theorem 5.5. The L2(11)-subalgebra of the Monster algebra VM is equal to X, the algebra corresponding to the unique Majorana representation of L2(11). □ Acknowledgements I would like to thank my supervisor Professor Alexander A. Ivanov for introducing me to the Monster algebra and to Majorana representations and for his guidance and advice on the case of L2 (11). My thanks also go to the EPSRC for supporting this research. References [1] W. Bosma, J. Cannon and C. Playoust, The magma algebra system I, the user language, J. Symbolic Comput. 24 (1997), 235-265. [2] J. H. Conway, A simple construction for the Fischer-Griess monster group, Invent. Mat. 79 (1985), 513-540. [3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, The atlas of finite simple groups, Clarendon Press, Oxford, 1985. [4] H. S. M. Coxeter, The abstract groups Gm'n'p, Trans. Amer. Math. Soc. 45 (1939), 73-150. [5] L. L. Dornhoff, Group representations theory, M. Dekker, New York, 1971-72. [6] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.12, 2008, http: //www.gap-system.org. [7] R. L. Griess, The Friendly Giant, Invent. Mat. 69 (1982), 1-102. [8] A. A. Ivanov, The monster group and Majorana involutions, CUP, Cambridge, 2009. [9] A. A. Ivanov, Majorana representation of A6 involving 3C-algebras, Bull. Math. Sci. (2011), online publication. [10] A. A. Ivanov, On Majorana representations of A6 and A7, Comm. Math. Phys. 307 (2011), 1-16. [11] A. A. Ivanov, D. V. Pasechnik, A. Seress, and S. Shpectorov, Majorana representations of the symmetric group of degree 4, J. Algebra 324 (2010), 2432-2463. [12] A. A. Ivanov and A. Seress, Majorana representations of A5, Math. Z. (2011), to appear. [13] A. A. Ivanov and S. Shpectorov, Majorana representations of L3(2), Adv. Geom. (2011), to appear. [14] W. M. Kantor, Automorphism groups of designs, Math. Z. 109 (1969), 246-252. [15] S. P. Norton, The uniqueness of the Fischer-Griess monster, Contemp. Math. 45 (1985), 271285. [16] S. P. Norton, The Monster algebra, some new formulae, Contemp. Math. 193 (1996), 297-306. [17] S. P. Norton, Anatomy of the Monster I, LMS Lect. Notes Ser. 249 (1998), 198-214. [18] A. Seress, Construction of 2-closed M-representations, Proc. International Symposium on Symbolic and Algebraic Computation (ISSAC '12) (2012), 311-318. [19] J. A. Todd, A combinatorial problem, J. Math, and Phys. 12 (1933), 321-333. /^creative , ars mathematica ^commons contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 105-121 On fat Hoffman graphs with smallest eigenvalue at least —3 Hye Jin Jang , Jack Koolen Department of Mathematics, POSTECH, Pohang 790-784, South Korea Akihiro Munemasa Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan Tetsuji Taniguchi Matsue College of Technology, Nishiikuma-cho 14-4, Matsue, Shimane 690-8518, Japan Received 31 October 2011, accepted 19 September 2012, published online 21 March 2013 Abstract We investigate fat Hoffman graphs with smallest eigenvalue at least -3, using their special graphs. We show that the special graph S (H) of an indecomposable fat Hoffman graph H is represented by the standard lattice or an irreducible root lattice. Moreover, we show that if the special graph admits an integral representation, that is, the lattice spanned by it is not an exceptional root lattice, then the special graph S-(H) is isomorphic to one of the Dynkin graphs An, Dn, or extended Dynkin graphs An or Dn. Keywords: Hoffman graph, line graph, graph eigenvalue, special graph, root system. Math. Subj. Class.: 05C50, 05C76 1 Introduction Throughout this paper, a graph will mean an undirected graph without loops or multiple edges. Hoffman graphs were introduced by Woo and Neumaier [5] to extend the results of Hoffman [3]. Hoffman proved what we would call Hoffman's limit theorem (Theorem 2.14) which asserts that, in the language of Hoffman graphs, the smallest eigenvalue of a fat Hoffman graph is a limit of the smallest eigenvalues of a sequence of ordinary graphs with increasing minimum degree. Woo and Neumaier [5] gave a complete list of fat indecomposable Hoffman graphs with smallest eigenvalue at least -1 - %/2. From their list, we E-mail addresses: laluz@postech.ac.kr (Hye Jin Jang), koolen@postech.ac.kr (Jack Koolen)munemasa@math.is.tohoku.ac.jp (Akihiro Munemasa), tetsuzit@matsue-ct.ac.jp (Tetsuji Taniguchi) find that only -1, -2 and -1 - %/2 appear as the smallest eigenvalues. This implies, in particular, that -1, -2 and -1 - %/2 are limit points of the smallest eigenvalues of a sequence of ordinary graphs with increasing minimum degree. It turns out that there are no others between -1 and -1 - %/2. More precisely, for a negative real number A, consider the sequences niX) = inf{Amin(r) | mindegr > k, Amin(r) > A} (k = 1, 2,... ), 6{kX) = sup{Amin(r) | mindegr > k, Amin(r) < A} (k = 1, 2,... ), where r runs through graphs satisfying the conditions specified above, namely, r has minimum degree at least k and r has smallest eigenvalue greater than (or less than) A. Then Hoffman [3] has shown that lim = -1, lim ej-_1) = -2, k —^^o k —^^o lim ni-1-72) = -2, lim e2) = -1 - V2. k—w k—w In other words, real numbers in (-2, -1) and (-1 - %/2, -2) are ignorable if our concern is the smallest eigenvalues of graphs with large minimum degree. Woo and Neumaier [5] went on to prove that lim nka) = -1 - A k—w where a « -2.4812 is a zero of the cubic polynomial x3 + 2x2 - 2x - 2. Recently, Yu [6] has shown that analogous results for regular graphs also hold. Woo and Neumaier [5, Open Problem 4] raised the problem of classifying fat Hoffman graphs with smallest eigenvalue at least -3. They also proposed a generalization of a concept of a line graph based on a family of isomorphism classes of Hoffman graphs. This enables one to define generalized line graphs in a very simple manner. As we shall see in Proposition 3.2, the knowledge of ^-saturated indecomposable fat Hoffman graphs gives a description of all fat Hoffman graphs with smallest eigenvalue at least For ^ = -3, this in turn should give some information on the limit points of the smallest eigenvalues of a sequence of ordinary graphs with increasing minimum degree. Also, using the generalized concept of line graphs, we can expect to give a description of all graphs with smallest eigenvalue at least -3 and sufficiently large minimum degree. Thus our ultimate goal is to classify (-3)-saturated indecomposable fat Hoffman graphs, as proposed by Woo and Neumaier [5]. The purpose of this paper is to begin the first step of this classification, by determining their special graphs for such Hoffman graphs having an integral reduced representation. One of our main result is that the special graph S-(H) of such a Hoffman graph H is isomorphic to one of the Dynkin graphs An, Dn, or extended Dynkin graphs An or Dn. We also show that, even when the Hoffman graph H does not admit an integral representation, its special graph S (H) can be represented by one of the exceptional root lattices En (n = 6,7, 8). This might mean that the rest of the work can be completed by a computer as in the classification of maximal exceptional graphs (see [1]). Indeed, if one attaches a fat neighbor to every slim vertex of an ordinary maximal exceptional graph, then we obtain a (-3)-indecomposable fat Hoffman graph. However, maximal graphs among (-3)-indecomposable fat Hoffman graphs represented in E8 are usually much larger (see Example 3.8 and the comment following it), so the problem is not as trivial as it looks. We plan to discuss in the subsequent papers the determination of these special graphs and the corresponding Hoffman graphs. 2 Hoffman graphs and their smallest eigenvalues 2.1 Basic theory of Hoffman graphs In this subsection we discuss the basic theory of Hoffman graphs. Hoffman graphs were introduced by Woo and Neumaier [5], and most of this section is due to them. Since the sums of Hoffman graphs appear only implicitly in [5] and later formulated by Taniguchi [4], we will give proof of the results that use sums for the convenience of the reader. Definition 2.1. A Hoffman graph H is a pair (H, p) of a graph H = ( V, E) and a labeling map p : V ^ {f, s}, satisfying the following conditions: (i) every vertex with label f is adjacent to at least one vertex with label s; (ii) vertices with label f are pairwise non-adjacent. We call a vertex with label s a slim vertex, and a vertex with label f a fat vertex. We denote by Vs = Vs(H) (resp. Vf (H)) the set of slim (resp. fat) vertices of H. The subgraph of a Hoffman graph H induced on Vs (H) is called the slim subgraph of H. If every slim vertex of a Hoffman graph H has a fat neighbor, then we call H fat. For a vertex x of H we define Nf (x) = Nf (x) (resp. Ns (x) = Ng (x)) the set of fat (resp. slim) neighbors of x in H. The set of all neighbors of x is denoted by N(x) = NH (x), that is N(x) = Nf (x) U Ns(x). In a similar fashion, for vertices x and y we define Nf (x, y) = Nf (x, y ) to be the set of common fat neighbors of x and y. Definition 2.2. A Hoffman graph Hi = (H1,p1) is called an (induced) Hoffman subgraph of another Hoffman graph H = (H, p), if H1 is an (induced) subgraph of H and p(x) = p1(x) for all vertices x of H1. Unless stated otherwise, by a subgraph we mean an induced Hoffman subgraph. For a subset S of Vs(H), we denote by ((S)) h the subgraph of H induced on the set of vertices S U J Nf (x) Vxes Definition 2.3. Two Hoffman graphs H = (H, p) and H' = (H', p') are called isomorphic if there exists an isomorphism from H to H' which preserves the labeling. An ordinary graph H without labeling can be regarded as a Hoffman graph H = (H, p) without any fat vertex, i.e., p(x) = s for all vertices x. Such a graph is called a slim graph. Definition 2.4. For a Hoffman graph H, let A be its adjacency matrix, A = As C CT O in a labeling in which the fat vertices come last. Eigenvalues of H are the eigenvalues of the real symmetric matrix B(H) = As - CCT. Let Amin(H) denote the smallest eigenvalue of H. Definition 2.5 ([5]). For a Hoffman graph H and a positive integer n, a mapping ^ V (H) ^ Rn such that WxU(y)) 'm if x = y G Vs(H), 1 if x = y G Vf (H), 1 if x and y are adjacent in H, 0 otherwise, is called a representation of norm m. We denote by A(H,m) the lattice generated by j^(x) | x G V (H)}. Note that the isomorphism class of A(H, m) depends only on m, and is independent of justifying the notation. Definition 2.6. For a Hoffman graph H and a positive integer n, a mapping ^ : Vs(H) ^ Rn such that {m — | Nf (x)| if x = y, 1 — |Nf (x, y) | if x and y are adjacent, — |Nf (x, y)| otherwise. is called a reduced representation of norm m. We denote by Ared(H, m) the lattice generated by {^(x) | x G Vs(H)}. Note that the isomorphism class of Ared(H, m) depends only on m, and is independent of justifying the notation. While it is clear that a representation of norm m > 1 is an injective mapping, a reduced representation of norm m may not be. See Section 4 for more details. Lemma 2.7. Let H be a Hoffman graph having a representation of norm m. Then H has a reduced representation of norm m, and A(H, m) is isomorphic to Ared(H, m) © Z|Vf as a lattice. Proof. Let ^ : V (H) ^ Rn be a representation of norm m. Let U be the subspace of Rn generated by {^(x) | x G Vf (H)}. Let p, p^ denote the orthogonal projections from Rn onto U, Urespectively. Then p^ o ^ is a reduced representation of norm m, p^(A(H,m)) = Ared(H,m),andp(A(H,m)) ^ Z|Vf□ Theorem 2.8. For a Hoffman graph H, the following conditions are equivalent: (i) H has a representation of norm m, (ii) H has a reduced representation of norm m, (iii) Amin(H) > —m. Proof. From Lemma 2.7, we see that (i) implies (ii). Let ^ be a reduced representation of H of norm m. Then the matrix B(H) + ml is the Gram matrix of the image of and hence positive semidefinite. This implies that B(H) has smallest eigenvalue at least —m and hence Amin(H) > —m. This proves (ii) (iii). The proof of equivalence of (i) and (iii) can be found in [5, Theorem 3.2]. □ From Theorem 2.8, we obtain the following lemma: Lemma 2.9. If G is a subgraph of a Hoffman graph H, then Amin(G) > Amin(H) holds. Proof. Let m := -Amin(H). Then H has a representation ^ of norm m by Theorem 2.8. Restricting ^ to the vertices of G we obtain a representation of norm m of G, which implies Amin(G) > -m by Theorem 2.8. □ In particular, if r is the slim subgraph of H, then Amin(r) > Amin(H). Under a certain condition, equality holds in Lemma 2.9. To state this condition we need to introduce decompositions of Hoffman graphs. This was formulated first by the third author [4], although it was already implicit in [5]. Definition 2.10. Let H be a Hoffman graph, and let H1 and H2 be two non-empty induced Hoffman subgraphs of H. The Hoffman graph H is said to be the sum of H1 and H2, written as H = H1 W H2, if the following conditions are satisfied: (i) V (H) = V (H1) u V (H2); (ii) { Vs (H1 ), Vs (H2 )} is a partition of Vs (H ) ; (iii) if x G Vs(Hi), y G Vf (H) and x ~ y, then y G Vf (H®); (iv) if x G Vs(H1), y G Vs(H2), then x and y have at most one common fat neighbor, and they have one if and only if they are adjacent. If H = H1 W H2 for some non-empty subgraphs H1 and H2, then we call H decomposable. Otherwise H is called indecomposable. Clearly, a disconnected Hoffman graph is decomposable. It follows easily that the above-defined sum is associative and so that the sum n H = y h i=1 is well-defined. The main reason for defining the sum of Hoffman graphs is the following lemma. Lemma 2.11. Let H be a Hoffman graph and let H1 and H2 be two (non-empty) induced Hoffman subgraphs of H satisfying (i)—(iii) of Definition 2.10. Let ^ be a reduced representation of H of norm m. Then the following are equivalent. (i) H = H1 W H2, (ii) (^(x),^(y)) = 0 for any x G Vs(H1) and y G Vs(H2). Proof. This follows easily from the definitions of H = H1 W H2 and a reduced representation of norm m. □ Lemma 2.12. If H = H1 W H2, then Amin(H) = min{Amin(H1), Amin(H2)}. Proof. Let m = - min{Amin(H1), Amin(H2)}. In view of Lemma 2.9 we only need to show that Amin(H) > —m. By Theorem 2.8, H® has a reduced representation ^ : V (H®) ^ Rni of norm m, for each i = 1,2. Define ^ : V (H) ^ Rni 0 Rn2 by ^(x) = ^1(x) 0 0 if x G V (H1), ^(x) =0 © ^2(x) otherwise. It is easy to check that ^ is a reduced representation of norm m, and the result then follows from Theorem 2.8. □ 2.2 Hoffman's limit theorem In this subsection, we state and prove Hoffman's limit theorem (Theorem 2.14) using the concept of Hoffman graphs. Lemma 2.13. Let G be a Hoffman graph whose vertex set is partitioned as Vi U V2 U V3 such that (i) V U V3 c VS(G), (ii) there are no edges between Vi and V3, (iii) every pair of vertices x G V2 and y G V3 are adjacent, (iv) V3 is a clique. Let H be the Hoffman graph with the set of vertices Vi U V2 together with a fat vertex f G V (G) adjacent to all the vertices of V2. If G has a representation of norm m, then H has a representation of norm , (m - 1)|V2| m + |Vs| + m - 1 • Proof. Let ^ : V (G) ^ Rd be a representation of norm m, and let fPi' P = I P2 be the | V(G) | x d matrix whose rows are the images of V(G) = Vi U V2 U V3 under Set u = — \ A ò(x) V|V3|(|V3| + m - 1) J? «i = 1 -I V| |Vs| + m - 1' m - 1 «2 : _ |Vs| + m - 1" Let j denote the row vector of length | V21 all of whose entries are 1. Then uuT = 1, (2.1) PiuT = 0, (2.2) P2UT = (1 - «i )jT, (2.3) «2 = 2«i - «2. (2.4) Fix an orientation of the complete digraph on V2, and let B be the |V2| x matrix defined by Ba,(ß,y) = Saß - SaY (a, ß, 7 G V2 , ß = 7)• Then BBt = |V2|I - J. (2.5) We now construct the desired representation of H, as the row vectors of the matrix Pi £2VW\I 0 D = ( P2 + £ijTu 0 £25 u 0 0 Then, using (2.1)-(2.5), we find Pi £2 y/\V2\I 0 P2 + £1 jT u 0 £25 u 0 0 DDT PT £2VW\I 0 PT + £iuTj uTN 0 £2B T 'pipT + £2\V2\I P2PT 0 fPi pT + £2\v2\i PiPT P2PT + (2£ 1 - £2 )J + £2(\V2\I - J) j Pi PT P2PT P2P2T + £2 \V2\1 jT 2P1 0 'pipT pìp2t -1p1 P1P2 P2PT P2PT 0 2P2 j I00 r i + £2\V2\ (0 i 0 1 / \0 0 0> 0 T1 Therefore, the row vectors of D define a representation of norm m + £2 \ V2 \ of the Hoffman graph H. □ Theorem 2.14 (Hoffman). Let H be a Hoffman graph, and let /2 ,...,/ G Vf (H). Let gni,...,nfc be the Hoffman graph obtained from H by replacing each / by a slim n-clique Kj, and joining all the neighbors of / with all the vertices of Kj by edges. Then Amin(©ni—nk) > Amin(H), (2.6) and lim Amin ( Gni ,...,nk )= Amin(H). ni (2.7) Proof. We prove the assertions by induction on k. First suppose k = 1. Let Mn = -Amin(©n). Let Hn denote the Hoffman graph obtained from H by attaching a slim n-clique K to the fat vertex /1, joining all the neighbors of /1 and all the vertices of K by edges. Then Hn contains both H and ©n as subgraphs, and Hn = H W H', where H' is the subgraph induced on K U {/}. Since Amin(H') = —1, Lemma 2.12 implies Amin(H) = Amin(Hn) < Amin(©"0 = -Mn. Thus (2.6) holds for k = 1. Since n is arbitrary and {-m^^^ is decreasing, we see that limn^ro Mn exists and Amin(H) < - lim Mn. (2.8) Since ©n has a representation of norm Mn, it follows from Lemma 2.13 that H has a representation of norm (Mn - 1)\nh(/)\ Mn +--:-:-. n + Mn - 1 0 1 j 0 1 j By Theorem 2.8, we have A (H) . (ßn - l)|Nn(f)| Amin (H ) > -ßn--;-;-, n + ßn - l which implies Amin (H) > - lim ßn. (2.9) n—^to> Combining (2.9) with (2.8), we conclude that (2.7) holds for k = l. Next, suppose k > 2. Let ©ni , . ,nk—i be the Hoffman graph obtained from H by replacing each fi (l < i < k - 1)bya slim nž-clique Ki, and joining all the neighbors of fi with all the vertices of Kl by edges. Then ©ni,...,nk is obtained from ©ni, . ,nk—i by replacing fk by a slim nk-clique Kk, and joining all the neighbors of fk with all the vertices of Kk by edges. Then it follows from the assertions for k =1 that Amin(©ni,...,nk ) > Amin(©ni,...,nk—i ), (2.10) and lim Amin(©ni,...,nk ) = Amin(©ni,...,nk—i ). nk—TO This means that, for any e > 0, there exists N1 such that nk > Ni 0 < Amin(Gni,...,nk) - Amin(©ni,...,nk—i) < e. By induction, we have Amin (©ni—nk —i ) > Amin(H), (2.11) and lim Amin (©ni,...,nk —i ) = Amin(H). (2.12) ni ,...,nk—i —to Combining (2.10) with (2.11), we obtain (2.6), while (2.11) and (2.12) imply that there exists N0 such that ni, . . . , nk-1 > No 0 < Amin (©ni,...,nk —i ) - Amin(H) < e. Setting N = max{N0, N1}, we see that ni,...,Uk > N 0 < Amin(©ni,...,nk ) - Amin(H) < 2e. This establishes (2.7). □ Corollary 2.15. Let H be a Hoffman graph. Let rn be the slim graph obtained from H by replacing every fat vertex f of H by a slim n-clique K (f ), and joining all the neighbors of f with all the vertices of K (f ) by edges. Then Amin (rn ) > Amin (H), and lim Amin(rn) = Amin(H). In particular, for any e > 0, there exists a natural number n such that, every slim graph A containing rn as an induced subgraph satisfies Amin (A) < Amin(H) + e. Proof. Immediate from Theorem 2.14. □ 3 Special graphs of Hoffman graphs Definition 3.1. Let ^ be a real number with ^ < — 1 and let H be a Hoffman graph with smallest eigenvalue at least Then H is called saturated if no fat vertex can be attached to H in such a way that the resulting graph has smallest eigenvalue at least Proposition 3.2. Let ^ be a real number, and let h be a family of indecomposable fat Hoffman graphs with smallest eigenvalue at least ^. The following statements are equivalent: (i) every fat Hoffman graph with smallest eigenvalue at least ^ is a subgraph of a graph H = l±| ! l=1 H® such that H® is a member of h for all i — 1,... ,n. (ii) every saturated indecomposable fat Hoffman graph is isomorphic to a subgraph of a member of h. Proof. First suppose (i) holds, and let H be a ^-saturated indecomposable fat Hoffman graph. Then H is a fat Hoffman graph with smallest eigenvalue at least hence H is a subgraph of H' — 1+1n=1 H®, where H® is a member of H for i — 1,..., n. Since H is ^-saturated, it coincides with the subgraph ((Vs (H)))h' of H'. Since H is indecomposable, this implies that H is a subgraph of H® for some i. Next suppose (ii) holds, and let H be a fat Hoffman graph with smallest eigenvalue at least Without loss of generality we may assume that H is indecomposable and saturated. Then H is isomorphic to a subgraph of a member of H, hence (i) holds. □ Definition 3.3. For a Hoffman graph H, we define the following three graphs S-(H), S+(H) and S (H) as follows: For e G {-, +} define the special e-graph Se — (Vs(H),Ee) as follows: the set of edges Ee consists of pairs {s1, s2} of distinct slim vertices such that sgn(^(s1), ^(s2)) — e, where ^ is a reduced representation of H of norm m. The graph S (H) :— S +(H) US- (H) — (Vs(H),E- U E+) is the special graph of H. Note that the definition of the special graph S (H) is independent of the choice of the norm m of a reduced representation It is easy to determine whether a Hoffman graph H is decomposable or not. Lemma 3.4. Let H be a Hoffman graph. Let Vs(H) — V1 U V2 be a partition, and set H® — ((V®))h for i — 1, 2. Then H — H1 W H2 if and only if there are no edges connecting V1 and V2 in S (H). In particular, H is indecomposable if and only if S (H) is connected. Proof. This is immediate from Definition 2.10(iv) and Definition 3.3. □ A A H(Amin = -1 H( 2), Amin = -2 H(3), Amin = -3 Figure 1. For an integer t > 1, let H( t) be the fat Hoffman graph with one slim vertex and t fat vertices. Lemma 3.5. Let t be a positive integer. If H is a fat Hoffman graph with Amin(H) > —t containing H(t) as a Hoffman subgraph, then H = H(t) W H' for some subgraph H' of H. In particular, if H is indecomposable, then H = H(t). Proof. Let x be the unique slim vertex of H(t). Let ^ be a reduced representation of norm t of H. Then ^(x) = 0, hence x is an isolated vertex in S (H). Thus H = H(t) W ((VS(H) \ |x}))h by Lemma 3.4. □ Lemma 3.6. Let H be a fat Hoffman graph with smallest eigenvalue at least —3. Let ^ be a reduced representation of norm 3 of H. Then for any distinct slim vertices x, y of H, (V>(x),V(y)) g{1,0, -1}. Proof. Since H is fat, we have (^(x),^(x)) < 2 for all x G Vs(H). Thus |(^(x), ^(y))| < 2 for all x, y G Vs(H) by Schwarz's inequality. Equality holds only if ^(x) = ±^(y) and (^(x),^(x)) = 2. The latter condition implies |N" (x)| = 1, hence |N" (x,y)| < 1. Thus (^(x), ^(y)) > -1, while (^(x), ^(y)) = 2 cannot occur unless x = y, by Definition 2.6. Therefore, | (^ (x), ^ (y )) | < 2, and the result follows. □ Let H be a fat Hoffman graph with smallest eigenvalue at least -3. Then by Lemma 3.6, the edge set of the special graph Se(H) is {{x,y} | x, y G Vs(H), (^(x),^(y)) = e1}, for e G {+, -}. Theorem 3.7. Let H be a fat indecomposable Hoffman graph with smallest eigenvalue at least -3. Then every slim vertex has at most three fat neighbors. Moreover, the following statements hold: (i) If some slim vertex has three fat neighbors, then H = H(3). (ii) If no slim vertex has three fat neighbors and some slim vertex has exactly two fat neighbors, then Ared(H, 3) ~ Zn for some positive integer n. (iii) If every slim vertex has a unique fat neighbor, then Ared(H, 3) is an irreducible root lattice. Proof. As the smallest eigenvalue is at least -3, every slim vertex has at most three fat neighbors. If |Nf (x)| = 3 for some slim vertex x of H, then H contains H(3) as a subgraph. Thus H = H(3) by Lemma 3.5, and (i) holds. Hence we may assume that |Nf (x)| < 2 for all x G Vs(H). Then for each x G Vs(H) we have ||^(x)||2 = 1 (resp. 2) if and only if |Nf (x)| = 2 (resp. |Nf (x)| = 1). Suppose that Ared(H, 3) contains m linearly independent vectors of norm 1. We claim that Ared(H, 3) can be written as an orthogonal direct sum Zm © A', where A' is a lattice containing no vectors of norm 1. Indeed, if x is a slim vertex such that ||^(x) ||2 = 2 and ^(x) G Zm, then ^(x) is orthogonal to Zm. This implies Ared(H, 3) = Zm © A' and ^(Vs(H)) C Zm U A'. If m > 0 and A' = 0, then the special graph S (H) is disconnected. This contradicts the indecomposability of H by Lemma 3.4. Therefore, we have either m = 0 or A' = 0. In the latter case, (ii) holds. In the former case, Ared(H, 3) = A' is generated by vectors of norm 2, hence it is a root lattice. Again by the assumption and Lemma 3.4, (iii) holds. □ We shall see some examples for the case (ii) of Theorem 3.7 in the next section. As for the case (iii), Ared(H, 3) is an irreducible root lattice of type An, Dn or En. If Ared(H, 3) is not an irreducible root lattice of type En, then it can be imbedded into the standard lattice, hence the results of the next section applies. On the other hand, if Ared(H, 3) is an irreducible root lattice of type En, then it is contained in the irreducible root lattice of type E8, and hence there are only finitely many possibilities. For example, Let r be any ordinary graph with smallest eigenvalue at least -2 (see [1] for a description of such graphs). Attaching a fat neighbor to each vertex of r gives a fat Hoffman graph with smallest eigenvalue at least -3. However, this Hoffman graph may not be maximal among fat Hoffman graphs with smallest eigenvalue at least -3. Therefore, we aim to classify fat Hoffman graphs with smallest eigenvalue at least -3 which are maximal in E8. This may seem a computer enumeration problem, but it is harder than it looks. Example 3.8. Let n denote the root system of type E8. Fix a G n. Then there are elements ßi G n (i = 1,..., 28) such that 28 {ß G n | (a, ß) = 1} = J {ßi, a - ßi}. i=1 Let V denote the set of 57 roots consisting of the above set and a. Then V is a reduced representation of a fat Hoffman graph H with 29 fat vertices. The fat vertices of H are fi (i = 0,1,..., 28), f0 is adjacent to a, and fi is adjacent to ßi, a - ßi (i = 1,..., 28). It turns out that H is maximal among fat Hoffman graphs with smallest eigenvalue at least -3. Indeed, no fat vertex can be attached, since the root lattice of type E8 is generated by V \ {y} for any y G V, and attaching another fat neighbor to y would mean the existence of a vector of norm 1 in the dual lattice Eg of E8. Since Eg = E8, there are no vectors of norm 1 in Eg. This is a contradiction. If a slim vertex can be attached, then it can be represented by S G n with (a, S) = 0. Then there exists i G {1,..., 28} such that (ßi, S) = ±1. Exchanging ßi with a - ßi if necessary, we may assume (ßi, S) = -1. This implies that ßi and S have a common fat neighbor. Since (ßi, a - ßi) = -1, ßi and a - ßi have a common fat neighbor. Since ßi has a unique fat neighbor, S and a - ßi have a common fat neighbor, contradicting (S, a - ßi) = 1. On the other hand, it is known that there is a slim graph r with 36 vertices represented by the root system of type E8 (see [1]). Attaching a fat neighbor to each vertex of r gives a fat Hoffman graph H' with smallest eigenvalue —3 such that Ared(H', 3) is isometric to the root lattice of type E8. The graph H' is not contained in H, so there seems a large number of maximal fat Hoffman graphs represented in the root lattice of type E8. 4 Integrally represented Hoffman graphs In this section, we consider a fat (—3)-saturated Hoffman graph H such that Ared(H, 3) is a sublattice of the standard lattice Zn. Since any of the exceptional root lattices E6, E7 and E8 cannot be embedded into the standard lattice (see [2]), this means that, in view of Theorem 3.7, Ared(H, 3) is isometric to Zn or a root lattice of type An or Dn. Note that Ared(H, 3) cannot be isometric to the lattice A1, since this would imply that H has a unique slim vertex with a unique fat neighbor, contradicting (—3)-saturatedness. The following example gives a fat (—3)-saturated graph H with Ared(H, 3) = Z. Example 4.1. Let H be the Hoffman graph with vertex set Vs(H) U Vf (H), where Vs(H) = Z/4Z, Vf (H) = {f | i G Z/4Z}, and with edge set {{0, 2}, {1, 3}} U {{i, fj} | i = j or j + 1}. Then H is a fat indecomposable (—3)-saturated Hoffman graph such that Ared(H, 3) is isomorphic to the standard lattice Z. Since S-(H) has edge set {{i, i + 1} | i G Z/4Z}, S- (H) is isomorphic to the Dynkin graph A3. Theorem 4.9 below implies that H is maximal, in the sense that no fat indecomposable (—3)-saturated Hoffman graph contains H. For the remainder of this section, we let H be a fat indecomposable (—3)-saturated Hoffman graph such that Ared(H, 3) is isomorphic to a sublattice of the standard lattice Zn. Let ^ be a representation of norm 3 of H. Then we may assume that ^ is a mapping from V(H) to Zn 0 ZVf (H), where its composition with the projection Zn ©ZVf (H) ^ Zn gives a reduced representation ^ : Vs(H) ^ Zn. It follows from the definition of a representation of norm 3 that = ^(S)+ ^ ef, f (s) n ^(s) = E ^(s)j ej, j=1 V>(s)j G{0, ±1}, and |{j | j G {1,. .. ,n}, V(s)j G {±1}}| = 3 — |nH(s)| < 2. (4.1) Lemma 4.2. If i G {1,...,n} and ^(s)® = 0 for some s G Vs(H), then there exist s1, s2 G Vs(H) such that ^(s^® = — ^(s2)® = 1. Proof. By way of contradiction, we may assume without loss of generality that i = n, and ^(s)n G {0,1} for all s G Vs(H). Let G be the Hoffman graph obtained from H by attaching anew fat vertex g and join it to all the slim vertices s of H satisfying ^(s)n = 1. Then the composition of ^ : Vs(H) = Vs (G) ^ Zn and the projection Zn ^ Zn-1 gives a reduced representation of norm 3 of G. By Theorem 2.8, G has smallest eigenvalue at least —3. This contradicts the assumption that H is (—3)-saturated. □ Proposition 4.3. The graph S (H) is connected. Proof. Before proving the proposition, we first show the following claim. Claim 4.4. Let s1 and s2 be two slim vertices such that ^(si)j = 1 and ^(s2)j = —1 for some i G {1,..., n}. Then the distance between s1 and s2 is at most 2 in S-(H). By (4.1), we have (^(s1), ^(s2)) G {0, —1}. If (^(s1), ^(s2)) = —1, then s1 and s2 are adjacent in S- (H ) by the definition, hence the distance equals one. If ( ^(s1),^(s2)) = 0, then there exists a unique j G {1,..., n} such that ^(s1)j = ^(s2)j = ±1. From Lemma 4.2, there exists a slim vertex s3 such that ^(s3)j = —^(s1)j-. If ^(s3)j = 0, then (^(sq), -0(s3)) G {±2} for some q G {1, 2}, which is a contradiction. Hence ^(s3)j = 0. This implies that (^(sj),^(s3)) = —1 for i = 1,2, or equivalently, s3 is a common neighbor of s1 and s2 in S-(H). This shows the claim. Since H is indecomposable, S (H) is connected by Lemma 3.4. Thus, in order to show the proposition, we only need to show that slim vertices s1 and s2 with (^(s1), ^(s2)) = 1 are connected by a path in S-(H). There exists i G {1,... ,n} such that ^(s1)i = ^(s2)j = ±1. From Lemma 4.2, there exists a slim vertex s3 such that ^(s3)j = — ^(s1 )j, and hence the distances between s 1 and s3 and between s3 and s2 are at most 2 in S- (H ) by Claim 4.4. Therefore, s 1 and s2 are connected by a path of length at most 4 in S- (H ). □ Lemma 4.5. Let H be a fat indecomposable (—3)-saturated Hoffman graph. Then the reduced representation of norm 3 of H is injective unless H is isomorphic to a subgraph of the graph given in Example 4.1. Proof. Suppose that the reduced representation ^ of norm 3 of H is not injective. Then there are two distinct slim vertices x and y satisfing ^(x) = ^(y). Then (^(x), ^(y)) = 0 or 1 . If (^(x), -0(y)) = 0, then ^(x) = ^(y) = 0, hence both x and y are isolated vertices, contradicting the assumption that H is indecomposable. Suppose (^(x), -0(y)) = 1. Since S-(H) is connected by Proposition 4.3, there exists a slim vertex z such that (^(x), ^(z)) = —1. Then we may assume ^(x) = e1 + ef + ef2, ^(y) = e1 + ef 3 + e f 4, ^(z) = —e1 + efi + ef3. If H has another slim vertex w, then = —e1 + e f2 + e f4 , and no other possibility occurs. Therefore, H is isomorphic to either the graph given in Example 4.1, or its subgraph obtained by deleting one slim vertex. □ Lemma 4.6. Suppose that s G Vs (H) has exactly two fat neighbors in H. Then the following statements hold. (i) for each f G Nf (s), |N5-(j5 )(s) n Ng(f )| < 2, (ii) |NS- (h )(s)| < 4, and if equality holds, then S -(H) is isomorphic to the graph D 4, (iii) if | NS— (h) (s) I = 3, then two of the vertices in Ns—(/)(s) have s as their unique neighbor in S- (H). Proof. Let ^ be the reduced representation of norm 3 of H. Since s has exactly two fat neighbors, (^(s), ^(s)) = 1. This means that we may assume without loss of generality ^(s) = ei. Let f G N/ (s). If t i and t2 are distinct vertices of Ns—(H)(s) n NH(f),then 1 > (0(ti),0(t2)) > (^(ti)+ ef ,V(*2)+ e/) = (^(ti ),^(t2)) + 1, Thus we have (^(ti), ^(t2)) < 0. Since ti,t2 G Ns— (H)(s), we have (^(s),^(ti)) = (^(s), ^(t2)) = -1, and hence we may assume without loss of generality that V>(ti) = -ei + e2, (4.2) V>(t2) = -ei - &2. (4.3) Now it is clear that there cannot be another t3 g Ns— (h)(s). Thus |NS— (H)(s) n NH (f)| < 2. This proves (i). As for (ii), let N/(s) = {f, f '}. Then |Ns—(H)(s)| < |Ns—(H)(s) n N/ (f )| + |Ns—(H)(s) n N/ (f ')| < 4 by (i). To prove (iii) and the second part of (ii), we may assume that {ti, t2} — Ns—(/)(s)n N/(f ). We claim that neither ti nor t2 has a neighbor in S-(H) other than s. Suppose by contradiction, that t3 = s is a neighbor in S-(H) of ti. By (4.2) (resp. (4.3)), f is the unique fat neighbor of ti (resp. t2). In particular, f is also a neighbor of t3. Observe 1 > (^(s),^(ts)) > Ms),V(ts)) + 1, 1 > (^(ti), ^(ts)) = (^(ti), ^(ts)) + 1 (i =1, 2). Thus (ei,V(ts)) < 0, (-ei ± e2,^(ts)) < 0. These imply (ei,V(ts)) = (e2,V(ts)) = 0. But then -1 = (V>(ti), V(ts)) = (-ei + e2, ^(ts)) = 0. This is a contradiction, and we have proved our claim. Now (iii) is an immediate consequence of this claim. Continuing the proof of the second part of (ii), if |NS— (/) (s) | =4, then we may assume ^(t'i) = -ei + es, ^(t2) = -ei - es, where {ti,t2} = Ns— (/)(s) n N/(f'). It follows that ti,t2,t'i,t2 are pairwise non-adjacent in S-(H). By our claim, none of ti,t2,t'i,t2 is adjacent to any vertex other than s in S-(H). Since S-(H) is connected by Proposition 4.3, S-(H) is isomorphic to the graph D 4. □ Lemma 4.7. Suppose that slim vertices s,t+,t share a common fat neighbor and that they are represented as ^(s) = ei + e2, ^(t±) = -ei ± e3. If there exists a slim vertex t with ^(t) = -e2 + e j for some j G {1, 2, 3}, then the vertices t± have s as their unique neighbor in S- (H). Proof. Note that each of the vertices s,t±,t has a unique fat neighbor. Since (^(s), ^(t± )) = (^(s), ^(t)) = -1, these vertices share a common fat neighbor f. Suppose that there exists a slim vertex t' adjacent to t- in S-(H). This means (^(t'), ^(t-)) = -1. Since f is the unique fat neighbor of t-, t' is adjacent to f, and hence (^(t'), ^(u)) < 0 for u G {t,t+, s}. This is impossible. Similarly, there exists no slim vertex adjacent to t+ in S-(H). □ Lemma 4.8. Suppose that s G Vs(H) has exactly one fat neighbor in H. Then the following statements hold: (i) |NS- (h) (s) | < 4, and if equality holds, then S -(H) is isomorphic to the graph D 4, (ii) if |Ns-(h)(s)| = 3, then two of the vertices in Ns-(h)(s) have s as their unique neighbor in S-(H). Proof. Let ^ be the reduced representation of norm 3 of H. Since s has exactly one fat neighbor, (^(s), ^(s)) = 2. This means that we may assume without loss of generality ^(s) = ei + e2. Let f be the unique fat neighbor of s. If t G NS- (H)(s), then t is adjacent to f, hence ^(t) G {-ei, -e2} U {-e, ± ej | 1 < i < 2 < j < n}. (4.4) If t,t' G NS-(H)(s) are distinct, then 1 > (^(t),^(t')) > (^(t)+ ef ,V(t') + ef ) = (V>(t),v(t')) + 1, Thus we have (^(t),^(t')) < 0. If |NS-(h)(s)| > 3, then by (4.4), we may assume without loss of generality that there exists t g Ns-(h)(s) with ^(t) = -ei + e3. Then ^(NS- (h) (s) \ {t}) is contained in {-e2 - e3}, {-e2, -ei - e3}, or {-ei - e3} U {-e2 ± ej} for some j with 3 < j < n. Thus |NS-(H)(s)| < 4, and equality holds only if ^(Ns-(h) (s)) = {-ei ± e3, -e2 ± ej} for some j with 3 < j < n. In this case, Lemma 4.7 implies that each of the vertices in NS-(H)(s) has s as a unique neighbor. This means that S- (H) contains a subgraph isomorphic to the graph D4 as a connected component. Since S-(H) is connected by Proposition 4.3, we have the desired result. To prove (ii), suppose |Ns-(h)(s)| — 3. Then we may assume without loss of generality that NS-(H)(s) — {t,t+,t-}, where ^(t±) — —e2 ± e4. Then by Lemma 4.7, the vertices t± have s as their unique neighbor in S- (H ). □ Theorem 4.9. Let H be a fat indecomposable (-3)-saturated Hoffman graph such that Ared(H, 3) is isomorphic to a sublattice of the standard lattice zn. Then S -(H) is a connected graph which is isomorphic to Am, Dm, Am or Dm for some positive integer m. Proof. From Proposition 4.3, S-(H) is connected. First we suppose that the maximum degree of S-(H) is at most 2. Then S-(H) — Am or S-(H) = Am for some positive integer m. Next we suppose that the degree of some vertex s in S-(H) is at least 3. From Lemma 4.6(ii) and Lemma 4.8(i), degS-(H)(s) < 4, and S -(H) = D 4 if degS-(H)(s) — 4. Thus, for the remainder of this proof, we suppose that the maximum degree of S- (H) is 3 and degs-(H) (s) — 3. It follows from Lemma 3.5 that if H has a subgraph isomorphic to H(3), then H — H(3), in which case S- (H) consists of a single vertex, and the assertion holds. Hence it remains to consider two cases: s is adjacent to exactly two fat vertices, and s is adjacent to exactly one fat vertex. In either cases, by Lemma 4.6(iii) or Lemma 4.8(ii), s has at most one neighbor t with degree greater than 1. Thus, the only way to extend this graph is by adding a slim neighbor adjacent to t. We can continue this process, but once we encounter a vertex of degree 3, then the process stops by Lemma 4.6(iii) or Lemma 4.8(ii). Thus, S-(H) is isomorphic to one of the graphs Dm or Dm. □ Example 4.10. Let n1,... ,nk be positive integers satisfying n® > 2 for 1 < i < k. Set mj — J2j=1 n® and £j — mj — j for j — 0,1,..., k. Let H be the Hoffman graph with Vs(H) — {v® | i — 0,1,..., mfc}, Vf (H) — {fj | j — 0,1,..., k + 1}, and E(H) — {{v®, v®'} | 1 < j < k, mj-1 < i + 1 < i' < mj} U {{v m j-1, Vmj + 1} | 1 < j < k} U {{fj,v®} | 1 < j < k, mj-1 < i < mj} U {{fo,vo}, {ffc+1,vmfc}}. Then H is a fat Hoffman graph with smallest eigenvalue at least —3, and S- (H) is isomorphic to the Dynkin graph Amfc+1. Indeed, H has a reduced representation ^ of norm 3 defined by ^(v®) = ( —1)jej if i — mj, 0 < j < k, ( —1)j (e®-j — e®-j-1) if mj < i < mj+1, 0 < j < k. Moreover, H is (—3)-saturated. Indeed, suppose not, and let H be a Hoffman graph obtained by attaching a new fat vertex f to H, and let ^ be a reduced representation of norm 3 of H. If f is adjacent to vmj for some j G {0,1,..., k}, then vmj has three fat neighbors in H, hence V'(vmj ) — 0. This is absurd, since (V'(vmj ), i/>(vmj±1)) — —1. If f is adjacent to v® with mj-1 < i < mj, then ^(v®)! — 1. Since (V>(v®-1), ^(v®)) — (^(v®^), ^(v®)) — -1 while (^/j(vi-1), V>(vi+1)) = 0, we may assume ^A(vi±1) = e1 ± e2, V'(vi) = -e1. Then i +1 < m j, and 0 = (i//(vi+2),i//(vi_1)) = (^(vi+2), e1 - e2) = (^(vi+2), 2e1 - (e1 + e2)) = 2(^//(vi+2),V'(vi)) - (lMvi+2),V'(vi+1)) = 1, which is absurd. We note that the graph S +(H) has the following edges: {{vm,--1,vm.+1} | 1 < j < k}. Example 4.11. Let H be the Hoffman graph constructed in Example 4.10 by setting n1 = 1, n2 = 2, and n3 = 1. Let Ho (resp. H1) be the Hoffman graph obtained from H by identifying the fat vertices f4 and f0 (resp. f4 and f1), and adding edges {v0, v2}, {v2, v4}. Then H0 and H1 are fat (-3)-saturated Hoffman graphs and S-(Hi) is isomorphic to the Dynkin graph A5 for i = 0,1. We note that S +(Hi) has two edges {v0, v2}, {v2, v4}. Examples 4.10 and 4.11 indicate that H is not determined by S± (H ). We plan to discuss the classification of fat indecomposable (-3)-saturated Hoffman graphs with prescribed special graph in subsequent papers. References [1] D. Cvetkovic, P. Rowlinson and S. K. Simic, Spectral Generalizations of Line Graphs — On Graphs with Least Eigenvalue — 2, Cambridge Univ. Press, 2004. [2] W. Ebeling, Lattices and Codes, Vieweg, 2nd ed. Friedr. Vieweg & Sohn, Braunschweig, 2002. [3] A. J. Hoffman, On graphs whose least eigenvalue exceeds —1 — \[2, Linear Algebra Appl. 16 (1977), 153-165. [4] T. Taniguchi, On graphs with the smallest eigenvalue at least — 1 — \[2, part I, Ars Math. Contemp. 1 (2008), 81-98. [5] R. Woo and A. Neumaier, On graphs whose smallest eigenvalue is at least —1 — \[2, Linear Algebra Appl. 226-228 (1995), 577-591 . [6] H. Yu, On the limit points of the smallest eigenvalues of regular graphs, Designs, Codes Cryp-togr. 65 (2012), 77-88. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 123-140 Convex cycle bases Marc Hellmuth Center for Bioinformatics, Saarland University Building E 2.1, D-66041 Saarbrücken, Germany Josef Leydold * Institute for Statistics and Mathematics, WU, Augasse 2-6, A-1090 Wien, Austria Peter F. Stadler Bioinformatics Group, Department of Computer Science; and Interdisciplinary Center of Bioinformatics, University of Leipzig, Härtelstraße 16-18, D-04107 Leipzig, Germany Max-Planck-Institute for Mathematics in the Sciences Inselstraße 22, D-04103 Leipzig, Germany Fraunhofer Institut f. Zelltherapie und Immunologie Perlickstraße 1, D-04103 Leipzig, Germany Inst. f. Theoretical Chemistry, University of Vienna Wahringerstraße 17, A-1090 Wien, Austria Center for non-coding RNA in Technology and Health, University of Copenhagen Gr0nnegardsvej 3, DK-1870 Frederiksberg, Denmark Santa Fe Institute, 1399 Hyde ParkRd., Santa Fe, NM 87501, USA Received 29 August 2011, accepted 30 January 2013, published online 4 April 2013 Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles bases that includes partial cubes and hence median graphs. Keywords: cycle basis, convex subgraph, isometric subgraph, Cartesian product, partial cubes Math. Subj. Class.: 05C15, 05C10 * corresponding author E-mail addresses: marc.hellmuth@bioinf.uni-sb.de (Marc Hellmuth), josef.leydold@wu.ac.at (Josef Leydold), studla@bioinf.uni-leipzig.de (Peter F. Stadler) Abstract 1 Introduction and basics The cycle space C (G) of a simple, unweighted, undirected graph G = (V, E) consists of all its Eulerian subgraphs (or generalized cycles), i.e., all the subgraphs of G for which every vertex has even degree. It is convenient in this context to interpret subgraphs of G as edge sets. The generalized cycles form a vector space over GF(2) with vector addition X e Y := (X U Y) \ (X n Y) and scalar multiplication 1 • X = X, 0 • X = 0, for X,Y e C (G). This vector space is generated by the elementary cycles of G, i.e., the connected subgraphs of G for which every vertex has degree 2. A basis B of the cycle space C is called a cycle basis of G = (V, E) [9]. The dimension of the cycle space is the cyclomatic number ^(G) (or first Betti number). For a connected graph we have ^(G) = |E | — |V | + 1. Notice that the cycle space of a graph is the direct sum of the cycle spaces of its 2-connected components. Cycle bases of graphs have diverse applications in science and engineering. Examples include structural flexibility analysis [27], electrical networks [11], chemical structure storage and retrieval systems [15], scheduling problems [36], graph drawing [33], andbiopoly-mer structures [34, 35]. Surveys and extensive references can be found in [19, 22, 28, 37]. A convexity space (V, C) [6] consists of a ground set V and a set C of subsets of V satisfying (C1) 0 e C, V e C, and (C2) K ', K " e C implies K ' n K " e C. For a simple, undirected graph G with vertex set V, every set p of paths on G defines a convexity space (V, C(p)) in the following way: A set of vertices K is p-convex, K e C(P), if and only if, for every path P e P with both end vertices in K, all vertices of P are contained in K. This construction is discussed in detail in [14]. Several special types of paths p have been studied in this context, most prominently the set of all paths [5], the set of all triangle paths [8], the set of all induced paths [13], and the set of all shortest paths [39]. We will be concerned here only with the latter definition of convexity, usually known as geodetic convexity, see Section 2 for a formal definition. Geodetically convex cycles play an important role in the theory of Cartesian graphs products and their isometric subgraphs. The absence of convex cycles longer than 4, for example, characterizes semi-median graphs [3]. Such long convex cycles furthermore play a role e.g. in Euler-type inequalities for partial cubes [31]. It appears natural, hence, to investigate whether there is a connection between the cycle space and the (geodetic) convexity space of a graph G = (V, E). Note that the cycle space is defined on the edge set, while the convexity space is defined on the vertex set. Intuitively, this connection is made possible by the fact that induced elementary cycles in G are characterized by either their vertex sets or their edge sets. Definition 1.1. A convex cycle basis of a graph G is a cycle basis that consists of convex elementary cycles. We briefly consider a generalized definition of convex cycle bases relaxing the requirement for elementary basis cycles in the final section. Cycle bases with special properties have been investigated in much detail in the literature. Examples include minimum cycle bases [2, 17, 19, 29, 44], (strictly) fundamental cycle bases [20, 32, 38], or (quasi) robust cycle bases [26, 40]. Here, we consider convex cycle bases. We show that convex cycle bases are not related to other types of cycle bases, we introduce a polynomial-time algorithm to compute a convex cycle basis for an arbitrary input graph, and we construct a class of graphs with convex cycles bases by means of Cartesian products that in particular includes partial cubes. 2 Geodetic convexity and characterization of convex cycles For a graph G we denote the vertex set and edge set of G by V(G) and E(G), respectively. Similarly, we write C(G) for the cycle space of G. An edge that joins vertices x and y is denoted by the unordered pair {x, y}. The lengths |P| and |C| of a path P and a cycle C in G, respectively, is the number of their edges. For simplicity, we will refer to a path with end vertices u and v as uv-path. The distance distG(u, v) between two vertices u and v of G is the length of a shortest uv-path. It is well known that this distance forms a metric on V. The set of all shortest uv-paths will be denoted by PG [u, v]. The cardinality of this set, i.e., the number of shortest uv-paths, will be denoted by Suv = |PG [u, v] |. A modification of Dijkstra's algorithm computing both the distance matrix of G and the matrix S is given in the appendix. A subgraph H of G is isometric if distH(u, v) = distG(u,v) holds for all u,v G V (H ). We say that H is a (geodetically) convex subgraph of G if for all u, v G V (H ), all shortest uv-paths P G PG[u, v] are contained in H. In the following, convex will always mean geodetically convex. The empty subgraph will be considered as convex. The intersection of convex subgraphs of G is again a convex subgraph of G [42]. Since H is an isometric subgraph of G if and only if H contains at least one P G PG[u, v] for every pair u, v G V (H ), we see that convex implies isometric. Furthermore, if H is an isometric subgraph of G, it is in particular an induced subgraph of G. Finally, the connectedness of G implies that all its isometric subgraphs are connected. Our first result characterizes elementary convex cycles. Lemma 2.1. Let C be an elementary cycle of G. If |C | is odd, then C is convex if and only if for every edge e = {x, y} in C there is a unique vertex z in C such that distG(x, z) = distG(y, z) = (|C| — 1)/2 and Sxz = Syz = 1. If |C | is even, then C is convex if and only if for every edge e = {x, y} in C there is a unique edge h = {u, v} in C such that (i) distG(x,u) = distG(y, v) = |C|/2 — 1, (ii) distG(x, v) = distG(y, u) = |C|/2, (iii) Sxu = Syv = 1, and (iv) Sxv = Syu = 2. Proof. Suppose C is convex. Consider two vertices p and q in C with distG (p, q) < | C| /2. If C is convex, then the unique shortest path between p and q must run along C, so that Spq = 1. Clearly, this condition characterizes convex cycles provided C is odd. The situation is more complicated for even cycles. Let us first suppose that C is convex and fix an arbitrary edge {x, y}. In an even elementary cycle there is a unique edge h = {u, v} satisfying (i) distC(x, u) = distC(y, v) = |C|/2 — 1, (ii) distC(x,v) = distC(y, u) = |C|/2. Isometry of C implies that properties (i) and (ii) are satisfied. The argument of the preceding paragraph shows that (iii) holds. For x, the only point in C at distance |C|/2 is v. Thus there are two paths P' and P" in C of length distG(x, v) = |C|/2. By the convexity of C, these paths are contained in C (so that C = P' U P") and must be the only shortest paths connecting x and v; hence consequently Sxv = 2. An analogous argument shows that Syu = 2. In order to prove the converse, consider an even elementary cycle C satisfying (i) through (iv). Again we fix an arbitrary edge {x, y} of C. Since C is even, there is a unique antipodal point v of x and a unique antipodal point u of y with distC(x,v) = distC(y, u) = |C|/2. We claim that {u, v} is the required edge. If this were not the case, then there would be some other edge with both endpoints closer to x along C than v that satisfies condition (ii). This is impossible, however, since for such a vertex v' we would have |C|/2 = distG(x, v') < distC(x, v') < distC(x,v) = |C|/2. We easily check that distC (x,u) = |C|/2 -1 and distC (y,v) = |C|/2 -1. By property (i), therefore, the paths from x to u and from y to v along C are shortest paths in G. Furthermore, the two paths from x to v along C via either u or y are also shortest paths in G by property (ii). Thus distC (x, q) = distG(x, q) for all vertices q in C. Repeating this argument for all x in C shows that C is isometric. By property (iii), the shortest path from x to u is unique. Since all sub-paths of shortest paths are again shortest path, this is also true for all vertices q in C along the shortest path from x to u. The same is true for all q in C along the unique shortest path from v to y. By property (iv), finally, there are exactly two shortest paths from x to v. We have already seen that two of these run along either half of the cycle C. The same is true for the two paths connecting y with u. Thus all shortest path connecting a vertex q in C with either x or y are contained in C. Repeating the argument for all edges {x, y} in C shows that C is convex. □ A direct consequence of Lemma 2.1 is that a cycle C in G can be efficiently tested for convexity provided both the distance matrix and the matrix S containing the number of shortest paths have been pre-computed: it suffices to verify, in constant time, the conditions of the lemma for each antipodal pair of edges or pair of edge and vertex, respectively. The test thus requires O( |C|) time provided that C is given as ordered list of its vertices. As a simple corollary of Lemma 2.1 we have Corollary 2.2. Let C be an elementary convex cycle of G. Then, for every e = {x, y} G C there is a vertex z in C such that C = P' U P'' U {x, y}, P' G PG [x, z], and P'' G PG [y, z]. A closely related, but much weaker, condition appears in the theory of minimal cycles bases [22]: Definition 2.3. A cycle C is edge-short if it contains an edge e = {x, y} and a vertex z such that C = Cxy,z := {x, y} U Pxz U Pyz where Pxz and Pyz are shortest paths. Corollary 2.4. If C is an elementary convex cycle of G then it is edge-short. 3 Convex cycle bases Corollary 2.2 sets the stage for enumerating all elementary convex cycles in a graph. The following result establishes an upper bound and provides a polynomial time algorithm for this purpose. Theorem 3.1. Any graph G = (V, E) contains at most |E||V| elementary convex cycles. These can be constructed and listed in O(|E||V|2) time. Proof. Every pair of an edge e = {x, y} and a vertex z specifies at most one elementary convex cycle in the following way: If distG(x, z) = distG(y, z) and Sxz = Syz = 1 we set Cez := Pxz U Pyz U {x, y}. If distG (x, z) = distG (y, z) + 1, Sxz = 2 and Syz = 1, then we choose a neighbor u of z such that distG(x, u) — distG(y, z), Sux — 1 and Suy — 2, and set Cez :— Pxu U {u, z} U Pyz U {x, y}. Note that the choice of u is unique if C is convex. The selection of these |E| |V | candidates thus requires O(\E | |V |2) time. In order to efficiently retrieve each candidate cycle in O(|C|) time given {x,y} and z we need the to know the predecessor nsu of u on the shortest path from s to u. Note that this information is needed only if Ssu — 1. The modified Dijkstra algorithm in the Appendix computes this array without changing the asymptotic complexity of the shortest path algorithm. Since each candidate cycle can then be checked for convexity in O(|C|) time, the total effort to extract all elementary convex cycles is in O(|E|| V|2). □ This algorithm outlined in the proof of Theorem 3.1 can be regarded as a variant of Vismara's construction of prototypes of candidates for relevant cycles [44]. The fact that the number of elementary convex cycles in G is bounded by | V| |E| immediately implies that a convex cycle basis can also be found in polynomial time: Corollary 3.2. For each graph G — (V, E) it can be decided whether G has a convex cycle basis, and if so, a convex cycle bases can be constructed, in O(\E |2 |V | p(G)2) time. Proof. Since the cycles of a graph form a matroid, the canonical greedy algorithm can be applied to find a maximum set of linearly independent elementary convex cycles, see e.g. [21]. G has a convex cycle basis if and only if this set has size p(G) — |E| - |V| + 1. For each of the at most | V| |E| candidate cycles, this requires a test of linear independence with a partial basis that is not larger than p(G) — |E| - |V| + 1, i.e., O(|E|). Applying Gaussian elimination for this purpose, the total effort is bounded by O(|E| |V|2) + O(|E|2 |V|m(G)2) — O(|E|2 |V|m(G)2) time. □ There are graphs that do not have a convex cycle basis. The complete bipartite graph K2,3 is the simplest counter example (see Fig. 1). None of its three cycles (all have length 4) is convex. Figure 1: None of the cycles in K2,3 is convex. 4 Relation of convex cycle bases to other types of cycle bases Although we have an efficient algorithm to test whether a graph has a convex cycle basis, it will be interesting to characterize the class of graphs that admit convex cycle bases. However, we first investigate the relation between convex cycle bases and other types of cycle bases. A procedure analogous to Corollary 3.2 was introduced in [22] for the purpose of retrieving minimal cycle bases from a candidate set of edge-short cycles. One would expect, therefore, that convex cycle bases and minimal cycles bases are closely related. Convex cycle bases of a graph need not have the same length. Consider the graph that is obtained from the cube Q3 where one edge is contracted. Then the four quadrangles and two triangles are convex and five of these form a convex cycle basis. Thus convex bases contain either exactly one or two triangles and thus may have different lengths. The length 1(B) of a cycle basis B is the sum of the lengths of its generalized cycles: ^(B) = J2ceB C|. A minimum cycle basis M is a cycle basis with minimum length. The generalized cycles in M are chord-less cycles (see [22]). Hence, we may consider elementary cycles instead of generalized cycles in the remaining part of this section. For the sake of completeness we note that a minimum cycle basis is a cycle basis in which the longest cycle has the minimum possible length [10]. The set R of relevant cycles of a graph is the union of its minimum cycle bases [41,44]. In analogy to convex cycle bases one may want to consider isometric cycle bases, i.e., cycle bases consisting of isometric cycles. Lemma 4.1. All relevant cycles of a graph are isometric. Thus every minimal cycle basis is an isometric cycle basis. Proof. We start from Lemma 2 of [44]: If P is a subpath of a relevant cycle C such that |P| < 1 |C|, then P is a shortest path. It follows that every relevant cycle is isometric, and hence every minimal cycle basis of G consists of elementary isometric cycles. □ Theorem 4.2. If G has a uniquely defined minimal cycle basis, then this minimal cycle basis is convex. Proof. Assume that G has a unique minimal cycle basis B. By Lemma 4.1 the cycles of B are necessarily isometric. Now suppose that C e B is not convex. Then there exist two vertices u,v e C and (at least) three edge disjoint uv-paths P, P' and P'' such that |P| > |P'| = |P"| and C = P U P'. Hence there are two cycles C1 = P U P" and C2 = P' U P" with |C| = |Ci| > IC21. By construction C, Ci, and C2 are linearly dependent and thus one of C1 or C2 cannot be represented as sum of cycles in B \ {C}. Hence we get a new cycle basis B' = (B\{C}) U{C '} where C ' is either C1 or C2. In either case we find l(B') < 1(B) a contradiction to our assumption that B is the unique minimal cycle basis. □ As a consequence, we can conclude that Halin graphs that are not necklaces [43] and outerplanar graphs [35] have a convex cycle basis. The converse of Theorem 4.2 is not true, however, as Figure 2 shows. This graph has a convex cycle basis but its minimal cycle basis is not uniquely defined. Even worse, none of its minimal cycle bases is convex. A cycle basis B = {C1,..., CM(-G)} of G is called fundamental [20, 46] if there is an ordering n such that for 2 < k < p(G): Fundamental cycle bases are obtained from ear decomposition, suggesting that there could be a relation between convex and fundamental cycles bases. Champetier's graph [4], however, has a cycle basis consisting entirely of triangles, which obviously is convex. On the other hand, this basis is not fundamental [1]. Conversely, fundamental cycle bases need not be convex, as shown, e.g., by the planar basis of K2 3. (4.1) Figure 2: The cyclomatic number of the graph is 7. All minimal cycle bases consist of the two triangles, all quadrangles that do not contain the upper dashed edge and two of the three quadrangles that contain the upper dashed edge (which also includes the outer cycle). However, two of these three quadrangles that contain the upper dashed edge are not convex. Hence none of the minimal cycle bases is a convex cycle basis. On the other hand there is a unique convex cycle basis that consists of all triangles, all quadrangles that do not contain the upper dashed edge, the outer quadrangle and the cycle of length 5 at the bottom. 5 Convexity in subgraphs and intersections This section contains some auxiliary results which we will need for our investigation of isometric subgraphs in Section 6 below. Lemma 5.1. Let M be an isometric (convex) subgraph of G and F C M be a subgraph of M. Then F is isometric (convex) in M if and only if it is isometric (convex) in G. Proof. If F is an isometric subgraph of G, then for each pair of vertices u, v e V (F), F contains a shortest uv-path. Since F C M, this path is also a shortest uv-path in M and hence F is isometric in M. If F is a convex subgraph of G, then it contains all shortest uv-paths which are also shortest paths in M and thus F is convex in M. Now assume that F is not isometric in G. Then there exist two distinct vertices u, v e V (F ) C V (M ) such that there are shortest uv-paths P in G with |P | < distF (u, v). At least one of these paths must be contained in M since M is an isometric subgraph of G. Thus F cannot be an isometric subgraph of M, either. If F is not convex in G then there exist two distinct vertices u, v e V (F ) C V (M ) such that there is at least one shortest uv-path P which is not contained in F. Since M is convex, P must be contained in M and thus F cannot be convex in M, either. □ Lemma 5.2. Let M be an isometric subgraph of G and F be a convex subgraph of G. Then F n M is convex in M. Proof. For each pair of vertices x, y e V (F ) n V (M ), F contains all shortest xy-path in G. Since M is an isometric subgraph of G it must contain at least one of these and thus the proposition follows. □ Lemma 5.3. Assume that G has a convex cycle basis. Let M be a convex subgraph of G that has a convex cycle basis BM. Then BM can be extended to a convex cycle basis BG of G. Proof. By Lemma 5.1 the cycles in BM are convex subgraphs of G. By assumption there exists a convex cycle basis B'G of G. By the Austauschsatz we can replace appropriate cycles in B'G by the cycles in BM. Thus we obtain a convex cycle basis BG of G which Figure 3 shows that the converse of this lemma is not true in general: a convex subgraph of a graph that has a convex cycle basis need not necessarily have a convex cycle basis. Figure 3: The cyclomatic number of this graph is |E| — | V| + 1 = 16 - 9 + 1 = 8. The three triangles and the five quadrangles that do not entirely consist of dashed edges form a convex cycle basis. The subgraph that consists of the dashed edges is convex but does not have a convex cycle basis (see Fig. 1). 6 Isometric subgraphs of Cartesian products In this section, we will be concerned with the Cartesian product GO H and its isometric and convex subgraphs. The Cartesian product has vertex set V (GOH ) = V (G) x V (H ); two vertices (xG, xH) and (yG, yH) are adjacent in GOH if {xG, yG} € E(G) and xH = yH, or {xH, yH} € E (H ) and xG = yG. For detailed information about product graphs we refer the interested reader to [18, 24]. For the Cartesian product GOH the subgraph Gv induced by all vertices (x, v) with x € V (G) and a fixed vertex v € V (H ) is called a layer of G (or G-layer) in GOH. The projection nG : GOH ^ G is the usual weak homomorphism defined as (x, y) € V(GOH) ^ x € V(G). Note that edges in G-layers are mapped into edges in G and edges in H-layers are mapped into vertices in G. There is a close relationship between (geodetic) convexity and Cartesian products, see [7] for a general result. The fundamental result for this purpose is the distance lemma. Proposition 6.1 (Distance Lemma, [23]). Let x = (xG, xH) and y = (yG, yH) be arbitrary vertices of the Cartesian product GOH. Then Moreover, if P is a shortest xy-path in GOH, then nG(P ) is a shortest xGy G-path in G. It seems natural that convexity properties of products also hold for layers and projections. BG B such that Bm C Bg as claimed. □ distGDH(x, y) = distG(xG,yG) + distH(xh, yH) . Lemma 6.2 ([24]). The layers Gv and Hw are convex subgraphs of the Cartesian product GOH. Moreover, if F is an isometric (convex) subgraph of GOH, then for all v € V (H) and w G V (G) the following holds: F n Gv and F n Hw are isometric (convex) subgraphs of F, Gv and Hw, respectively. Corollary 6.3. Let M be an isometric subgraph of GDH. If (xG, xH) and (yG, yH) are two vertices in M with xG = yG, then there exists a shortest xHyH-path in M n HXG. Moreover, all shortest (xG, xH)(yG, yH)-paths in M are contained in HXG. Another consequence of the distance lemma is the following auxiliary result. Lemma 6.4. Let P be a shortest xy-path in GDH. Then nG(P) is a path with |nG(P)| = distG(xG, vg) = v£H |P n Gv |, where the last term is the total number of edges of P in G-layers. The result holds analogously for nH (P ). Proof. If (w, xH) and (w, yH ) are two distinct points of P, then by Corollary 6.3 all shortest xH vh -paths are contained in layer Hw. Consequently, there cannot be two distinct edges e1 and e2 in G with nG(e1) = nG(e2) that belong to P since otherwise P also must contain two shortest paths in different H-layers that connect corresponding vertices of these edges, that is, P would contain a cycle. Hence all vertices of nG(P) have degree 2 except its end vertices which have degree 1 (or 0 in the case where nG(P) is a single vertex). Thus nG(P) is path of length |nG(P)| = distG(xG,yG) = J2veH |P n Gv |, as claimed. □ Lemma 6.5. For every isometric (convex) subgraph F of GDH, nG(F) is an isometric (convex) subgraph of G. Proof. Let x = (xG,xH) and y = (yG,yH) be two vertices in F. If F is isometric in GDH, then there exists a shortest xy-path P in F. By the distance lemma, nG(P) is a shortest xGyG-path in G and contained in nG(F). Thus nG(F) is an isometric subgraph of G. Now if nG(F) is not convex in G, then there exists a shortest xGyG-path PG in G that is not contained in nG(F). Let PH be a shortest xHyH-path in H. Then P = PGD{xh} U PHD{yG} is a shortest xy-path as its length is |P| = distG(xG, yG) + distH(xH, yH) = distGnH(x, y). However, by construction P cannot be contained in F and hence F is not convex in GDH. Consequently, if F is convex in GDH, then nG(F) is convex in G, as claimed. D On the other hand convexity and isometry properties of factors are also propagated to their Cartesian product. The following result is well-known and holds for more general notions of convexity. Lemma 6.6 ([7]). If F and M are convex subgraphs of G and H, respectively, then F DM is a convex subgraph ofGDH. The last lemma also holds for isometric subgraphs. Lemma 6.7. If F and M are isometric subgraphs of G and H, respectively, then F DM is an isometric subgraph of GDH. Proof. Immediate corollary of the distance lemma. □ We now want to extend convex cycle bases of two graphs G and H to a cycle basis of their Cartesian product GDH. Let TG and TH denote spanning trees of G and H, respectively. Let Bn = {eDf : e G E(G), f G Th } U {eDf : e G Tg, f G E (H )} . (6.1) Then for fixed vertices v g V (H) and w G V (G) and respective cycle basis BG and BH is a cycle basis of GDH [25]. Theorem 6.8. Let G and H be two graphs that have convex cycle bases BG and BH, respectively. Then their Cartesian product GDH has a convex cycle basis that can be constructed using Eq. (6.2). Proof. Notice that all quadrangles in Bn are convex subgraphs in GDH. By Lemma 5.1 Cv is a convex cycle in GDH. Thus we get a convex cycle basis of GDH by means of basis (6.2) when both BG and BH are convex cycle basis. D Remark 6.9. An analogous statement for the strong product (see [18]) is not true, as the strong product of an elementary cycle and an edge K2 shows. We have seen in Figure 3 that a convex subgraph of a graph that has a convex cycle basis does not necessarily have a convex cycle basis. However, a more restrictive property appears to propagate under the formation of Cartesian products: we consider the class of graphs for which every convex subgraph has a convex cycles basis. Theorem 6.10. Let G be a graph that has a convex cycle basis. Then every isometric subgraph M of GDK2 with nG(M) = G has a convex cycle basis. For the proof of this theorem we need some intermediate results. Lemma 6.11. Let C be an isometric elementary cycle in GDH. Then one of the following holds: (1) nG(C) = Ki, i.e., a single vertex, or (2) nG(C) = K2, i.e., a single edge, or (3) nG(C) is an isometric elementary cycle in G. Proof. Notice that ^g(C) = Uv£V(H) ^g(C n Gv). Let x = (xG,xff ) and y = (yG,yH) be two vertices in C with xG = yG and xH = yH. Analogously to the proof of Lemma 6.4 no vertex v in nG ( C) can have degree greater than 2. Now if C C Hw for some w G V ( G), then nG(C) = {w} = K1, i.e., case (1). If there is a vertex x where nG(x) has degree 1, then there exist two distinct vertices u, v G V (H ) such that nG(C n Gu) and nG(C n Gv ) have a common edge e. However, this only can happen if nG(C) = {e} = K2, i.e., for case (2). Otherwise, there would be two vertices y' and y" in C so that nG (y') = nG (y") is adjacent to nG(x) with vertex degree larger than 1 in the projection, contradicting isometry of C. If we have neither case (1) nor case (2), then all vertices of nG(C) have degree 2 and hence nG(C) is an elementary cycle which is isometric in G by Lemma 6.5, i.e., case Now let C be an elementary cycle in G and M be an isometric subgraph of GDK2. Let Z (C, M ) denote the set of elementary cycles C ' C M that are convex in M and satisfy nG (C') = C. We set Z (C, M ) = 0 if no such cycle exists. {Cv : C G Bg} U {Cw : C G BH} U Bn (6.2) (3). D Lemma 6.12. Let M be an isometric subgraph of GQK2 and let C e G be a convex elementary cycle with C C nG(M). Then Z (C, M) is non-empty. Proof. First notice that CQK2 is a convex subgraph of GQK2 by Lemma 6.6. M' = M n (CQK2) is isometric in CQK2 by Lemma 5.1 and convex in M by Lemma 5.2. Let M1 and M2 denote the respective intersections of M' with the two K2-layers of CQK2. If M1 = C (or M2 = C) then M1 (M2) is a convex elementary cycle in CQK2 by Lemma 6.2, and thus also in M' by Lemma 5.2. Otherwise, both M1 and M2 are paths of length |MŽ| < 1 |C| for i = 1, 2, since M' is isometric. As nG(M') = C, nC(M1) U nC(M2) = C. Consequently, as M' is isometric, M' is an elementary cycle that is trivially convex in M '. In all cases Z (C, M ') is non-empty. Since Lemma 5.1 implies that Z (C, M ') C Z (C, M ), the proposition follows. □ Remark 6.13. The arguments in the proof of Lemma 6.12 together with the distance lemma also show that the elements of Z (C, M) form the set of all shortest cycles C in M with the property nG (C') = C. Proof of Theorem 6.10. Let EG be a convex cycles basis of G. Let En be as in (6.1) and define BZ be a set of cycles that contains exactly one cycle C' e Z (C, M) for each C e Bg. By Lemma 6.12 all these sets Z (C, M ) are non-empty. Clearly, the cycles in En UBZ are linearly independent and thus form a cycle basis of GQK2. Now let BM be the set of all cycles in En U EZ that are contained in M. By construction all cycles in BM are convex subgraphs of M and EZ C BM. Thus it remains to show that |BM| = ^(M). Let mG and mK2 denote the numbers of edges in (GQK2) \ M that lie in G-layers and K2 -layers, respectively. Let n be the number of vertices in (GQK2) \ M. Since nG(M) = G and M is an isometric subgraph of GQK2 we find that mK2 = n. Thus ^(M) = (|E (GQK2)| - m g - mK2 ) - (|V (GQK2 )| - n) + 1 = |E(GQ K2)| - |V (GQK2)| + 1 - mG = m(GQK2) - mG. On the other hand, there are exactly mG cycles in En that are not contained in M and hence |BM | = ^(GQK2) - mG = ^(M), i.e., EM is a cycle basis of M. This finishes the proof of the theorem. □ We easily can generalize Theorem 6.10 to arbitrary isometric subgraphs of GQK2. Theorem 6.14. Let G be a graph such that every isometric subgraph has a convex cycle basis. Then every isometric subgraph of GQK2 also has a convex cycle basis. Proof. Let H be an isometric subgraph of GQK2. By Lemma 6.5, G' = nG (H) is an isometric embedding into G and thus has a convex cycle basis by our assumptions. Hence by Theorem 6.10 every isometric subgraph M of G'QK2 C GQK2 has a convex cycle basis. □ Theorem 6.14 has quite strong implications. A d-dimensional hypercube is the d-fold product of K2 by itself, Qd = Qd=1K2. Partial cubes are isometric subgraphs of Qd and form a very rich graph class that contains hypercubes, trees, median graphs, tope graphs of oriented matroids, benzenoid graphs, tiled partial cubes, netlike partial cubes, and flip graphs of point sets that have no empty pentagons; see [30, 31] and references therein. As K2 has a convex cycle basis (namely 0) we immediately obtain the following results by a recursive application of Theorem 6.14. Theorem 6.15. Partial cubes have a convex cycle basis. Theorem 6.16. Let G be a graph such that every isometric subgraph has a convex cycle basis and let Q be any partial cube. Then every isometric subgraph of GQQ has a convex cycle basis. Proof. Let G be as claimed. Theorem 6.14 implies that every isometric subgraph of GQK2Q • • • DK2 = GQQn has a convex cycle basis. Lemma 6.7 implies that GQQ is an isometric subgraph of GQQn. Moreover, Lemma 5.1 implies that every isometric subgraph of GQQ is an isometric subgraph of GQQn and thus, has a convex cycles basis. □ Figure 4 shows that the class covered by Theorem 6.16 is much larger than the class of partial cubes. Recall that partial cubes are characterized by the so-called Djokovic-Winkler-Relation ©: Two edges e = {u, v} and f = {x, y} are in relation ©, (ef ) G ©, if dist(u, x) + dist(v, y) = dist(u, y) + dist(v, x). A graph is a partial cube if and only if it is bipartite and the relation © is an equivalence relation [47]. Figure 4: Observe that (e1e2) G © and (e2e3) G ©, but (e1e3) G ©. Thus © is not an equivalence relation. Therefore, this bipartite graph is not a partial cube. However, it has a convex cycle base consisting of the three planar faces. It seems natural that Theorem 6.16 should remain true also for a more general type of Cartesian products. We state this as Conjecture 6.17. Let G1 and G2 be graphs such that each of their isometric subgraphs have convex cycle bases. Then every isometric subgraph of G1QG2 has a convex cycle basis. A further step towards a proof of this conjecture is given by the following special case: Theorem 6.18. Let G be a graph such that every isometric subgraph has a convex cycle basis and let Cn be an elementary cycle. Then every isometric subgraph of GQCn has a convex cycle basis. Notice that this theorem is an immediate corollary of Theorem 6.16 if n is even since cycles of even length are partial cubes [30, 45]. The proof of the general case is present after Lemma 6.19 below. For this purpose we first have to introduce a graph operation for the case when C is a cycle of odd length. So assume that C = C2k-1 for some integer k > 2. First fix three vertices u,v,w G V (C ) with {u, v}, {v, w} G E (C2k-1). Create a new cycle C' = C2k by splitting vertex v, that is, replace v by two vertices v' and v" and the edges {u, v}, {v, w} by three edges {u, v'}, {v', v''}, {v'', w}. This splitting operation can be generalized to subgraphs F of GQC. In essence, we replace F n Gv by (F n Gv )QK2. In more detail, we introduce the graph operations T and its converse T* as follows: For a fixed vertex v g C, and any subgraph F C GDC, we obtain the subgraph T(F) C GDC' by splitting all vertices (x, v) G F with x G G in the following way: Replace vertex (x, v) by (x, v') and (x, v''), and replace the edges {(x, u), (x, v)}, {(x, v), (x, w)}, and {(x, v), (y, v)}, when present, by the corresponding edges {(x,u), (x, v')}, {(x, v'), (x, v'')}, {(x, v''), (x,w)}, {(x,v'), (y, v')} and {(x, v''), (y, v'')}. Conversely, for a subgraph F' C BDC' we obtain the subgraph T*(F') C GDC by contracting all edges {(x, v'), (x,v'')} G E(GDC') and remove possible double edges. This construction in particular has the property that T(GDC) = GDC' and T* (GDC') = GDC. Lemma 6.19. Let C = C2k-1 be an elementary cycle of odd length 2k — 1. If P is a shortest xy-path in GDC, then T(P) contains a shortest x'y'-path P' in GDC' where x' and y' are vertices in T(x) and T(y), resp. Proof. Let x = (xG,xC) and y = (yG,yC) be two vertices in GDC and let x' = (x'G, x'Ci) and y' = (yG, yC' ) be two vertices in T(x) and T(y), resp. Let P' be a shortest x'y'-path in T(P). We have to show that P' is also a shortest x'y'-path in GDC'. Observe that Lemma 6.4 implies that |nG(P)| = |nG(P')| and |nc (P)| = |nc (P')| — J(P') where J(P') = 1 if (P') contains edge {v',v''} and J(P') = 0 otherwise. Moreover, distC (xC,yC) < k — 1 and distC/ (x'C/, yC' ) < k. Now suppose that P' is not a shortest x'y'-path in GDC'. Then there exists a x'y'-path P'' that is strictly shorter than P', that is, |nc (P'')| < |nc (P')| < k. As P is a shortest xy-path we have |nC(T*(P''))| = |nC(T*(P'))| = |nc(P)| < k —1. Again |nc(T*(P''))| = |nc(P'')| — J(P''). Consequently nc (P'') must contain edge {v', v''} while nc (P') must not. Therefore nc (P'') n ic (P') = C'. However |nc (P'')| + |nc (P')| < k + k = 2k = |C'|, a contradiction. This completes the proof. □ Proof of Theorem 6.18. Let C be an odd cycle. Thus C' is even and hence a partial cube. Lemma 6.19 implies that T(M) is an isometric subgraph of GDC' if M is an isometric subgraph in GDC. In this case, T(M) has a convex cycle basis B'. Now consider a convex cycle D' G B'. Lemma 6.11 implies that T*(D') is either an elementary cycle or T*(D') is a single edge in layer Gv. The latter happens if and only if D' contains edges {(x, v'), (x, v'')} and {(y, v'), (y, v'')}. In this case D' must be a convex quadrangle. There are |E(M n Gv )| quadrangles of this type, and they form an linearly independent set Q of convex cycles. Thus we can assume, w.l.o.g., that they all are contained in B'. Lemma 6.19 implies that T* (D') is a convex subgraph of M. Thus let B := {T*(D')|D' g B' and T*(D') is an elementary cycle} . The cycles in B are linearly independent: Consider any linear combination of the form J2i AjT* (D') = 0. It follows that there is a corresponding linear combination J2i ^D' = J2j £jQj, where Qj G Q is a quadrangle that is contracted to 0 by T*. Since B' is linearly independent by assumption, all £j and Ai must be 0, however. It remains to show that |B| = ^(M). Observe that T(M) contains the subgraph induced by vertices (x, v') and (x, v'') if (v,x) G V (M ) for some x G G. Otherwise T(M) contains none of these two vertices. Thus we find for the cyclomatic number m(M) = ^(T(M)) — |E(M n Gv)|. On the other hand |B| = |B'| — |E(M n Gv)| = m(t(M)) — |E(M n Gv)| = ^(M). This completes the proof. □ 7 Convex Eulerian graphs that are not cycles Convex cycles need not be elementary, even though they are necessarily connected when G is connected. Furthermore, the elementary cycles whose union forms convex Eulerian subgraph need not be convex themselves. An example is the K2 4, which can be decomposed into two elementary but not convex squares. In fact, the sum of convex cycles typically is not convex: Lemma 7.1. Let C\ and C2 be two convex cycles in G. If C 1 © C2 is 2-connected, then C i © C2 is not convex. Proof. If Ci © C2 is 2-connected, then it contains at least two distinct vertices u, v e V(C1) n V(C2). Since C1 n C2 is also convex, it contains the set of all shortest uv-path which cannot be empty as u — v. Consequently, C1 © C2 — (C1 U C2) \ (C1 n C2) cannot contain any of these shortest path and is thus not convex. □ If C1 © C2 is convex for two convex cycles C1 and C2, then C1 © C2 — C1 U C2 and connected (but not 2-connected). 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Plotkin, Mathematical basis of ring-finding algorithms in CIDS, J. Chem. Doc. 11 (1971), 60-63. [42] E. Sampathkumar, Convex sets in a graph, Indian J. pure appl. Math. 15 (1984), 1065-1071. [43] P. F. Stadler, Minimal cycle bases of Halin graphs, J. Graph Theory 43 (2003), 150-155. [44] P. Vismara, Union of all the minimum cycle bases of a graph, Electr. J. Comb. 4 (1997), #R9, printed version J. Comb. 4 (1997), 73-87. [45] P. M. Weichsel, Distance regular subgraphs of a cube, Discrete Mathematics 109 (1992), 297 -306. [46] H. Whitney, On abstract properties of linear dependence, Am. J. Math. 57 (1935), 509-533. [47] P. M. Winkler, Isometric embedding in products of complete graphs, Discr. Appl. Math. 8 (1984), 209-212. Appendix A: Modified Dijkstra Algorithm A shortest path algorithm that keeps track of the multiplicity of shortest paths and keeps some backtracing information is required as a pre-processing step in the computation of convex cycle bases. We use a modified version of Dijkstra's approach [12]. Let ^(x, y) denote the length of the edge {x, y} in G, dxy — distG(x, y) is the length and Sxy is the number of shortest paths in between x and y, nsx is the predecessor of x along the unique shortest path from s to x, and nsx — 0 otherwise. Q denotes a priority queue sorted by dsx for fixed s. Input: G — (V, E, /* an edge-weighted graph */ Output: Matrices [Sxy], [dxy], and [nxy]. 1: for all s e V do 2: /* Modified Dijkstra algorithm */ 3: for all v e V do 4: dsv — to; Ssv — 0; nsv — 0 5: dss — 0; Sss — 0; nss — s 6 7 8 9 10 11 12 13 14 15 16 Q ^ V ; while (Q — 0) do u := vertex with smallest d. 'su- if (dsu — to) then break /* G not connected */ remove u from Q for all neighbors v e Q n N(u) of u do t := dsu + ^(u,v) if (dsv — t) then Ssv :— Ssv + 1; ns v — 0 /* more than one shortest path */ if (dsv > t) then 17: dsv : t; Ssv : 1; nsv u The algorithm runs in O(|V|(|E| + |V| log |V|)) when the min-priority queue Q is implemented by means of a Fibonacci heap [16]. The modifications do not change the asymptotic complexity of the algorithm. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 141-151 From spanning forests to edge subsets Martin Trinks * Hochschule Mittweida, Faculty Mathematics / Sciences / Computer Science, Technikumplatz 17, 09648 Mittweida, Germany Received 16 September 2011, accepted 13 February 2013, published online 8 April 2013 We give some insight into Tutte's definition of internally and externally active edges for spanning forests. Namely we prove, that every edge subset can be constructed from the edges of exactly one spanning forest by deleting a unique subset of the internally active edges and adding a unique subset of the externally active edges. Keywords: Spanning forests, edge subsets, Tutte polynomial. Math. Subj. Class.: 05C30, 05C31 1 Introduction The Tutte polynomial originally defined by a sum over spanning forests using (the number of) internally and externally active edges [12], can also be given as a sum over edge subsets [14, Equation (9.6.2)]. We show how both representations, as sum over spanning forests and as sum over edge subsets, are directly connected to each other. Namely we prove, that every edge subset can be constructed from the edges of exactly one spanning forest by deleting a unique subset of the internally active edges and adding a unique subset of the externally active edges. While seeking a generalization to matroids we observed that the statement is already given by Bjorner [4, Proposition 7.3.6]. It seems that this result is not well known in graph theory. Hence we state it explicitly in the special case of graphs and verify it graph-theoretically. We apply this in some direct proofs for the equivalence of different representations of the Tutte polynomial, the chromatic polynomial, the reliability polynomial and the weighted graph polynomial. Definition 1.1. A graph G = (V, E) is an ordered pair of a set V, the vertex set, and a multiset E, the edge set, such that the elements of the edge set are one- and two-element subsets of the vertex set, e G (1) U (2) for all e G E. ♦Supported by European Social Fond grant 080940498. E-mail address: trinks@hs-mittweida.de (Martin Trinks) Abstract For a graph G = (V, E ), we denote the number of connected components of G by k(G) and refer to G with the edge e g E deleted and with the edge f e (1) U (2) added by G-e and G+f, respectively. Definition 1.2. Let G = (V, E) be a graph and A C E an edge subset of G. A graph G (A) = ( V, A) is a spanning subgraph of G. A tree T = ( V, A) is a spanning tree of G. A forest F = (V, A) is a spanning forest of G, if k(G) = k(F ). The set of spanning trees and the set of spanning forests of the graph G are denoted by T (G) and F (G), respectively. While the term "spanning tree" is unambiguous, the term "spanning forest" is not, because not every spanning subgraph, which is a forest, is a "spanning forest". A spanning forest is the union of spanning trees of each connected component. In the following we consider graphs G = (V, E) with a linear order < on the edge set E. This linear order can be represented by a bijection ß : E ^ {1,..., |E|} for all e, f e E with e < f ^ ß(e) <ß(f ). (1.1) Definition 1.3 (Section 3 in [12]). Let G = (V, E) be a graph with a linear order < on the edge set E and F = (V, A) e F (G) a spanning forest of G. An edge e e A is internally active in F with respect to G and <, if there exists no edge f e E \ A, such that e < f and F-e+ f e F (G). We denote the set of internally active edges and the number of internally active edges of F with respect to G and < by Ej(F, G, <) and i(F, G, <), respectively. An edge e in the spanning forest F is internally active, if it is the maximal edge of all edges in the cut crossed by itself (connecting the vertices in the connected components arising by deleting e from F). In other words, the edge e can not be replaced by a greater edge (not in the spanning forest), such that F remains a spanning forest. Definition 1.4 (Section 3 in [12]). Let G = (V, E) be a graph with a linear order < on the edge set E and F = (V, A) e F (G) a spanning forest of G. An edge f e E \ A is externally active in F with respect to G and <, if there exists no edge e e A, such that f < e and F-e+ f e F (G). We denote the set of externally active edges and the number of externally active edges of F with respect to G and < by Ee(F, G, <) and e(F, G, <), respectively. An edge f not in the spanning forest is externally active, if it is the maximal edge of all edges in the cycle closed by itself. In other words, there is no greater edge (in the spanning forest), which can be replaced by f, such that F remains a spanning forest. Definition 1.5 (Section 3 in [12]). Let G = (V, E) be a graph with a linear order < on the edge set E. The Tutte polynomial is defined as T (G, x, y) = £ x^Oy^O. (1.2) F eF (G) The primal usage of "a linear order on the edge set" seems to be by Whitney [18, Section 7]. Internally and externally active edges were probably first defined by Tutte [12, Section 3] to state the Tutte polynomial. This polynomial was originally introduced under the name "dichromate" for connected graphs [12, Equation (13)] and extended to disconnected graphs by the multiplicativity with respect to components [12, Equation (18)]. It was shown, that the value of the polynomial is independent of the linear order on the edge set [12, page 85-88]. For some background to the definition of internally and externally active edges and the Tutte polynomial we refer to [1, 9, 15]. For surveys on the Tutte polynomial and its applications we refer to [5, 7, 11]. 2 Main theorem The spanning forests and their internally and externally active edges can be used to generate all edge subsets. We use the disjoint union , the union of pairwise disjoint sets, in the statement of this main theorem below to indicate its bijectivity. Theorem 2.1. Let G = (V, E) be a graph with a linear order < on the edge set E. Then U U {(Af \ Ai) U Ae} = U {A} = 2e. (2.1) F=(V,Af )eF(G) AiCEi(F,G,<) ac_E AeCEe(F,G,<) c Proof. We prove that the function m : {(Af ,Ai,Ae) | F = (V,Af ) G F (G), Ai C Ei (F, G, <), Ae C Ee(F,G,<)} ^ 2E with m((Af ,A ,Ae)) = (Af \ Ai) U Ae is a bijection. First, we show that the function m is injective by an indirect proof. Assume it is not, that means there are two different triples A1 = (Aj,, Aj, Aj) and A2 = (Aj, A2, A2e), such that m(Aj) = m(A2) = A. If A1 = A2, then Aj = Aj, because otherwise Aj = Af \ A = Af \ A = Aj and Aj = A \ Aj = A \ Af = Ae and the triples would not be different. As Aj and Af are the edges of different spanning forests, there is an edge g G A j \ A2 . Furthermore, for any choice of g, there is an edge h G Af \ A j, such that (V, Aj )-g+h, (V, A2 )~h+g G F (G). (There is at least one edge in the path connecting the incident vertices of g in (V, Af ), which is in the cut crossed by g in (V, Aj). These conditions ensure that we can "compare" the edges g and h, because g is in the cycle closed by adding h to A j and, equivalently, in the cut crossed by h in A2, and vice versa.) We distinguish whether g (g G Af but g G Af ) and h (h G Aj but h G A2) are in A or not: • Case 1: g G A, h g A: We have a contradiction by - g G A ^ g G Ae ^ h < g, - h G A ^ h G Aj ^ g < h. • Case 2: g g A, h G A: We have a contradiction by - g G A ^ g G Ae ^ h < g, - h GA ^ h G A2 ^ g < h. • Case 3: g G A, h g A: We have a contradiction by - g GA ^ g G Aj ^ h k(F-e+f ) = k(F-e) = k(F) - 1. (2.4) Proof. The first part, k(F) = k(F+f ) > k(F-e) = k(F) - 1, follows directly from the definition of a spanning forest. The idea to prove the rest, k(F-e+f ) = k(F-e), is already used in the case distinction in the proof of Theorem 2.1: The edge f can not reconnect the connected components arising from the deletion of e, because otherwise each of the two edges must be greater than the other. □ 3 Applications of the main theorem As an application of Theorem 2.1 we prove the equivalence of representations using sums over spanning forests/trees (spanning forest/tree representation) and sums over edge subsets (edge subset representation) for the Tutte polynomial, the chromatic polynomial, the reliability polynomial and (a derivation of) the weighted graph polynomial. 3.1 Edge subset representation of the Tutte polynomial The edge subset representation of the Tutte polynomial was first given by Tutte stating the relation to the dichromatic polynomial [13, Equation (21)]. In this article, the dichromatic polynomial is defined by an edge subset representation and it is shown, that it satisfies recurrence relations [13, Equations (18) - (20)] analogous to the recurrence relations satisfied by the Tutte polynomial [12, Equations (18) - (20)]. Theorem 3.1 (Equation (9.6.2) in [14]). Let G = (V, E) be a graph with a linear order < on the edge set E. The Tutte polynomial has the edge subset representation T (G, x, y) = £ (x - l)k(G)(-l) \ A' \ (-1) \ A= \ F=(V,Af )eF(G) A'=Af\Ai AiCEi(F,G,<) AeCEe(F,G,<) = £ £ Xk(GA>-)(-l)\A' \ £ (-1)A \ F =(V,Af )eF(G) A'=Af\Ai Ae CES(F,G,<) AiCEi(F,G,<) = £ £ xk(G(A'>)(-1)\A' \(1 - 1)e(F,G,<) F=(V,Af )eF(G) A'=Af\Ai AiCEi(F,G,<) = £ £ xk(G)(-1)\A'\. F=(V,Af )eF(G) A'=Af\Ai e(F,G,<)=0 AiCEi(F,G,<) Second, we analyze the contribution of the internally active edges Ai C Ei(F, G, <) to the term xk(G^A>)(-1)\A\: Each internally active edge e G Ei(F,G, <) contributes (independently) the factor -x if e G Ai (the number of connected components is increased by 1), and the factor 1 otherwise: x(G, x) = £ £ xk(GA\Ai>)(-1) \ Af\Ai \ F=(V,Af )eF(G) AiCEi(F,G,<) e(F,G,<) = 0 £ £ xk(G/Af>)xJAi\(-1) \ Af \ (-1) \ Ai \ F=(V,Af )eF(G) AiCEi(F,G,<) e(F,G,<) = 0 = £ xk(G)(-1) \ V \-k(G) £ (-x) \ Ai \ FeF(G) AiCEi(F,G,<) e(F,G,<) = 0 = (-1)\V \(-x)k(G) £ (1 - x)i(F,G,<). □ F €F(G) e(F,G,<)=0 The proof above also "includes" the Broken-cycle Theorem [18, Section 7], [6, Theorem 2.3.1]: The edge subsets not including any broken cycle are exactly the edge subsets resulting from spanning forests having no externally active edges by deleting a subset of internally active edges. Hence the Broken-cycle Theorem can be stated as x(G, x) = £ £ xk(G/A'>)(-1)\A' \ (3.3) F=(V,Af )eF(G) A'=Af\Ai e(F,G,<)=0 AiCEi(F,G,<) = £ £ x \V\—\A'\(-1) \ A' \. (3.4) F=(V,Af )eF(G) A'=Af\Ai e(F,G,<)=0 AiCEi(F,G,<) The connection between the spanning forest representation and the Broken-cycle Theorem is also given in [1]. 3.3 Spanning tree representation of the reliability polynomial The set of connected spanning subgraphs of a connected graph can be enumerated from the spanning trees by only adding externally active edges. We apply this insight to obtain a spanning tree representation of the reliability polynomial. For a statement S, let [S] equal 1, if S is true, and 0 otherwise [8]. Lemma 3.4 (Section 5, Item (19) in [16]). Let G = (V, E) be a graph with a linear order < on the edge set E. The generating function (in the indeterminant y) for the number of connected spanning subgraphs S (G, y) has the spanning tree representation S(G,y)= E [k(G{A)) = 1]y'A' (3.5) ACE = y|V 1-1 E (1 + y)e(T'G'<). (3.6) t er (G) Proof. We start by applying Corollary 2.2: S (G, y) = £ [k(G(A)) = 1]y|A| ACE = E E [k(G(A)) = 1]y|A|. F=(V,Af )eF(G) A=(Af\Ai)UAe AiCEi(F,G,<) AeCEe(F,G,<) The spanning subgraph G (A) is connected only if the graph G is connected, that means the spanning forests are spanning trees with | V | — 1 edges, and if no (internally active) edge is deleted from the spanning tree. It follows: S(G, y) = E E y|A| T=(V,At)er (G) A=AtUAe AeCEe(T,G,<) = E E yIA t'y'Ae| T=(V,At)er(G) AeCEe(T,G,<) = E y'A t ' E y'Ae' T=(V,At )er (G) AeCEe(T,G,<) = y'V ' -1 E (1 + y)e(T'G'<). D t er (G) The probability, that all vertices of a graph are connected, if all edges of the graph are independently available with a probability p, is a polynomial in p, the reliability polynomial R(G,p) [7, 16]. Definition 3.5. Let G = (V, E) be a graph. The reliability polynomial is defined as R(G,p) = E [k(G(A)) = 1] p ' A '(1 — p)' E\A '. (3.7) ACE Theorem 3.6 (Section 5, Item (15) in [16]). Let G = (V, E) be a graph with a linear order < on the edge set E. The reliability polynomial R(G,p) has the spanning tree representation R(G,p) = (1 - p) \ E \ - \ V \+V \ V \ -1 E (1_ p)U,<). (3.8) T eric) K ' Proof. We rewrite the definition of the reliability polynomial using S(G,y): R(G,p) = E [k(G(A» = 1]p \ A \(1 - p) \ E\A \ ACE \ A \ E [k(G(A» = 1] (1 - p) \ ACE ^ (1 - p) \ E \ S G,- p 1 - p j From this the statement follows directly by Lemma 3.4. □ 3.4 Spanning forest representation of a derivation of the weighted graph polynomial For the graph polynomials above it was possible to derive a spanning forest/tree representation that depends only on the number of internally and externally active edges, independently of the corresponding edge sets. Obviously, this is not possible for every graph polynomial, also not for those having an edge subset representation. The graph polynomial U'(G, x, y), a derivation of the weighted graph polynomial U (G, x,y) [10], is an example where only the contribution of the externally active edges can be summed up. Definition 3.7. Let G = (V, E) be a graph and X = (xi,..., x\ V \ ). The graph polynomial U '(G, x, y) is defined as U'(G,x,y)= Ell xki(G(A>)y \ A \ , (3.9) ACEi=1 where ki(G) denotes the number of connected components including exactly i vertices. Theorem 3.8. Let G = (V, E) be a graph with a linear order < on the edge set E and x = (x1,..., x \ V \ ). The (derivation of the) weighted graph polynomial U '(G, x, y) has the spanning forest representation u '(g, x, y) = E E n xki(G(A>)y \ A\(1 + y)e(F'G'<), (3.10) F=(V,Af )eF(G) A=Af\Ai i=1 AiCEi(F,G,<) where ki (G) denotes the number of connected components including exactly i vertices. E Proof. We start by applying Corollary 2.2 and then sum up the contribution of the externally active edges (as in the proofs above): U '(G, x, y) = £n xki(G(A>) y'A| ACEi=1 = £ £ n xki(G)y 'a ' (1+y)e(F'G'<). □ F=(V,Af )eF(G) A=Af\Ai i= 1 AiCEi(F,G,<) References [1] R. A. Bari, Chromatic polynomials and the internal and external activities of Tutte, in: J. A. Bondy and U. S. R. Murty (eds.), Graph Theory and Related Topics, Academic Press, London, 1979, pp. 41-52, Proceedings of the conference held in honour of Professor W. T. Tutte on the occasion of his sixtieth birthday, University of Waterloo, July 5-9, 1977. [2] N. L. Biggs, Algebraic graph theory, volume 2, Cambridge University Press, Cambridge, 1993. [3] G. D. Birkhoff, A determinant formula for the number of ways of coloring a map. The Annals of Mathematics, 14 (1912), 42-46, 1912, JSTOR:1967597.6. [4] A. Björner, Homology and shellability of matroids and geometric lattices, in: N. White (ed.), Matroid Applications, volume 40 of Encyclopedia of mathematics and its applications, chapter 7, Cambridge University Press, Cambridge, 1992, pp. 226-283, doi:10.1017/ CBO9780511662041.008. [5] T. H. Brylawski and J. G. Oxley, The Tutte polynomial and its applications, in: N. 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Welsh, A weighted graph polynomial from chromatic invariants of knots, Annales de l'institut Fourier 49 (1999), 1057-1087, http://aif.cedram.org/ item?id=AIF_19 99_4 9_3_105 7_0. [11] A. D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, in: Surveys in combinatorics, volume 327, 2005, pp. 173-226, doi:10.1017/ CBO9780511734885.009. [12] W. T. Tutte, A contribution to the theory of chromatic polynomials, Canadian Journal of Mathematics 6 (1954), 80-91, doi:10.4153/CJM-1954-010-9. [13] W. T. Tutte, On dichromatic polynomials, Journal of Combinatorial Theory, 2 (1967), 301320, doi:10.1016/S0021-9800(67)80032-2. [14] W. T. Tutte, Graph Theory, volume 21 of Encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, 2001. [15] W. T. Tutte, Graph-polynomials, Advances in Applied Mathematics 32 (2004), 5-9, doi:10. 1016/S019 6-8858(03)00041-1. [16] D. J. A. Welsh and C. Merino, The Potts model and the Tutte polynomial, Journal of Mathematical Physics 41 (2000), 1127-1152, doi:10.1063/1.533181. [17] H. Whitney, The coloring of graphs, Proceedings of the National Acadamy of Sciences of the USA, 17 (1931), 122-125, http://www.ncbi.nlm.nih.gov/pmc/articles/ PMC10 7 600 7. [18] H. Whitney, A logical expansion in mathematics, Bulletin of the American Mathematical Society 38 (1932), 572-579, http://projecteuclid.org/euclid.bams/ 1183496087. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 153-160 Revised and edge revised Szeged indices of graphs Morteza Faghani, Ali Reza Ashrafi * Department of Mathematics, Faculty of Mathematical Science, University ofKashan, Kashan 87317-51167, I. R. Iran 4= Received 31 October 2011, accepted 19 August 2012, published online 8 April 2013 Abstract The revised Szeged index is a molecular structure descriptor equal to the sum of products [nu(e) + n0(e)/2] x [nv(e) + n0(e)/2] over all edges e = uv of the molecular graph G, where n0(e) is the number of vertices equidistant from u and v, nu(e) is the number of vertices whose distance to vertex u is smaller than the distance to vertex v and nv (e) is defined analogously. In this paper, new formula for computing this molecular descriptor is presented by which it is possible to reprove most of results given in [M. Aouchiche and P. Hansen, On a conjecture about the Szeged index, European J. Combin. 31 (2010), 16621666]. We also present an edge version of this graph invariant. At the end of the paper an open question is presented. Keywords: Szeged index, edge Szeged index, revised Szeged index, edge revised Szeged index. Math. Subj. Class.: 05C12 1 Introduction We first describe some notations which will be kept throughout. Let G be a simple graph with vertex set V (G) and edge set E (G). If e = uv G E (G) then d(u, v) stands for the distance between u and v in G. A topological index is a graph invariant applicable in chemistry. A topological index x is called distanced-based, if x is related to the distance function d(-, -). The first use of a distance-based topological index occurred in the year 1947 in a seminal paper by an American chemist Harold Wiener [14]. Suppose G is a connected graph and e = uv G E (G). The quantities n0(e), nu(e) and nv (e) are defined to be the number of vertices equidistant from u and v, the number of vertices whose distance to vertex u is smaller than the distance to vertex v and the number * Corresponding author. E-mail addresses: m faghani@pnu.ac.ir (Morteza Faghani), ashrafi@kashanu.ac.ir (Ali Reza Ashrafi) of vertices closer to v than u, respectively. Similarly, the quantities m0(e), mu(e) and mv (e) are defined to be the number of edges equidistant from u and v, the number of edges whose distance to vertex u is smaller than the distance to vertex v and the number of edges closer to v than u, respectively. Here, for an edge e = xy and vertex u, the distance between e and u is defined as dG(e, u) = Miu{dG(x, u), dG(y, u)}. The Szeged, edge Szeged, edge-vertex Szeged, vertex-edge Szeged, revised Szeged and edge revised Szeged indices of G are defined as follows: Sz(G) = X^[uu(e) x Uv(e)], e=uv Sze(G) = X^[mu(e) x mv(e)], e=uv Szev (G) = 1/^E[mu(e) x Uv (e) + mv (e) x Uu(e)], e=uv Szve(G) = 1/2 E [mu(e) x Uu(e) + mv (e) x Uv (e)], e=uv Sz*(G) = E [(Uu(e) + Uo(e)/2) x (Uv(e) + Uo(e)/2)], e=uv sz:(G) = E [(mu(e) + mo(e)/2) x (mv(e) + mo(e)/2)] e=uv It is worth mentioning here that the Szeged index was introduced by Ivan Gutman [4] and the name Szeged index was given in [5]. For the mathematical properties of this topological index we refer to [3, 9, 10]. The concept of edge Szeged index was introduced in [6] and mathematical properties of this graph invariant are studied in [2, 7, 8]. The revised Szeged index was introduced by Milan Randic [13] as a modification of the classical Wiener index. Nowadays the scientists prefer the name revised Szeged index for this distance-based topological index. The interested readers can consult [1, 11, 12, 15] for mathematical properties of this new topological index. Throughout this section graph means finite simple connected graph. The notation is standard and can be taken from the standard books on graph theory. 2 Main results In this section, we first present a new formula for computing revised Szeged index of graphs. Then apply this new formula to reprove all results given by Aouchiche and Hansen [1]. We also present an edge version of the revised Szeged index and extend the results given in the mentioned article to this new graph invariant. We begin by an example. Example 2.1. Suppose Gi = Kn, G2 = C" and G3 = Wn denote the complete, cycle and wheel graphs of order n, and G4 = Km,n is the complete bipartite graph with partitions of size m and U, respectively. Then, • If e = uv G E(G1) then mu = mv = n — 2 and m0 = "2-5"+8. Therefore, Sze(Gi) = "("-12("-2)2 andSz*(Gi) = "3("-1)3. • Suppose e = uv is an arbitrary edge of G2. If n = 2k +1, then mu = mv = k and so mo = 1. Therefore, Sze(G2) = (2k + 1)k2 = "("-1)2 and Sz*^) = "3. If n = 2k then mu = mv = k — 1 and so m0 = 2. This implies that Sze(G2) = n(k — l)2 = ^^ and Sz*^) = ^. • Consider the n—vertex wheel graph G3, n > 5. If e = uv is an edge of G3 such that the vertex v is the center of G3, then mu = 3, mv =2n — 7 and m0 = 3. If both of u and v are not the center of G3, then mu = 3 , mv = 3 and m0 = 2n — 8. Therefore Sze(G3) = (n — 1)(4n — 5) and Sz*(G3) = (n — 1)(n2 + 5n — 73/4). • Suppose G4 = , x + y = n, is the complete bipartite graph containing an arbitrary edge e = uv, where deg(u) = x and deg(v) = y. Then we have mu = x—1, mv = y—1, m0 = xy— x—y+2. This implies that Sze(G4) = xy(x—1)(y—1) and Sz*(Gi) = f (x2y2 — x2 — y2 + 2xy). Theorem 2.2. Let G be an n-vertex and m-edge graph. Then Sz*(G) = ^n! — 1 ^ (nu + nv) + 1 Sz(G) e=uv Proof. Since nu(e) + «v (e) = n — no(e) we have: 02 nu«v + y(nu + nv ) + 4 n0 Sz*(G) = ]T[K + «f )(nv + ^ ) e=uv = Er e=uv = E nu«v + -((n — (nu + «v )))(n„ + «v ) + -(n — («u + «v ))2 E n, \ 1 / V.; nu«v + -(nu + nv ) — (n„ + «v ) 2 2 n2 1 1 2 + -4 — 2 «(«u + «v ) + 4 («u + «v ) E n2 1 2 2 nu«v + 4 — 4(«u + «v + 2(nu)(nv )) 1 mn2 1 2 2 _Sz(G) + — — 4 JJ[nu + nv]. proving the result. The next Corollary is already known result that stated and proven in [1]. □ Corollary 2.3. Sz(G) < Sz*(G) < ^. mn 4 22 Proof. Since nu + nv < n, (nu + nv)2 < n2. So, J2e=uv [nu + nv]2 < mn2 and therefore uv m42 — |E e=uv [nu+nv ]2 = m42 — iE e=uv K +«2 ] — 1 Sz(G) > 0. Now, Theorem 2.2 implies that Sz(G) < Sz* (G), the left hand side of inequality. The right hand side is a direct consequence of Theorem 2.2 and the following inequality: ^Sz(G) — 4 ]T [nu + nv] = — 1 ]T [nu — n]2 < 0. By a similar argument as Theorem 2.2, one can prove: Theorem 2.4. Let G be an n-vertex and m-edge graph. Then Sz*(G) = m! - 4 E [mU + mV] + 2Sze(G). e=uv Corollary 2.5. Sze(G) < Sz* (G) < ^ Proof. The proof is similar to the proof of Corollary 2.3 and so omitted. □ Suppose G is a connected graph and u is a vertex of G. Define D(u, G) = [d,Q(u,x)\. xev(G) The graph G is called distance-balanced (or transmission-regular according to [1]) if for every u, v e V (G), D(u, G) = D(v, G). Similarly for a vertex u and an edge e = xy define De(u, G) = Ye£E(G) [dG(e, u). A graph G is called edge-distance-balanced if for every vertices u, v e V (G), De(u, G) = de (v, G). Theorem 2.6. Suppose u and v are vertices of a connected graph G. Then mu = mv if and only if De(u, G) = de(v, G). Proof. Let e = uv be an arbitrary edge of G. We partition the edge set of G into three parts as follows: • M (u) is the set of all edges that are closer to u than v. • M (v) is the set of all edges that are closer to v than u. • M (o) is the set of all edges that are equidistant from u and v. Suppose mu(e) = |M(u)|, mv (e) = |M(v)| and m0(e) = |M(o)|. Then we have : De (u, G) = E dG(e,u) e£E(G) = E dG(e, u) + E dG(e,u)+ ^ dG(e,u) e£M (u) e£M(v) e£M (0) = E dG(e, u) + E (1 + dG(e,v))+ ^ dG(e,u) e£M (u) e£M(v) e£M (0) = E dG(e, u) + mv (e) + ^ dG(e, v) + ^ dG(e,u). e£M (u) e£M (v) e£M (0) A similar argument shows that De(v,G)= E dG(e, u) + mu(e)+ ^ dG(e, v) + ^ dG(e,v). e£M(u) e£M(v) e£M(0) But De(u,G) - De(v,G) = mv(e) - mu(e) and so mu(e) = mv(e) if and only if De(u, G) = De(v, G). This complete our argument. □ Corollary 2.7. If Sze(G) = m then G is an edge-distance-balanced graph. Proof. If Sze(G) = m then by Corollary 2.5, Sz*e(G) = mL. Thus 1 Sze (G) — 1 E [mU + m2 ] = — 4 E [mu — mv]2 =0- uveE(G) uveE(G) Therefore mu = mv. Now Theorem 2.6 implies that G is an edge-distanced-balanced graph. □ In the end of this paper, we compute an exact formula for the edge revised Szeged index of Cartesian product of graphs. To do this, we assume that G and H are connected graphs with vertex sets V(G) = {ui, u2,..., ur} and V(H) = [v\, v2,..., vs}. We also assume that |E(G)| = ei and |E(H )| = e2. Then by definition V (G x H ) = V (G) x V (H ) and we have: E (G x H ) = {(u, v)(a, b) | [u = a, vb e E (H )] or [ua e E(G), v = b]}. Clearly, |E(G x H)| = |V(G)||E(H)| + |V(H)||E(G)|. To compute the edge revised Szeged index of G x H we partition the edge set of this graph into the following parts: Am = {(um,x)(um,y) | xy e E (H )} ;1 < m < r, Bt = {(a, vt)(b, vt) | ab e E (G)} ; 1 < t < s. Theorem 2.8. (See [15, Lemmas 2 and 3]). With above notations we have: (a) If e = (um, v j )(um,vq ) e Am) then m(um,vj)(e) = |V(G)|mvj (vjvq) + |E(G)|nv3- (vjvq) = rmvó (H) + ei«,^. (H), m(um,vq ) (e) = |V ^^q (v j vq )+ |E(G)Kj (v j vq ) = rmvq (H ) + e1n„, (H ). (b) If e = (ui,vt)(up,vt) e Bt then m(ui,vt)(e) = |V (H )|mui (uu) + ^(H )|n^ (uu) = sm^ (G) + eyn^ (G), m(up,vt)(e) = ^(H)|mup(uu) + ^(H)|n^(uu) = smup(G) + e2«up (G). Theorem 2.9. With notation of Theorem 2.8, the edge revised Szeged index of Cartesian product of G and H can be computed as follows: Sz'*(G x H) = 2r3Sze(H)+ r2eiSzev(H) + 2re\Sz(H) + 1 re2(re2 + sei)2 r2eiSzve(H) — 1 r3 £ [m2x(H ) + m2y (H)] xyeE(H) 1 re2 £ [«x (H )+ n2y (H )] + 1 s3 Sze (G) + s2e2Szev (G) xyeE(H) + 1 se\Sz(G) + 1 sei(sei + re2)2 — s2e2Szve (G) 4 s3 £ [m2a(G) + m2(G)] — 4 se2 £ [n2a(G) + «2(G)]. abeE(G) abeE(G) Proof. Let e = (um,x)(um, y) e Am. Then mo(e) = re2 + se i - r(mx(H ) + my (H )) - ei(nx(H) + ny(H)). Set, A= B = m(Um,x)(e) + m(a,vt )(e) + mo(e) 2 mo(e) 2 e) + m(um,y m(b,vt) (e) + mo(e) 2 mo (e) 2 Then we have: A = 2 r2mx(H )my (H ) + 1 ei r(nx(H )my (H ) + ny (H)mx(H )) + 1 e2nx(H )ny (H ) + 4(re2 + sei)2 - 2 ei r(nx(H )mx(H ) + ny (H )my (H )) 1 1 - —r2(mx(H) + m2y(H)) - -e^H) + n2(H)). Thus, E (um,x)(um,y)eA„ m(u (e) + (e) / \ , mo(e) m(um,y)(e) + —2"" 1 1 1/2r2Sze(H) + eirSzev(H) + -e2Sz(H) + -e2(re2 + sei) 2 4 - eirSzve(H) - 4r2 ]T [mx(H)+ m^(H)] xyEE(H) - 1 e2 E [nx(H ) + ny (H)]. xyeE(H) Using a similar argument for the edge e = (a, vt)(b, vt) e Bt, we have: (2.1) B m(a e) + mo (e) m(b,v t)(e) + mo (e) 1 = öS mui(G)mup(G) + -e2s [na(G)mb(G) + nb(G)ma(G)] 2 2 + 1 e\na(G)nb(G) + 4(sei + re2)2 - 2e2S [na(G)ma(G) + nb(G)mb(G)] - 1 s2 [ma(G) + m2(G)] - 1 e2 [na(G) + n2(G)] . So, E (a,vt)(b,vt)eBt m(a,v t)(e) + mo(e) 2 m(b,v t)(e) + mo(e) 2 2 s2 Sze (G) + e2 sSzev (G) + 2 e2 Sz(G) + 1 ei(sei + re2)2 - e2sSzve(G) - 1 s2 E [ma(G) + m2(G)] (2.2) abeE(G) - 1 e2 E [na(G) + n2(G)] . abeE(G) X m 2 2 2 2 Now multiplying Eq. (2.1) by r and Eq. (2.2) by s and summation of these values, the formula given in the theorem will be obtained. □ 3 Conclusions Some of mathematicians recently focus on the revised Szeged index of graphs. In this paper a new formula for computing this topological index is presented by which it is possible to reprove some earlier results. We also investigate an edge version of this interesting topological index. We proved that the edge version of this graph invariant is more complicated than its vertex version. In the case of vertex version, it is easy to find an exact formula for the Cartesian product of graphs but in the edge version it is too difficult. In Theorem 2.6 and Corollary 2.7, it is proved that Sze(G) = m implies that G is an edge-balanced-distance graph. We end the paper with the following open question: Question: Characterize graphs G such that Sze(G) = m3/4. Acknowledgements The authors are indebted to the referees for their suggestions and helpful remarks. This research has been supported by the research affair of the University of Kashan, I. R. Iran under grant number 159020/4. References [1] M. Aouchiche and P. Hansen, On a conjecture about the Szeged index, European J. Combin. 31 (2010), 1662-1666. [2] E. Chiniforooshan and B. Wu, Maximum values of Szeged index and edge- Szeged index of graphs, Electronic Notes Discrete Math. 34 (2009), 405-409. [3] K. Ch. Das and I. Gutman, Estimating the Szeged index, Appl. Math. Lett. 22 (2009), 16801684. [4] I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes New York 27 (1994), 9-15. [5] I. Gutman, P. V. Khadikar, P. V. Rajput and S. Karmarkar, The Szeged index of polyacenes, J. Serb. Chem. Soc. 60 (1995), 759-764. [6] I. Gutman and A. R. Ashrafi, The edge version of the Szeged index, Croat. Chem. Acta 81 (2008), 277-281. [7] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi and I. Gutman, The edge Szeged index of product graphs, Croat. Chem. Acta 81 (2008), 277-281. [8] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi and S. G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009), 1149-1163. [9] S. Klavžar and I. Gutman, A theorem on Wiener-type invariants for isometric subgraphs of hypercubes, Appl. Math. Lett. 19 (2006), 1129-1133. [10] T. Mansour and M. Schork, The vertex PI index and Szeged index of bridge graphs, Discrete Appl. Math. 157 (2009), 1600-1606. [11] T. Pisanski and M. Randic, Use of the Szeged index and the revised Szeged index for measuring network bipartivity, Discrete Appl. Math. 158 (2010), 1936-1944. [12] T. Pisanski and J. Žerovnik, Edge-contributions of some topological indices and arboreality of molecular graphs, Ars Mathematica Contemporanea 2 (2009), 49-58. [13] M. Randic, On generalization of Wiener index for cyclic structures, Acta Chim. Slovenica 49 (2002), 483-496. [14] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), 17-20. [15] R. Xing and B.Zhou, On the revised Szeged index, Discrete Appl. Math. 159 (2011), 69-78. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 161-173 Petersen-colorings and some families of snarks Jonas Hägglund Department of Mathematics and Mathematical Statistics, Umeà University SE-901 87 Umeà, Sweden Eckhard Steffen Paderborn Institute for Advanced Studies in Computer Science and Engineering, Paderborn University, Zukunftsmeile 1, 33102 Paderborn, Germany Received 1 December 2011, accepted 19 November 2012, published online 15 April 2013 In this paper we study Petersen-colorings and strong Petersen-colorings on some well known families of snarks, e.g. Blanuša snarks, Goldberg snarks and flower snarks. In particular, it is shown that flower snarks have a Petersen-coloring but they do not have a strong Petersen-coloring. Furthermore it is proved that possible minimum counterexamples to Jaeger's Petersen-coloring conjecture do not contain a specific subdivision of K3 3. Keywords: Petersen colorings, strong Petersen colorings, snarks Math. Subj. Class.: 05C15, 05C21, 05C70 1 Introduction We study finite graphs G with vertex set V(G) and edge set E(G). If we distinguish an initial and a terminal end for every edge e, then we obtain a directed graph. For S c V (G), the set of edges with initial end in S and terminal end in V (G) - S is denoted by w+(S). We write w- (S) = w+(V (G) - S) and (S) = w+(S) u w-(s). If S consists ofasingle vertex v we also write wG(v) instead of wG({v}). Subsets of E (G) of the form wG(S) for S c V (G) are called cocycles of G. If R c E(G), then G[R] denotes the graph with vertex set V (G) and edge set R. Given graphs G and H, we say that f : E(G) ^ E(H) is a H-coloring of G if it is a proper edge-coloring and for every v g V (G) there exists a v' g V (H ) such that f (wG(v)) c wH(v'). That is, adjacent edges in G are mapped to adjacent edges in H. If H is the Petersen graph, we say that G has a Petersen-coloring. E-mail addresses: jonas.hagglund@math.umu.se (Jonas Hägglund), eckhard.steffen@uni-paderborn.de (Eckhard Steffen) Abstract Jaeger [6] studied nowhere-zero flow problems on graphs where the set of flow values are certain subsets of some Abelian group. He showed that a number of problems in graph theory such as the cycle double cover conjecture [9, 12] and Fulkerson's conjecture [2] (i.e. that every bridgeless cubic graph has six perfect matchings such that every edge is in precisely two of them) can be formulated in terms of such flows. He posed the following conjecture which would imply both previously mentioned conjectures, and many others, see [8]. Conjecture 1.1 (Petersen Coloring Conjecture [6]). Every bridgeless cubic graph has a Petersen-coloring. In [5] an even more specific notion is introduced. Associate to G a directed graph dG with vertex set V (dG) = V (G) u E (G), and to every edge e = xy in G correspond two directed edges ex and ey with initial end e and terminal ends x and y, respectively. We say ex is opposite to ey and vice versa. Let G and G' be two graphs. A mapping ^ from E(dG) to E(dG') is compatible, if for any two opposite edges ei and e2 in dG, ^(e1) and ^(e2) are opposite edges in dG'. For a cubic graph G the set of triples of edges of dG of the form wdG(v) is denoted by T + (dG), where v is a trivalent vertex in dG. T-(dG) is the set of triples of the form {e-, e-, e-} where {e1, e2, e3} g T+(dG) and e- is opposite to ej. Let G and G' be two cubic graphs. A dG'-coloring of dG is a compatible mapping Y from E(dG) to E(dG') which maps every triple of T +(dG) to a triple of T +(dG') u T- (dG'). For the particular case when dG has a dG'-coloring and G' is the Petersen graph, we say that G is strongly Petersen-colorable. Clearly, strongly Petersen-colorable graphs satisfy the Petersen-coloring conjecture and hence Fulkerson's and the cycle double cover conjecture as well. Jaeger [5] noticed that moreover these graphs also satisfy Tutte's 5-flow- and the orientable cycle double cover conjecture. All these conjectures are trivially true for 3-edge-colorable cubic graphs. Hence we focus on bridgeless cubic graphs, which are not 3-edge-colorable; so called snarks. Snarks are of major interest in graph theory since they are potential counterexamples to many hard conjectures. Brinkmann et al. [1] generated all snarks with at most 36 vertices and they disproved a couple of conjectures concerning these graphs. The paper also gives an overview on conjectures which are related to snarks. In [11] it is shown that cubic graphs with high cyclic connectivity have a nowhere-zero 5-flow. This result can also be considered as a first approximation to a conjecture of Jaeger and Swart [7] who conjectured that every cyclically 7-edge connected cubic graph has a nowhere-zero 4-flow. The paper is organized as follows. The next section delivers Jaeger's characterizations of Petersen-colorable and strongly Petersen-colorable graphs, [5]. We show that type 1 Blanuša snarks have a strong Petersen-coloring while flower snarks do not have such a coloring. We study the structure of a minimum counterexample to the Petersen-coloring conjecture and finally we show that the flower-, the Goldberg-, and all Blanuša snarks have a Petersen-coloring. 2 Normal 5-edge-colorings Let G be a cubic graph and ^ : E (G) ^ {1,2,3,4, 5} be a proper 5-edge-coloring. An edge e = xy in G is poor if |^>(w(x)) u ^(w(y))| = 3 and it is rich if |^>(w(x)) u ^(w(y))| = 5. If every edge in G is either rich or poor, then ^ is a normal 5-edge-coloring. Jaeger characterizes Petersen-colorable and strongly Petersen-colorable graphs in terms of normal 5-edge-colorings. Theorem 2.1. [5] A cubic graph is Petersen-colorable if and only if it has a normal 5-edge-coloring. Theorem 2.2. [5] A cubic graph is strongly Petersen-colorable if and only if it has a normal 5-edge-coloring, and the set of poor edges forms a cocycle. If ^ is a normal 5-edge-coloring of a graph G, such that the set of poor edges forms a cocycle, then we call ^ a strong normal 5-edge-coloring. Jaeger [5] stated that cubic graphs with strong normal 5-edge-coloring do not contain a triangle (cf. Proposition 4.1). 3 Strong Petersen-colorings 3.1 Blanuša snarks The generalized Blanuša snarks were introduced by Watkins in [13]. Let A be the graph formed by removing two adjacent vertices from the Petersen graph. The generalized Blanuša snarks of type 1 are formed by joining n copies of the graph A as depicted in Figure 1 and one copy of the graph P2. Figure 1: The generalized Blanuša snark of type 1. Theorem 3.1. Every generalized Blanuša snark of type 1 with an odd number of A-blocks is strongly Petersen-colorable. Proof. Let G2n-1 be a Blanuša snark of type 1 formed by blocks Ai,..., A2n+1, P2 and let ^ be the coloring of the even respectively odd blocks as shown in Figure 2. Then it is easy to see that ^ is a normal edge-coloring where the set of poor edges is the set U{u(V(Aj))}2" and hence a cocycle. It now follows from Theorem 2.2 that G2n-1 is strongly Petersen-colorable. □ 3.2 Flower snarks In this section we will show that flower snarks do not have a strong Petersen-coloring. Let G be a graph which has a normal 5-edge-coloring. We first study possible partitions of the edge set of C6 (the cycle of length 6) into rich and poor edges. We denote the set of rich edges with R. Figure 2: A normal edge-coloring ^ of a generalized Blanuša snark of type 1 where the only poor edges are the diagonal edges between the blocks A1,..., A2n+1. Lemma 3.2. Let G be a cubic graph that has a strong normal 5-edge-coloring. If C6 is a subgraph of G, then the connected components of C6[E(C6) n R] are either C6 or two paths of length 2 or two isolated edges and two isolated vertices or six isolated vertices. Proof. Let G be a cubic graph that has a strong normal 5-edge-coloring and that contains C6 as a subgraph. Then the set of poor edges forms a cocycle by Theorem 2.2 and therefore, it partitions V(G) into two sets S and S' such that the following two conditions are satisfied: C1: If e = vw is a poor edge, then v e S if and only if w G S'. C2: If e = vw is a rich edge, then either v,w G S or v,w G S. Taken into account these two conditions, it is easy to see that the following claim is true. Claim 3.3. The number of rich (poor) edges in C6 is even. Figure 3: C6 Let the edges of C6 be labeled as indicated in Figure 3. Claim 3.4. The rich edges do not induce a path of length 4. Proof. Assume that e^e2,e3,e4 are rich. W.l.o.g. we may assume that 0(e^ = 1, 0(f1) = 2, 0(ee) = 3, 0f = 4, ^(e2) = 5. Then ^(f3),^(e3),^(fi),^(ei) = 5. Hence 5 g {0(f5), 0(e5)}. But on the other hand {0(e5), 0(f5)} = {1, 3} or = {2,3}, a contradiction. Claim 3.5. The rich edges do not induce a path of length 3 and an isolated edge in Ce. Proof. Assume that e1,e2,e3,e5 are rich. W.l.o.g. we may assume that ^(e1) = 1, 0(A) = 2, 0(ee) = 3, 0f = 4, 0(e2) = 5. This implies, that {^(e4), f)} = {1,4} and {0(e4), 0(f5)} = {4,5}; hence 0(e4) = 4. Thus 0(f5) = 5 = 0(f4), a contradiction. Claim 3.6. The rich edges do not induce precisely one path of length 2 in Ce. Proof. Assume that e1, e2 are rich. W.l.o.g. we may assume that ^(e1) = 1, ^(f1) = 2, 0(ee) = 3, 0f = 4, 0(e2) = 5. This implies that 3 g {0(f5), 0^)} and 5 g {0(e4), 0(f4)}. On the other hand we have that {0(e5), 0(ee), 0(fe)} = {1,2,3} and hence 5 g {0(e4),0(f5)}. But then 0(e4) = 3, 0(f4) = 5 and therefore 5 g {0(ee), 0(f5)}, a contradiction. □ For the further study we will go a little bit more into the details of possible (strong) normal 5-edge-colorings. Lemma 3.7. Let G be a cubic graph that has a normal 5-edge-coloring 0. If Ce is a subgraph of G and all its edges are rich, then E (Ce) is partitioned into three color classes, say 0-1(1), 0-1(2), 0-1(3), such that ej, ej+3 g 0-1(i), for i =1, 2, 3. Proof. Clearly, at least three colors appear at the edges of Ce since for otherwise there are two edges of the same color with distance 1, contradicting the fact that all edges are rich. If more than three colors appear at the edges of Ce, then there is a path of length 4, say e1, e2, e3, e4, whose edges are colored pairwise differently, say 0(ej) = i. W.l.o.g. we may assume that 0(f2) = 4 and 0(f3) = 5. Thus 0(f4) = 1, and since all edges are rich, it follows that {0(e5), 0(f5)} = {2,5}, {0(ee), 0(f1)} = {3,5}, and hence {0(ee), 0(fe)} = {1, 3} and {0(e5), 0(fe)} = {2,4}, a contradiction. It is easy to see that a coloring as stated in the claim exists. □ Lemma 3.8. Let G be a cubic graph that contains Ce as a subgraph and 0 be a strong normal 5-edge-coloring. If precisely two edges of Ce are rich, then they receive the same color. Proof. It follows from Lemma 3.2 that there are two non-isomorphic distributions of the rich edges. 1) The distance between the rich edges in Ce is 2. Assume that e1, e4 are rich. W.l.o.g. we may assume that 0(e1) = 1, 0(f1) = 2, 0(ee) = 3, 0(f2) = 4, 0(e2) = 5. Assume to the contrary 0(e4) = 1. Case 1: 0(e5) = 1. Then it follows that 0(f3) = 1 and 0(f4) = 1, contradicting the fact that e4 is rich. Case 2: 0(e5) = 1, i.e. 0(e5) = 2, and hence 0(fe) = 0(f5) = 1 and 0(e4) = 3. But 3 g {0(e2), 0(f3)}, a contradiction. 2) The distance between the rich edges in Ce is 1. Assume that e1, e3 are rich. W.l.o.g. we may assume that 0(e1) = 1, 0(f1) = 2, 0(ee) = 3, 0(f2) = 4, 0(e2) = 5. Assume to the contrary 0(e3) = 1. Then 0(e3) = 4 and hence 4 g {0(e5), 0(f5)}, and therefore in any case 4 g {^(e5), $(f6),^(e6)}. But on the other hand {^(e5), $(f6), ^(e6)} = {1, 2,3 }, a contradiction. □ Figure 4: Cg Let Cg be the graph of Figure 4 without the edges f1, f3, f4, f6. Our objective is to reduce the number of non-isomorphic partitions of the edge set of Cg into rich and poor edges to the five partitions shown in Figure 5. Figure 5: Five types of non-isormorphic partitions of E (Cg) into rich and poor edges. (The rich edges are bold.) Lemma 3.9. Let G be a cubic graph that has a strong normal 5-edge-coloring. If Cg is a subgraph of G and Ep, Er is a partition of the edges of E(Cg) into poor and rich edges, then this partition is isomorphic to one of the types in Figure 5. Proof. The result follows by case checking along the number r of rich edges in Cg. Let the edges of Cg be labeled as in Figure 4. It contains three C6 - with edge sets {e1, e2,..., e6}, {e1,f2, fo, f5, e5, e6}, and {e2, e3, e4, f5, f0, f2} - which share pairwise a path of length 3. r = 0: We obtain a partition of type A of Figure 5. r = 1: Then there is a C6 with an odd number of rich edges, contradicting Lemma 3.2. r = 2: By Lemma 3.2 any of the three C6 has either no rich edge or two rich edges, which induce two isolated edges. Now it is easy to see that type B of Figure 5 is the only solution (up to isomorphism). r = 3: By Lemma 3.2 any of the three C6 has two rich edges, which induce two isolated edges. It is easy to see that types C and D are the only possible solutions. r = 4: The matching number of Cg is 4. If r = 4 and the four rich edges induce a matching, then there is a C6 that contains an odd number of rich edges, a contradiction. Thus, by Lemma 3.2, we can assume that there is a C6 such that the rich edges induce two paths of length 2. The only realizable partition is of type E of Figure 5 (up to isomorphism). 5 < r < 8: It is easy to see that Lemma 3.2 can not be satisfied for all three C6 of Cg. r = 9: In this case, we obtain a contradiction to Lemma 3.7. □ The following lemma easily follows from Lemma 3.8. Lemma 3.10. Let G be a cubic graph that has a strong normal 5-edge-coloring. If Cg is a subgraph of G and the edges of E(Cg) are partitioned into poor and rich edges as shown in Figure 5 B, C or D, then the three rich edges receive the same color. The flower snarks are invented by Isaacs [4]. They are cyclically 6-edge connected and have girth 6, if k > 3. For k > 1, the flower snark J2k+1 has vertex set V(J2k+1) = {ai, bi, ci, di|i = 0,1,..., 2k} and edge set E(J2fc+i) = {bjOj, 6jCj, bidi; ajai+i; Cjdi+i; dici+1\i = 0,1..., 2k} (indices are added modulo 2k + 1). Theorem 3.11. For every k > 1, the flower snark J2k+1 is not strongly Petersen-colorable. Figure 6: Substructure of J2k+1 Proof. We show that the flower snarks do not have a strong normal 5-edge-coloring. Then the result follows with Theorem 2.2. Assume to the contrary that J2k+1 has a strong normal 5-edge-coloring Let Cg be the graph as indicated in Figure 4. The flower snark J2k+1 can be considered as the union of 2k +1 copies D0,..., D2k of Cg, where Di and Di+1 share precisely the subgraph which is induced by one vertex of degree 3 and its neighbors (indices are added modulo 2k + 1); see Figure 6. By Lemma 3.9, the five partitions of the edges of C6g shown in Figure 5 are the only non-isomorphic types of possible partitions of the edges of C6g into rich and poor edges. 1) There is i g {0 ... 2k} such that Di is of type E. Since Di shares with Di+1 a vertex of degree 3 with its three incident edges, it follows that Di+1 is of type E as well. Hence all Di are of type E and therefore all edges of the inner cycle of length 2k + 1 are poor, contradicting our assumption, that J2k+1 has a strong normal 5-edge-coloring. Thus all Di are not of type E. 2) There is i g {0... 2k} such that Di is of type D. Then Di+1 can be of any other type different from E. We may assume that the edge bici is rich. Hence ci-1di and ai-1ai are rich, too. All the other edges of Di are poor. If Di+1 is of type C or D, then it follows, that two different rich edges, one of Di and one of Di+1 are adjacent. By Lemma 3.10, they all have the same color, contradicting the fact that ^ is a coloring. Thus Di+1 is of type B. On the other hand, Di-1 shares with Di the vertex bi-1 of degree 3 which is incident to three poor edges. As above, it follows that Di-1 cannot be of type D; thus it is of type A. Since the number of the Di is odd it follows that the types A, B, C and D cannot combined to get a coloring of J2k+1. Thus all Di are not of type D. 3) There is i g {0... 2k} such that Di is of type A. Since Di shares with Di+1 a vertex of degree 3 with its three incident edges, it follows that Di+1 is of type A as well. Not all Dj can be of type A since then J2k+1 has no rich edges and therefore it is 3-edge-colorable, a contradiction. Thus all Di are of type B or C. 4) There is i g {0... 2k} such that Di is of type B or C. It follows that Di+1 is of type B or C. It turns out, that in any case the two rich edges which are adjacent to the trivalent vertices bi and bi+1 are of the form bici, bi+1di+1 or bidi, bi+1ci+1. This implies that eventually two edges bj cj and bj dj are rich, contradicting the fact that every Di is of type B or C. Since the five types of Figure 5 are the only possible strong normal 5-edge-colorings of Cg and no combination of them yields a strong normal 5-edge-coloring of J2 k+1, it follows with Theorem 2.2 that J2k+1 has no strong Petersen-coloring. □ 4 Structure of a possible minimum counterexample to the Petersen-coloring conjecture Jaeger [6] showed that a possible minimum counterexample to the Petersen-coloring conjecture must be cyclically 4-edge connected snark. If G contains a triangle, then let G- be the graph obtained from G by contracting the triangle to a single vertex. Clearly, every normal 5-edge-coloring of G- can be extended to one of G. On the hand, if a cubic graph G has a normal 5-edge-coloring then this coloring can be extended to any graph which is obtained from G by expanding a vertex to a triangle. The following proposition is a reformulation of Proposition 15 in [5]. Lemma 4.1. Let ^ be a normal 5-edge-coloring of a bridgeless cubic graph G. If there is an edge e which is contained in a triangle, then e is poor. Proof. Let e1 = v1v2, e2 = v2v3, e3 = v3v1 be the edges of a triangle T in G and let fi be the edge which is incident to vi and not an edge of T. Assume that e1 is rich, then |^(w(v1))) u ^(w(v2))| = 5 and hence e1, e2, e3, f1, f2 and f3 have to receive pairwise different colors; contradicting the fact that ^ is a 5-edge-coloring. □ Consider K3,3 with partition sets {u, v, w} and {v1, v2, v3}. Let K3,3 be the graph obtained from K3,3 by subdividing the edges uvi and wvi by vertices ui and wi, respectively. Graph K|,3 is shown in Figure 7. Figure 7: It is easy to see that the statements of this section are also true if we consider Fulkerson-colorings (i.e. a cover with six perfect matchings such that every edge is contained in precisely two of them) instead of Petersen-colorings. Theorem 4.2. If G is a minimum counterexample to the Petersen-coloring conjecture (or to the Fulkerson conjecture), then it does not contain K|,3 as a subgraph. Proof. Let ^ be a normal 5-edge-coloring of G, and assume that K|,3 is a subgraph of G. Remove the vertices u and w and add edges WjWj, for i = 1, 2, 3, to obtain a cubic graph G'. Since G is cyclically 4-edge connected it follows that G' is bridgeless. Thus G' has a normal 5-edge-coloring by induction hypothesis. Since uj, vj, wj span a triangle in G' (i = 1, 2,3), it follows by Lemma 4.1 that edge ujwj receives the same color as vvj. Thus is extendable to a normal 5-edge-coloring of G, a contradiction. The statement follows with Theorem 2.1. The proof for the Fulkerson conjecture is similar. □ This also yields a method to generated cubic graphs with normal 5-edge-colorings from smaller ones (with normal 5-edge-coloring). Let v be a vertex of a cubic graph with normal 5-edge-coloring and let wi, w2, w3 be the neighbors of v. Expand wj to a triangles Tj with vertex set {wj,1, wj,2, wj,3} such that v, wj,1 are incident, to obtain a graph G1. Then ^ can be extended to a normal 5-edge-coloring on G1. By Lemma 4.1 it follows that ^1(vwj1) = ^1(wj,2wj 3). Hence edges wj 2wj 3 can be removed and two vertices can be added so that we obtain a K3 3 as a subgraph and a normal 5-edge-coloring of the new graph. We will use this fact, to prove Conjecture 1.1 for flower snarks. 5 Petersen-colorings for some families of snarks 5.1 Flower snarks If a cubic graph G contains a K|,3 and we reduce it to a smaller graph G' as in the proof of Theorem 4.2, then G' contains three triangles. If we contract these three triangles to single vertices we obtain a new cubic graph G* that has 8 vertices less than G. Let us say that G is K|,3-reducible to G*. Theorem 4.2 can be reformulated as follows: Theorem 5.1. Let G be a cubic graph that is K33-reducible to a graph H. If H has a Petersen-coloring, then G has a Petersen-coloring. The following lemma is a simple consequence of Lemma 4.2 of [10]. Lemma 5.2. For k > 1, the flower snark J2k+3 is K33-reducible to J2k+1. Since J3 can be reduced to the Petersen graph by contracting the triangle to a single vertex, Theorem 5.1 and Lemma 5.2 imply the following theorem. Theorem 5.3. For all k > 1, the flower snark J2k+1 has a Petersen-coloring. 5.2 Goldberg snarks Let k > 5 be a odd integer. The Goldberg snark [3] Gk is formed from k copies B1,.. of the graph B in Figure 8 and the edges [aiai+1, cibi+1, eidi+1} for each i g {1,2,.. where indices are added modulo k. , Bk ., k} Figure 8: A block B in the Goldberg snark. Theorem 5.4. Every Goldberg snark Gk, where k > 5 is odd, has a Petersen-coloring. Proof. Let Gk be a Goldberg snark. Then Gk can be constructed from one 3-block (see Figure 10) and k-3 2-blocks (see Figure 9). Using the normal 5-edge-colorings provided in Figure 9 and 10 it is easy to see that it will give a normal 5-edge-coloring of Gk. □ 5.3 Blanuša snarks Let G be a Blanuša snark of type 1 as defined in Section 3.1. If we color the blocks A1,..., Ar-1 as in figure 11 and Ar and C1 as in figure 12 and 13, it is easy to see that we have a normal edge coloring of all such graphs. The generalized Blanuša snarks of type 2 are formed by joining r copies of A and one copy of C2 (see Figure 14). Once again it is straightforward to see that all such graphs has normal edges colorings by coloring A1,..., Ar-2 as in Figure 11, Ar-1 as in Figure 13 and finally C2 as in Figure 14. From this we get the following theorem. Theorem 5.5. All generalized Blanuša snarks of type 1 and 2 have Petersen-colorings. 4 4 Figure 9: A 2-block in the Goldberg snark with a normal 5-edge-coloring. Figure 10: A 3-block in the Goldberg snark with a normal 5-edge-coloring. Figure 11: Block Aj in the generalized Blanuša snark. Figure 12: Block Ar in the generalized Blanuša snark. Figure 13: Block P2 in the generalized Blanuša snark. Figure 14: Block C2 in the generalized Blanuša snark. References [1] G. Brinkmann, J. Goedgebeur, J. Hägglund and K. Markström, Generation and properties of snarks (2012), arXiv:120 6.6690v1. [2] D. R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971), 168-194. [3] M. K. Goldberg, Construction of class 2 graphs with maximum vertex degree 3, J. Combin. Theory Ser. B 31 (1981), 282-291, doi:10.1016/00 95-8956(81)90030-7. [4] R. Isaacs, Infinite families of nontrivial trivalent graphs which are not Tait colorable, Amer. Math. Monthly 82 (1975), 221-239. [5] F. Jaeger, On five-edge-colorings of cubic graphs and nowhere-zero flow problems, Ars Combin. 20 (1985), 229-244. [6] F. Jaeger, Nowhere-zero flow problems, in: Selected topics in graph theory, 3, Academic Press, San Diego, CA, pp. 71-95, 1988. [7] F. Jaeger and T. Swart, Conjecture 1 and 2, in: M. Deza and I. G. Rosenberg (eds.), Combinatorics 79, volume 9, 1980. [8] D. Kràl, E. Mäcajovä, O. Pangräc, A. Raspaud, J.-S. Sereni and M. Škoviera, Projective, affine, and abelian colorings of cubic graphs, European J. Combin. 30 (2009), 53-69, doi: 10.1016/j.ejc.2007.11.029. [9] P. D. Seymour, Sums of circuits, in: Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977), Academic Press, New York, 1979, pp. 341-355. [10] E. Steffen, Classifications and characterizations of snarks, Discrete Math. 188 (1998), 183-203, doi:10.1016/S0012-365X(97)00255-0. [11] E. Steffen, Tutte's 5-flow conjecture for highly cyclically connected cubic graphs, Discrete Math. 310 (2010), 385-389. [12] G. Szekeres, Polyhedral decompositions of cubic graphs, Bull. Austral. Math. Soc. 8 (1973), 367-387. [13] J. J. Watkins, On the construction of snarks, Ars Combin. 16 (1983), 111-124. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2013) 175-199 Constructions for large spatial point-line (nk ) configurations Gabor Gevay * Bolyai Institute, University of Szeged, Aradi vértanuk tere 1, H-6720 Szeged, Hungary Received 31 October 2011, accepted 12 April 2012, published online 19 April 2013 Highly symmetric figures, such as regular polytopes, can serve as a scaffolding on which spatial (nk ) point-line configurations can be built. We give several constructions using this method in dimension 3 and 4. We also explore possible constructions of pointline configurations obtained as Cartesian products of smaller ones. Using suitable powers of well-chosen configurations, we obtain infinite series of (nk ) configurations for which both n and k are arbitrarily large. We also combine the method of polytopal scaffolding and the method of powers to construct further examples. Finally, we formulate an incidence statement concerning a (IOO4) configuration in 3-space derived from the product of two complete pentalaterals; it is posed as a conjecture. Keywords: Spatial configuration, Platonic solid, regular 4-polytope, product of configurations, incidence statement. Math. Subj. Class.: 51A20, 51A45, 51E30, 52B15 1 Introduction By a (pq, nk ) configuration we mean a set consisting of p points and n lines such that k of the points lie on each line and q of the lines pass through each point [7, 16, 19, 21]. If, in particular, p = n, then q = k; in this case the notation (nk ) is used, and we speak of a balanced configuration [16]. We consider configurations embedded either in Euclidean or projective spaces. In the last decades, there has been a revival of interest in point-line configurations; the developments and results are summarized in the quite recent research monograph by Branko Grünbaum [16]. This book deals predominantly with planar configurations. However, as the author notices in the Postscript, a "...seemingly safe guess is that there will be interest in higher-dimensional analogues of the material described in this book". * Supported by the TAMOP-4.2.1/B-09/1/KONV-2010-0005 project. E-mail address: gevay@math.u-szeged.hu (Gabor Gévay) Abstract In fact, the theory was and has not been restricted to planar configurations. Research in higher dimensions go back to Cayley, Cremona, Veronese and others [4, 7, 10, 16]. One of the most well known spatial configurations is Reye's (124,163) configuration in projective 3-space [19, 25]. Its construction is based on the ordinary cube; the cube serves, so to say, as a "scaffolding": once the configuration has been built, the underlying cube is deleted. Moreover, the configuration inherits its symmetry (at least, in this case, in a combinatorial sense). We apply the same building principle in several of our constructions. For an underlying polytope, we choose a regular polytope in dimension 3 or 4. In all but one of these cases, the geometric symmetry group of the configuration will be the same as that of the underlying polytope. We note that the term geometric symmetry group, or briefly, symmetry group, is meant in the usual sense; i.e., it denotes the group of isometries of the ambient (Euclidean) space that map the configuration onto itself. The combinatorial counterpart of this notion is the group of automorphisms, i.e., the group of incidence-preserving permutations of points and lines (both among themselves); Coxeter simply calls it the group of the configuration [7]. (The distinction between these two types of groups will play some role in Section 5.) In Section 4 we apply the notion of a product, which can be considered as the Cartesian product of configurations. Using this tool, we construct two infinite series and a finite class of (nk) configurations which are powers of smaller configurations. In this way very high n and k values can be attained (and so can the dimension of the space spanned by these configurations). We think this may be interesting for future research; for, as it is emphasized just recently, little is known in general on the existence of such large configurations [3]. We summarize our results in the following theorem. Theorem 1.1. There exist (nk ) configurations which form the following classes: 1. infinite series of type they have the symmetry group of a regular tetrahedron, cube and dodecahedron, respectively, and each spans the Euclidean space E3; 2. infinite series of type (I8(t + 1)3), (36(t + 1)3) and (90(t +1)3), (t = 1, 2,... ); they span the projective space P2k; 3. infinite series of type (((2k)2fc-2(2k +1)k)2fc) , (k = 2, 3,... ); they span the projective space P2k; 4. finite class of types (2403 ), (7683) and (28 8003), with the symmetry group of a regular 4-simplex, a regular 4-cube, or a regular 120-cell, respectively; each spans the Euclidean space E4; 5. finite class of types (14 4004 ) C E6 (5 832 2006) C E9 (3 317 760 0008) C E12 ((2.43 • 1010)10 ) C E15 (218 • 312 • 56)i2 ) C E18 each full-dimensional in the given Euclidean space; 6. sporadic examples of type (I8O3), (6O4), (54O4) and (7804 ), with the symmetry group of a regular dodecahedron, all of them spanning the Euclidean space E3. In Sections 2-4 below, we construct all these configurations, and thus their existence is proved; the formula numbers of them at the location where they are actually constructed are as follows: 1: (3.8); 2: (4.1); 3: (4.2); 4: (3.14); 5: (4.5); 6: (3.10), (2.1), (2.2) and (2.3). In each case, an essential part of the constructive proof is to exclude unintended incidences (i.e., incidences that do not belong to the given configuration, cf. [16], Section 2.6). We emphasize that this possibility has been checked in each case. However, in all but one case this part of the proof was omitted, to save space (the one exception is the case of type (2.2), where the problem has been indicated in Remark 2.2). In stating our results, we avoid using the term "d-dimensional" for a configuration that we construct in some space of dimension d. The reason is that the dimension of a configuration C is defined as the largest dimension of the space that is spanned by C (see p. 24 and Section 5.6 in [16]). Thus, we can only state that each configuration in our theorem above spans the given space (sometimes we also say, equivalently, that the configuration is full-dimensional in the given space). Investigating the actual dimension of our configurations is beyond the scope of this paper. We think that the symmetry group is an essential property of our configurations realized in some Euclidean space. Although we did not investigate, we believe that several of these configurations have the maximal symmetry which is possible in the given space. (We note that e.g. for convex polytopes a question like this is far from trivial [12, 17].) It would also be interesting to know that for dd < d, whether there is a configuration in Ed whose symmetry group is a subgroup of the orthogonal group of Ed and which is not symmetrically realizable in Ed . We consider these and other related questions as a possible subject of future study. Finally, in Section 5 we present an incidence conjecture. It is suggested by one of the new configurations that we found. This is a (1004) configuration in projective 3-space consisting of four quintuples of complete pentalaterals (thus there are altogether 400 incidences, 100 for each quintuple). Informally, the conjecture states that the incidences belonging to three quintuples of the complete pentalaterals imply the remaining 100 incidences. 2 First examples of spatial configurations Before presenting our more detailed constructions, we remark that there are spatial pointline configurations which need little or no construction. They are there in the zoo of geometric figures which have been known for a long time; one just has to realize them. The first configuration presented here is just such an example. It is a nice (6O4 ) (2.1) configuration, provided by a polyhedron called the great icosidodecahedron. The lines of the configuration are spanned by the edges, and half of its points are the vertices of this polyhedron. (a) (b) Figure 1: The icosidodecahedron (a), and the great icosidodecahedron (b). The great icosidodecahedron is one of the 53 non-regular non-convex uniform polyhedra [8, 18, 22] (we note that a polyhedron is called uniform if its faces are regular polygons and its symmetry group is transitive on its vertices). It also occurs in [6], given by its Coxeter's symbol j5^ j ; this indicates that it has triangles and pentagrams (i.e. {5/2} star-polygons, see [6]) as faces. The number of these faces is 20 and 12, respectively. Its name refers to its close relationship with its convex hull, the icosidodecahedron (one of the Archimedean solids). In particular, its 30 vertices coincide with those of the icosidodeca-hedron. The mutual position of its two pentagram faces in non-parallel planes is of two kinds (just like that of the faces of the regular dodecahedron): the angle between them is either arctan 2, or n - arctan 2. The angle between the planes of two such faces sharing a common vertex is the acute angle. In addition to its vertices, the edges of a pentagram have five other intersection points (these points can be called "internal vertices" if, instead of a pentagram, we speak of a—complete—pentalateral, cf. Section 5 below). These "internal vertices" of the pentagrams do not belong to the vertex set of the polyhedron (and, in strict sense, not even to that of the pentagram). But two pentagrams in planes with obtuse angle between them share such an "internal vertex". Taking into account these latter points as Figure 2: The (604) configuration associated with the great icosidodecahedron. well, we have a system consisting of 12 x 5 = 60 edges, and 30+30 points, the latter all tetravalent. Replacing the edges by the lines that are spanned by them, we obtain directly the configuration (2.1). The symmetry properties of the underlying polyhedron imply that this configuration has two orbits of points and a single orbit of lines. This relatively high degree of symmetry makes it particularly interesting. The same configuration is also provided by two other types of polyhedra in the same natural way. Namely, both the great icosihemidodecahedron and the great do- decahemidodecahedron j^0/3 j has a system of vertices and edges coinciding with that of the great icosidodecahedron; and, the other 30 points of the configuration are provided likewise. With the details omitted, we just remark that all three types of these polyhedra can be derived from the regular dodecahedron. We note that a much more simple example can be obtained from the ordinary cube, in the following way. Let the points be the 8 vertices of the cube, the 6 centres of the faces of the cube and the centre of the cube. As lines, choose the 12 diagonals of the faces of the cube, plus the 3 lines between the centres of two opposite faces of the cube. Thus we obtain a (153) configuration. Our next two examples require some more steps of construction. We start from two planar configurations. The first is a (254) configuration, due to Jürgen Bokowski ([15], Figure 4; see also [16], Figure 3.3.13). The other is closely related to this and is due to Branko Griinbaum ([15], Figure 9). These configurations are shown in our Figure 3. Figure 3: Two planar configurations. Put 12 copies of the (254) configuration onto the faces of a regular dodecahedron so that the vertices of the pentagonal "frame" of the configuration coincide with the midpoints of edges of the dodecahedron. Then delete the edges of the dodecahedron and all the external edges of the 12 pentagonal frames. Thus we obtained a system, which is not a configuration; however, for each of its lines there are precisely four points incident with it. A system like this deserves to be the subject of a new definition. Definition 2.1. A set consisting of p points and n lines is called a semiconfiguration if either of the following conditions hold: (1) each point is incident with precisely q lines; or (2) each line is incident with precisely k points. The type of a semiconfiguration is denoted by (pq, n*) or (p*,nk), respectively. If one wants to specify which version is actually used, one may call it a P-semiconfiguration or an L-semiconfiguration; thus, the abbreviation refers to the fact that the incidences are uniformly distributed among the points or the lines, respectively. Clearly, a system is a configuration if and only if it is both P-semiconfiguration and L-semiconfiguration. Using this notion, we see that the system we obtained in the present step is an L-semiconfiguration of type (270*, 2404). It contains a class of 120 trivalent points, a class of 120 tetravalent points, and a class of 30 tetravalent points. These classes are distinguished by the position of their points; in fact, they are transitivity classes with respect to the symmetry group of the dodecahedron. To obtain a balanced configuration, take a second, concentric and homothetic copy of this semiconfiguration; thus we have an "outer shell" and an "inner shell" of points. Finally, connect the trivalent points of these shells by radial lines. Thus we have 60 new lines, all incident with four points. At the same time, the 2 X 120 trivalent points turned into tetravalent. Hence we obtained a configuration of type (5404 ). (2.2) It has 6 orbits of points and 7 orbits of lines. It is shown in Figure 4. Although this figure, due to the large number of its elements, is necessarily somewhat crowded, the two shells can be distinguished; the radial lines are indicated by orange colour. Remark 2.2. The relative size of the outer and the inner shell must be chosen carefully in order to avoid unintended incidences. Clearly, there are infinitely many choices. Using the (354) configuration, and proceeding analogously, we obtain a (7804) (2.3) configuration. It has 8 orbits of points and 13 orbits of lines. We note that one may find several analogous cases on the basis of our examples (2.2) and (2.3). For example, a geometrically different but completely analogous (5404) configuration can be obtained by starting from another planar (254) configuration that is shown in [2], Figure 4a. Furthermore, starting from the same planar configurations P with pentagonal symmetry as above, one can also obtain geometrically different examples if one chooses other points of P to tack onto the edges of the dodecahedron's face, and delete the appropriate lines (the radial lines will also be different in these cases). The full icosahedral symmetry can also be reduced so as to obtain a chiral configuration in this construction as well. One just has to replace the starting planar configuration by a suitable chiral one. What is more, even movable spatial examples can be obtained in Figure 4: A (5404) configuration. this way, too; see a beautiful construction in [1] for movable planar (n4) configurations (the example with 10-fold rotational symmetry given there in Figure 4 may be a possible candidate to this purpose). We do not pursue this idea here. 3 Classes of configurations based on regular polytopes First we construct three infinite series of balanced configurations, so that we make use of the structure and symmetry properties of Platonic solids. (We note that these series have already been mentioned in [13], p. 327). In what follows, TP denotes a Platonic solid whose 1-skeleton is a trivalent graph, i.e. TP is a tetrahedron, cube, or dodecahedron. It is well known that the Petrie polygon of these polytopes is a (regular, skew) quadrangle, hexagon, or decagon, respectively. (We recall that the Petrie polygon of a regular 3-polytope is a skew polygon such that any two consecutive edges, but no three, belong to a face of the polytope [6].) Given a Petrie polygon, consider for each of its vertices the third edge emanating from it but not belonging the to Petrie polygon. Take a point on each of these edges such that it subdivides, but not bisects, the edge in an arbitrary but fixed ratio; moreover, it is closer to the endpoint of the edge belonging to the Petrie polygon than to the other endpoint. Connect these points by straight line segments in the cylic order induced by the Petrie polygon. What is obtained is again a regular skew polygon, clearly having the same number of edges as the Petrie polygon we started from. We shall call it a P -polygon. The vertices of a P-polygon, together with the lines spanned by their edges, form a (p2 ) configuration, where p is 4, 6 or 10 according as TP is the tetrahedron, the cube or the dodecahedron, respectively. Clearly, the number of Petrie polygons and P-polygons is the same in a given TP, that is 3, 4 or 6, respectively. It follows that taking the disjoint union of all these P-polygons (more precisely, the corresponding (p2) configurations), one obtains a (non-connected) configuration whose type is (122), (242) or (6O2). (3.1) We call this configuration a P-system (see Figure 5). (a) The whole system. (b) Edge crossing of two P-polygons. Figure 5: The P-system (242) in the cube. We emphasize that when constructing the P-system, the same ratio is used in the definition of each P-polygon. Thus it follows from the construction that the P-system inherits the (geometric) symmetry of TP, i.e. it has the same symmetry group. Moreover, note that both its points and its lines form a single transitivity class. Looking at this configuration more closely, we find that it can be extended to form a non-balanced but connected configuration. For, observe that each edge e of TP belongs to precisely two distinct Petrie polygons. Moreover, these Petrie polygons are mirror images of each other with respect to the mirror plane of TP containing the edge e. It follows that the edges of the corresponding P-polygons cross each other in the vicinity of e in a point lying in that mirror plane. Furthermore, this point also lies in the mirror plane that is perpendicular to the former plane and bisects e. This amounts to saying that this point is on the line connecting the centre of TP with the midpoint of e. (For an example of such a crossing point in the cube, see Figure 5b, where it is indicated by red colour). The number of these crossing points equals the number of the edges of TP. Hence, adding them to the P-system, we obtain a new configuration, which we shall call a P-configuration. Its type is The special position of the crossing points provides the possibility of a further extension of this configuration, as follows. Shrink the 1-skeleton of the original TP until the midpoints of its edges coincide with the crossing points, and add it to the configuration. Then, remove the vertices of this skeleton, and replace each of its edges by the line spanned by it. The new structure that is obtained is a subfiguration. A subfiguration (pq ,nk) is defined as a set of points and lines with incidences as in the definition of configurations, but with the difference that each of the p points is incident with at most q of the n lines, and each line is incident with at most k of the points [16]. If we want to emphasize that the number of the missing incidences is s (in comparison to a (pq, nk) configuration), we say that it is an #s-subfiguration. (We note that in the converse case the notion of a superfiguration has also been introduced, in a similar way, in [16]). Thus, the subfiguration that is obtained is of type with s equal to 24, 48 or 120, respectively. (Here the superscript refers to the missing incidences; we use this notation to avoid confusion with a configuration of type (nk ).) One half of the missing incidences of this subfiguration belong to the one subsystem, and the other half of them belongs to the other subsystem, of which it is composed. For example, in the cubic case there are 24 missing incidences because the points of the P-system are of valency two, instead of three; and there are 24 other missing incidences, since all the 8 vertices of the cube skeleton (which are of valency three, and have been removed) are missing. We shall call these two kinds of defective points type A and type B, respectively. Likewise, the corresponding subsystems will be referred to as type A and B, respectively. Due to the equal number of the defective points of the two types in the two subsystems, this subfiguration can serve as a repetitive unit; hence we shall call it an R-unit. The repetition is meant in the following way. Take a copy of an R-unit R\, and shrink it so as to obtain a homothetic copy R2, such that the points of type A of R2 fit onto the corresponding lines in the subsystem of type B of Ri ; then take the union R1 U R2. Using Figure 5a, (I82,123), (362,243) or (902,6O3). (3.2) (I83)-, (363)" or (9O3)-, (3.3) in the cubic case this can simply be conceived as if the brown lines belonged to R1 and the green lines belonged to R2. Observe that the new figure obtained in this way is again an #s-subfiguration such that s remains the same; for, half of the defects both in R1 and R2 have been repaired, but the other half in both of them remained. At the same time, both the number of points and lines have been doubled. The operation that we applied here is not simply a disjoint union; for, new incidences occured, and (in our particular case) the result is a connected structure. Thus we think it is appropriate to fix these properties in a separate definition. As that will refer not only to configurations, first we give a common name for all the four related types of structures used in this paper: we shall call such a structure an X-figuration, where "X-" may mean either "con", "semicon", "sub" or "super". Definition 3.1. By the incidence sum1 of X-figurations F1 and F2 we mean the X-figuration F which is the disjoint union of F1 and F2, together with a specified set I C?1 x L2 U V2 xL1 of incident point-line pairs, where V denotes the point set and L denotes the line set of Fi, for i = 1,2. We denote it by F1 ©/ F2. Note that F1 and F2 may form distinct incidence sums depending on the set I ; we do not consider here such cases; on the other hand, if the set I is fixed and is clear from the context (as in our present case), it can be omitted from the operation symbol. Accordingly, in the present step of our construction we obtained the subfiguration of the form R1 © R2. Furthermore, it is clearly seen that the process by which we obtained R1 © R2 from R1 can be repeated arbitrary many times. Thus, let A be a shrinking factor defined by the equality R2 = AR1, and set R = R1. Then, starting with R, we obtain after t — 1 steps the subfiguration t 0 Ai-1R, (3.4) i= 1 which is still an #s-subfiguration with s = 24,48 or 120, and is of type ((18t)3)-, ((36t)3)- or ((90t)3)-, (3.5) respectively. Finally, we have to extend this subfiguration, so as to obtain a configuration. First we construct a unit which, when added, closes the structure "outside". This construction also is analogous for each of the three types of TP; we explain it in the case of the tetrahedron. Start from the 1-skeleton of a regular tetrahedron, and take the midpoints of its edges. Add these points to the structure, so that one obtains a spatial graph with 10 vertices and 12 edges, such that four vertices are trivalent and six vertices are bivalent. For each bivalent vertex, take a line connecting it to the centre of the tetrahedron, and shift the vertex along this line outwards, each to the same extent; simultaneously, the edges incident to these vertices are stretched, and remain straight line segments. Although any ratio would serve our purpose, we note that if the distance of these shifted vertices from the centre is twice that of the original, then the angle between any two adjacent edges is arccos( — 1 /3). This is the famous "tetrahedral bond angle" in organic chemistry, and the figure that we obtained is precisely the carbon skeleton of a hydrocarbon molecule called adamantane2, well known 1 The present (improved) version of this definition was proposed by TomaZ Pisanski. 2 The name refers to the fact that this is a repetitive unit of the diamond crystal lattice. to chemists. This skeleton is shown in Figure 6 (by courtesy of H. Ramezani, from [24]). A related figure can be obtained from the 1-skeleton of either kind of TP in an analogous way; so we shall call each of them an adamantane skeleton. (Note that they have the symmetry of the type of TP from which it has been derived). Figure 6: The adamantane skeleton. We extend the adamantane skeleton in the following way. Take the 1-skeleton of a TP of suitable size and position (and of the corresponding type) such that the midpoints of its edges coincide with the bivalent vertices of the adamantane skeleton. Form the union of these figures, then replace each of its edges by the line spanned by it. We obtain a P-semiconfiguration of type (143,18*), (283,36,) or (703,90,). (3.6) This semiconfiguration will serve as a closure unit so as to close our construction "outside". In fact, observe that it has defective lines, i.e., lines that are incident with two points, instead of three (the lines corresponding to the original half-edges of TP). The number of these lines is 12, 24 or 60, respectively, and this is the same as the number of the missing incidences. On the other hand, it is half of the number of missing incidences of our subfiguration (3.4) (these latter come from points of type A). Furthermore, by taking a copy of a suitable size of this closure unit, one can form the incidence sum of it with the subfiguration (3.4). The result is a semiconfiguration of type ((18t + 14)3, (18(t + 1)),), ((36t + 28)3, (36(t + 1)),) or ((90t + 70)3, (90(t + 1)),). (3.7) The very last step of our construction is to close our system "inside". This is very simple, since 4, 8 or 20 points are missing from the smallest R-unit of the subfiguration (3.4) (points of type B, each representing three incidences). These are nothing else than the vertices of a tetrahedron, cube or dodecahedron, respectively. We just add these points to the system, and our construction is ready, resulting in three infinite series of balanced configurations, whose type is (18(t +1)3), (36(t +1)3) and (90(t + 1)3), (t = 1,2,... ). (3.8) Figure 7: A (723) configuration: the cubic case with t =1 of (3.8). We note that the latter "closure units", consisting merely of points (but with fixed mutual position), can also be conceived as semiconfigurations. Denoting them by CI, and those in (3.6) by CO, we see that our configurations of type (3.8) can be described (and in fact, have been constructed) in the following form: Co © Ai-1fij © CI, (3.9) where the middle term is the subfiguration (3.4). We emphasize that throughout the construction, the original symmetry of the TP we started from is preserved. Thus the (geometric) symmetry group of all of the configurations which we obtained here is equal to that of the corresponding Platonic solid. Our next construction provides a sporadic example. In this construction we apply some structural elements that have already been constructed above. Start from the compound of five tetrahedra, which can be obtained by inscribing these tetrahedra in a regular dodecahedron [6] (see Figure 8). This is made possible by the property that the set of vertices of the dodecahedron can be partitioned into five quadruples such that within a quadruple, the vertices are at a distance 3 from each other (regarded in the graph of the dodecahedron). The same compound also determines a partition of the set of edges of the dodecahedron into five sextuples in the following way. Consider a compound inscribed in a dodecahedron of a fixed size. Apply a dilation to this compound. It is chosen so that the following condition holds. Let ABC D be any path of length 3 in the graph of the dodecahedron, and let M be the midpoint of the corresponding tetrahedron edge AD in the compound. Then the dilate M' of M coincides with the midpoint of the edge BC of the dodecahedron. Thus, each sextuple of the dodecahedron edges corresponds Figure 8: The compound of five tetrahedra inscribed in a dodecahedron. to the set of edges of a tetrahedron in the compound. A consequence is that one can inscribe a (tetrahedral) adamantane skeleton in each sextuple of the edge-midpoints of the dodecahedron so that the bivalent vertices of the adamantane skeleton coincide with these edge-midpoints. In such an inscribed adamantane skeleton, we inscribe a P-configuration (3.2) constructed previously in this section, in the sense that the points of the latter fit onto the edges of former; moreover, this is performed so that the original local tetrahedral symmetry is preserved. Then we add a tetrahedron skeleton to this structure, so that the midpoints of the tetrahedron edges coincide with the "crossing points" of the P-configuration (cf. Figure 5b). Replace each edge by the corresponding line; thus, taking into account all the points, lines and incidences, we obtained a (32*, 303) semiconfiguration inscribed in a sextuple of edges of the dodecahedron. By inscribing altogether five copies of this semiconfiguration into the 1-skeleton of the dodecahedron in the same way (and replacing its 30 edges by lines), a configuration of type (1803) (3.10) is obtained (it is mentioned in [13], too). The symmetry group of the compound of tetrahedra which we started from is the rotation group T of the tetrahedron; thus this compound is a chiral figure, i.e. it has no mirror symmetry. Our new configuration inherited this symmetry group, so it is a chiral configuration. Its set of points decomposes into 6 orbits, while there are 4 orbits of lines. These latter orbits are indicated with different colours in Figure 9. Note, in addition, that the structure of this configuration is closely related to those described above in this section. In fact, the orbits correspond to those of the tetrahedral case of (3.8), with t = 1. The only difference is that the outermost tetrahedral orbit is replaced by a dodecahedral orbit, and the others are multiplied by five. Our last class constructed here is based on certain regular 4-polytopes. We start from a TP that we used above. Take a P-system inscribed in it (that is, inscribed in the sense that the points of the configuration lie on the edges of TP). Then take a smaller homothetic copy of TP in concentric position, and also inscribe a P-system in this copy. The smaller P-system is chosen so that it is not the homothetic copy of the larger one, but each triple of their vertices in the vicinity of a vertex of the TP (determining it) is relatively at a smaller distance from that vertex, than the triple of vertices of the larger P-system is from the corresponding vertex of the larger TP. If the two P-systems were homothetic copies of each other (with respect to their common centre), then the lines connecting their corresponding points would meet all in a common point (in fact, in the centre). However, due to our particular choice, these lines meet now in threes, forming altogether 4, 8 or 20 points of intersection (depending on the type of TP). This is a consequence of the threefold rotational symmetries of TP. Thus for each such triples of lines in the vicinity of a given vertex v of TP there is a point of intersection which lies on the axis of rotation connecting v with the centre, and this point also is located in the vicinity of v. We shall not use these points later, they just served to explain the location of the connecting lines. On the contrary, we need the connecting lines in the following, so we shall call them c-lines. Another condition for the c-lines is that they are not perpendicular to the edges of TP (this can also be ensured by a suitable choice of the P-systems). Figure 9: A (1803) configuration derived from the compound of five tetrahedra. The two copies of the P-system, together with the c-lines, form a configuration whose type is (243,362), (483,722) or (I2O3, I8O2) (3.11) (see (3.1) above in this section for the type of the P-system). By adding the crossing points of the lines of the P-systems, discussed in the first construction of this section, one obtains an s-subfiguration of type (363)-, (723)- or (I8O3)- (3.12) with s = 24,48 or 120, respectively (note that the missing incidences are equally distributed between the points and the lines). Let now P be a regular 4-polytope whose facets are of type TP. Thus P is a regular 4-simplex, a regular 4-cube, or a regular 120-cell. It has 5, 8, or 120 facets, respectively [6]. We put a copy of the subfiguration (3.12) in each of the facets of P, so that each of the points of such a subfiguration is in the interior of the facet, and the whole system preserves the original symmetry of P. This results in a (non-connected) subfiguration of type (I8O3)-, (5763 )- or (2I6OO3)-, (3.13) respectively. We convert it to a connected structure as follows. Consider a facet F of P, and a c-line connecting two points of the P-systems whithin F. Clearly, this c-line intersects the edge of F, which is in the vicinity of these points (since it is within the local mirror plane of the facet lying on that edge). There are three facets of P meeting in a common edge; thus the point of intersection of the c-lines is triva-lent. There are two such points on each edge of P. Thus, by adding all these points to our structure (3.13), the number of (trivalent) points increases by 20, 64 or 2400, respectively. Note that with this completion the number of the missing incidences has been halved. The other half is supplied as follows. Take the 1-skeleton of TP, and put a pair of its copies in each facet of P. The size and location of these copies is such that for each of them there is a P-system in (3.13) in which the crossing points coincide with the midpoints of the edges in TP. Finally, replace each of the edges by the line spanned by it. In this way we supplied not only the rest of the missing incidences, but completed the structure by 40, 128 or 4800 points, and by 60, 192 or 7200 lines, respectively. As a result, we obtained three new balanced configurations in E4 whose type is (24O3), (7683) and (28 8OO3). (3.14) Note that in each step of the construction the original symmetry was preserved, thus the symmetry group of these configurations is equal to that of the regular 4-polytope we started from. In each of these three configurations of type (n3) the number n is twice the order of the corresponding symmetry group. Furthermore, in all three cases, there are 7 orbits of points and 5 orbits of lines. 4 Cartesian product of point-line configurations We explore here the following notion. Definition 4.1. Let C1 be a (pq, mk) configuration in an Euclidean space Ei and C2 be an (rs, nk) configuration in an Euclidean space E2. Observe that these two configurations have the same number k of points on each line. The Cartesian product of C1 and C2 is the {(pr)(q+s), (pn + rm)fc) configuration C1 x C2 in E1 x E2 whose point set is the Cartesian product of the point sets of C1 and C2 and where there is a line incident to two points (x1, x2) and (y1, y2) if and only if either x1 = y1 and there is a line incident to x2 and y2 in C2, or x2 = y2 and there is a line incident to x1 and y1 in C1. We emphasize that the incidence degree of the lines of the two configurations C1 and C2 have to coincide. Therefore, in terms of abstract algebra, this product is merely a partial operation on the set of configurations (it is not defined for any pair of configurations). This shows that, when applied to configurations, the analogy of this kind of product with the classical Cartesian product of other objects (like polytopes, graphs, etc.) is not complete, in strict sense. On the other hand, one observes that if the incidence degrees differ, then this product can still be defined, and it results in a semiconfiguration (see Definition 2.1). Furthermore, the definition of the product can also be extended to semiconfigurations. Thus, the larger set of semiconfigurations will be closed under this product, and the partial operation extends to a total operation. Hence using the term Cartesian product is still justified, in this sense. A consequence of the definition that if both C1 and C2 is full-dimensional in E1 resp., in E2, then C1 x C2 is also full-dimensional in E1 x E2. We note, however, that one cannot say that in the product the dimensions of C1 and C2 are added (see the remark on the dimension of a configuration in the Introduction). Thus, we do not think that our definition of product would automatically imply the additivity of dimension of configurations. We remark that our motivating example is the spatial version of the Gray configuration consisting of 27 points and 27 lines. Actually, it provided the intuitive idea for the definition above, see Figure 10. We note that the (273) Gray configuration can in fact be decomposed into the product of three (31,13) configurations; however, to visualize the intuitive idea we think the decomposition given in Figure 10 is better. More generally, the ((kfc)fc) generalized Gray configuration is the kth power of the (k1,1k ) configuration. (For the Gray configuration and the generalized Gray configuration, see [23]). configuration Figure 10: The Gray configuration as a product. Accordingly, we formulated the definition above in the context of Euclidean geometry. However, an analogous construction also works in projective spaces, which can be described as follows. First recall that a d-dimensional (real) projective space Pd can be defined as the set of one-dimensional (linear) subspaces of Rd+1. Given Pd in this way, let {e1,..., ek, ek+1,..., ed+1} be a basis in the corresponding vector space Rd+1. Now we have a projective space Pk given as the set of one-dimensional subspaces of the vector space spanned by the basis {e1,..., ek, ek+1}, and a projective space Pd-k determined analogously by the basis {ek+1,..., ed+1}. In this case we say that Pd is decomposed to the direct sum of the spaces Pk and Pd-k. More generally, let Pk and P1 be two projective spaces. If there are projective isomorphisms Pk = Pk and P1 = P1 such that Pk and P1 form a direct sum decomposition of a space Pk+', then Pk+1 is said to be the direct sum of the spaces Pk and P1. It is not hard to see that this definition determines a unique bijec-tion from the Cartesian product Pk x P1 to Pk+1; thus the points in Pk+1 can uniquely be represented by pairs (P, Q) with P g Pk, Q e P1. Now given the configurations C1 and C2, embedded in P1 and P2, respectively, both full-dimensional, the point set of their product consists of pairs (P1,P2) with P1 e P1, P2 e P2; furthermore, two points (P1, P2) and (Q1, Q2) are connected by a line in the product if and only if either P1 = Q1 and P2 and Q2 are connected in C2, or P2 = Q2 and P1 and Q1 are connected in C1. Clearly the product configuration is full-dimensional in the direct sum of P1 and P2, and its type is determined by the types of C1 and C2 in the same way as before. It is clear that the product of a configuration with itself can be repeated, i.e. it can be raised to a power; given a configuration of a suitable type, this may provide a balanced configuration (as we have seen above in the case of generalized Gray configurations). In what follows we give some classes of such examples. Examples: Class 1. Consider n lines in the projective plane P2 in general position, i.e. such that no more than two of them intersect in one point. Together with all their points of intersection, they form a configuration ((")2, , which we call a complete n-lateral. (We note that it has already appeared in this context in [21], see p. 85, Satz 21.) Taking (2k+1)-laterals (k = 2, 3,... ), we have the following infinite series of balanced configurations obtained as powers: (102,54)2 = (1004) c (212,7e)3 = (92616) C (362,98)4 = (1 679 6168) C (552,1110)5 = (50328437510) C complete complete complete complete 5-lateral: 7-lateral: 9-lateral: 11-lateral: P4 P6 P 10 The general element of this series can be given as '2k + 1 2 , (2k + 1) 2k 2k + 1 2 2k (4.1) and is full-dimensional in the projective space P2k. Examples: Class 2. Again, let k = 2,3,..., and start from the simple configuration ((2k)1,12k) consisting of 2k points and a single projective line. Raise it to the power 2k - 2 so as to obtain a k 2 configuration of type (((2k)2k-2)2k-2 , ((2k)2k-3(2k - 2))2k) . Then form the product of this configuration with the complete (2k + 1)-lateral. The result is a balanced configuration of type (((2k)2k-2(2k +1)k)2k ) , (4.2) which spans the projective space P2k. We note that in this series the number of points grows faster than in the former one. For comparison, we give the type of the first four members: (I6O4), (27 2166), (9437184g), ((5.5 • 10%) . Examples: Class 3. The method of scaffolding polytopes and raising to powers can also be combined to obtain balanced configurations. Here we construct in this way a finite class of examples in Euclidean space. We start from the well-known Archimedean solid, the rhombicosidodecahedron [9] (see Figure 11). It can obtained from the regular dodecahedron by truncation [5, 11]; thus it is bounded by 12 pentagons, 20 triangles and 30 squares, originating from the faces, vertices and edges of the dodecahedron, respectively. Its 60 vertices can be given in the following form: (±1, ±1, ±t3)c , (±T, ±2t, ±t2)c , (O, ±(2 + T), ±T2)c , (4.3) where the superscript denotes that all cyclic permutations of the coordinates are to be taken, and t denotes the golden mean: t = 1 (1 + %/5). Figure 11: The rhombicosidodecahedron with a regular pentalateral. Figure 12: The (3602, 6012) configuration based on the rhombicosidodecahedron. We use the rhombicosidodecahedron (briefly, RID) as a scaffolding to construct a class of configurations whose types are as follows: (1202,6O4); (1802,60s); (24O2,60g) ; (3OO2,6O10) ; (3602,6O12). (4.4) Observe that in the boundary complex of the RID, the link (we use this term following e.g. [26], p. 237) of a pentagonal face forms a regular decagon. Connecting by straight lines the vertices of this decagon that are pairwise at a distance 3 from each other, one obtains a (102,54) configuration, which is a regular complete pentalateral (see the definition of a complete n-lateral in our examples of Class 1; now we are in E3, and this figure is regular in Euclidean sense, i.e. its symmetry group is the dihedral group D5). Figure 11 shows which one of the two possible positions of such a pentalateral is chosen (it can also be seen that five of its points are inside the RID). Clearly there are altogether 12 such regular pentalaterals, and they form a single orbit under the action of the symmetry group of the RID (this group is obviously the full icosahedral group Ih). Hence we obtain a system of 60 lines, which together with the vertices of the pentalaterals form a (1202, 604) configuration (see the first type of (4.4)). It turns out, however, that there are altogether 360 intersection points of these 60 lines, so that the whole set of points and lines forms a (3602,6012) configuration (the last type of (4.4)). This can be explained using the symmetry properties of the regular dodecahedron or, equivalently, of the RID. First, recall that a regular dodecahedron has altogether 15 mirror planes. For each edge of the dodecahedron, there are precisely three mirror planes in special position: one lies on it, one is its perpendicular bisector and one is parallel to it. The others intersect it obliquely. The 60 lines are parallel in pairs to the 30 edges of the dodecahedron (and, none of them lie on a mirror plane). Hence, for each of these lines, too, there are precisely 12 mirror planes in oblique position. Because these planes are mirror planes, their intersections with the lines provide points in which precisely two of the 60 lines meet. This is equivalent to the fact that on a given line no two of the 12 intersection points coincide. For, the coincidence means that more than two planes (not perpendicular to each other) meet in such a point, which implies that more than two lines meet in that point. But such multiple intersection does not occur here; this can be visually checked in a model constructed by a dynamic geometry software3. Figure 12 shows a screenshot of this model. (a) The convex hull of point class (1) (b) The convex hull of point class (2) K (c) The convex hull of point class (5) (d) The convex hull of point class (6) Figure 13: Supporting polytopes for the (3602, 60i2) configuration. The 360 points can be partitioned into 6 classes with respect to their distance from the origin, each containing 60 points; they also form orbits under the action of the group Ih. Because of this latter property, the convex hull of each of them is a vertex-transitive polytope (in fact, in each case it is combinatorially equivalent to an Archimedean solid, see Figure 13). In this way these polytopes form a nested sequence, and are particularly suitable for visualizing the structure of our (3602, 60i2) configuration. Hence we call them 3Euler 3D developed by Tamas Petro. http://www.mozaik.info.hu/Homepage/Mozaportal/MPeuler3d.php supporting polytopes of this configuration. In Figure 13 they are shown in the order of growing size. The convex hull of class (3) is just the RID we started from; and class (4) is a homothetic copy of class (2) (actually, t times larger), so we did not repeat the corresponding polytope in the figure. Note that these six polytopes fall by two into three combinatorial types. Observe that these 6 classes of points can be switched in and out independently of each other; hence, all but last of the five types in list (4.4) above can be realized as more than one geometrically distinct configuration. (Among them, isomorphism may occur; we did not investigate this possibility.) By raising them to a suitable power, one obtains balanced configurations of the following types: (1202,6O4)2 (I8O2, 60e)3 (2402,60s)4 (3002, 60io)5 (3602, 60i2)6 (14 4004) (5 832 2006) (3 317760 0008) ((2.43 ■ 10l°)io ((218 ■ 312 ■ 56)12) C E C E6 C E9 C E12 C E15 C E18 (4.5) Due to the geometric differences we mentioned just above, a number of geometrically distinct cases occur here as well. For example, even for the (14 400)4 configuration, this amounts to 125 geometrically distinct cases (possibly not all combinatorially distinct). 5 An incidence conjecture Recall our definition of a complete n-lateral in the preceding section (examples of Class 1). For the case n = 5 we use the term complete pentalateral. The points of this configuration we shall also call vertices. The following properties of complete pentalaterals are well known (cf. [21], pp. 85-86, Satz 21 and Aufgabe 3b). Proposition 5.1. There is a unique complete pentalateral in the projective plane P2 up to combinatorial equivalence. It decomposes P2 into one pentagonal, five quadrangular and five triangular regions. We shall call the vertices of the complete pentalateral belonging to the pentagonal region internal vertices, while the other external vertices. The existence and uniqueness of the pentagon guarantees that such a distinction is indeed possible: Proposition 5.2. The partition of the set of vertices of the complete pentalateral to internal and external vertices is well-defined. The structure of the tiling of P2 just described is shown in Figure 14. Figure 14a also illustrates that the group of a complete pentalateral is isomorphic to D5, i.e. to the symmetry group of a regular pentagon (the latter in Euclidean sense). Recall that the group of a configuration is defined as the group of the permutations (both the points and lines among themselves) preserving incidences [7]. We have seen that squaring a complete pentalateral results in a configuration (1004) in projective 4-space (cf. Class 1 in the preceding section). This configuration can nicely be visualized by projecting it into three dimensions and restricting ourselves to Euclidean space. To this end, a useful tool is the Schlegel diagram [14,26]. In fact, there are 10 copies Figure 14: The decomposition of the projective plane by a complete pentalateral, in two versions, with the pentagonal region shaded. The internal and external vertices are indicated by black and white vertices, respectively. of the complete pentalateral in the configuration (1004) such that they can be inscribed in the 10 pentagonal 2-faces of the Cartesian product of two pentagons, which is a 4-polytope. The Schlegel diagram of this latter polytope is depicted in Figure 15, while the image of the (1004) configuration is shown in Figure 16. Figure 15: Schlegel diagram of the Cartesian product of two pentagons. The following conjecture is motivated by the three-dimensional image of the (1004) configuration. We will denote a complete pentalateral determined by lines li,... ,l5 by P (li,..-,ls). Conjecture 5.3. Let be given in the projective space P3 25 lines, aij (i,j = 1,..., 5) such that they form five complete pentalaterals: Ai = P(aii, .. ., ai5), ..., A5 = P(a5i, .. ., a55). Assume that the following conditions hold: Figure 16: Twenty pentalaterals in E3: a (1OO4) configuration. 1. the external vertices of the pentalaterals Ai form the external vertices of complete pentalaterals Bj = P (b1j,..., b5j ), as follows: aij H a.ij+2 = bij fi bi+2,j; 2. the internal vertices of the pentalaterals Ai form the external vertices of complete pentalaterals Cj = P (c1j,..., c5j ), as follows: aij f aij+i = Cij f Ci+2,j (indexing is meant modulo 5). Then there is a quintuple of complete pentalaterals Di such that their vertices coincide with the internal vertices of the pentalaterals Bj and Cj, as follows: bij f bi+ij = dij f di,j+2 and Cj f ci+i,j = dj f di,j+i. In some particular cases this conjecture is known to be true. In these cases the pentalaterals are embedded in Euclidean 3-space, every Ai is in distinct and pairwise parallel planes (these planes can simply be conceived as "horizontal planes"), while every Bj and Cj are in planes perpendicular to the former ones (thus they can be conceived as being in "vertical position"). In addition, every Di is in "horizontal" planes, too. The cases are as follows: Case A. The pentagons determined by the Ais and Dis are all regular (in Euclidean sense), and they have a common axis of rotation (of order five). In this case the conditions of the conjecture can easily be satisfied by suitably scaling the A^ and by suitably chosen shapes and sizes of the Bjs. Just this case is shown in our Figure 16 above. Here the lines of the pentalaterals A^ Bj, Cj and A, are distinguished by black, blue, red and green colour, respectively. Observe that each of these colour classes represents 100 incidences. Thus, our conjecture can also be formulated that the incidences belonging to any three of the colour classes imply the remaining 100 incidences. Case B. All the pentalaterals Aj are homothetic copies of a pentalateral A0. Furthermore, the external vertices of A0 (hence those of all the Ažs) are inscribed in a circle. This case is visualized in an interactive model made using Mathematica [20]. In this model it is possible to move the external vertices of A0 (and simultaneously, all the corresponding vertices of the Ajs) along a circle, while all the incidences required by the conjecture are preserved. These cases provide some support for the conjecture. We remark that any projective transformation preserves the conjecture. We also remark that Case A also illustrates the fact that the automorphism group of this configuration is larger than or isomorphic to the group D5 X D5 X C2. Here the first factor corresponds to the group of the pentalaterals Aj and A, the second factor to that of the pentalaterals Bj and Cj, while the last term is responsible for interchanging the "horizontal" and "vertical" quintuples of pentalaterals. More generally, one expects that given a configuration C with group G, the group of its pth power is larger than or isomorphic to the semi-direct product Gp x Sp, where the first term is a direct power of G, and the second term is the symmetric group of degree p. 6 Acknowledgements I would like to express my gratitude to the (anonymous) referees for the very careful reading of the manuscript; their valuable comments and suggestions improved the presentation of the paper in many respects. In exploring many of the configurations and in preparing most of the figures in this paper the software Euler 3D proved to be an indispensable tool. I am indebted to its developer Tamas Petro for providing it to me. My special thanks go to Janos Karsai and Lajos Szilassi for preparing the Mathematica notebook on the the model supporting Conjecture 5.3. References [1] L. W. Berman, A new class of movable (n4) configurations, Ars Math. Contemp. 1 (2008), 44-50. [2] L. W. Berman and B. Grünbaum, Deletion constructions of symmetric 4-configurations. Part I., Contrib. Discrete Math. 5 (2010), 18-33. [3] J. Bokowski and V. Pilaud, On the generation of topological (nk )-configurations, Proc. 23rd Canadian Conference on Computational Geometry, Fields Institute, University of Toronto, 2011, 1-6. [4] W. B. Carver, On the Cayley-Veronese class of configurations, Trans. Amer. Math. Soc. 6 (1905), 534-545. [5] H. S. M. Coxeter, Regular and semi-regular polytopes. I. Math. Z. 46 (1940), 380-407. Reprinted in: F. A. Sherk, P. McMullen, A. C. Thompson and A. Ivic Weiss (eds.), Kaleidoscopes. Selected Writings ofH. S. M. Coxeter, Wiley, New York, 1995, 251-278. [6] H. S. M. 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Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929. [22] P. W. Messer, Closed-form expressions for uniform polyhedra and their duals, Discrete Comput. Geom. 27 (2002), 353-375. [23] T. Pisanski, Yet another look at the Gray graph, New Zealand J. Math. 36 (2007), 85-92. [24] H. Ramezani, G. A. Mansoori and M.R. Saberi, Diamondoids-DNA nanoarchitecture: from nanomodules design to self-assembly, J. Comput. Theor. Nanoscience 4 (2007), 96-106. [25] B. Servatius and H. Servatius, The generalized Reye configuration, Ars Math. Contemp. 3 (2010), 21-27. [26] G. M. Ziegler, Lectures on Polytopes, Springer, New York, 1995. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 201-213 Distinguishing graphs with infinite motion and nonlinear growth Johannes Cuno * Institut fur Mathematische Strukturtheorie, Technische Universität Graz Steyrergasse 30/III, 8010 Graz, Austria Wilfried Imrich Department Mathematik und Informationstechnologie Montanuniversitat Leoben, 8700 Leoben, Austria The distinguishing number D(G) of a graph G is the least cardinal d such that G has a labeling with d labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs G with infinite motion and growth o(n2/ log2 n) is either 1 or 2, which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs G of arbitrary cardinality are 2-distinguishable if every nontrivial automorphism moves at least uncountably many vertices m(G), where m(G) > |Aut(G)|. This extends a result of Imrich et al. to graphs with automorphism groups of arbitrary cardinality. Keywords: Distinguishing number, automorphisms, infinite graphs. Math. Subj. Class.: 05C25, 05C63, 05C15, 03E10. *All three authors, namely Johannes Cuno, Wilfried Imrich, and Florian Lehner, acknowledge the support of the Austrian Science Fund (FWF), project W1230-N13. E-mail addresses: cuno@math.tugraz.at (Johannes Cuno), imrich@unileoben.ac.at (Wilfried Imrich), f.lehner@tugraz.at (Florian Lehner) Florian Lehner Institut fur Geometrie, Technische Universitat Graz Kopernikusgasse 24/IV, 8010 Graz, Austria Received 11 May 2012, accepted 19 March 2013, published online 22 April 2013 Abstract 1 Introduction Albertson and Collins [1] introduced the distinguishing number D(G) of a graph G as the least cardinal d such that G has a labeling with d labels which is only preserved by the trivial automorphism. This seminal concept spawned many papers on finite and infinite graphs. We are mainly interested in infinite, locally finite, connected graphs of polynomial growth, see [8], [15], [13], and in graphs of higher cardinality, see [9], [11]. In particular, there is one conjecture on which we focus our attention, the Infinite Motion Conjecture of Tom Tucker. Before stating it, we introduce the notation m(fy) for the number of elements moved by an automorphism fy, and call m(fy) the motion of fy. In other words, m(fy) is the size of the set of vertices which are not fixed by fy, that is, the size of its support, supp(fy). The Infinite Motion Conjecture of Tom Tucker. Let G be an infinite, locally finite, connected graph. If every nontrivial automorphism of G has infinite motion, then the distinguishing number D(G) of G is either 1 or 2. For the origin of the conjecture and partial results compare [13]. The conjecture is true if Aut(G) is countable, hence we concentrate on graphs with uncountable group. The validity of the conjecture for graphs with countable group follows from either one of two different results in [10]. One of them replaces the requirement of infinite motion by a lower and upper bound on the size of the automorphism group. It asserts that every infinite, locally finite, connected graph G whose automorphism group is infinite, but strictly smaller than 2N°, has countable group, infinite motion, and distinguishing number 2. For a precise formulation see Theorem 4.1. The proof is not easy and follows from results of either Halin [6], Trofimov [14], or Evans [3]. The other one relaxes the condition of local finiteness and requires that the group is at most countable. It asserts that countably infinite, connected graphs with finite or countably infinite group and infinite motion are 2-distinguishable, no matter whether they are locally finite or not, see Theorem 4.2. The proof is short and elementary. For uncountable connected graphs with countable motion the Infinite Motion Conjecture need not be true. We turn to this case in Section 4, suggest a version of the conjecture for uncountable connected graphs, and prove its validity under a bound on the size of the automorphism group. 2 Preliminaries Throughout this paper the symbol N denotes the set {1, 2,3,...} of positive integers, whereas the symbol N0 refers to the set {0,1,2, 3,...} of non-negative integers. Let G be a graph with vertex set V(G). Let X be a set. An X-labeling l of G is a mapping l : V(G) ^ X. For us X will mostly be the set {black, white}. In this case, we speak of a 2-coloring of G. Let l be an X-labeling of G. Consider an automorphism fy G Aut(G). If, for every v G V(G), l(fy(v)) = l(v), we say that l is preserved by fy. If this is not the case, we say that l breaks fy. An X-labeling l of G is called distinguishing if it is only preserved by the trivial automorphism. The distinguishing number D(G) of G is the least cardinal d such that there exists a distinguishing X-labeling of G with |X | = d. Given a group A equipped with a homomorphism fy : A ^ Aut(G), we say that A acts on G. Moreover, we say that A acts nontrivially on G if there is an a G A such that fy(a) moves at least one vertex of G. By abuse of language we write a(v) instead of 0(a)(v) and say that an X-labeling l of G is preserved by a G A if it is preserved by 0(a) G Aut(G). The ball with center v0 G V(G) and radius r is the set of all vertices v G V(G) with dG(v0, v) < r and is denoted by BG (r), whereas S G (r) stands for the set of all vertices v G V(G) with dG(v0, v) = r. We call it the sphere with center v0 G V(G) and radius r. If G is clear from the context, we just write Bv0 (r) and Sv0 (r) respectively. For terms not defined here we refer to [7]. Although our graphs are infinite, as long as they are locally finite, all balls and spheres of finite radius are finite. The number of vertices in B^ (r) is a monotonically increasing function of r, because r BG(r)I = E|SG0(i)\ and |SG0(i)| > 1. i=0 Nonetheless, the growth of |bG (r) | depends very much on G, and it is helpful to define the growth rate of a graph. We say that an infinite, locally finite, connected graph G has polynomial growth if there is a vertex v0 g V(G) and a polynomial p such that V r G N0 : |bG0 (r)| < p(r) . It is easy to see that this implies that all functions |BG(r) | are bounded by polynomials of the same degree as p, independent of the choice of v g V(G). In this context it should be clear what we mean by linear and quadratic growth. Observe that the two-sided infinite path has linear growth, and that the growth of the grid of integers in the plane is quadratic. We say that G has exponential growth if there is a constant c > 1 such that V r G N0 : |BvG0 (r)| > cr . Notice that homogeneous trees of degree d > 2, that is, infinite trees where every vertex has the same degree d, have exponential growth. For the distinguishability of such trees and tree-like graphs, see [16] and [9]. We are mainly interested is the distinguishability of infinite, locally finite, connected graphs of polynomial growth. For us, the following lemma will be helpful. Lemma 2.1. Let A be a finite group acting on a graph G. If a coloring of G breaks some element of A, then it breaks at least half of the elements of A. Proof. The elements of A that preserve a given coloring form a subgroup. If some element of A is broken, then this subgroup is proper and thus, by Lagrange's theorem, cannot contain more than half of the elements of A. □ If the action is nontrivial, then we can always find a coloring that breaks at least one element. Hence, we have the following result. Lemma 2.2. Let G be a graph. If A is a finite group acting nontrivially on G, then there exists a 2-coloring of G that breaks at least half of the elements of A. The proof of Lemma 2.2 is based on the fact that A is a group. But a very similar result holds for any finite family of nontrivial automorphisms, as the following lemma shows. Lemma 2.3. Let G be a finite graph. If A is a finite set equipped with a mapping < : A ^ Aut(G) \ {id}, then there exists a 2-coloring of G that breaks <(a) for at least half of the elements of A. Proof. Let V(G) = {v\, v2,..., vn}. For every k e {1,2,..., n}, let Ak be the set of all a e A with supp(<(a)) C {vi, v2,..., vk}. We show by induction that the assertion holds for all Ak and, in particular, for A. Because A1 is the empty set, the assertion is true for A1. Suppose it is true for Ak-1. Then we can choose a 2-coloring of G that breaks 4>(a) for at least half of the elements of Ak-1. This remains true, even when we change the color of vk. Notice that, for every a e Ak \ Ak-1, <(a) either maps vk into a white vertex in {v1, v2,..., vk-1} or into a black vertex in {v1, v2,..., vk-1}. Depending on which of the two alternatives occurs more often, we color vk black or white such that this 2-coloring also breaks <(a) for at least half of the elements of Ak \ Ak-1 and, hence, for at least half of the elements of Ak. □ If every nontrivial automorphism of a graph G has infinite motion, we say that G has infinite motion. For such graphs the following result from [10] will be of importance. Lemma 2.4. Let G be an infinite, locally finite, connected graph with infinite motion. If an automorphism < e Aut(G) fixes a vertex v0 e V(G) and moves at least one vertex in SVo (k), then, for every i > k, it moves at least one vertex in SVo (i). 3 Graphs of nonlinear growth In [10], it was shown that infinite, locally finite, connected graphs with infinite motion and linear growth have countable automorphism group, and therefore distinguishing number either 1 or 2. If the growth rate of such graphs becomes nonlinear, then the automorphism group can become uncountable. This holds, even if the growth rate becomes only slightly nonlinear. Theorem 3.1. Let e > 0. Then there exists an infinite, locally finite, connected graph G with uncountable automorphism group, infinite motion, and nonlinear growth function g : No ^ No such that, for sufficiently large n e N0, g(n) is bounded from above by n1+e. Proof. We construct G from T3, that is, the tree in which every vertex has degree 3. First, choose an arbitrary vertex v0 e V(T3). Our strategy is to replace the edges of T3 by paths such that, for sufficiently large n e N0, g(n) = | BV0 (n) | < n1+e. For every i e N0, there are 3 • 2® edges from SV03 (i) to SV03 (i + 1). If we replace them by paths of the same length, then the cardinality of the balls BV0 (n) grows linearly with slope 3 • 2® from S^3 (i) to S^3 (i + 1). Observe that, given any affine linear function h : N0 ^ N0, there is a number nh e N such that, for all n > nh, h(n) < n1+e. In particular, we may consider the functions h® : N0 ^ N0 defined by h®(x) = 3 • 2® • x + 1, and choose numbers n® e N such that, for every n > n®, h®(n) < n1+e. As illustrated in Figure 1, for every i e N0, we replace the edges from ST3 (i) to ST3 (i + 1) by paths of length nj+1. For every i e N and every vertex v e V(G) on such a path from ST3 (i) to S^O3 (i + 1), we have dG(v, v0) > n® and, hence, g(dG(v, v0)) < 3 • 2® • dG(v,v0) + 1 = h®(dG(v,v0)) < dG(v,v0)1+e Figure 1: Replacing the egdes of T3 by paths. So, for every n > ni, g(n) is bounded from above by n1+e. Every automorphism of T3 that fixes v0 induces an automorphism of G. It is easy to see that this correspondence is bijective. Thus, Aut(G) is uncountable. Furthermore, G inherits infinite motion from T3. Since Aut(G) is uncountable, the result of [10] mentioned at the beginning of Section 3 implies that G cannot have linear growth. □ Though we cannot assume that the automorphism groups of our graphs are countable, we prove that infinite, locally finite, connected graphs with infinite motion and nonlinear, but moderate, growth are still 2-distinguishable, that is, they have distinguishing number either 1 or 2. Our construction of a suitable coloring consists of several steps. In Lemma 3.2 we color a part of the vertices in order to break all automorphisms that move a distinguished vertex v0. In Lemma 3.3 we show how to color some of the remaining vertices in order to break more automorphisms. Iteration of this procedure yields a distinguishing coloring, as shown in Theorem 3.4. Lemma 3.2. Let G be an infinite, locally finite, connected graph with infinite motion and v0 G V(G). Then, for every k G N, one can 2-color all vertices in BVo (k + 3) and SVo (Ak + 4), A G N, such that, no matter how one colors the remaining vertices, all automorphisms that move v0 are broken. Proof. If k = 1, then we color v0 black and all v G V(G) \ {v0} white, whence all automorphisms that move v0 are broken. So, let k > 2. First, we color all vertices in SVo (0), SVo (1), and SVo (k + 2) black and the remaining vertices in BVo (k + 3) white. Moreover, we color all vertices in SVo (Ak + 4), A G N, black and claim that, no matter how we color the remaining vertices, v0 is the only black vertex that has only black neighbors and only white vertices at distance r G {2,3,..., k + 1}, see Figure 2. It clearly follows from this claim that this coloring breaks every automorphism that moves v0. It only remains to verify the claim. Consider a vertex v G V(G) \ {v0}. If v is not in SVo (1), then it is easy to see that v cannot have the aforementioned properties. So, let v be in SVo (1) and assume it has only vo ♦ Sv„ (1) Sv„ (0) Sv„ (k + 4) Sv„ (k + 2) Sv„ (2k + 4) Figure 2: Breaking all automorphisms that move v0. black neighbors and only white vertices at distance 2. Then it cannot be neighbor to any vertex in Sv„ (2), but must be neighbor to all vertices in Bv„ (1) except itself. Therefore, the transposition of the vertices v and v0 is a nontrivial automorphism of G with finite support. Since G has infinite motion, this is not possible. □ Lemma 3.3. Let G be an infinite, locally finite, connected graph with infinite motion and v0 G V(G). Moreover, let e > 0. Then there exists a k G N such that, for every m G N and for every n G N that is sufficiently large and fulfills |Sv„ (n)| < (1 + e)log2 n (3.1) one can 2-color all vertices in Sv„ (m + 1),Sv„ (m + 2),... ,Sv„ (n), but not those in Sv„ (Xk + 4), A G N, such that all automorphisms that fix v0 and act nontrivially on Bv„ (m) are broken. The coloring and the meaning of the variables m, n, and k is illustrated by Figure 3. Proof. First, choose a k G N that is larger than 1 + 1. Then k - 1 1 > k 1 + e Let m G N. By (3.2), there is an n0 G N such that k - 1 1 V n > no : (n — m) •- > n •--+ 1. k 1 + e (3.2) (3.3) Let n G N be sufficiently large, that is, n > n0, and assume it fulfills (3.1). Then, the number of spheres Sv„ (m+1), Sv„ (m+2),..., Sv„ (n) that are not of the type Sv„ (A k+4), A g N, is at least (n — m) k1 > 1 1+e +1 > 1+e (3.4) n n n Svo (m) Svo (n) Figure 3: Breaking all automorphisms that fix v0 and act nontrivially on BVo (m). Our goal is to 2-color the vertices in these spheres in order to break all automorphisms that fix v0 and act nontrivially on BVo (m). Let Aut(G, v0) be the group of all automorphisms that fix v0. Every ^ G Aut(G, v0) induces a permutation ^|BVo (n) of the vertices in BVo (n). These permutations form a group A. If a and r are different elements of A, then arG A acts nontrivially on BVo (n). By Lemma 2.4, it also does so on SVo (n), which means that a and r do not agree on SVo (n). Therefore, the cardinality of A is at most |SVo (n) | !, for which the following rough estimate suffices for our purposes: |Svo(n)|! < |Svo(n)||SvoM"1 < () ^^" n / n N (3.5) < n (1 + E)1og2 n -1 = ^ (1 + e)1og2 n l0g2 n < 2 1+7-1 . It is clear that, if an element a G A that acts nontrivially on BVo (m) is broken by a suitable 2-coloring of some spheres in BVo (n), then all ^ G Aut(G, v0) with ^|BVo (n) = a are broken at once. So it suffices to break all a G A that act nontrivially on BVo (m) by a suitable 2-coloring of some spheres in BVo (n) in order to ensure that all ^ G Aut(G, v0) that act nontrivially on BVo (m) are broken. Before doing this, let us remark that any element a G A that acts nontrivially on the ball BVo (m), also acts nontrivially on every sphere SVo (m + 1),..., SVo (n). This is a consequence of Lemma 2.4, and implies that we can break a by breaking the action of a on any one of the spheres SVo (m + 1),..., SVo (n). Now, consider the subset S C A of all elements that act nontrivially on BVo (m). As already remarked, every a G S acts nontrivially on every sphere SVo (m + 1),..., SVo (n). Hence, we can apply Lemma 2.3 to break at least half of the elements of S by a suitable coloring of SVo (m +1). What remains unbroken is a subset S' C S of cardinality at most |S|/2. Now, we proceed to the next sphere. We can break at least half of the elements of S' by a suitable coloring of SVo (m + 2). What still remains unbroken, is a subset S" C S of cardinality at most |S |/4. Iterating the procedure, but avoiding spheres of the type SVo (Ak + 4), A G N, we end up with the empty subset 0 C S after at most log2 |S| + 1 < log2 | A| + 1 < steps, see (3.5). This is less than the number of spheres not of the type SVo (Ak + 4), A G N, between Sv0 (m + 1) and Sv0 (n), see (3.4). Thus, we remain within the ball BVo (n). Hence, all s G S and, therefore, all ^ G Aut(G, v0) that act nontrivially on Bv0 (m) are broken, and we are done. □ Theorem 3.4. Let G be an infinite, locally finite, connected graph with infinite motion and v0 G V(G). Moreover, let e > 0. If there exist infinitely many n G N such that n |Svo (n)|<-—^-, (3.6) (1 + e)log2 n then the distinguishing number D(G) of G is either 1 or 2. Proof. Consider the k G N provided by Lemma 3.3. First, we use Lemma 3.2 to 2-color all vertices in BVo (k + 3) and in SVo (Ak + 4), A G N, such that, no matter how we color the remaining vertices, all automorphisms that move v0 are broken. Let m1 = k + 3. Among all n G N that satisfy (3.6) we choose a number n1 G N that is larger than m1 and sufficiently large to apply Lemma 3.3. Hence, we can 2-color all vertices in SVo (m1 + 1), SVo (m1 + 2),..., SVo (n1), except those in SVo (Ak + 4), A G N, such that all automorphisms that fix v0 and act nontrivially on BVo (m1) are broken. Next, let m2 = n1 and choose an n2 G N to apply Lemma 3.3 again. Iteration of this procedure yields a 2-coloring of G. If an automorphism ^ G Aut(G) \ {id} moves v0, then it is broken by our coloring. If it fixes v0, consider a vertex v with ^(v) = v. Since G is connected and m1 < m2 < m3 < ..., there is an i G N such that v is contained in BVo (m,). Hence, ^ acts nontrivially on BVo (m,) and is again broken by our coloring. □ Corollary 3.5. Let G be an infinite, locally finite, connected graph with infinite motion and v0 G V(G). Moreover, let e > 0. If there exist infinitely many n G N such that 2 n2 |Bvo (n)|< ——-, (3.7) (2 + e) log2 n then the distinguishing number D(G) of G is either 1 or 2. In particular, the Infinite Motion Conjecture holds for all graphs of growth o(n2/ log2 n). Proof. Let n1 < n2 < n3 < ... be an infinite sequence of numbers that fulfill (3.7). Notice that, for every k G N, nk i n 2 nk g (T+iPg-i > * l*o <"k>l > g |Svo(01 . <3-8> Since (JC (1 + 2)log2 J (2 + e)log2 nj , (35) we infer that nk / ■ \ lim £ (77—^1-: -|SV0 (i)U = , (3.10) V (1+ 2 )log2 : J and that, for infinitely many i G N, |Svo (i)| < ^-:. (3.11) (1 + 2)log2 i Hence, we can apply Theorem 3.4 to show that the distinguishing number D(G) of G is either 1 or 2. □ A result similar to Theorem 3.4 can also be obtained for graphs with countably many ends1, none of which grows too fast. Readers not familiar with the notion of ends may safely skip the rest of this section, as the result is not used elsewhere in the paper. Theorem 3.6. Let G be an infinite, locally finite, connected graph with countably many ends and infinite motion. Moreover, let v0 G V(G) and e > 0. For an end u of G let S" (n) be the set of vertices in SVo (n) that lie in the same connected component of G \ BVo (n — 1) as u. If, for every end u, there are infinitely many n G N such that i i n I SV (n)| < --r-, (3.12) 1 Vo () < (1+ e ) log 2 n' V ; then the distinguishing number D(G) of G is either 1 or 2. Proof. Basically the proof consists of three steps. First we color part of the vertex set in order to break all automorphisms that move v0. In the second step we break all automorphisms in Aut(G, v0) that do not fix all ends of the graph by coloring some other vertices. Finally, we color the remaining vertices to break the rest of the automorphisms. In order to break all automorphisms that move v0 we apply Lemma 3.2, just as in the proof of Theorem 3.4. The only difference is that we choose k twice as large as proposed by Lemma 3.3, because we would like to color some additional spheres in the second step of the proof before applying an argument similar to that in Lemma 3.3. For the second step consider the spheres SVo (^i+ik + 4), A g N. We wish to color those spheres such that every automorphism that fixes v0 and preserves the coloring also fixes every end of G. It is not hard to see that the sets S" ( k+4), u an end of G, A G N, carry the following tree structure. Consider v0, the root, which is connected by an edge to S" (3k + 4) for each end u. For every end u of G and every A G N, draw an edge from S^0 ( k + 4) to SV ( k + 4). To see that this is indeed a tree just notice that if SV1 (n) = S"2 (n), then, for every m < n, S"1 (m) = S^2(m). So there cannot be any circles. By construction, this tree structure is infinite, locally finite, and does not have any endpoints. Next, notice that every automorphism ^ G Aut(G, v0) that does not fix all ends also acts as an automorphism on this tree structure. By [16], the distinguishing number of infinite, locally finite trees without endpoints is at most 2. Therefore it is possible to 2-color the sets S" (k + 4), u an end of G, A G N, such that every such automorphism 1 Ends were first introduced by Freudenthal [4] in a topological setting, but here the definition of Halin [5] is more appropriate. For an accessible introduction to ends of infinite graphs see [2]. is broken. It is also worth noting that so far we did not use the countability of the end space of G, nor did we use the growth condition on the ends. Let us turn to the third step of the proof. So far we have colored the ball Bv° (k + 3) and the spheres Sv° ( |k + 4), A > 2, in a way that color preserving automorphisms fix v0 and move every S"° (n) into itself. Consider such an automorphism fy, which acts nontrivially on G. If we remove the fixed points of fy from G, then the infinite motion of G implies that the resulting graph has only infinite components. Hence, there is a ray in G which contains no fixed point of fy. The image of this ray must lie in the same end w. Thus, there is an index n0, such that, for every n > n0, fy acts nontrivially on S"0 (n). Let (wj)ieN be an enumeration of the ends of G. Choose a function f : N ^ N such that, for every i G N, f -1(i) is infinite. Assume that all spheres up to Sv° (m) have been colored in the first i - 1 steps. In the i-th step we would like to color some more spheres in order to continue breaking all automorphisms in Aut(G, v0) that act nontrivially on each of the spheres S"/(i) (n), n > m. This can be done by exactly the same argument as the one used in the proof of Lemma 3.3. As we already mentioned, every automorphism that was not broken in the first two steps acts by nontrivially on the rays of some end. Since, in the procedure described above, every end is considered infinitely often, it is clear that every such automorphism will eventually be broken. This completes the proof. □ 4 Graphs with higher cardinality If a graph G has trivial automorphism group, then G is obviously 1-distinguishable, that is, D(G) = 1. From now on we assume that our graphs G have nontrivial automorphism group. In this case, the motion m(G) of G is defined as m(G) = min m(fy). (4.1) 0eAut(G)\{id} As already mentioned, the Infinite Motion Conjecture does not hold for graphs of higher cardinality. An example is the Cartesian product G = Kn ^ Km of two complete graphs on infinitely many vertices n and m with 2n < m. By [9], G has motion n, but D(G) > n. The question arises whether one can adapt the Infinite Motion Conjecture to graphs of higher cardinality. The starting point is [12, Theorem 1]. It asserts that a finite graph G is 2-distinguishable if m(G) > 2log2 |Aut(G)|. However, a second look at the proof shows that the inequality sign can be replaced by >. For details see Section 5. For finite graphs we thus infer that m(G) > 2log2 |Aut(G)| implies D(G) = 2, (4.2) which can also be written in the form m(G) |Aut(G)| < 2 — implies D(G) = 2 . Notice that 2m(G = 2m(G) if m(G) is infinite. We are thus tempted to conjecture for graphs G with infinite motion that |Aut(G)| < 2m(G) implies D(G) = 2. We formulate this conjecture as the Motion Conjecture. Let G be a connected graph with infinite motion m(G) and |Aut(G)| < 2m(G). Then D(G) = 2. How does this compare with the Infinite Motion Conjecture? It asserts that the distinguishing number of a locally finite, connected graph G is 2 if m(G) is infinite. Since locally finite graphs are countable, the condition that m(G) is infinite is equivalent to m(G) = H0. Furthermore, for countable graphs we have |Aut(G)| < H^ = 2n° . Hence, for countable graphs, and thus also for locally finite, connected graphs with infinite motion, the inequality of the Motion Conjecture is automatically satisfied, which means that the Infinite Motion Conjecture is a special case of the Motion Conjecture. Now, let us focus on the two results from [10] that imply the validity of the Infinite Motion Conjecture for graphs with countable group. Theorem 4.1. Let G be a locally finite, connected graph that satisfies H0 < |Aut(G)| < 2No. Then |Aut(G)| = Ho, m(G) = Ho, and D(G) = 2. Notice that the only thing that is required here, besides local finiteness and connectedness, is an upper and a lower bound on the size of Aut(G). And it turns out, that Aut(G) is countable, even without the continuum hypothesis. Even infinite motion and D(G) = 2 are consequences of this restriction on the size of the automorphism group. Theorem 4.2. Let G be a countably infinite, connected graph that satisfies the conditions |Aut(G)| < m(G) and m(G) = H0. Then D(G) = 2. Here, without local finiteness, one cannot drop the assumption of infinite motion. If we assume that Aut(G) has smaller cardinality than the continuum, then we can ensure 2-distinguishability if the continuum hypothesis holds, but we do not know whether this is really necessary. Corollary 4.3. Let G be a countably infinite, connected graph with infinite motion. If the continuum hypothesis holds, and if |Aut(G)| < 2m(G), then D(G) = 2. The next theorem shows that Theorem 4.2 also holds for graphs of higher cardinality and uncountable motion. Theorem 4.4. Let G be a connected graph with uncountable motion. Then |Aut(G)| < m(G) implies D(G) = 2. Proof. Set n = | Aut(G) |, and let Z be the smallest ordinal number whose underlying set has cardinality n. Furthermore, choose a well ordering - of A = Aut(G) \ {id} of order type Z, and let a0 be the smallest element with respect to -. Then the cardinality of the set of all elements of A between a0 and any other a G A is smaller than n < m(G). Now we color all vertices of G white and use transfinite induction to break all automorphisms by coloring selected vertices black. INDUCTION BASE By the assumptions of the theorem, there exists a vertex v0 that is not fixed by a0. We color it black. This coloring breaks a0. Induction step Let ß g A. Suppose we have already broken all a - ß by pairs of distinct vertices (va, a(va)), where va is black and a(va) white. Clearly, the cardinality of the set R of all (va, a(va)), a - ß, is less than m(G) > n. By assumption, ß moves at least m(G) vertices. Since there are still n vertices not in R, there must be a pair of vertices (vß, ß(vß)) that does not meet R. We color vß black. This coloring breaks ß. □ Corollary 4.5. Let G be a connected graph with uncountable motion. If the general continuum hypothesis holds, and if |Aut(G)| < 2m(G), then D(G) = 2. Proof. Under the assumption of the general continuum hypothesis 2m(G) is the successor of m(G). Hence | Aut(G) | < m(G), and the assertion of the corollary follows from Theorem 4.4. □ 5 The Motion Lemma of Russell and Sundaram In order to show that a finite graph G is 2-distinguishable if m(G) > 2log2 |Aut(G)|, Russell and Sundaram [12] first defined the cycle norm of an automorphism If ^ = (viivi2 . . .Vl(l )(v21 . . . V2i2) . . . (vkl . . . Vklk) , then the cycle norm c(^) of ^ is k c(^) = E(1i - !). i=i The cycle norm c(^) is related to graph distinguishability as follows: Let G be randomly 2-colored by independently assigning each vertex a color uniformly from {black, white}. Then the probability that every cycle of ^ is monochromatic is 2- c(^). In this case, ^ preserves the coloring so chosen. Further, they define the cycle norm c( G) of a graph G as c(G) = min c(^). 0eAut(G)\{id} We now reprove Theorem 2 of [12] with > instead of >. Because c(^) > m(^)/2 and thus c(G) > m(G)/2 we infer from Theorem 5.1 below that G is 2-distinguishable if m(G) > 2log2 |Aut(G)|. We propose to call this result "Motion Lemma of Russell and Sundaram". Actually, the only difference from the original proof is the insertion of the middle term in (5.2). Theorem 5.1. Let G be a finite graph, and c(G)log d > log |Aut(G)|. Then G is d-distinguishable, that is, D(G) < d. Proof. Let x be a random d-coloring of G, the probability distribution being given by selecting the color of each vertex independently and uniformly in the set {1,..., d}. For a fixed automorphism ^ G Aut(G) \ {id} consider the probability that the random coloring X is preserved by / i \ c(0) / i \ c(G) Prx[Vv : x(#v)) = x(v)] = ^J < (^J . (5.1) Collecting these events yields the inequality Prx[3 $ G Aut(G) \ {id} V v : x(^(v)) = x(v)] < (|Aut(G)| - 1) ( d )c(G) < |Aut(G)| ( 1 )c(G) . . By hypothesis the last term is at most 1. Thus there exists a coloring x such that, for every ^ G Aut(G) \ {id}, there is a v for which x(^(v)) = x(v), as desired. □ Acknowledgement We thank the referees for their comments and remarks, as they contributed considerably to the readability of the paper. Furthermore, we are grateful to Norbert Sauer and Claude Laflamme for their suggestions pertaining to Theorem 4.4 and Corollary 4.5. References [1] M. O. Albertson and K. L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996), R18. [2] R. Diestel, Graph theory, 3rd ed., Graduate Texts in Mathematics 173, Springer, Berlin, 2006. [3] D. M. Evans. A note on the automorphism groups of countably infinite structures, Arch. Math. (Basel) 49 (1987), 479-483. [4] H. Freudenthal, Über die Enden diskreter Raume und Gruppen, Comment. Math. Helv. 17 (1945), 1-38. [5] R. Halin, Über unendliche Wege in Graphen, Math. Annalen 157 (1964), 125-137. [6] R. Halin, Automorphisms and endomorphisms of infinite locally finite graphs, Abh. Math. Sem. Univ. Hamburg 39 (1973), 251-283. [7] R. Hammack, W. Imrich and S. Klavzar, Handbook of product graphs, second ed., Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2011. [8] W. Imrich, J. Jerebic and S. Klavzar, The distinguishing number of Cartesian products of complete graphs, European J. Combin. 29 (2008), 922-929. [9] W. Imrich, S. Klavzar and V. Trofimov, Distinguishing infinite graphs, Electron. J. Combin. 14 (2007), R36. [10] W. Imrich, S. M. Smith, T. Tucker and M. E. Watkins, Infinite motion and the distinguishing number of graphs, in preparation. [11] C. Laflamme, L. Nguyen Van The and N. W. Sauer, Distinguishing number of countable homogeneous relational structures, Electron. J. Combin. 17 (2010), R20. [12] A. Russell and R. Sundaram, A note on the asymptotics and computational complexity of graph distinguishability, Electron. J. Combin. 5 (1998), R23. [13] S. M. Smith, T. Tucker and M. E. Watkins, Distinguishability of infinite groups and graphs, Electron. J. Combin. 19 (2012), P27. [14] V. I. Trofimov, Groups of automorphisms of graphs as topological groups, Russian, Mat. Zametki 38 (1985), 378-385. [15] T. Tucker, Distinguishing maps, Electron. J. Combin. 18 (2011), P50. [16] M. E. Watkins and X. Zhou, Distinguishability of locally finite trees, Electron. J. Combin. 14 (2007), R29. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 215-235 Isomorphic tetravalent cyclic Haar graphs Hiroki Koike University of Primorska, UP IAM, Muzejski trg 2, SI6000 Koper, Slovenia Istvan Kovacs * University of Primorska, UP IAM and UP FAMNIT Muzejski trg 2, SI6000 Koper, Slovenia * Received 31 January 2012, accepted 20 April 2013, published online 26 April 2013 Abstract Let S be a subset of the cyclic group Zn. The cyclic Haar graph H(Zn, S) is the bipartite graph with color classes Z+ and Z-, and edges {x+, y-}, where x, y G Zn and y - x G S .In this paper we give sufficient and necessary conditions for the isomorphism of two connected cyclic Haar graphs of valency 4. Keywords: Graph isomorphism, cyclic Haar graph, 4-BCI-group. Math. Subj. Class.: 20B25, 05C25, 05C60 1 Introduction Let S be a subset of a finite group G. The Haar graph H (G, S) is the bipartite graph with color classes identified with G and written as G+ and G-, and the edges are {x+, y-}, where x, y G G and yx-1 G S. Haar graphs were introduced for abelian groups by Hladnik, Marusic and Pisanski [6], and were redefined under the name bi-Cayley graphs in [17]. A Haar graph H (G, S ) is called cyclic if G is a cyclic group. In this paper we consider the problem of giving sufficient and necessary conditions for the isomorphism of two cyclic Haar graphs. This is a natural continuation of the isomorphism problem of circulant digraphs which has been solved by Muzychuk [12]. It appears in the context of circulant matrices under the name bipartite Adam problem [16], and also in the context of cyclic configurations [2, 6]. The symbol Zn denotes the additive group of the ring Z/nZ of residue classes modulo n, and Z*n denotes the multiplicative group of units in Z/nZ. Two Haar graphs H(Zn, S) and H (Z n,T ) are called affinely equivalent, written as H (Z n,S) =aff H (Zn ,T ), if S * Corresponding author. E-mail addresses: hiroki.koike@upr.si (Hiroki Koike), istvan.kovacs@upr.si (Istvan Kovacs) can be mapped to T by an affine transformation, i.e., aS + b = T for some a e Z*n and b e Zn. It is an easy exercise to show that two affinely equivalent cyclic Haar graphs are isomorphic as usual graphs. The converse implication is not true in general, and this makes the following definition interesting (see [17]): we say that a subset S C Zn is a BCI-subset if for each T C G, H (Zn, S) = H (Z n,T) if and only if H (Zn, S) =aff H (Z n,T). Wiedemann and Zieve proved in [16, Theorem 1.1] that any subset S of Zn is a BCI-subset if |S| < 3 (a special case was proved earlier in [3]). However, this is not true if |S| > 4 (see [6, 16]), hence the first nontrivial case of the isomorphism problem occurs when |S| =4. In this paper we settle this case by proving the following theorem: Theorem 1.1. Two connected Haar graphs H(Zn, S) and H(Zn, T) with |S| = |T| =4 are isomorphic if and only if there exist a\,a2 e Z*n and b\,b2 e Zn such that (1) aiS + bi = T ; or (2) aiS + bi = {0, u,v,v + m} and a2T + b2 = {0, u + m, v, v + m}, where n = 2m, Zn = (u, v), 2 | u, 2u | m and u/2 = v + m/(2u)(mod m/u). Remark 1.2. A group G is called an m-BCI-group if every subset S of G with |S| < m is a BCI-subset (see [7, 17]). In this context [16, Theorem 1.1] can be rephrased as Zn is a 3-BCI-group for any number n; and Theorem 1.1 says that Zn is not a 4-BCI-group if and only if n is divisible by 8. This refines [16, Theorem 7.2] in which it is proved that, if Zn contains a non-BCI-subset of size k, k e {4, 5}, then n has a prime divisor less or equal to 2k(k - 1). Our approach towards Theorem 1.1 is group theoretical, we adopt the ideas of [1, 11]. In short terms the initial problem is transformed to a problem about the automorphism group of the graphs in question. Theorem 1.1 is proven in two steps: first it is settled for graphs H(Zn, S) with S satisfying additional conditions (see Theorem 3.1); then it is shown that, if S is not a BCI-subset, then it is affinely equivalent to a set satisfying the conditions of Theorem 3.1 (see Theorem 4.1). We conclude the introduction with the following modification of Theorem 1.1: Theorem 1.3. Two connected Haar graphs H(Zn, S) and H(Zn, T) with |S| = |T| =4 are isomorphic if and only if there exist ai,a2 e Z*n and bi,b2 e Zn such that (1) aiS + bi = T ; or (2) aiS + bi = {0, u, v, v + m} and a2T + b2 = {0, u + m, v, v + m}, where n = 2m, Zn = (u, v), 2 | u, 2u | m. Proof. In view of Theorem 1.1 it is sufficient to prove that H(Zn, S) = H(Zn, T) if aiS + bi = {0, u, v, v + m} and a2T + b2 = {0, u + m, v, v + m}, where n = 2m, Zn = (u, v), 2 | u, 2u | m and u/2 = v + m/(2u)(mod m/u). In fact, we are going to show below that there exist a e Zn and b e Zn such that a • {0, u, v, v + m} + b = {0, u + m, v, v + m}. Then (a-iaai) • S + a-i(abi + b - b2) = T, and so H(Zn, S) = H(Zn, T). Let us consider the following system of congruences: ux = —u + m(mod n) and vx = —u + v(mod n). (1.1) By the first congruence, using also that 2u | m, x may be written in the form x = (n/u)y — 1 + m/u. Plugging this in the second one, we obtain (vn/u)y = 2v — u — vm/u(mod n), which has an integer solution in y exactly when gcd(vn/u, n) | (2v — u — vm/u). Then gcd(vn/u, n) = n/ugcd(u,v), and since Zn = {u,v),n/u and gcd(u,v) are coprime. Since gcd(u, v) is clearly a divisor of 2v — u — vm/u, a solution in y exists if and only if n/u | (2v — u — vm/u), i.e., u = 2v — vm/u(mod 2m/u) (recall that n = 2m). On the other hand, one of the initial assumptions is u/2 = v + m/(2u)(mod m/u), and so u = 2v + m/u(mod 2m/u). We conclude that (1.1) has an integer solution if —vm/u = m/u(mod 2m/u). Now, the latter congruence holds because of the conditions 2 | u, 2 | n, and Zn = (u, v). Let a be a solution of (1.1). It follows from the above argument that gcd(a, m/u) = 1. Notice that, since 2u | m, 2 { a. Let d = gcd(a, u). By (1.1), av = —u + v(mod n), implying that d | v, and so d =1. We see that gcd(a, 2m) = 1, i.e., a G Zn. Choosing b = u + m, we finally get by (1.1), a • 0 + b = u + m, au + b = 0, av + b = v + m, and a(v + m) + b = v, as required. □ Theorem 1.3 becomes especially interesting when we compare it with the solution of the isomorphism problem of trivalent circulant digraphs. In fact, the same conditions can be derived from Muzychuk's general algorithm presented in [12]: two connected Cayley digraphs Cay(Zn, S) and Cay(Zn, T) with |S| = |T| =3 are isomorphic if and only if there exist ai, a2 G Z*n such that • a1S = T ; or • aiS = {u, v, v + m} and a2T = {u + m, v, v + m}, where n = 2m, Zn = (u, v), 2 | u, and 2u | m. The natural question arises whether this phenomenon holds also for graphs of larger valencies. 2 A Babai type theorem In this paper every group, graph and digraph is finite. For a (di)graph r, the symbols V(r), E(r) and Aut(r) denote the set of its vertices, (directed) edges and the full group of its automorphisms, respectively. Regarding terminology and notation in permutation group theory we follow [5]. Let S be a subset of a group G. The Cayley digraph Cay(G, S) is the digraph with vertex set G, and its directed edges are (x, y), where x, y G G and yx-1 G S. Two digraphs Cay(G, S) and Cay(G, T) are called Cayley isomorphic, written as Cay(G, S) =cay Cay(G, T), if T = Sv for some group automorphism ^ G Aut(G). It is clear that such an automorphism induces an isomorphism between Cay(G, S) and Cay(G, T), and thus Cayley isomorphic digraphs are isomorphic as usual digraphs. It is also well-known that the converse implication is not true, and this makes sense for the following definition (see [1]): a subset S C G is a CI-subset if for each T C G, Cay(G, S) = Cay(G, T) if and only if Cay(G, S) =cay Cay(G, T). The following equivalence was proved by Babai [1, 3.1 Lemma]. Theorem 2.1. The following are equivalent for every Cayley digraph Cay(G, S). (1) S is a CI-subset. (2) Every two regular subgroups of Aut(Cay(G, S)) isomorphic to G are conjugate in Aut(Cay(G, S)). Theorem 2.1 essentially says that the Cl-property of a given subset S depends entirely on the automorphism group Aut(Cay(G, S)). In this section we prove analogous results for cyclic Haar graphs. Let V = V(H(Z„, S)) be the vertex set of the Haar Graph H(Z„, S). Throughout this paper c and d denote the permutations of V defined by c : xe ^ (x +1)e and d : xe ^J (n - x)+ if £ = + (2.1) I (n — x)+ if e = —, where x G Zn and e G {+, —}. It follows immediately that both c and d are automorphisms of any Haar graph H(Zn, S). Denote by C the group generated by c, and by D the group generated by c and d. The group D acts regularly on V, and D is isomorphic to D2n. Thus H(Zn, S) is isomorphic to a Cayley graph over D, and so Theorem 2.1 can be applied. The following corollary is obtained. Corollary 2.2. The implication (1) ^ (2) holds for every Haar graph H (Zn, S). (1) S is a BCI-subset. (2) Every two regular subgroups of Aut(H(Zn, S)) isomorphic to D are conjugate in Aut( H(Zn,S)). However, we do not have equivalence in Corollary 2.2 as it is shown in the following example. Example 2.3. Let r = H (Zio, {0,1,3,4}). Using the computer package Magma [4] we compute that r is edge-transitive and its automorphism group Aut(r) = D20 x Z4. Furthermore, Aut(r) contains a regular subgroup X which is isomorphic to D20 but X = D20, hence (2) in Corollary 2.2 does not hold. On the other hand, we find that for every subset T C Z10 with 0 G T and |T| = 4, the corresponding Haar graph H(Z10,T) = r exactly when H(Z10,T) =aff r. Thus {0,1,3,4} is a BCI-subset, so (1) in Corollary 2.2 holds. □ Example 2.3 shows that the isomorphism problem of cyclic Haar graphs is not a particular case of the isomorphism problem of Cayley graphs over dihedral groups. We remark that the latter problem is still unsolved, for partial solutions, see [1, 13, 14]. Nonetheless, the idea of Babai works well if instead of the regular subgroup D we consider its index 2 cyclic subgroup C. We say that a permutation group G < Sym(Z+ U Z-) is bicyclic if G is a cyclic group which has two orbits: Z+ and Z-. By a bicyclic group of a Haar graph r = H(Zn, S) we simply mean a bicyclic subgroup X < Aut(r). Obviously, C is a bicyclic group of any cyclic Haar graph, and being so it will be referred to as the canonical bicyclic group. Let Iso(r) denote the set of all isomorphisms from r to any other Haar graph H(Zn, T), i.e., Iso(r) = {/ G Sym(V) : rf = H(Zn,T) for some T C Zn}. And let Ciso(T) denote the isomorphism class of cyclic Haar graphs which contains r, i.e., Ciso(r) = {rf : f e iso(r)}. Lemma 2.4. Let r = H(Zn, S) be a connected Haar graph and f be in Sym(V). Then f e iso(r) if and only if fCf-1 is a bicyclic group of r. Proof. Let f e iso(r). Then f Cf-1 < Aut(r). Clearly, f Cf-1 is a cyclic group. Since the sets Z+ and Z- are the color classes of the connected bipartite graph r, f preserves these color classes, implying that Orb(fCf-1, V) = {Z+, Z-}. The group fCf-1 is a bicyclic group of r. Conversely, suppose that fCf-1 is abicyclic group of r. Then C = f-1(f Cf-1)f < Aut(rf ). Because that Orb(fCf-1, V) = {Z+, Z-}, the graph rf is connected and bipartite with color classes Z+ and Z-. We conclude that rf = H (Z n,T ) for some T C Zn, so f e iso(r). The lemma is proved. □ Lemma 2.4 shows that the normalizer NSym(V)(C) C Iso(H(Zn,S)). The group NSym(V) (C) is known to consist of the following permutations: where r e Z*n and s,t e Zn. Note that, two Haar graphs H (Zn, S ) and H (Z n,T ) are from the same orbit under NSym(V)(C) exactly when H(Zn,S) =aff H(Zn,T). Let Caff (r) denote the affine equivalence class of cyclic Haar graphs which contains the graph r = H (Zn, S), i.e., Caff (r) = {rv : ^ e NSym(V )(C )}. It is clear that the isomorphism class Ciso(r) splits into affine equivalence classes: Our next goal is to describe the above decomposition with the aid of bicyclic groups. Let X be a bicyclic group of a connected graph r = H(Zn, S). Then is also a bicyclic group for every g e Aut(r), hence the full set of bicyclic groups of r is the union of Aut(r)-conjugacy classes. We say that a subset S C Iso(r) is a bicyclic base of r if the subgroups £C£-1, £ e S, form a complete set of representatives of the corresponding conjugacy classes. Thus every bicyclic group X can be expressed as Remark 2.5. Our definition of a bicyclic base copies in a sense the definition of a cyclic base introduced by Muzychuk [11, Definition, page 591]. Theorem 2.6. Let r = H(Zn, S) be a connected Haar graph with a bicyclic base S. Then (rx + s) if e = +, (rx + t)+ if e = —, (2.2) Ciso(r) = Caff(roo ••• ùCaff(rfc)*. X = g£C(g£) 1 for a unique £ e S and g e Aut(r). Ciso(r) = U «esCaff (r« ). Proof. It follows immediately that, Ciso(r) D U Caff (r«). (2.3) 1Here we mean that ciSo(r) = caff (ri) u • • • u caff (rfc) and caff (ri) n caff (r^) = 0 for every i,j € {1,..., k}, i = j. We prove that equality holds in (2.3). Pick S G Cisc,(T). Then S = rf for some f G Iso(r). By Lemma 2.4, fCf-1 is a bicyclic group of r, hence fCf-1 = g£C(g£)-1, e G S, g g Aut(r). Thus f-1g£ = h, where h G NSym(V)(C). Then s = rf = r9«h-1 = (r« )h-1. This shows that S G Caff (r« ), and so Ciso(r) C y Caff (r«). «es In view of (2.3) the two sides are equal. Moreover, if Caff (r«1 ) n Caff (r«2 ) = 0 for £i ,£2 G S, then r«1 = r«2h for some h G NSym(V)(C). Hence £2h£-1 = g for some g G Aut(r), and so eiCe-1 = g-1 ^hCh-Vg = g-1 (SC-1^ The bicyclic subgroups £1C£-1 and £2C£-1 are conjugate in Aut(r), hence £1 = £2 follows from the definition of the bicyclic base S. We obtain that Caff (r«1 ) n Caff (r«2 ) = 0 whenever £1, £2 G S, £1 = £2, and so Cisc(r) = Q«eSCaff (r«). The theorem is proved. s □ As a direct consequence of Theorem 2.6 we obtain the following corollary, analog of Theorem 2.1. Corollary 2.7. The following are equivalent for every connected Haar graph H (Zn, S ). (1) S is a BCI-subset. (2) Any two bicyclic groups of H (Zn, S) are conjugate in Aut(H (Zn, S )). In our last proposition we connect the BCI-property with the CI-property. For aE G V, in what follows Aut(H(Zn, S))ae denotes the vertex stabilizer of ae in Aut(H(Zn, S)). Proposition 2.8. Suppose that r = H(Zn, S) is a connected Haar graph such that for some a G Zn, Aut(r)0+ = Aut(r)a-. Then the following are equivalent. (1) S is a BCI-subset. (2) S — a = {s — a : s G S} is a CI-subset. Proof. For sake of simplicity we put A = Aut(r) and G = Aut(r){Z+}, i.e., the setwise stabilizer of the color class Z+ in Aut(r). Obviously, X < G for every bicyclic group X of r. Since A = G x (d) and d normalizes C, it follows that the conjugacy class of subgroups of A containing C is equal to the conjugacy class of subgroups of G containing C. Using this and Theorem 2.6, we obtain that S is a BCI-subset if and only if every bicyclic group is conjugate to C in G. Let W = {0+, a-} and consider the setwise stabilizer A{W}. Since A0+ = Aa-, A0+ < A{W}. By [5, Theorem 1.5A], the orbit of 0+ under A{W} is a block of im-primitivity (for short a block) for A. Denote this block by A and the induced system of blocks by S (i.e., S = {Ag : g G G}). Consider the element g = dca from D. We see that g switches 0+ and a-, hence A{W} = A0+ (g). Therefore, A = (0+)A{W} = (0+)Ao+ (g) = (0+)(g) = W, and so S = { {x+, (x + a)-} : x G Zn }. Define the mapping p : S ^ Zn by p : {x+, (x + a)-} ^ x, x G Zn. Now, an action of A on Zn can be defined by letting g G A act as xa = xv-lgv, x G Zn. For g G A we write g for the image of g under the corresponding permutation representation, and for a subgroup X < A we let X = {x : x G X}. In this action of A the subgroup G < A is faithful. Also notice that, a subgroup X < G is a bicyclic group of r if and only if X is a regular cyclic subgroup of G. In particular, for the canonical bicyclic group C, C = (Zn)right, where (Zn)right denotes the group generated by the affine transformation x ^ x + 1, x G Zn. Pick g G G and (x, x + s - a) G Zn x Zn, where s G S. Then g maps the directed edge (x+, (x + s)-) to a directed edge (y+, (y + q)-) for some y G Zn and q G S. Since S is a system of blocks for G, g maps (x + s - a)+ to (y + q - a)+, and so g maps the pair (x, x + s - a) to the pair (y,y + q - a). We have just proved that g leaves the set { (x, x + s - a) : x G Zn, s G S } setwise fixed. As the latter set is the set of all directed edges of the digraph Cay(Zn, S - a), G < Aut(Cay(Zn, S - a)). For an automorphism h of Cay(Zn, S - a), define the permutation g of V by „ f(xh)+ if e = +, g : xe ^ < x€ Zn, e € {+,-}. g \ ((x - a)h + a)- if e = -, ^ } The reader is invited to check that the above permutation g is an automorphism of r. It is clear that g G G and g = h; we conclude that G = Aut(Cay(Zn, S - a)). Now, the proposition follows along the following equivalences: (1) ^^ Every bicyclic group of r is conjugate to C in G ^^ Every regular cyclic subgroup of G is conjugate to C in G ^ (2). The last equivalence is Theorem 2.1. □ Remark 2.9. Let us remark that the equality Aut(r)0+ = Aut(r)a- does not hold in general. For example, take r as the incidence graph of the projective space PG(d, q) where d > 2 and q is a prime power (i.e., r is the bipartite graph whose color classes are identified by the set of points and the set of hyperplanes, respectively, and the edges are defined by the incidence relation of the space). It is well-known that PG(d, q) admits a cyclic group of automorphisms (called a Singer subgroup) acting regularly on both the points and the hyperplanes. This shows that r is isomorphic to a cyclic Haar graph, and we may identify the set of points with Z+, and the set of hyperplanes with Z-, where n = (qd - 1)/(q - 1). The automorphism group Aut(T) = PrL(d + 1, q) x Z2; and as PrL(d + 1, q) acts inequivalently on the points and the hyperplanes, Aut(r)0+ cannot be equal to Aut(r)a- for any a e Zn. 3 Haar graphs H(Z2m, {0, u, v,v + m}) In this section we prove Theorem 1.1 for Haar graphs H(Zn, S) satisfying certain additional conditions. Theorem 3.1. Let n = 2m and S = {0, u, v, v + m} such that (a) Zn = (u,v); (b) 1 < u < m, u | m; (c) Aut(H (Zn, S))0+ leaves the set {0-,u-} setwise fixed. Then H (Zn, S) = H (Zn, T ) if and only if there exist a e Z*n and b e Zn such that (1) aT + b = S ; or (2) aT + b = {0, u + m, v, v + m}, and 2 | u, 2u | m, u/2 = v + m/(2u)(mod m/u). By (b) of Theorem 3.1 we have 2u < m. We prove the extremal case, when 2u = m, separately. Notice that, in this case the conditions in (2) of Theorem 3.1 that 2 | u, 2u | m and u/2 ^ v + m/(2u)(mod m/u) can be replaced by one condition: u = 2(mod 4). Lemma 3.2. Let S be the set defined in Theorem 3.1. If 2u = m, then H(Zn, S) = H (Zn, T ) if and only if there exist a e Zn and b e Zn such that (1) aT + b = S ; or (2) aT + b = {0, u + m, v, v + m} and u = 2(mod 4). Proof. Let d = gcd(n, v). Because of (u, v) = Zn we have that gcd(u, v, n) = 1, i.e., gcd(n/4, v) = 1, and this gives that d e {1,2,4}. Note that, if d =1, then necessarily 2 { u. Let us write v = vid, where gcd(vi, n) = 1. Let v-1 denote the inverse of vi in the group Zn. Then the following hold in Zn (here we use that u = n/4): v-1v = d, v-1(v + m) = d + m and v-1u e {u, 3u}. We conclude that S is affinely equivalent to one of the sets Sj(d), i e {1,2} and d e { 1 , 2, 4} , where S1(d) = {0, u, d, d +2u} or S2(d) = {0, 3u, d, d + 2u}. The lemma follows from the following claims: (i) H (Zn ,S1(1)) = H (Zn,S2(1)). (ii) H (Zn, S1(1)) =aff h (Zn, S^d)) for d e {2,4}; (iii) H(Zn,S1(d)) =aff H(Zn,S2(d)) ^ d e {2,4} or (d =1 and u = 2(mod 4)); u-2 u-1 u-1 0 0 1 12 u-2 u-1 u-1 0 0 1 12 Figure 1: Haar graphs H(Zn,S1(l)) and H(Z;,S2(1)). (i): Define the mapping f : V ^ V by „ £ IV if x G {0, 1,....u - 1} U {2u, .. . .3u - 1}, f : x ^ < I (x + 2u)£ otherwise. We leave for the reader to verify that f is in fact an isomorphism from H(Zn, Si(1)) to H(Zn, S2(1)) (compare the graphs in Figure 1; here the white vertices represent the color class Z+, while the black ones represent the color class Z-). (ii): Since d G {2,4}, u is an odd number. For d G {2,4} define rd G Z; as follows: _ J2 + u if u = 1(mod4), _ f4 + u if u = 3(mod 4), r2 [2 + 3u if u = 3(mod4), r4 [4 + 3u if u = 1(mod4). It can be directly checked that rdS1(1)+ u = S1(d), so H (Zn,S1(1)) =aff H (Zn,S1(d)) for d G {2, 4}. (iii): If u is odd, then (2u+1)Si(d) = S2(d), hence H (Z„,Si(d)) =aff H (Z„,S2(d)). Since u is odd whenever d G {2, 4}, we are left with the case that d = 1 and u is even. If also u = 0(mod 4), then (u + 1)Si(1) + 3u = S2(1), and again H(Z„,Si(1)) =aff H (Zn,S2(1)). Suppose that d =1 and u = 2(mod 4). We finish the proof by showing that in this case H(Zn, Si(1)) =^aff H(Zn, S2(1)). Suppose that, there is an affine transformation ^ : x ^ rx + s, r G Z; and s G Zn, which maps the set Si(1) to Si(2). Then - (1+2u)^ = 2u in Zn. This implies that {1,1 + 2u}^ = {1,1 + 2u} and {o, u}^ = {0, 3u}, and hence r + s G {1,1 + 2u} and r{0,u} + s = {0, 3u}. A direct analysis shows that the above equations cannot hold if u = 2(mod 4). Thus H (Zn, Si(1)) =aff H (Zn, S2(1)), this completes the proof of (iii). □ Now, we turn to the case when 2u = m. Recall that the canonical bicyclic group C is generated by the permutation c defined in (1). For a divisor I | n, C will denote the u + 2 u + 2v + m 2v + u + u+m Jcb + u + m 2v + 2u 2u+m 2V+ 2u + m Vo Vi V2 Figure 2: The Haar graph H(Zn, S) subgroup of C generated by cg. It will be convenient to denote by Sg the partition of V into the orbits of Cg, i.e., Sg = Orb(Cg, V). Furthermore, we set nn,g for the homomorphism Vn,£ : Zn ^ Zg defined by nn,g(1) = 1. Observe that, if Sg is, in addition, a system of blocks for the group A = Aut(H(Zn, S)), then an action of A can be defined on H(Zg, nn,g(S)) by letting g g A act as for x g Zg and for e, e' g {+, -}, (x£)g = y£ {z£ : z g nn,g(xUg = K : z g ^JM}. (3.1) We denote by A(^) the corresponding kernel, and by góe the image of an element g g G. Note that, if X is a bicyclic group of H(Zn, S), then XSe = {x^ : x g X} is a bicyclic group of H(Zg,nn,g(s)). Let S = {0, u, v, v + m} be the subset of Zn defined in Theorem 3.1. Let S be the partition of V defined by S = {X U X^°-0 : X g Orb(Cu,V)}, (3.2) where ^1,0,0 is defined in (2.2). We write S = {V0,..., VM-1}, where V = {(iv + ju)+, (iv + ju)_ : j G {0,1,. .., (n/u) - 1}}. A part of H(Zn, S) is drawn in Figure 2 using the partition S. White and black colors represent again the color classes Z+ and Z_, respectively. For i G {0,1,..., u - 1}, let e® be the involution of V defined by (x + m)£ if x£ G Vi, x£ otherwise. It is clear that each e® G Aut(H(Zn, S)), and also that e®ej = eje® for all i, j G {0,1,..., u - 1}. Let E = (e0, e1,..., e„_1). Thus E < Aut(H(Zn, S)) and E = Z£. For a subset I C {0,1,..., u - 1} let e/ be the element in E defined by e/ = f]ie/ e®. The following lemma about imprimitivity systems of blocks (systems of blocks for short) will be used throughout the paper. Lemma 3.3. Let r = H(Zn, R) be a Haar graph and suppose that R* C R such that the point stabilizer Aut(r)0+ fixes setwise R-, and let d = |(R* — R*)|, where R - R = {ri — r2 : ri, r2 e R*}. Then the partition n of V defined by n = {X U X^-r.-r : X e Orb(Cn/d, V)}, where r e R*, is a system of blocks for Aut(r).2 Proof. For short we set A = Aut(r). Since R- is fixed setwise by A0+, we may write R* = Ri U---U Rfc, where R- is an A0+ -orbit for every i e {1,2,..., k}. Choose an arc (0+, r-) of r where we fix an element r e Ri for every i e {1,..., k}. We claim that, the orbital graph of A containing (0+, r-) is self-paired, and in fact it is equal to the Haar graph H(Zn, Rj) (for a definition of an orbital graph, see [5]). Define A as the color preserving subgroup of A. Then A = A x (^-i,0,0). Also, A = A0+ C, as C is transitive on Z+. Then the orbit of the arc (0+, r-) under A is (0+,r-)A = (0+,r-)Ao+ cW-1,0,0) = {(0+,rj-) : rj e Rj}0«-1.».») = {(j+, (j + rij)-) : rj e Rj,j e Zn}W-1.°.°> = {(j+, (j + rj)-): rj e Rj,j e Zn} U {((—j)-, (—j — rj) +): rj e Rj,j e Zn}, = {(j+, (j + rj)-): rj e Rj,j e Zn} U {((j + rj)-,j + ) : rj e Rj,j e Zn}, which is clearly equal to the set of arcs of H(Zn, Rj). The claim is proved. Since H(Zn,Rj) is an orbital graph, A < Aut(H(Zn,Rj)). Combining this with H (Zn, R*) = Uk=iH (Zn, Rj), we have that A < Aut(H (Zn, R*). Let S be the connected component of H(Zn,R*) which contains 0+. Obviously, the set W of vertices contained in S is a block for A. It is easy to verify that W = X U X^1r -r where X is the orbit of 0+ under Cn/d. The lemma follows. □ Lemma 3.4. Let S be the set defined in Theorem 3.1. If 2u = m, then the stabilizer Aut(H (Zn, S))0+ is given as follows. (1) If u = 2v(mod m/u), then Aut(H (Zn, S))0+ = E0+. (2) If u = 2v(mod m/u), then Aut(H(Zn, S))0+ = E0+ x F for a subgroup F < Aut(H(Zn, S))0+, |F| = 2. Proof. For short we set r = H(Zn, S) and A = Aut(r). Consider the partition S defined in (3.2). Applying Lemma 3.3 with R = S, R* = {0, u} and r = 0, we obtain that S is a system of blocks for A. The quotient graph r/S is a u-circuit if u > 2 and a 2-path if u = 2. Let g e A0+. Then g fixes the directed edge (V0, Vi) of r/S, hence it must fix all sets V. Thus A0+ < A^), where A^) is the kernel of the action of A on S. Consider the action of A0+ on V0. The corresponding kernel is A(V0), the pointwise stabilizer of V0 in A, and the corresponding image is a subgroup of Aut(r[V0]), where 2 Notice that, n does not depend of the choice of the element r E R» r[V0] is the subgraph of r induced by V0. Using that 2u = m, we show next that A(Vo ) = E0+. It is clear that A(Vo) > E0+. We are going to prove that A(Vo) < E0+ also holds. Let g G A(Vo). Then for a suitable element e G (e1), the product g e fixes pointwise V0 and fix the vertex v- from block V (see Figure 2). Thus ge acts on V as the identity or the unique reflection of the circuit r[VL] that fixes v-. If this action is not the identity, then ge switches v+ and (v + n — u)+, and so it must switch (v + u)- and (v + n — u)-. On the other hand, since (v + u)- is connected to u+ G V0, it follows that (v + u)- can only be mapped to (v + u + m)-, and so (v + n — u)- = (v + u + m)-, contradicting that 2u = m. We conclude that ge acts as the identity also on V1. Continuing in this way, we find that ge' is the identity with a suitable choice of e' G E0+, hence g = e'. The equality A(Vo) = E0+ together with Aut(r[V0]) = D4u imply that |A0+ : E0+1 < 2. Moreover, |A0+ : E0+1 = 2 holds exactly when A0+ contains an involution g for which g : 0- ^ u-. In the latter case A0+ = E0+ x (g) as g centralizes E (to see this, observe that g is in the kernel A(Ä), and acts on every block V as an element of D2n/u, whereas E acts on V as the center Z(D2n/u).) We settle the lemma by proving the following equivalence : A0+ = E0+ x Z2 ^^ u = 2v(mod m/u). (3.3) Suppose first that A0+ = E0+ x (g), where g G A0+ and g : 0- ^ u-. By (c) of Theorem 3.1, {v-, (v + m)-}Ao+ = {v-, (v + m)-}. Applying Lemma 3.3 with R = S, R = {v, v + m} and r = v, we obtain that the set B = {0+, m+, v-, (v + m)-} is a block for A. The induced graph r[B] is a 4-circuit (again, see Figure 2). Denote by A{g} the setwise stabilizer of B in A, and by Agg} the permutation group of B induced by A{B}. As r[B] is a 4-circuit, Agg} < D8. This gives that {0+, m+} is a block for Agg}, and therefore it is also a block for A. We conclude that Sm = {X : X g Orb(Cm, V)} is a system of blocks for A. Consider the action of A on H(Zm, nn,m(S)) defined in (3.1). Then E < A(Äm), while g G A(^m). This implies that gSm is an automorphism of H(Zm, nn,m(S)) which normalizes its canonical bicyclic group. This means that gSm = 2,0,s = idV, r2 = 1 in Zn, and we calculate that (cVr,0,s)2 sends x+ to (x + r(r + 1)i) + for every x e Zn. That Z+ is an orbit of X is equivalent to the condition that gcd(n, r + 1) = 2. Using this and that r2 — 1 = (r — 1)(r + 1) = 0(mod n), we find that n/2 divides r — 1, so r = 1 or r = n/2 + 1. Since r = 1, we have that r = n/2 + 1 and 8 | n. Then (^y,0,s)2 sends x-to (x + (n/2 + 2)s) Since (^y,0,s)2 = idV, we obtain that s = 0 or s = n/2. On the other hand, it can be directly checked that, if 8 | n, r = n/2 + 1 and s e {0, n/2}, then the permutation c^r 0,s generates a bicyclic subgroup of (D, yv,0,s). Obviously, this bicyclic subgroup cannot be C. The lemma is proved. □ Everything is prepared to prove the main result of the section. Proof of Theorem 3.1. The case that 2u = m is settled already in Lemma 3.2, hence let 2u = m. We consider the action of A = Aut(H(Zn, S)) on the system of blocks defined in (3.1). We claim that the corresponding image Ađm has a unique bicyclic subgroup (which is, of course, Cóm). This is easy to see if A0+ = E0+, because in this case Ađm = (DA0+ = Dđm. Let A0+ = E0+. Then A0+ = E0+ x F for some subgroup F, |F| = 2. By (3.4), Ađm = (Dđm, ¥V,o,nn m(u)). Also, r = — 1(mod m/u), hence r =1 in Zm. By Lemma 3.5, Ađm contains more than one bicyclic subgroup if and only if 8 | m, r = m/2 + 1 and nn,m(u) G {0, m/2}. In the latter case u G {m, m/2}, which is impossible as u < m/2. Hence ASm contains indeed a unique bicyclic subgroup. We calculate next the number of bicyclic groups of H(Zn, S); we denote this number by B. In fact, we are going to derive the following formula: B =l>-2 if 2 | u and 2 { (m/u), (38) Ì2"-1 otherwise. Let g G G such that (g) is a bicyclic group of H(Zn, S). Since G = DA0+, g can be written as g = xy with x G D and y G A0+. Since (g) is a bicyclic group, g fixes the color classes setwise, implying that x g C. The image (g)đm is also a bicyclic subgroup of ASm, hence by the previous paragraph, (g)đm = CSm. Now, since x G C, ySm G CSm, from which ySm = idsm. We conclude that x = c® G C for some i G {1,..., n - 1} with gcd(i, m) = 1, and y G E0+, and so y = e/ for a subset I C {1,..., u - 1}. Obviously, the product ^(n)B calculates the number of elements g G G such that (g) is a bicyclic group of H(Zn,S), where ^ denotes the Euler's totient function. Therefore, ^(n)B is equal to the number of elements in the form c®e/ that i G {1,..., n - 1}, gcd(i, m) = 1, I C {1,..., u - 1}, and (c®e/) is a bicyclic group of H(Zn, S). Let us pick c®e/ with i G {1,..., n - 1}, gcd(i, m) = 1, and I C {1,..., u - 1}. It is easily seen that e/c® = c®e/+j, where I + i = {x + i : x G I}, here the addition is taken modulo u. Using this and induction on u, it follows that (c®e/)" = c"®e/e/+® • • • e/+(„-i)®. Since gcd(i, m) = 1 and u | m, gcd(i, u) = 1, from which e/e/+® • • • e/+(u-i)i = (e0ei • • • e^i)1' cm|/| Thus (c®e/)" = cu(®+ m 1/1 ). This and gcd(i,u) = 1 show that (c®e/) is a semiregular group. Therefore, (c®e/) is a bicyclic group if and only if c®e/ is of order n, or equivalently, gcd (i + m|11, ^) =1. (3.9) uu Notice that, since gcd(i, m) = 1, the greatest common divisor above is always equal to 1 or 2. Suppose at first that 2 | (m/u). Then 2 | m and i is odd. Hence (3.9) always holds. We obtain that the number of elements in A which generate a bicyclic group is ^(n)2u-:i, and so B = 2u-iL, as claimed in (3.8). Suppose next that 2 \ (m/u). Now, if 2 | u, then 2 | m, hence 2 \ i, and so (3.9) holds if and only if |11 is even. We deduce from this that B = 2u-2, as claimed in (3.8). Finally, if 2 \ u, then 2 \ m, and in this case (3.9) holds if and only if gcd(i, n) = 1 and |11 is even, or gcd(i, n) = 2 and |11 is odd. We calculate that B = 2u-1, and this completes the proof of (3.8). Let S be a bicyclic base of H(Zn, S). By (3.5) and (3.8) we obtain that, |S| > 1 if and only if |A : Na(C)| = 2u-2 andB = 2"^. This happens exactly when (2 | u and (u ^ 2v(mod m/u) or u/2 = v(mod m/u)^ and (2 { u or 2u | After some simplification, |S| > 1 ^^ 2 | u, 2u | m and u/2 = v + m/(2u)(mod m/u). Suppose that |s| > 1. Then A contains exactly 2n-1 bicyclic subgroups, 2n-2 of which are conjugate to C. These 2n-1 subgroups are enumerated as: (ce/), I C {1,..., u - 1}. For i e {1,..., u - 2}, ej ce j = ce{ii+1}. We can conclude that the bicyclic subgroups split into two conjugacy classes: {(ce/) : I C {1,..., u - 1}, |1 | is even } and {(ce/) : I C {1,.. .,u - 1}, |1 | is odd }. In particular, |S| = 2. Choose £ from Sym(V) which satisfies £c£-1 = ce1 and £ : 0+ ^ 0+, 0- ^ 0-. Then S can be chosen as S = {idV, £}. Also, {v-, (v + m)-}« = {v-, (v + m)-}, and since (ce1)u+m = cu, (u-)« = (0-)(cei)"+m« = (0-)«c"+m = (u + m)-. Thus H(Zn, S) « = H(Zn, {0, u + m, v, v + m}). The theorem follows from Theorem 2.6. □ 4 Proof of Theorem 1.1 Theorem 1.1 follows from Theorem 3.1 and the following theorem. Theorem 4.1. Let H(Zn, S) be a connected Haar graph such that |S| = 4 and S is not a BCI-subset. Then n = 2m, and there exist a e Zn and b e Zn such that aS + b = {0, u, v, v + m} and the conditions (a)-(c) in Theorem 3.1 hold. Before we prove Theorem 4.1 it is necessary to give three preparatory lemmas. For an element i e Zn, we denote by |i| the order of i viewed as an element of the additive group Zn. Thus we have |i| = n/ gcd(n, i). Lemma 4.2. If R = {i,n - i, j} is a generating subset of Zn with |i| odd, then R is a CI-subset. Proof. For short we set A = Aut(Cay(Zn, R)) and denote by A0 the stabilizer of 0 e Zn in A. Clearly, A0 leaves R setwise fixed. If A0 acts on R trivially, then A = Zn, and the lemma follows by Theorem 2.1. If A0 acts on R transitively, then Cay(Zn, R) is edge-transitive. This condition forces that R is a CI-subset (see [10, page 320]). We are left with the case that R consists of two orbits under A0. These orbits must be {i, n - i} and {j}. It is clear that A0 leaves the subgroups (i) and (j) fixed; moreover, the latter set is fixed pointwise, and since |i| is odd, (i) consists of (|i| - 1)/2 orbits under A0, each of length 2, and one orbit of length 1. We conclude that Zn = (i) x (j), and also that A is permutation isomorphic to the permutation direct product ((Z|j|)rjght x (n)) X (Z|j| )rjght. For l e {|i|, |j|}, (Z£)right is generated by the affine transformation x ^ x + 1, and n is the affine transformation x ^ -x. We leave for the reader to verify that the above group has a unique regular cyclic subgroup. The lemma follows by Theorem 2.1. □ Lemma 4.3. Let n = 2m and R = {i, n - i, j, j + m} be a subset of Zn such that (a) R generates Zn; (b) |i| is odd; (c) the stabilizer Aut(Cay(Zn, R))0 leaves the set {i, n — i} setwise fixed. Then R is a CI-subset. Proof. For short we set A = Aut(Cay(Zn, R)). Let T be a subset of Zn such that Cay(Zn, R) = Cay(Zn, T) and let f be an isomorphism from Cay(Zn, R) to Cay(Zn, T) such that f (0) = 0. Let us consider the subgraphs ri = Cay(Zn, {i, n - i}) and r2 = Cay(Z;, {j,j + m}). By condition (c), the group A preserves both of these subgraphs, that is, A < Aut(r^) for I G {1, 2}. As f is an isomorphism between two Cayley graphs, f (Zn)rightf-i < A. Then f(Z;)rightf-i < A < Aut(r), implying that f maps r to a Cayley graph Cay(Zn, Tft) for both I G {1,2}. Clearly, T = Ti U T2. It was proved by Sun [15] (see also [10]) that every subset of Zn of size 2 is a CI-set. Using this, it follows from Cay(Zn, {i,n - i}) = Cay(Zn,T\) that T = a{i, n - i} for some a G Z;. Letting ti = ai, we have T = {ti,n - ti} suchthat |i| = |ti|. In the same way, T2 = a'{j,j + m} for some a' G Z;, and letting t2 = a'j, we have T2 = {t2, t2 + m} with |t21 = |j|. Since f (0) = 0, f maps {i, n - i} to T\ = {ti, n - ti} and {j, j + m} to T2 = {t2, t2 + m}. We claim that the partition of Zn into the cosets of (m) is a system of blocks for Aut(r2), hence also for the group A < Aut(r2). Let us put A = Aut(r2). Then A0 leaves the set T = {j, j+m} setwise fixed. Thus the setwise stabilizer A{T} of the set T in A can be written as A{T} = A{T} n A = A{T} n Ao(Z;)right = a4o(a4{t} n (Zn)right) = A0(mright). Here (Zn)right is generated by the affine transformation x ^ x +1, and mright is the permutation x ^ x + m for every x G Zn. Thus A0(mright) is a subgroup of A which clearly contains A0. By [5, Theorem 1.5A], the orbit of 0 under the group A0 (mright) is a block for A. Now, the required statement follows as the latter orbit is equal to 0A0 {mright > =0Ìmright) = (m) . Since the partition of Zn into the cosets of (m) is a system of blocks for A, the isomorphism f induces an isomorphism from Cay(Zm, nn,m(R)) to Cay(Zm, nn,m(T)), we denote this isomorphism by f. Note that, f (0) = 0 for the identity element 0 G Zm. The set nn,m(R) satisfies the conditions (a)-(c) of Lemma 4.2, hence it is a CI-subset. This means that f is equal to a permutation x ^ rx for some r G Z^. Let s G Z; such that nn,m(s) = r. Then nn,m(si) = n;,m(s)n;,m(i) = nn,m(ti), and so the following holds in Zn: si = ti or si = ti + m. (4.1) The order |ti| = |i| is odd by (b), implying that |ti| = |ti + m|, and so si = ti holds in (4.1). We conclude that sR = T, so R is a CI-subset. The lemma is proved. □ Lemma 4.4. Let n = 2m and S = {0, u, v, v + m} such that (a) S generates Zn; (b) 1 < u < n, u | n but u { m; (c) Aut(H (Zn,S))0+ leaves the set {0-,u-} setwise fixed. Then S is a BCI-subset. Proof. Let S be the partition of V defined in (3.2). Applying Lemma 3.3 with R = S, R = {0, u} and r = 0, we obtain that S is a system of blocks for A = Aut(H(Zn, S)). Thus the stabilizer A0+ leaves the set V0 setwise fixed, and we may consider the action of A0+ on V0. The subgraph of H(Zn, S) induced by the set V0 is a circuit of length 2n/u, thus A0+ fixes also the vertex on this circuit antipodal to 0+. We find that this antipodal vertex is (u/2 + m)_. Therefore, A0+ = A(m+„/2)-, and thus S is a BCI-subset if and only if S - u/2 + m is a CI-subset of Zn, see Proposition 2.8. The latter set is S - u/2 + m = { u/2 + m, -u/2 + m, v - u/2, v - u/2 + m }. Since u { m, u is even and the order |u/2 + m| is odd. Lemma 4.3 is applicable to the set S - u/2 + m (choose i = u/2 + m and j = v - u/2), it gives us that S - u/2 + m is a CI-subset. This completes the proof. □ Proof of Theorem 4.1. Let S be the subset of Zn given in Theorem 4.1. We deal first with the case when the canonical bicyclic group C is normal in A = Aut(H(Zn, S)). Case 1. C < A. By Theorem 2.6, there is a bicyclic group X of H(Zn, S) such that X = C. Since C < A, X is generated by a permutation in the form c®^r,0,s, r G Zn, s G Zn, andord(yv,0,s) > 2. The permutation yv,0,s acts on both Z+ and Z_ as an affine transformation. This fact together with the connectedness of H(Zn, S) imply that, ^y,0,s acts faithfully on S_. Thus ord(^y,0,s) < 4. Suppose that ord(yv,0,s) = 4. We may assume without loss of generality that S_ can be obtained as S_ = { (0_) 2. We prove first that Cn/ni < A. Applying Lemma 3.3 with R = S, R* = S1 and r = s1 G S1, we obtain that the partition Ó = {X U X^1,'i,-'i : X G Orb(Cn/ni, V)}, m m is a system of blocks for A. Let us consider the action of A^) (the kernel of A acting on S) on the block of S which contains 0+. Denote this block by A, and by A' the block which contains s- for some s e S2. Notice that, the subgraph of H(Zn, S) induced by any block of S is a circuit of length 2n1, and when deleting these circuits, the rest splits into pairwise disjoint circuits of length 2n2. Let S denote the unique (2n2)-circuit through s-. Now, suppose that g e A(đ) which fixes A pointwise. If V (S) n A = {0+}, then g must fix the edge {0+, s-}, and so fixes also s-. If V (S) n A = {0+}, then |V (S) n A| = n2 > 2. This implies that g fixes every vertex on S, in particular, also s-. The block A' has at least n1 vertices having a neighbor in A, hence by the previous argument we find that all are fixed by g. Since n1 > 2, A' is fixed pointwise by g. It follows, using the connectedness of H(Zn, S), that g = idV, hence that A(đ) is faithful on A. Thus Cn/ni is a characteristic subgroup of A(Ä), and since A^) < A, Cn/ni < A. Let G be the unique normal subgroup of A that fixes the color classes Z+ and Z-. We consider N = G n CA(Cn/ni ). Then C < N and N < A. Pick g e N0+ such that g acts non-trivially on S-. Since N centralizes Cn/ni, g fixes pointwise the orbit of 0+ under Cn/ni, and hence also A. Then g2 fixes S- pointwise, and so also A'. We conclude that g2 = idV, and that either N = C, or N = C x (g). The case N = C is impossible because C ^ A. Let N = C x (g). Then (S-)g = S- (for both i e {1, 2}), hence S, is a union of orbits of g. As g normalizes C and fixes 0+, g = y>r,0,s. Recall that ord(g) = 2. If r =1, then by Lemma 3.6, either C is the unique cyclic subgroup of N, or 8 | n, r = n/2 +1 and s = 0 or s = n/2. In the former case C is characteristic in N, and since N < A, C < A, a contradiction. Therefore, we are left with the case that r = 1 (and so s = n/2), or 8 | n, r = n/2 + 1 and s = 0 or s = n/2. Then every orbit of g is of length 1 or 2, and if it is of length 2, then is in the form {je, (j + m)e} as we proved in Case 1. Since n, > 2, we see that S- must be fixed pointwise by g for both i e {1,2}. This, however, contradicts that g was assumed to act non-trivially on S-; and so n1 = 2. This means that 2 | n, say n = 2m, and the group generated by the set S1 — S1 = {x — y : x, y e S1} is equal to {0, m}. Then we can write S1 = {s, s + m}. It can be proved as before that there exist a e Zn and b e Zn such that aS2 + b = {0, u} for some divisor u of n. Then, letting v = as + b, we get aS1 + b = {v, v + m}. We finish the proof of this case by showing that the set R = aS + b = {0, u, v, v + m} satisfies the conditions (a)-(c) of Theorem 3.1. (a): As H(Zn, S) is connected, H(Zn, R) is also connected. This implies that {u, v} is a generating set of Zn. (c): Since S1 and S2 are left fixed setwise by A, Aut(H(Zn,R))0+ leaves the set {0-, u- } setwise fixed. (b): If u =1, then Aut(H(Zn, {0, u})) < D4n. But then C < A, which is a contradiction. 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ARS MATHEMATICA CONTEMPORANEA 7 (2014) 237-246 Small cycles in the Pancake graph Elena Konstantinova * Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia Department of Mathematics, Yeungnam University, 712-749, South Korea Alexey Medvedev Sobolev Institute ofMathematics, Novosibirsk, 630090, Russia Central European University, Budapest, 1051, Hungary Received 17 June 2011, accepted 23 April 2013, published online 26 April 2013 Abstract The Pancake graph is well known because of the open Pancake problem. It has the structure that any /-cycle, 6 < / < n!, can be embedded in the Pancake graph Pn, n > 3. Recently it was shown that there are exactly n independent 6-cycles and n!(n - 3) distinct 7-cycles in the graph. In this paper we characterize all distinct 8-cycles by giving their canonical forms as products of generating elements. It is shown that there are exactly n!(n3+i2ni-i03n+i76) distinct 8-cycles in Pn,n > 4. A maximal set of independent 8-cycles contains n of these. Keywords: Cayley graphs, Pancake graph, cycle embedding, small cycles. Math. Subj. Class.: 05C15, 05C25, 05C38, 90B10 1 Introduction The Pancake graph Pn = (Symn,PR),n > 2, is the Cayley graph on the symmetric group Symn of permutations n = [n1n2... nn], where ni = n(i) for any 1 < i < n, with the generating set PR = [ri g Symn : 2 < i < n} of all prefix-reversals r reversing the order of any substring [1, i], 2 < i < n, of a permutation n when multiplied on the right, i.e. [n1... nini+1... nn]ri = [ni... n1ni+1... nn]. It is a connected vertex-transitive (n-1)-regular graph of order n!. This graph is well known because of the combinatorial Pancake problem which was posed in [4] as the problem of finding its diameter. The problem is still *The research was supported by grant 12-01-00448 of the Russian Foundation of Basic Research. The corresponding author. E-mail addresses: e konsta@math.nsc.ru (Elena Konstantinova), an medvedev@yahoo.com (Alexey Medvedev) open. Some upper and lower bounds [5, 6] as well as exact values for 2 < n < 19 [1,2] are known. One of the main difficulties in solving this problem is the complicated cycle structure of the Pancake graph. This graph is hamiltonian [11] and the following result is also known. Theorem 1.1. [7, 10] All cycles of length l, where 6 < l < n!, can be embedded in the Pancake graph Pn, n > 3, but there are no cycles of length 3,4 or 5. An explicit description of cycles is gradually being developed. The first results concerning cycle characterization in the Pancake graph were obtained in [8] where the following cycle representation via a product of generating elements was used. A sequence of prefix-reversals Cl = ri0 ... ril_1, where 2 < j < n, and j = ij+1 for any 0 < j < l - 1, such that nri0 ... ril_1 = n, where n g Symn, is called a form of a cycle Cl of length l. A cycle Ci of length l is also called an l-cycle, and a vertex of Pn is identified with the permutation which corresponds to this vertex. It is evident that any l-cycle can be presented by 21 forms (not necessarily different) with respect to a vertex and a direction. The canonical form Cl of an l-cycle is called a form with a lexicographically maximal sequence of indices io ... il-i. By using this description, the results characterizing 6- and 7-cycles were obtained in [8]. Theorem 1.2. [8] The Pancake graph Pn,n > 3, has n6 independent 6-cycles of the canonical form C6 = rir2rir2rlr2 and n!(n — 3) distinct 7-cycles of the canonical form C7 = rkrk-1rkrk-1rk-2rkr2, where 4 < k < n. Each of the vertices of Pn belongs to exactly one 6-cycle and 7(n — 3) distinct 7-cycles. The main result of this paper is the following theorem. Theorem 1.3. Each of vertices of Pn, n > 4, belongs to N = n3+12n2-10in+176 distinct 8-cycles of the following canonical forms: C1 — rk rj ri rj rk rk-j+i ri rk—j+i, 2 < i 4. Corollary 1.5. A maximal set of independent 8-cycles in Pn,n > 4, contains n of these. The proof of Theorem 1.3 is based on the hierarchical (recursive) structure of the Pancake graph which can be presented as follows. The graph Pn,n > 3, is constructed from n copies of Pn_i(i), 1 < i < n, such that each Pn-1(i) has the vertex set Vi = {[ni... nn_1i], where nk g {1,..., n}\{i} : 1 < k < n - 1}, |Vi| = (n - 1)!, and the edge set Ei = {{[n .. .nn_ii], [n .. .nn_ii] ró} : 2 < j < n - 1}, |Ei| = (n_i}2(n_2). Any two copies Pn_i(i),Pn_i(j),i = j, are connected by (n - 2)! edges presented as {[in2...n„_ij], [jn„_i...n2i]}, where [in...n„_ij]r„ = [jn„_i...^i]. Prefix-reversals rj, 2 < j < n - 1, define internal edges in all n copies Pn_i(i), 1 < i < n, and the prefix-reversal rn defines external edges between copies. Copies Pn_i(i), or just Pn_i when it is not important to specify the last element of permutations belonging to the copy, are also called (n - 1)-copies. Since P3 = C6 and due to the hierarchical structure, P4 has four copies of P3, each of which obviously cannot contain 8-cycles. However, P4 has 8-cycles consisting of paths within copies of P3 together with external edges between these copies. In general, any 8-cycle of Pn, n > 4, must consist of paths within subgraphs that are isomorphic to Pk_i for some 4 < k < n, joined by external edges between these subgraphs. Hence, all 8-cycles of Pn, n > 4, could be found recursively by considering 8-cycles within each Pk, 4 < k < n, consisting of vertices from some copies of Pk_i. This approach is used in the proof of Theorem 1.3. To get the main result, we also need some technical lemmas concerning paths of length three between vertices of a given form. So, in the next section we introduce additional notations and prove two small lemmas. In Section 3 we prove Theorem 1.3 and its corollaries. 2 Technical lemmas A segment [ni... nj] of a permutation n = [ni... ni... nj ... nn] consists of all elements that lie between ni and nj inclusive. Any permutation can be written as a sequence of singleton and multiple segments. We use characters from {p, q, s, t} to denote singletons and characters from {a, ß, 7, A, B, C} to denote multiple segments. If n = [aß], where a = [nin2... ni] and ß = [ni+i... nn], then nr|a| = [aß], where |a| is the number of elements in a segment a, and a is the inversion of a segment a. It is obvious that a = a. Note that we allow empty segments where this does not contradict the initial definitions. An independent set D of vertices in a graph is called an efficient dominating set if each vertex not in D is adjacent to exactly one vertex in D [3]. It is known [9] that Dp = {[pn2 ...nn] : nj g {1,..., n}\{p}, 2 < j < n}, |Dp| = (n - 1)!, 1 < p < n, are efficient dominating sets in Pn, n > 3. Let us note that external edges of Pn are incident to vertices from different efficient dominating sets of Pn. The distance d = d(n, t) between two vertices n, t in Pn is defined as the least number of prefix-reversals transforming n into t , i.e. nri1 n2 ...rid = t . The next lemma gives a full list of paths of length three between two vertices of the same efficient dominating set. Lemma 2.1. Two permutations n, t g Dp, 1 < p < n, are at distance three from each other in Pn, n > 3, if and only if 1) either t = nrjrirj-, 2 ^ i < j ^ n, where n = [AB7], t = [ABy]; 2) or t = nrjriri_j+i, 2 ^ j < i ^ n, where n = [pABy], t = [pBAy]. Proof. We consider n g Dp such that n = [paqßk], nj = q. Let us find other vertices from Dp being at the distance three from n. Let ni = nrj = [qapßk], where nj = p, 2 < j < n. An application of a prefix-reversal ri, 2 < i < n, i = j, to the permutation ni gives us two cases: either i < j or i > j. 1) If i < j then n2 = n1ri = [a2qa1pßk], where n2 = p, a = a\a2 and |a2| = i — 1, and then t = n2rj = [pa1qa2ßk]. Hence t = nrj rirj and we get n = [ABj], t = [ABj] by setting A = pa1, B = a2q, 7 = ßk. Note, that using rj is the only way to restore p to the first position and thus to end at an element of Dp after reaching n2. 2) If i > j then n2 = n1ri = [ß1paqß2k], where n2_j+1 = p, ß = ß1ß2 and |ß1| = i — j, and then t = n2ri-j+1 = [pß1 aqß2k]. Hence t = nrjriri-j+1 and we get n = [pABy], t = [pBAy] by setting A = aq, B = ß1, 7 = ß2k. Note, that using ri_j+1 is the only way to restore p to the first position and thus to end at an element of Dp after reaching n2. □ The next lemma gives a description of paths of length three defined on internal edges of the graph between vertices of given forms. Lemma 2.2. Let permutations n and t be presented as: 1) n = [y1ABy2] and t = [y1ABy2], where |y1| > 1, |A| > 2. Then: a) there exists a unique path of length three: t = nr|7i| + |A|r|A|r|7l| + |A|, (2.1) provided that either |y1| > 2 and |A| > 2, or |y1| = 1 and |A| > 3; b) there are two paths of length three: t = nr2 r3r2, t = nr3r2r3, (2.2) provided that |y1 | = 1 and |A| = 2; 2) n = [Y1ABY2] and t = [71BA72], where |y1 | > 0, |A| > 1, |B| > 1. Then: a) there is a unique path of length three: t = nr|7i| + 2r2r|7i|+2, (2.3) provided that |y1| > 2, and |A| = |B| = 1; b) there is a unique path of length three: T = nr|Yi | + | A|r|Yi | + | A| + |B| r|Yi | + |B|, (2.4) provided that |y1| = 1, and |A| = 1 or |B| = 1; c) there are two paths of length three: t = nr2r3r2 = nr3r2r3, (2.5) provided that |y1| = |A| = |B| = 1; d) there is a unique path of length three: T = nr|A| r|A| + |B|r|B|, (2.6) provided that |y1| =0 and |A| > 2, |B| > 2. There are no other paths of length three between n and t of these types. Proof. 1) If n = [yiaby2] and t = [71AB72], then (2.1) is checked by a direct verification: [71AB72] |y-1-|a| [atTb72] —' [ay?by2] iy1—|a| [y1aby2]. Suppose that there is one more path of length three. Then these two paths should form a 6-cycle. In part (a), either |y1| > 2 and |A| > 2, or |y1| = 1 and |A| > 3, so r|Yl|+|A| = rm for some m > 4, but by Theorem 1.2, no 6-cycle includes rm with m > 4 in its form. Therefore, the given path is the only one in this case. In part (b), |y1| = 1 and |A| = 2, so m = 3 and the condition of Theorem 1.2 holds, hence we obtain two distinct paths of stated forms (2.2). 2) If n = [y1aby2] and t = [y1bay2], and |y1 | > 2, |A| > 1, |B| > 1, then there is the following path of length four between these vertices: n = [y1aby2] r 1 ^ 1 A 1 [atIby2] r 1 Yl' +4 1+1 B 1 ^71^72]r 1 1 B 1 [7IBAY2] r^'[Y1BA72]= T. (2.7) Suppose there is also a path of length three between n and t. By Theorem 1.1, there are no cycles of length 3 or 5, and hence no paths of lengths 3 and 4 exist between two fixed vertices unless the paths are disjoint. This means that these two paths should form a 7-cycle, including the sequence rm+arm+a+brm+brm, where |y1| = m, |A| = a and |B| = b. By Theorem 1.2, this is possible only in the case when m = k — 2, k > 4, and a = b =1, which implies that a unique path of length three has the form rm+2r2rm+2 that corresponds to (2.3). Putting |y1 | = 1 in (2.7), a path t — nr|7l |+|A|r|7l|+|A|+|B|r|7l|+|£|, corresponding to (2.4), is obtained. Taking |y1| = m, |A| = a and |B | = b, the obtained path is presented as rm+arm+a+brm+b. Suppose that there is one more path of length three between n and t. Then these two paths should form a 6-cycle. By Theorem 1.2, this is possible only in the case when m = a = b = 1, which gives us the paths t = nr2r3r2 and t = nr3r2r3, corresponding to (2.5). Putting |y1| =0 in (2.7), a path t = nr|A|r|A|+|B|r|B|, corresponding to (2.6) with |A| > 2, |B| > 2, is obtained. Suppose there is one more path of length three between n and t. Then these two paths should form a 6-cycle. By the conditions of Lemma, |A| + |B| > 4, hence r|A| + |B| = rm for some m > 4, but by Theorem 1.2, no 6-cycle includes rm with m > 4 in its form. Therefore, the given path is the only one in this case. If |A| = 1 or |B| = 1, then the path above is transformed into a 2-path or an edge. This completes the proof of the lemma. □ 3 Proof of Theorem 1.3 To find all 8-cycles passing through the same vertex in Pn, n > 4, we use its hierarchical structure by considering recursively 8-cycles within each copy Pk, 4 < k < n, consisting of vertices from copies of Pk-1. It is assumed that any copy of Pk-1 has at least two vertices, since each vertex has a unique external edge. We obtain canonical forms of 8-cycles and count their numbers. Case 1: an 8-cycle within Pk has vertices from two copies of Pk-1 Suppose that an 8-cycle is formed on vertices from copies Pk-1(p) and Pk-1(s), 1 < p = s < k. It was shown in [8] that if two vertices n and t, belonging the same (k — 1)-copy, are at the distance at most two, then their external neighbours n and t should belong to distinct (k — 1)-copies. Hence, an 8-cycle cannot occur in situations when its two (three) r> Pk-l(p) case 1 case 2 Figure 1: (4 + 4)-situation. vertices belong to one copy and six (five) vertices belong to another one. Therefore, such an 8-cycle must have four vertices in each of the two copies. (4 + 4)-situation. Suppose that four vertices of such an 8-cycle belong to a copy Pk-1(s), and other four vertices belong to a copy Pk-1(p). Herewith, four vertices of Pk-1 (s) should form a path of length three whose endpoints should be adjacent to vertices from Pk-1(p), which means both vertices should belong to the efficient dominating set Dp. So, one vertex of Pk-1(s) that is adjacent to a vertex of Pk-1(p) must have the form [paqßs]. By Lemma 2.1, it is not hard to see that this gives rise to two possible forms for the remaining vertices of Pk-1(s). These are given in Figure 1, where |a| = j - 2 and so |ß| = k - j - 1. In the first case we also have a = a1a2 and |a2| = i - 1 > 1, while in the second case we also have |ß11 = i - j > 1 and |ß21 = k - i - 1. Denote y1 = sß, A = qO", B = a1, y2 = p, where |y1| = |ß| + 1 > 1, |A| > 2, |B| > 0, then in the first case ,nP4 have the forms [y1ABy2] and [y1ABy2]. By Lemma 2.2 (case 1a), there is a unique path of length three between these permutations if |y1| = |ß| + 1 = k - j > 1 and |A| = |a2| + 1 = i > 3, or k - j > 2 and i > 2, and by Lemma 2.2 (case 1b), there are two distinct paths if k - j = 1 and i = 2. Hence, such an 8-cycle has the form Cg = rk-j+irirk-j+irkrjrirjrk, with 2 < i < j < k - 1, 4 < k < n, the canonical form of which corresponds to (1.1). The case of k - j = 1 and i = 2 by symmetry gives one additional form Cg = rk-1r2rk-1rkr2r3r2rk, the canonical form of which corresponds to (1.2). Denote 71 = sß2, A = ß1, B = qa, 72 = p, where = |ß21 + 1 > 1, |A| = |ß11 > 1, |B| = |a| + 1 > 1, then in the second case we have npi = [y1ABy2], nP4 = [y1BAy2]. By Lemma 2.2 (case 2a), there is a unique path of length three between npi and nP4 if |Y11 = k - i > 2, |A| = |ß11 = i - j = 1 and |B| = |a| + 1 = j - 1 = 1. Hence, j = 2, 1 = 3, and for k > 5 an 8-cycle has the form rk-1r2rk-1rkr2r3r2rk, corresponding again to the canonical form (1.2). By Lemma 2.2 (case 2b), there also exists a unique path of length three between npi and nP4 if Y | = k - i = 1, |A| = |ßx| = i - j > 1, |B| = |a| + 1 = j - 1 > 1. This means that i = k -1, and such an 8-cycle has the form Cg = rkrk-j-rk-1rjrkrk-j-rk-1rj, 2 < j < k - 2, the canonical form of which corresponds to (1.3), if we set j = i. So, there Figure 2: (2 + 2 + 4)-situation. is a unique path of length three under the conditions listed, unless |A| = |B| = 1 when by Lemma 2.2 (case 2c) this path is not unique. So, k = 4, j = 2, i = 3 and 8-cycles take forms r2r3r2r4r3r2r3r4 and r3r2r3r4r3r2r3r4, corresponding to forms (1.2) and (1.1). Thus, all 8-cycles occurring in the case of two copies are found. Case 2: an 8-cycle within Pk has vertices from three copies of Pk_1 Suppose an 8-cycle is formed on vertices from copies Pk-1(p), Pk-1(q), Pk-1(s), where 1 < p = q = s < k. There are following possible situations in this case. (2 + 2 + 4)-situation. The distribution of vertices among the copies is presented by Figure 2. Let nsi = [paqßs\ where nsii = q with |a| = i — 2, |ß| = k — i — 1. Then nS2, nPi, nP2, nqi and nq4 are straightforward to define. Vertices nqi and nq4 differ in the order of segments sß and pa, hence they have the forms [y1ABy2\ and [y1BAy2\, where Y1 is empty, A = sß, B = pa, y2 = q and |A| = |ß| + 1 > 1, |B| = |a| + 1 > 1. By Lemma 2.2 (case 2d), between nqi and nq4 there exists a unique path of length three provided that |A| = k — i > 2, |B| = i — 1 > 2, and no path of this length if |A| = 1 or |B| = 1. Thus, an 8-cycle has the form C4 = rkrk-i+1rkri_1rk_1rk_irkri, where 3 < i < k — 2, k > 5, the canonical form of which corresponds to (1.4). (2 + 3 + 3)-situation. The distribution of vertices among the copies is presented by Figure 3. Let nsi = [paqßs\, where |a| = i — 2 and |ß| = k — i — 1. Then nS2, npi and nqi are straightforward to define. Since nP3 and nq3 are joined by an external edge, np3 = q. Moreover, npi and nP3 should be joined by a path of length two that can be obtained by two ways: pi r 7; _ -, \ [ß2sß1qap\ ^ [qß1sß2ap\ = np3, where |ß21 = 0. n i = [sßqap\ ^ < 2 I [a2qßsSTp\ ^ [qä^ßsOip] = np'3, where |a2| = 0. From the other side, nq'3 and nP3 are joined by an external edge, hence n\3 = p, and there Figure 3: (2 + 3 + 3)-situation. should be a path of length two between nqi and nq3 such that: nqi _ ^ f [ßisßipaq] ^ [pßisß^aq] _ nq3, where \ß2 \ _ 0. I [ö7pßsa2q] ^ [pa\ßsa2q] _ nq3, where \«i\ _ 0. Analysis of non-empty segments in these permutations shows that external edges occur between: np3 and nq2, if |a2| _ 0, \ß1\ _ 0; np3 and nq1, if \a1\ _ 0, \ß1\ _ 0; np3 and nq'3, if \ß\ _ 0. There is no external edge between nP3 and nq3 since they have the same order of elements in the segment sß2. Since \a\ _ i - 2, \ß\ _ k - i - 1, then using the edge between nP3 and nq'3, where \a2\ _ 0, \ß1\ _ 0, we have \a\ > 1, \ß\ > 1, and such an 8-cycle has the form Cg _ rkrk-irk-i+1 rkri-1rk-1rkr®, with 3 ^ i ^ k — 2, 5 ^ k ^ n, the canoni- 2 1 cal form of which corresponds to (1.5). Using the external edge between nP3 and nq3, where \a1\ _ 0, \ß1\ _ 0, we have \a\ > 1, \ß\ > 1, and such an 8-cycle has the form rkrk-1ri-1rkrk-i+1rk-irkr®, where 3 < i < k - 2, the canonical form of which also corresponds to the form (1.5). Finally, using the external edge between nP3 and nq3, where \ß\ _ 0, we have i _ k - 1, \a1\ _ j > 1, \a2\ _ k - 3 - j > 1, so there is one more 8-cycle of the form Cg _ rkrk-j-1rk-j-2rkrj+1rj+2rkrk-1, where 1 < j < k - 4, 5 < k < n, the canonical form of which corresponds to (1.6), if we put j _ i - 1. Thus, all 8-cycles occurring in the case of three copies are found. Case 3: an 8-cycle within Pk has vertices from four copies of Pk-1 The distribution of vertices among the copies is presented by Figure 4. Let nqi _ [satßpYq], where \a\ > 0, \ß\ > 0, \y\ > 0. There are two cases. 1) Suppose that nqi is adjacent to nsi, and nq2 is adjacent to nti. Since there is only one cycle edge within each copy, hence this edge is uniquely defined and all vertices' labels are straightforward to obtain (see Figure 4, case 1). Thus, we end up with nPi _ [satßqYp], nP2 _ [tasßqYp]. If an 8-cycle does exist, then nPi, nP2 should be incident to the same Figure 4: (2 + 2 + 2 + 2)-situation. internal edge, and hence, there should exist a prefix-reversal transforming nPl into nPl, namely, f|a|+2. If we set |a| = i - 2, |ß| = j - i -1, |y | = k - j -1, where 2 < i < j < k, then such an 8-cycle is presented by the form Cg = rkrk-j+1rkfifkfk-j+1fkfi, where 2 < i < j ^ k - 1, 4 < k < n, the canonical form of which corresponds to (1.7). 2) Suppose that nqi is adjacent to nSl, and nqi is adjacent to nPl (see Figure 4, case 2), then we end up with n*1 = [saqYpßt], nt2 = [pßqYsat]. In this case, an internal edge between vertices n*1 and nt2 does exist only if |a| = |ß| = |y| = 0, which means that k = 4 and such an 8-cycle takes the form (1.8). Therefore, all canonical forms for 8-cycles in Pn, n > 4, are obtained. Now we count the total number N = J2g=1 NCj of distinct 8-cycles passing through a given vertex, where NCi corresponds to the number of distinct 8-cycles described by the canonical form Cg, 1 < i < 8. Let us note that any canonical form of an /-cycle describes / cycles (not necessarily distinct) for a given vertex. Among all canonical forms (1.1)-(1.8), there is the only one, namely the form (1.5), which describes eight distinct 8-cycles. In other cases, identical forms occur. For example, from the canonical form C88 = f4f3f4f3f4f3f4f3 one can get two forms, namely, f4f3f4f3f4f3f4f3 and f3f4f3f4f3f4f3f4 which are identical because they describe the same 8-cycle. Thus, the canonical form C88 gives the only 8-cycle, hence, NC| = 1. In other cases, it can be shown in the same manner (by taking into account identical forms) that the numbers NCi, 1 < i < 7, are given as follows: NCi = (n-3)(n^2)(n-1), nC| = 4(n - 3), NCj = (n - 2)(n - 3), nc4 = NOJ = 2(n - 3)(n - 4), Nc| = 4(n - 3)(n - 4), Ncj = ("-3)(n-2)(n-1). Thus, the total number is given by N = n3 + 12n2 - 103n + 176 = 2 ' which completes the proof of the theorem. □ The total number of distinct 8-cycles in Pn, n > 4, is given by n!(nJ+12n21-103n+176) since there are n! vertices in the graph each of which belongs to N distinct 8-cycles. This proves Corollary 1.4. A maximal set of independent 8-cycles in Pn, n > 4, contains f of these, since P4 has three independent 8-cycles, and there are fj copies of P4, each of which consists of exactly three independent 8-cycles. This proves Corollary 1.5. Acknowledgment The authors thank the referees for their critical remarks and valuable suggestions, which helped to improve the quality and clarity of the manuscript. References [1] S. Asai, Y. Kounoike, Y. Shinano and K. Kaneko, Computing the diameter of 17-pancake graph using a PC cluster, LNCS 4128 (2006), 1114-1124. [2] J. Cibulka, On average and highest number of flips in pancake sorting, Theoretical Computer Science 412 (2011), 822-834. [3] I. J. Dejter, O. Serra, Efficient dominating sets in Cayley graphs, Discrete Appl. Math. 129 (2003), 319-328. [4] H. Dweighter, E 2569 in: Elementary problems and solutions, Amer. Math. Monthly 82 (1975) 1010. [5] W. H. Gates and C. H. 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Zaks, A new algorithm for generation of permutations, BIT 24 (1984), 196-204. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 247-262 Fat Hoffman graphs with smallest eigenvalue at least — 1 — t Akihiro Munemasa Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan Yoshio Sano Division of Information Engineering, Faculty of Engineering, Information and Systems University ofTsukuba, Ibaraki 305-8573, Japan Tetsuji Taniguchi Matsue College of Technology, Matsue 690-8518, Japan Received 30 November 2011, accepted 14 March 2012, published online 30 April 2013 In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least -1 - t, where t is the golden ratio, can be described by a finite set of fat (-1 - t)-irre-ducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at least -1 - t is an H-line graph, where H is the set of isomorphism classes of maximal fat (-1 - t)-irreducible Hoffman graphs. It turns out that there are 37 fat (-1 - t)-irreducible Hoffman graphs, up to isomorphism. Keywords: Hoffman graph, line graph, graph eigenvalue, special graph. Math. Subj. Class.: 05C50, 05C75 1 Introduction P. J. Cameron, J. M. Goethals, J. J. Seidel, and E. E. Shult [1] characterized graphs whose adjacency matrices have smallest eigenvalue at least -2 by using root systems. Their results revealed that graphs with smallest eigenvalue at least -2 are generalized line graphs, except a finite number of graphs represented by the root system E8. Another characterization for generalized line graphs were given by D. Cvetkovic, M. Doob, and S. Simic [3] E-mail addresses: munemasa@math.is.tohoku.ac.jp (Akihiro Munemasa), sano@cs.tsukuba.ac.jp (Yoshio Sano), tetsuzit@matsue-ct.ac.jp (Tetsuji Taniguchi) Abstract by determinig minimal forbidden subgraphs (see also [4]). Note that graphs with smallest eigenvalue greater than -2 were studied by A. J. Hoffman [5]. Hoffman [6] also studied graphs whose adjacency matrices have smallest eigenvalue at least -1 - a/2 by using a technique of adding cliques to graphs. R. Woo and A. Neumaier [12] formulated Hoffman's idea by introducing the notion of Hoffman graphs. A Hoffman graph is a simple graph with a distinguished independent set of vertices, called fat vertices, which can be considered as cliques of size infinity in a sense (see Definition 2.1, and also [8, Corollary 2.15]). To deal with graphs with bounded smallest eigenvalue, Woo and Neumaier introduced a generalization of line graphs by considering decompositions of Hoffman graphs. They gave a characterization of graphs with smallest eigenvalue at least -1 - a/2 in terms of Hoffman graphs by classifying fat indecomposable Hoffman graphs with smallest eigenvalue at least -1 - a/2. This led them to prove a theorem which states that every graph with smallest eigenvalue at least -1 - a/2 and sufficiently large minimum degree is a subgraph of a Hoffman graph admitting a decomposition into subgraphs iso-morphic to only four Hoffman graphs. In the terminology of [12], this means that every graph with smallest eigenvalue at least -1 - a/2 and sufficiently large minimum degree is an H-line graph, where H is the set of four isomorphism classes of Hoffman graphs. For further studies on graphs with smallest eigenvalue at least -1 - a/2, see the papers by T. Taniguchi [10, 11] and by H. Yu [13]. Recently, H. J. Jang, J. Koolen, A. Munemasa, and T. Taniguchi [8] made the first step to classify the fat indecomposable Hoffman graphs with smallest eigenvalue -3. However, it seems that there are so many such Hoffman graphs. A key to solve this problem is the notion of special graphs introduced by Woo and Neumaier. A special graph is an edge-signed graph defined for each Hoffman graph. Although non-isomorphic Hoffman graphs may have isomorphic special graphs, it is not difficult to recover all the Hoffman graphs with a given special graph in some cases. In this paper, we introduce irreducibility of Hoffman graphs and classify fat (-1 - t)-irreducible Hoffman graphs, where t := 1+2/^ is the golden ratio. This is a somewhat more restricted class of Hoffman graphs than those considered in [8], and there are only 37 such Hoffman graphs. As a consequence, every fat Hoffman graph with smallest eigenvalue at least -1 - t is a subgraph of a Hoffman graph admitting a decomposition into subgraphs isomorphic to only 18 Hoffman graphs. In the terminology of [12], this means that every fat Hoffman graph with smallest eigenvalue at least -1 - t is an H-line graph, where H is the set of 18 isomorphism classes of maximal fat (-1 - t)-irreducible Hoffman graphs. 2 Preliminaries 2.1 Hoffman graphs and eigenvalues Definition 2.1. A Hoffman graph H is a pair (H, m) of a graph H and a vertex labeling M : V (H ) ^ {slim, fat} satisfying the following conditions: (i) every vertex with label fat is adjacent to at least one vertex with label slim; (ii) the vertices with label fat are pairwise non-adjacent. Let V (H) := V (H ), V s(H) := M-1(slim), Vf (H) := M-1(fat), and E(H) := E(H ). We call a vertex in V s(H) a slim vertex, and a vertex in Vf (H) a fat vertex of H. We represent a Hoffman graph H also by the triple ( Vs (H ), Vf (H ), E(H )). For a vertex x of a Hoffman graph H, we define NH (x) (resp. NH (x)) to be the set of fat (resp. slim) neighbors of x in H. The set of all neighbors of x is denoted by Nh (x), that is, Nfi(x) := nH(x) U NH (x). A Hoffman graph H is said to be fat if every slim vertex of H has a fat neighbor. A Hoffman graph is said to be slim if it has no fat vertex. Two Hoffman graphs H = (H, p) and H' = (H', p') are said to be isomorphic if there exists an isomorphism from H to H' which preserves the labeling. A Hoffman graph H' = (H', p') is called an induced Hoffman subgraph (or simply a subgraph) of another Hoffman graph H = (H, p) if H' is an induced subgraph of H and p(x) = p'(x) holds for any vertex x of H'. The subgraph of a Hoffman graph H induced by Vs(H) is called the slim subgraph of H. Definition 2.2. For a Hoffman graph H, let A be its adjacency matrix, 'As C ^ A = ^CT O in a labeling in which the fat vertices come last. The eigenvalues of H are the eigenvalues of the real symmetric matrix B(H) := As - CCT. We denote by Amin(H) the smallest eigenvalue of B(H). Remark 2.3. An ordinary graph H without vertex labeling can be regarded as a slim Hoffman graph H. Then the matrix B(H) coincides with the ordinary adjacency matrix of the graph H. Thus the eigenvalues of H as a slim Hoffman graph are the same as the eigenvalues of H as an ordinary graph in the usual sense. Example 2.4. Let HI, HII, and H m be the Hoffman graphs defined by V s(Hi) = {vi}, V f (Hi) = {fi}, E (Hi) = {{vi, fi}}, V s(Hii) = {vi}, Vf (Hii) = {fi, f2}, E (Hii) = {{vi, fi}, {vi,f2}}, V s(Hiii) = {vi, v2}, Vf (Hiii) = {fi}, E (Hiii) = {{vi, fi}, {v2, fi}} (see Figure 1). Note that Amin (Hi) = -1 and Amin(Hii) = Amin(Hiii) = -2. A V Hi Hii Hiii Figure 1: The Hoffman graphs HI, Hn, and HIII Lemma 2.5 ([12, Lemma 3.4]). The diagonal entry B(H)xx of the matrix B(H) is equal to -|Nf (x)|, and the off-diagonal entry B(H)xy is equal to Axy — |Nf (x) n Nf (y)|. Lemma 2.6 ([12, Corollary 3.3]). If G is an induced Hoffman subgraph of a Hoffman graph H, then Amin(©) > Amin(H) holds. In particular, if r is the slim subgraph of H, then Amin(r) > Amin(H). 2.2 Decompositions of Hoffman graphs Definition 2.7. A decomposition of a Hoffman graph H is a family {H®}n=i of Hoffman subgraphs of H satisfying the following conditions: (i) V (H) = Un=i V (tf); (ii) Vs(Hi) n Vs(Hj) = 0 if i = j; (iii) if x G Vs(Hi), y G Vf (H), and {x,y} g E(H),then y g V(Hi); (iv) if x g Vs(Hi), y G Vs(Hj), and i = j, then |Nf(x) n Nf (y)| < 1, and |Nf (x) n Nf (y) I = 1 if and only if {x, y} G E (H). If a Hoffman graph H has a decomposition {Hi}n=1, then we write H = 1+1 "=1 Hi. Example 2.8. The (slim) complete graph Kn is precisely the slim subgraph of the Hoffman graph H = 1+1 "=1 Hi where each Hi is isomorphic to Hi, sharing the unique fat vertex. Ordinary line graphs are precisely the slim subgraphs of Hoffman graphs H = 1+1 "=1 Hi, where each Hi is isomorphic to Hii. The (slim) cocktail party graph CP(n) = Knx2 is precisely the slim subgraph of the Hoffman graph H = 1+1 "=1 Hi where each Hi is isomorphic to HIII, sharing the unique fat vertex. Generalized line graphs are precisely the slim subgraphs of Hoffman graphs H = l+)"=1 Hi, where each Hi is isomorphic to Hn or Hm (see [12]). Definition 2.9. A Hoffman graph H is said to be decomposable if H has a decomposition {Hi}n=1 with n > 2. We say H is indecomposable if H is not decomposable. Example 2.10. A disconnected Hoffman graph is decomposable. Definition 2.11. Let a be a negative real number and let H be a Hoffman graph with Amin(H) > a. The Hoffman graph H is said to be a-reducible if there exists a Hoffman graph H' containing H as an induced Hoffman subgraph such that there is a decomposition {Hi}2=1 of H' with Amin(Hi) > a and Vs(Hi) n Vs(H) = 0 (i = 1,2). We say H is a-irreducible if Amin(H) > a and H is not a-reducible. A Hoffman graph H is said to be reducible if H is Amin(H)-reducible. We say H is irreducible if H is not reducible. Lemma 2.12 ([8, Lemma 2.12]). If a Hoffman graph H has a decomposition {Hi}"=1, then Amin(H) = min{Amin(Hi) | 1 < i < n}. In particular, an irreducible Hoffman graph is indecomposable. Example 2.13. For a non-negative integer t, let K1t be the connected Hoffman graph having exactly one slim vertex and t fat vertices, i.e., Km = (Vs(KM), Vf (Km), E(Km)) = ({v}, {/1,..., ft}, {{v, fi} I i = 1,... t}). Then K1jt is irreducible and Amin(K1jt) = -t. Example 2.14. By Example 2.13, the Hoffman graphs HI (= K1;1) and Hn (— K1j2) are irreducible. The Hoffman graph HIII is also irreducible. Example 2.15. Let Hiv be the Hoffman graph defined by Vs(Hiv) = {vi, v2}, Vf (Hiv) = {fi, /2} and E (Hiv ) = {{vi ,«2}, {vi,fi}, {v2 ,/2}}. The Hoffman graph HIV is indecomposable but reducible. Indeed, it is clear that HIV is indecomposable. Let H' be the Hoffman graph obtained from HIV by adding a new fat vertex f3 and two edges {vi, f3} and {v2, f3}. The Hoffman graph H' is the sum of two copies of HII, where the newly added fat vertex is shared by both copies, that is, H' is decomposable. Furthermore, Amin(Hn) = Amin(HIV) = -2. Hence HIV is reducible. Proposition 2.16. Let G be a slim graph with at least two vertices. If G has maximum degree k, then G is (-k)-reducible. Proof. Let G be a slim graph with maximum degree k. We define a Hoffman graph H by adding a fat vertex for each edge e of G and joining it to the two end vertices of e. Note that G is the slim subgraph of H. For each slim vertex x G Vs(H), let Hx be the Hoffman subgraph of H induced by {x} U Nf (x). Then Hx is isomorphic to the Hoffman graph K^degcO) defined in Example 2.13, and we can check that H = 1+1xeVs(f) Hx. Since the maximum degree of G is k, Amin(Hx) = - degG(x) > -k. Thus G is (-^-reducible. □ Definition 2.17. Let H be a family of isomorphism classes of Hoffman graphs. An H-line graph is an induced Hoffman subgraph of a Hoffman graph which has a decomposition {H*}?=i such that the isomorphism class of H® belongs to H for all i = 1,..., n. 2.3 The special graphs of Hoffman graphs Definition 2.18. An edge-signed graph S is a pair (S, sgn) of a graph S and a map sgn : E (S ) ^ {+, -}. Let V (S ) := V (S ), E+(S ) := sgn-i(+), and E-(S ) := sgn-i(-). Each element in E +(S) (resp. E-(S)) is called a (+)-edge (resp. a (-)-edge) of S. We represent an edge-signed graph S also by the triple (V(S), E+(S), E-(S)). An edge-signed graph S' = (S', sgn') is called an induced edge-signed subgraph of an edge-signed graph S = (S, sgn) if S' is an induced subgraph of S and sgn(e) = sgn'(e) holds for any edge e of S'. Two edge-signed graphs S and S' are said to be isomorphic if there exists a bijection 0 : V (S ) ^ V (S ') suchthat {u, v} G E+(S ) if and only if {0(u),0(v)} G E+(S ' ) and that {u, v} G E-(S) if and only if {0(u),0(v)} G E-(S'). An edge-sined graph S is said to be connected (resp. disconnected) if the graph (V (S ), E +(S) U E-(S)) is connected (resp. disconnected). Example 2.19. A connected edge-signed graph with at most two vertices is isomorphic to one of the edge-signed graphs Si,i, S2,i, and S2,2, where V (Si,i) = {vi}, E+(Si,i) = 0, E-(Si,i) = 0, V (S2,i) = {vi,v2}, E+(S2,i) = {{vi,v2}}, E-(S2,i) = 0, V (S2,2) = {vi,v2}, E+(S2,2) = 0, E-(S2,2) = {{vi, v2}}. (see Figure 2 in which we draw an edge-signed graph by depicting (+)-edges as full lines and (-)-edges as dashed lines). Sl,l S2,1 S2,2 Figure 2: The connected edge-signed graphs with at most two vertices Definition 2.20. The special graph of a Hoffman graph H is the edge-signed graph S (H) :=(V (S (H)),E+(S (H)), E-(S (H))) where V (S (H)) := V s(H) and E+(S (H)) := {{u, v} | u, v G V s(H), u = v, {u, v} G E (H), Nf (u) n Nf (v) = 0}, E -(S (H)) := {{u, v} | u, v G Vs (H), u = v, {u, v} G E (H), Nf (u) n Nf (v) = 0}. Lemma 2.21 ([8, Lemma 3.4]). A Hoffman graph H is indecomposable if and only if its special graph S (H) is connected. Definition 2.22. For an edge-signed graph S, we define its signed adjacency matrix M (S ) by (l if {u, v} G E+(S), (M(S))uv = < -1 if {u,v} G E-(S), 0 otherwise. We denote by Amin(S) the smallest eigenvalue of M (S). We remark that P. J. Cameron, J. J. Seidel, and S. V. Tsaranov studied the eigenvalues of edge-signed graphs in [2]. Lemma 2.23. If S' is an induced edge-signed subgraph of an edge-signed graph S, then Amin(S') > Amin (S). Proof. Since M (S ') is a principal submatrix of M (S ), the lemma holds. □ Lemma 2.24. Let H be a Hoffman graph in which any two distinct slim vertices have at most one common fat neighbor. Then M (S (H)) = B(H) + D(H), where D(H) is the diagonal matrix defined by D(H)xx := |Nf (x) | for x G Vs (H). Proof. This follows immediately from the definitions and Lemma 2.5. □ Lemma 2.25. If H is a fat Hoffman graph with smallest eigenvalue greater than -3, then Amin (S (H)) > Amin(H) + 1. Proof. If some two distinct slim vertices of H have two common fat neighbors, then H contains an induced subgraph with smallest eigenvalue at most -3. This contradicts the assumption by Lemma 2.6. Thus the hypothesis of Lemma 2.24 is satisfied. Since H is fat, the smallest eigenvalue of M (S (H)) = B(H) + D(H) is at least Amin(H) + 1 by [7, Corollary 4.3.3], proving the desired inequality. □ 3 Main Results 3.1 The edge-signed graphs with smallest eigenvalue at least —t Definition 3.1. Let p, q, and r be non-negative integers with p + q < r. Let Vp, Vq, and Vr be mutually disjoint sets such that |Vj| = i where i e {p, q, r}. Let Up and Uq be subsets of Vr satisfying |Up| = p, |Uq| = q, and Up n Uq = 0. Let Qp,q,r be the edge-signed graph defined by V( Qp,q,r ) := Vp U Vq U Vr, E+ ( Qp,q,r ) := {{u,v}| u e Up, v e Vp} U {{v, v'} | v,v' e Vr ,v = v'}, E— ( Qp,q,r ) := {{u,v}| u e Uq ,v e Vq } (see Figure 3 for an illustration). Figure 3: Q3,2,6 Lemma 3.2. For any non-negative integersp, q, and r with p+q < r, Am;n(Qp,q,r ) > -t. Proof. Let M (Qr,r,2r ) Multiplying I 0 xI 0 0 I 0 - xI 0 0 I 0 0 0 0 I 0 0 I 0 0 0 0 -I 1 0 J -1 J 0 -I J J -I from the left to xi - M(Qr,r,2r), we find |xi - M (Qr,r,2r )| = (-1) r (x2 + x - 1)i -(x2 + x - 1)i (x2 + x - 1)i xJ -(x2 + x - 1)i + xJ xJ (x2 + x — 1)r i i ( ) xJ (x2 + x - 1)i - xJ (x2 + x - 1)r |(x2 + x - 1)i - 2xJ| (x2 + x - 1)2r-i(x2 - (2r - 1)x - 1). In particular, we obtain Amin(Qr,r,2r) = -t. Since p < r and q < r, Qr,r,2r has an induced edge-signed subgraph isomorphic to Qp,q,r. By Lemma 2.23, Amin(QPiq,r) > Example 3.3. Let 71 and 72 be the edge-signed triangles having exactly one (+)-edge and exactly two (+)-edges, respectively, i.e., V(71) = V(72) = {v1, v2, v3}, E+(71) = E-(72) = {{vi,v2}}, and E-(Ti) = E+(72) = {{vi,v3}, {v2,v3}} (see Figure 4). For ei, £2, £3 € {1, -1} and S e {0, ±1}, let Si(ei, £2, £3), S2(£i,£2,S), £3^1, £2,S), and S4(£i, £2) be the edge-signed graphs in Figure 5, where an edge with label 1 (resp. -1) represents a (+)-edge (resp. a (-)-edge) and an edge with label 0 represents a non-adjacent pair. It can be checked that each of the edge-signed graphs 72, Si(£i, £2, £3), S2(£i, £2, S), S3(£i, £2, S), S4(£i, £2) has the smallest eigenvalue less than -t. Theorem 3.4. Let S be a connected edge-signed graph with Amin(S) > -t. Assume that S does not contain an induced edge-signed subgraph isomorphic to 7i. Then either S is isomorphic to Qp,q,r for some non-negative integers p, q, r with p + q < r, or S has at most 6 vertices and is isomorphic to one of the 15 edge-signed graphs in Figure 6. Proof. By using computer [14], we checked that the theorem holds when |V (S)| < 7. We prove the assertion by induction on | V (S) |. Assume that the assertion holds for | V (S) | = n (> 7). Suppose that |V (S)| = n + 1. It follows from Problem 6(a) in Section 6 of [9] that there exists a vertex v which is not a cut vertex of S. Then S - v is connected, where S - v is the edge-signed subgraph induced by V (S )\{v}. Since Amin(S-v) > Amin(S) > -t, the inductive hypothesis implies that S - v is isomorphic to Qp,q,r for some p, q, r with p + q + r = n. Thus S is the edge-signed graph obtained from Qp,q,r by adding the vertex v and signed edges between v and some vertices in Qp,q,r. Note that r > 4 since n = p + q + r > 7 and p + q < r. □ Figure 4: The edge-signed triangles Ti and 72 Figure 5: Edge-signed graphs with smallest eigenvalue less than —t We claim that either v is adjacent to only one vertex of Vr, or to all the vertices of Vr. Note that S cannot contain any of the edge-signed graphs 72, Si(ei, e2, e3), S2(e1, e2, S), S3(ei, e2, S), S4(e1, e2) in Example 3.3. If v is adjacent to none of the vertices of Vr, then S contains S4(e1, e2) as an induced edge-signed subgraph, a contradiction. If the number of neighbors of v in Vr is at least 2 and less than r, then S contains S3(e1, e2, S) as an induced edge-signed subgraph, a contradiction. Thus the claim holds. Now, if v is adjacent to only one vertex of Vr, then the unique neighbor of v in Vr is in Vr \ (Up U Uq). Indeed, otherwise we would find S2(e1, e2, S) as an induced edge-signed subgraph, a contradiction. Also, v is adjacent to none of the vertices of Vp U Vq since otherwise we would find S1(e1, e2, e3) as an induced edge-signed subgraph, a contradiction. Thus S is isomorphic to Qp+1,q,r or Qp,q+1,r. Suppose that v is adjacent to all the vertices of Vr. Since Vr is a clique consisting of (+)-edges only, the assumption implies that v is incident with at most one (—)-edge to Vr. If there is a vertex of Vr joined to v by a (—) -edge, then we find 72 as an induced edge-signed subgraph, a contradiction. Thus all the edges from v to Vr are (+)-edges. Now v is adjacent to none of the vertices of Vp U Vq since otherwise we would find S3(e1, e2,0) as an induced edge-signed subgraph, a contradiction. Thus S is isomorphic to Qp,q,r+1. Hence the theorem holds. □ Lemma 3.5. The smallest eigenvalues of the signed adjacency matrices of the edge-signed graphs in Figure 6 are given as follows: Amin(S ) —%/2 1-^17 2 1+ t -1.414213 if Sg{S3,1, S4,4}, -1.561553 if S E {S4,5, S5,5, S5,6}, -1.601679 if S = S6,3, -1.618034 otherwise, T # S34 & 4,5 i......: n & 4,1 4 4,2 & 4,3 4 4,4 5. 5,1 5 5,2 5. 5,3 5 5,4 5 5,5 —1 — T. ' □ Lemma 3.8. Let S be a connected edge-signed graph with three vertices. Let D be a 3 x 3 diagonal matrix with diagonal entries 1 or 2 such that at least one of the diagonal entries is 2. Then M (S ) — D has the smallest eigenvalue less than —1 — Proof. This can be checked by a direct calculation. Lemma 3.9. Let H be a fat indecomposable Hoffman graph with smallest eigenvalue at least — 1 — t . If some slim vertex of H has at least two fat neighbors, then the special graph S (H) of H is isomorphic to Qo,o,i, Qi,o,i, or Qo,i,i. Proof. In view of Lemma 2.6, it suffices to show that every fat indecomposable Hoffman graph with three slim vertices, in which some slim vertex has two fat neighbors, has the smallest eigenvalue less than —1 — t. Let H be such a Hoffman graph. Then S (H) is connected by Lemma 2.21 and B(H) — M (S (H)) — D for some diagonal matrix D with diagonal entries 1 or 2 such that at least one of the diagonal entries is 2. Then we have a contradiction by Lemma 3.8. □ Example 3.10. Let Hxvi and Hxvn be the Hoffman graphs in Figure 7. The special graphs of Hxvi and Hxvii are Qi,o,i and Qo,i,i, respectively, and Am;n(Hxvi) — Amin(Hxvii) — — 1 — t . iUiV Hxvi Hxvii Figure 7: Fat indecomposable Hoffman graphs Lemma 3.11. Let H be a Hoffman graph in which every slim vertex has at most one fat neighbor. Then the special graph S (H) of H does not contain an induced edge-signed subgraph isomorphic to Ti. Proof. Suppose that the special graph S (H) of H contains Ti — ({vi, v2, v3}, {{vi, v2}}, {{vi, v3}, {v2, v3}}) as an induced edge-signed subgraph. Since v3 is incident to a (—)-edge, v3 must have a fat neighbor. Since every slim vertex of H has at most one fat neighbor, v3 has a unique fat neighbor f. Then f is adjacent to vi and v2. This is a contradiction to {vi,v2}G E+(Ti). □ Lemma 3.12. Let H be a Hoffman graph in which every slim vertex has exactly one fat neighbor. If the special graph S (H) of H is isomorphic to Qp,q,r for some non-negative integers p, q, r, then H is an induced Hoffman subgraph of a Hoffman graph H' with Vs(H') — Vs(H) which has a decomposition {H®}r= such that Hl is isomorphic to Hxvi, Hxvii, or Hii for all i — 1,..., r. In particular, if r > 2, then H is ( — 1 — t)-reducible. Proof. By the assumption, V s(H) — V (S (H)) is partitioned into Vp U Vq U Vr as Definition 3.1. Consider the Hoffman graph H' defined by Vs(H') :— Vs(H), Vf (H') :— Vf (H) U {f *}, and E(H') :— E (H) U {{v,f *} | v G Vr}, where f * is anew fat vertex. Note that H is an induced Hoffman subgraph of H' with Vs(H) — Vs(H). Then H' has a decomposition {Hi}r=i with H® = Hxvi for 1 < i < p, H® — Hxvii for p < i < p + q, and H® = HII for p + q < i < p + q + r (see Examples 2.4 and 3.10). Since r > 2 and each of the Hoffman graphs H® has the smallest eigenvalue at least —1 — t, it follows that H is (—1 — t )-reducible. □ Theorem 3.13. Let H be a fat indecomposable Hoffman graph with smallest eigenvalue at least -1 - t. Then the following hold: (i) If some slim vertex of H has at least two fat neighbors, then the special graph S (H) of H is isomorphic to Qo,o,i, Qi,o,i, or Qo,i,i. (ii) If every slim vertex of H has exactly one fat neighbor, then the special graph S (H) of H is isomorphic to Qp,q,r for some non-negative integers p, q, r with p + q < r or one of the 15 edge-signed graphs in Figure 6. Proof. The statement (i) follows from Lemma 3.9. We show (ii). Suppose that every slim vertex of H has exactly one fat neighbor. By Lemma 2.21, S (H) is connected, and by Lemma 2.25, S (H) has smallest eigenvalue at least -t . Moreover, by Lemma 3.11, S (H) does not contain an induced edge-signed subgraph isomorphic to 7i. Now Theorem 3.4 implies that S (H) is isomorphic Qp,q,r or one of the 15 edge-signed graphs in Figure 6. □ Corollary 3.14. Let H be a fat (-1 - t)-irreducible Hoffman graph. Then the special graph S (H) of H is isomorphic to Q0,0,i, Qi,0,i, Q0,i,i, or one of the 15 edge-signed graphs in Figure 6. Proof. Since H is (-1 - t)-irreducible, H is indecomposable. If some slim vertex of H has at least two fat neighbors, then the statement holds by Theorem 3.13 (i). Suppose that every slim vertex of H has exactly one fat neighbor. By Theorem 3.13 (ii), S (H) is isomorphic to Qp,q,r for some non-negative integers p, q, r, or one of the 15 edge-signed graphs in Figure 6. Since H is (-1 - t)-irreducible, the former case occurs only for r =1 by Lemma 3.12. Hence the corollary holds. □ 3.3 The classification of fat Hoffman graphs with smallest eigenvalue at least — 1 — t Hh Hii H4,2 H4,3 o3> H2 4,3 h1 4,4 H1 4,5 «C H2 4,5 Figure 8: The fat (-1 - t)-irreducible Hoffman graphs with 3 or 4 slim vertices Lemma 3.15. Let H be a fat indecomposable Hoffman graph with smallest eigenvalue at least -1 - t. If the number of slim vertices of H is at most two, then H is isomorphic to one of Hi, Hii, Hin, Hiv, Hxvi, and Hxvii. Hl 5,1 H2 5,1 H3 5,1 A H1 5,2 H1 5,3 H1 5,4 A H2 5,4 H3 5,4 H4 5,4 tC m> H1 5,5 H2 5,5 H3 5,5 H1 5,6 Figure 9: The fat (-1 - t)-irreducible Hoffman graphs with 5 slim vertices Proof. Straightforward. □ Lemma 3.16. Let H be a fat (-1 — t )-irreducible Hoffman graph. If S (H) is isomorphic to Si,j in Figure 6, then H is isomorphic to Hk,j for some k in Figures 8, 9, and 10. Proof. By Lemma 3.9, every slim vertex has exactly one fat neighbor. It is then straightforward to establish the lemma. □ Theorem 3.17. Let H be a fat ( — 1 — t )-irreducible Hoffman graph. Then H is isomorphic to one of Hi, Hii, Hiii, Hxvi, Hxvii, and the 32 Hoffman graphs given in Figures 8, 9, and 10. Proof. By Corollary 3.14, the special graph S (H) of H is isomorphic to Qo,o,i, Qi,o,i, Qo,i,i, or one of the 15 edge-signed graphs in Figure 6. If the number of slim vertices of H is at most two, then the statement holds by Lemma 3.15 and Example 2.15. If the number of slim vertices of H is at least three, then the statement holds by Lemma 3.16. □ Theorem 3.18. Let H be the set of isomorphism classes of the maximal members of the 37 fat ( — 1 — t)-irreducible Hoffman graphs given in Theorem 3.17, with respect to taking induced Hoffman subgraphs. More precisely, H is the set of isomorphism classes of HXVi, HXVii, Hl,i, H4,3, Hs,2, H5,3, H5,6, and the 11 Hoffman graphs in Figure 10. Then every fat Hoffman graph with smallest eigenvalue at least —1 — t is an H-line graph. Proof. It suffices to show that every fat indecomposable Hoffman graph H with smallest eigenvalue at least —1 — t is an H-line graph. First suppose that some slim vertex of H has two fat neighbors. Then by Lemma 3.9, H has at most two slim vertices, and by Lemma 3.15, H is isomorphic to one of Hi, Hii, Hiii, Hiv, Hxvi, and Hxvii. Since Hi and Hn are induced Hoffman subgraphs of HXVi, and Hm is an induced Hoffman subgraph of HXVii, they are H-line graphs. Note that HiV is also an H-line graph since Example 2.15 shows that HiV is an induced Hoffman subgraph of a Hoffman graph having a decomposition into two induced Hoffman subgraphs isomorphic to Hi. Thus the result holds in this case. Next suppose that every slim vertex of H has exactly one fat neighbor. Then by Theorem 3.13 (ii), S (H) is isomorphic to Qp,q,r for some non-negative integers p, q, r with p + q < r or one of the 15 edge-signed graphs in Figure 6. In the former case, H is an H-line graph by Lemma 3.12. In the latter case, Lemma 3.16 implies that H is an H-line graph since H contains all the maximal members of the isomorphism classes of Hoffman graphs Hk,j. □ References [1] P. J. Cameron, J. M. Goethals, J. J. Seidel and E. E. Shult, Line graphs, root systems and elliptic geometry, J. Algebra 43 (1976), 305-327. [2] P. J. Cameron, J. J. Seidel and S. V. Tsaranov, Signed graphs, root lattices, and Coxeter groups, J. Algebra 164 (1994), 173-209. [3] D. Cvetkovic, M. Doob and S. Simic, Generalized line graphs, J. Graph Theor. 5 (1981), 385399. [4] D. Cvetkovic, P. Rowlinson and S. K. Simic, Spectral Generalizations of Line Graphs — On graphs with least eigenvalue —2, Cambridge University Press, Cambridge, 2004. [5] A. J. Hoffman, On limit points of the least eigenvalue of a graph, Ars Combinatoria 3 (1977), 3-14. [6] A. J. Hoffman, On graphs whose least eigenvalue exceeds —1 — \f2, Linear Algebra Appl. 16 (1977), 153-165. [7] R. A. Horn and C. R. Johnson, Matrix Analysis, Corrected reprint of the 1985 original, Cambridge University Press, Cambridge, 1990. [8] H. J. Jang, J. Koolen, A. Munemasa and T. Taniguchi, On fat Hoffman graphs with smallest eigenvalue at least —3, Ars Math. Contemp. 7 (2014), 105-121. [9] L. Lovasz, Combinatorial Problems and Exercises, 2nd edition, North-Holland, 1993. [10] T. Taniguchi, On graphs with the smallest eigenvalue at least —1 — y/2, part I, Ars Math. Contemp. 1 (2008), 81-98. [11] T. Taniguchi, On graphs with the smallest eigenvalue at least —1 — \f2, part II, Ars Math. Contemp. 5 (2012), 239-254. [12] R. Woo and A. Neumaier, On graphs whose smallest eigenvalue is at least —1 — \f2, Linear Algebra Appl. 226-228 (1995), 577-591. [13] H. Yu, On the limit points of the smallest eigenvalues of regular graphs, Designs Codes Cryp-togr. 65 (2012), 77-88. [14] The Magma Computational Algebra System for Algebra, Number Theory and Geometry, URL: http://magma.maths.usyd.edu.au/magma/. ars mathematica contemporanea Author Guidelines Papers should be prepared in LTpX and submitted as a PDF file. Articles which are accepted for publication have to be prepared in LTpX using class file amcjou.cls and bst file amcjou.bst (if you use BibTEX). 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Patrick Fowler, Sheffield, UK Tomaž Pisanski, Ljubljana and Koper, SI Guest Editors ARS MATHEMATICA CONTEMPORANEA k ivi v E- ANNOUNCEMENT PhD Fellowship ("Young Researcher" position) at the University of Primorska, Slovenia The University of Primorska announces three "Young researcher" positions under the supervision of • Dragan Marusic (Algebra and Discrete Mathematics); • Stefko Miklavic (Algebra and Discrete Mathematics); • Enes Pasalic (Cryptography). Applicants should have a BSc or equivalent training (by September 2014). Applicants for "Young researcher" positions in mathematics are expected to enroll in the PhD program at UP FAMNIT. The positions are for 3 and 1/2 years and include a tuition fee waiver. The holder is expected to complete his/hers PhD training in 4 years. The deadline for applications will be in June 2014 (the exact date yet to be confirmed). 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For any additional information please contact Olga Kaliada at Phone: +386 5 611 7599) Fax: +386 5 611 7592 Email: olga.kaliada@upr.si "Young Researcher position" University of Primorska, UP IAM Muzejski trg 2 6000 Koper Slovenia Photos: Miha Perosa Symmetries of Graphs and Networks (SYGN IV) and 2014 PhD Summer School in Discrete Mathematics Rogla, Slovenia, June 29 - July 5, 2014 http://www.famnit.upr.si/sl/konference INVITED SPEAKERS AT SYGN IV WORKSHOP: Marston Conder (University of Auckland, New Zealand), Shaofei Du (Capital Normal University, China), Yan Quan Feng (Beijing Jiaotong University, China), Michael Giudici (University of Western Australia, Australia), Isabel Hubard (IMATE-UNAM, Mexico), Gyorgy Kiss (Eotvos Lorand University, Hungary), Jin Ho Kwak (POSTECH, South Korea), Young Soo Kwon (Yeungnam University, South Korea), Cai-Heng Li (University of Western Australia, Australia) Luis Martinez Fernandez (University of the Basque Country, Spain), Joy Morris (University of Lethbridge, Canada), Tamas Szonyi (Eotvos Lorand University, Hungary), Paul Terwilliger (University of Wisconsin-Madison, USA), Steve Wilson (Northern Arizona University, USA), Dave Witte Morris (University of Lethbridge, Canada). Summer School Programme: Aimed at bringing PhD students to several open problems in the active research areas, four minicourses (6 hour of lectures each) will be given on the following topics: - Construction techniques for graph embeddings (Mark Ellingham, Vanderbilt University, USA), - Combinatorial designs (Mariusz Meszka, AGH University of Science and Technology, Poland), - Some topics in the theory of finite groups (Primoz Moravec, University of Ljubljana, Slovenia), - Symmetric key cryptography and its relation to graph theory (Enes Pasalic, University of Primorska, Slovenia). In addition to lectures, time will also be devoted to workshop sessions and students presentations. Venue: Both events will take place at Rogla, a highland in the north-eastern part of Slovenija, located 130 km by road from Slovenian capital Ljubljana. At around 1500m above sea level, the beautiful natural scenery of Rogla provides pleasant climate conditions and stimulating working environment. 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First Announcement 8th Slovenian International Conference on Graph Theory Organized by IMFM - Institute of Mathematics, Physics and Mechanics in collaboration with UL FMF - University of Ljubljana, Faculty of Mathematics and Physics UL Pef - University of Ljubljana, Faculty of Education UM FNM - University of Maribor, Faculty of Natural Sciences and Mathematics UP FAMNIT - University of Primorska, Faculty of Mathematics, Natural Sciences and Information Technologies Important dates Duration: June 21 - 27, 2015 Conference venue The conference will take place in Kranjska Gora, Slovenia Additional information Latest information on the conference will be available at the conference web page kg15.imfm.si Inquiries about the conference should be addressed by e-mail to kranjska.gora2015@gmail.com Scientific committee Sandi Klavžar • Dragan Marušič • Bojan Mohar • Tomaž Pisanski Printed in Slovenia by Birografika Bori d.o.o. Ljubljana