ARS MATHEMATICA CONTEMPORANEA Volume 8, Number 2, Spring/Summer 2015, Pages 235-444 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 What do we publish in AMC? Recently we had to decline a submission to AMC of a paper that we were unable to handle. We did not even send it to a quick assessment, let alone to referees. The reason was not the quality of the paper, but its content and scope, which were far outside the interests of our editors, referees and authors. Our declared scope is clear: we are looking for high quality papers that cover at least two different subject fields, at least one of which is within discrete mathematics. Some easy statistical analysis tells us to what extent this has been realised in practice, and we hope this will be also of help to authors who are considering AMC as a potential venue for their papers. In the first seven years of AMC we have published 204 papers, totalling 2872 pages. This is a little more than 14 pages per paper. The minimum average page length so far was 12.05, in volume 1, while the maximum average page length was 16.20, in volume 5. Approx. 43% of these papers used at least two different 2-digit MSC codes, while 52% used at least two different 3-letter MSC codes. A little over 85% of the papers used both primary and secondary classifications. A large majority of the primary 2-digit classification codes came under 05 (Combinatorics), followed by 52 (Convex and discrete geometry), 51 (Geometry) and 20 (Group theory and generalizations). In decreasing order, the other 2-digit primary codes were 57 (Manifolds and cell complexes), 68 (Computer science), 06 (Order, lattices, ordered algebraic structures), 91 (Game theory, economics, social and behavioral sciences), 01 (History and biography), 92 (Biology and other natural sciences) and 47 (Operator theory). The ten most frequent 3-letter classification codes were: 05C (Graph theory), 05E (Algebraic combinatorics), 20B (Permutation groups), 52B (Polytopes and polyhedra), 05B (Designs and configurations), 51E (Finite geometry and special incidence structures), 52C (Discrete geometry), 57M (Low-dimensional topology), 92E (Chemistry), 20F (Structure and classification of infinite and finite groups), and 06A (Ordered sets). The most frequent pairs of 5-letter MSC codes declared in a paper are shown as edges in the figure below, which comes from some analysis performed by Vladimir Batagelj using Pajek. The threshold value for inclusion of a pair as an edge was set to 2. Among the 5-letter classifications by far the most frequent were 05C25 (Graphs and abstract algebra), 05C10 (Planar graphs; geometric and topological aspects of graph theory), followed by 05C15 (Colorings of graphs and hypergraphs), 05E18 (Group actions on combinatorial structures), 05C12 (Distance in graphs), 20B25 (Finite automorphism groups of algebraic, geometric and combinatorial structures), 05C50 (Graphs and linear algebra), 05C76 (Graph operations), 05C75 (Structural characterization of families of graphs), 05C45 (Eulerian and Hamiltonian graphs), 05E30 (Association schemes, strongly regular graphs), and 05C85 (Graph algorithms). These figures show that we publish mostly papers in algebraic and topological graph theory, with discrete and convex geometry also having significant presence in AMC. We note, however, that papers with less frequent MSC codes still play an important role in world mathematics. According to MathSciNet at the time of writing of this editorial, the most highly cited paper in 05C76 (Graph operations) was published in our journal. Dragan Marušic and Tomaž Pisanski Editors In Chief Contents Super connectivity of direct product of graphs Jin-XinZhou.................................235 Mixed fault diameter of Cartesian graph bundles II Rija Erveš, Janez Žerovnik..........................245 An atlas of subgroup lattices of finite almost simple groups Thomas Connor, Dimitri Leemans......................259 Further biembeddings of twofold triple systems Diane M. Donovan, Terry S. Griggs, James G. Lefevre, Thomas A. McCourt.............................267 Edmonds maps on the Fricke-Macbeath curve Rubén A. Hidalgo...............................275 Counterexamples to a conjecture on injective colorings Borut Lužar, Riste Škrekovski........................291 Spherical tilings by congruent quadrangles: Forbidden cases and substructures Yohji Akama, Nico Van Cleemput......................297 Embedded graphs whose links have the largest possible number of components Stephen Huggett, Israa Tawfik........................319 Alternating plane graphs Ingo Althöfer, Jan Kristian Haugland, Karl Scherer, Frank Schneider, Nico Van Cleemput .............................. 337 On global location-domination in graphs Carmen Hernando, Merce Mora, Ignacio M. Pelayo ............. 365 Quartic integral Cayley graphs Marsha Minchenko, Ian M. Wanless.....................381 Strongly light subgraphs in the 1-planar graphs with minimum degree 7 Tao Wang...................................409 The multisubset sum problem for finite abelian groups Amela MuratoviC-RibiC, Qiang Wang ....................417 Chiral covers of hypermaps Gareth A. Jones ................................ 425 The Cayley isomorphism property for groups of order 8p Gäbor Somlai.................................433 Volume 8, Number 2, Spring/Summer 2015, Pages 235-444 ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 235-244 Super Connectivity of direct product of graphs* Jin-Xin Zhou Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China Received 15 July 2012, accepted 14 June 2014, published online 23 January 2015 For a graph G, k(G) denotes its connectivity. A graph G is super connected, or simply super-к, if every minimum separating set is the neighborhood of a vertex of G, that is, every minimum separating set isolates a vertex. The direct product Gi x G2 of two graphs Gi and G2 is a graph with vertex set V(Gi x G2) = V(Gi) x V(G2) and edge set E(Gi x G2) = {(ui,v1)(u2,v2) | uiu2 G E(G1),viv2 G E(G2)}. Let Г = G x Kn, where G is a non-trivial graph and Kn(n > 3) is a complete graph on n vertices. In this paper, we show that Г is not super-K if and only if either к(Г) = nK(G), or Г = K^ x K3(l> 0). Keywords: Super connectivity, direct product, vertex-cut. Math. Subj. Class.: 05C40, 05C76 1 Introduction Throughout this paper only undirected simple connected graphs without loops and multiple edges are considered. Unless stated otherwise, we follow Bondy and Murty [4] for terminology and definitions. Let G = (V (G), E (G)) be a graph. For two vertices u, v G V (G), u ~ v means that u is adjacent to v and uv is the edge incident to u and v in G. The set of vertices adjacent to the vertex v is called the neighborhood of v and denoted by NG(v), i.e., NG(v) = {u | uv G E (G)}. The degree of v is equal to |NG (v)|, denoted by dG (v). The number 6(G) = min{dG(v) | v G V (G)} is the minimum degree of G. For a subset S С V (G), the subgraph induced by S is denoted by G[S ]. As usual, Km,m, (m is a positive integer) denotes the complete bipartite graph; Km,m - mK2 denotes the graph obtained by removing a 1-factor from Km,m; Kn denotes the complete graph on n vertices; and Zn denotes the ring of integers modulo n. *This work was supported by the National Natural Science Foundation of China (11271012). E-mail address: jxzhou@bjtu.edu.cn (Jin-Xin Zhou) Abstract A separating set of a graph G is a set of vertices whose deletion either disconnects G or reduces G to the trivial graph K1. The connectivity of the graph G is the minimum number of vertices in a separating set of G, and will be denoted by k(G). In particular, к(Кп) = n - 1, and k(G) =0 if and only if G is disconnected or a Ki. Clearly, k(G) < 6(G). A graph G with minimum degree 6(G) is maximally connected if 6(G) = k(G). An interconnection network is often modeled as a graph G, where V(G) is the set of processors and E(G) is the set of communication links in the network. The connectivity k(G) of G is an important measurement for fault-tolerance of the network, and the larger k(G) is, the more reliable the network is. As more refined indices of reliability than connectivity, super connectivity was proposed in [2, 3]. A graph G is super connected, super-к, for short if every minimum separating set isolates a vertex of G. The direct product G1 x G2 of two graphs G1 and G2 is defined as the graph with vertex set V(G1) x V(G2) and edge set {(u1,v1)(u2,v2) | u1u2 e E(G1),v1 v2 e E(G2)}. The direct product is also called the Kronecker product, tensor product, cross product, categorical product, or conjunction. As an operation on binary relations, the direct product was introduced by Whitehead and Russell in their Principia Mathematica [21]. It is also equivalent to the direct product of the adjacency matrices of the graphs (see [20]). As one of the four standard graph products [11], the direct product has been studied from several points of view (see, for example, [1, 6, 8, 12, 13, 15]). The connectivity of the direct product of graphs has also been investigated in several recent publications. For example, Bresar and Spacapan [7] obtained an upper bound and a lower bound on the edge-connectivity of the direct products with some exceptions, and they also obtained several upper bounds on the vertex-connectivity of the direct products of graphs. Mamut and Vumar [14] proved that к(Кт x Kn) = (m - 1 )(n - 1) where m > n > 2. In [9], it was shown that if n > 3 and G is a bipartite graph, then k(G x Kn) = min{n«(G), (n - 1)6(G)}, and furthermore, the authors also conjectured that this is true for all nontrivial graph G. Later, this conjecture was confirmed independently by Wang and Wu [17] and Wang and Xue [18]. More recently, several papers dealing with the super-connectivity of direct product of graphs were published. Guo et al. [10] showed that for a bipartite graph G with k(G) = 6(G), G x Kn(n > 3) is super-к. In [19], the authors generalized this result by showing that for a nonbipartite graph G with k(G) = 6(G), G x Kn(n > 3) is super-к. In [19], the authors also pointed out that Guo et al.'s result is not true when G = > 1) and n = 3, and they also claimed that except for this case, Guo et al.'s statement is true. The aim of this article is to determine all graphs G such that G x Kn(n > 3) is not super-к. The following is the main result. Theorem 1.1. Let Г = G x Kn, where n > 3 and G is a non-trivial graph. Then Г is not super-к if and only if one of the following happens. (1) G has a minimum separating set T so that T x V(Kn) is a minimum separating set of Г. In particular, к(Г) = n^G). (2) Г = K£/ x K3(£> 0). From Theorem 1.1 we can immediately obtain the following corollaries. Corollary 1.2. [17, 18] Let Г = G x Kn, where n > 3 and G is a non-trivial graph. Then к(Г) = min{n^G), (n - 1)6(G)}. Corollary 1.3. [10, 19] For a maximally connected graph G, G x Kn(n > 3) is not super-K if and only if n = 3 and G = Ki,l(£ > 0). 2 Proof of Theorem 1.1 We start by introducing some notations. Notations. • Г := G x Kn, where n > 3 and G is a non-trivial graph. • V(G) := {ui | i G Zm}. • V(Kn) := Z«. • Vi := {m} x V (K«), i G Zm. • S: a minimum separating set of Г. • Г - S := (JS— ri, where each ri is a connected component of Г - S. • Wi := V(ri). In the following Lemmas 2.1-2.5, we assume that Г is not super connected, and S is a minimum separating set of Г with each component Г of Г - S having at least two vertices. By the definition, we can obtain the following easy facts. Lemma 2.1. (1) £(Г) = (n - 1)S(G). (2) For any i G Zm, Vi is an independent subset of V(Г). (3) If ui0 is adjacent to vil in G, then (ui0 , j) ~ (uil, k)(in Г) if and only if j = к. In particular, Г[^0 U Vil ] = Kn,n - nK2. (4) Let T be a separating set of G. Then T x V(Kn) is also a separating set of Г. In particular, |S| = к(Г) < min{nK(G), (n - 1)J(G)}. (5) s > 2 and |Wi| > 2 for each i G Zs. Lemma 2.2. For each (ui, j) G S, (ui, j) has at least one neighbor in Wi for each i G Zs. Proof. Suppose to the contrary that (ui,j) has no neighbors in Wi for some i G Zs. Set S' = S - {(ui, j)}. Then Wi must be a component of Г - S'. This implies that S' is also a separating set of Г, contrary to the minimality of S. □ Lemma 2.3. For two components Wk, Wi, if there exist (ui, i') G Wk and (uj, j') G W^ such that m ~ Uj (in G), then i' = j' and Wk П Vi = {(ui, i')} and We П Vj = {(uj, j')}. Proof. Since ui ~ Uj (in G), it follows from Lemma 2.1 (3)that U Vj ] = Kn,n - nK2. As Г - S is disconnected, there are no edges between Wk and Wi. Consequently, i' = j' and Wk n Vi = {(ui; i')} and Wi П Vj = {(uj, j')}. □ Lemma 2.4. Assume that for each Vi there exists at most one Wj such that Vi n Wj = 0. Then S = T x V(Kn), where T is a minimum separating set of G. In particular, к(Г) = nK(G). Proof. We shall first show the following two claims. Claim 1 If there exists an i e Zm such that V П S = 0 and V C S, then | Vi П S | = n - 1. Furthermore, for each Wj, there is a V£ such that | Wj П V«| = 1. By the assumption, there is a unique j e Zs such that Vi П Wj = 0. By Lemma 2.2, for each vertex, say (ui, i'), in Vi П S, there is at least one neighbor, say (u«, l'), in each Wt with t = j. By Lemma 2.3, |V П Wj | = |V« П Wt| = 1. From our assumption we know that |Vi n S| = |V« n S| = n - 1. Claim 2 For each i e Zm, either Vi n S = 0 or Vi C S. Suppose on the contrary that there exists an i e Zm such that Vi n S = 0 and Vi C S. For each j e Zs, let Qj = {l e Zm | |V« n Wj| = 1}, and set nj = |Qj|. By Claim 1, nj > 0. Without loss of generality, assume that n0 < ni < ... < ns_i. Assume that W0 C |J«Шо V«. Then for each (ui, l) e W0, we have |Vi n W0| = 1. Combining this with Lemma 2.3, we have for a fixed (ui, l) e W0, if uk — ui(in G), then | Vk n S| > n - 1. As n > 3, one has |S| > |Vi n S| + ^ |Vj n S| > n - 1 + J(G)(n - 1) >к(Г). u j eNa(ui) A contradiction occurs. Now assume that Wo C U«en0. Let U = |JVi, Zo = U£eo0 V, and Zi = U^eni V«. Set T = U U Z0. Clearl}?, |Z0 n W0| = n0. Since n1 > n0 and n > 3, one has |Z0 nJW0| = n0 < n1(n - 1) = |Z1 n S|. Then |T| = |U| + |Zo| = |U| + |Zo n S| + |Zo n Wo| < |U| + |Zo n S| + |Zi n S| < |S|. Since S is a minimum separating set, Г - T is connected. So there is an edge between Wo \ T and V(Г) \ (T U Wo). We may assume that (v, j) e Wo \ T is adjacent to (vs,k) e V(Г) \ (T U Wo). Obviously, (vs,k) e S \ T. Since U = (JVVi and T = U UZo, one has Vs C S. If Vs nWo = 0, then by Claim 1, we must have |VsnWo| = 1 and so Vs C Zo C T. This contradicts the fact that (vs, k) e S \ T. Consequently, Vs n Wo = 0. It follows that Vs n Wt = 0 for some t > 0. Since (vi, j) - (vs, k), by Lemma 2.3, |Vi n Wo| = 1 and so Vi C Zo C T. This contradicts the fact that (vi, j) e Wo \ T. Now we are ready to finish the proof. From Claim 2 it follows that S = T x V (Kn) for some subset T of V(G). Since n > 3, T is a separating set of G (see [20]). So, |S| > K(G)n. However, by Lemma 2.1 (4), |S| < n«(G). Hence, |S| = n«(G). □ Lemma 2.5. Assume that there exist a Vi and two different Wj0, Wj such that Vi n Wk = 0 with k = jo, j1. Then n = 3 and G = K«,«(l > 0). Proof. Recall that Wk = V(Гк ) with k = jo or j^. We shall finish the proof by the following claims. Claim 1 V(Г) = Wj0 U Wjl U S, | Vi n Wj01 = |Vi n Wjl | = 1 and | Vi n S| = n - 2. By Lemma 2.1 (2), V is an independent subset, and by Lemma 2.1 (5), | Wk | > 2 with k = j0 or j i. It follows that V П Wk C Wk. Since j is connected, there exist (ui,to) G Vi n Wjo and (uio,t0) e Wjo \ (V П Wj) such that K,to) - Ko,t'0). Similarly, there exist (ui,t1) e Vi n Wj1 and (ui1 ,ti) e Wj1 \ (Vi n Wj1 ) such that (ui,t1) — (ui1, ti). From Lemma 2.3 we obtain that t'0 = t1, V n Wjo = {(ui,t0)} and Vio n Wjo = {К,, ti)}, and ti = to, Vi n Wji = {(ui,ti)} and Vii nWj, = {(u^ ,to)} (see Figure 1). In particular, we have |Vi n Wjo | = | Vi n Wj11 = 1. Figure 1: Explanation of the proof of Claim 1 It follows that | Vi n S| < n - 2. If | V n S| < n - 2, then we would have Vi n Wj = 0 for some j = j0, ji Take (ui,t) e Vi n Wj. Clearly, t = t0,t1. This forces that (ui, t) — (uio, t1), contrary to the fact that j and Г j are two distinct components. Thus, |Vi n S| = n - 2. At last, we shall show that s = 2. Suppose to the contrary that s > 2. Since | Vi n S| = n - 2 > 0, we can take (ui; j) e Vi n S. By Lemma 2.2, (ui; j) has a neighbor, say (uk, j') in each Wj with j = joj! Since t0 = tb either (ufc, j ') — (ui,t 0) or (ufc, j') — (ui,t1). This is again contrary to the fact that j, Г j and Г are three distinct components. Thus, s = 2 and hence V(Г) = Wjo U Wj1 U S. By Claim 1, we may assume that Vi n Wjo = {(ui; t0)} and Vi n Wj = {(ui; t1)}. Claim 2 For each (uj-, e Nr((ui,t0)) n Wjo, |Vj n S| = n - 1 or n - 2. There is at least one (uio ,^0) e Nr((ui,t0)) n Wjo suchthat |Vio n S | = n - 2. Take (uje Np((ui,to)) n Wjo. From Claim 1 we see that Vi n Wj1 = {(ui,t1)}. By Lemma 2.3, we have | Vj n Wjo | = 1, implying | Vj n S| < n - 1. If |Vj n S| < n - 1 then we must have Vj n Wj^ = 0. By Claim 1, we have |Vj n S| = n - 2. Therefore, IV n S| = n - 1 or n - 2. Suppose that for each (uj, e Nr ((ui, t0 )) n Wjo, we have | Vj n S | = n - 1. Noting that n > 3, one has |S| > |Vi n S| + ^ n S| > n - 2 + J(G)(n - 1) >к(Г), Uj eNa(ui) a contradiction. Thus, there is at least one (uio, 4i) e Nr ((ui; t0)) n Wjo such that |Vio n S| = n - 2. o o o Now we know that Claim 2 holds. Since (uio,^0) e Nr((ui,t0)) n Wjo, it follows from Lemma 2.3 that Vio n Wjo = {(uio, t1)} and Vio n W^^ = {(uio, t0)} (see Figure 2). By the arbitrariness of Vi, Claims 1,2 also hold if we replace Vi by Vio. Figure 2: Explanation of Claims 1,2 For the convenience of statement, we shall use the following notations in the remainder of the proof. Notations (1) (2) (3) (4) (5) (6) Ni = Nr((ui,to)) n Wjo, Nio = Nr((uio,ti)) n Wjo, = {k G Zm | Vk n Ni = 0}, üio = {k G Zm | Vk n Ni o = 0}, | k G a,Vk n Nr((ui,to))= 0}, | k G ^,Vk n Nr((uio,ti)) = 0}. Ai = {k G Zm Aio = {k G Z. It is easy to see that NG(ui) = {uk | k G Qi U Ai} and NG(uio) = {uk | k G Qio U Aio}. Hence, + |Ai| = dcM and | + |Aio| = dc^). Claim 3 |Ni| = and |Nio| = К|. By Claim 1, for each k G we have |Vk n Wjo | = 1. It follows that |Ni| = Similarly, |Nio | = |O<01. Claim 4 Both Ni and Nio are independent subsets of V(Г). Take any two vertices, say (ui1, t), (ui2, t') in Ni. Since Vi n Wj = {(ui, t1)}, from Lemma 2.3 it follows that t = t' = t1. So, (ui1, t) is not adjacent to (ui2, t'). Therefore, Ni is an independent subset of V(Г). Similarly, Nio is also an independent subset. Claim 5 (U keQ.UA, Vk ) n (U^q Vk)= 0 and (U keQin UAi Vk) n (U^q, Vk) = 0. Suppose that (UkEQ,uA, Vk) n (UkeQ,n Vk) = 0. Take (uj,t) G (U keQ.UA, Vk ) n (UkeQ,0 Vk). Then VjnNio = 0, implying that Vj nWjo = 0. Assume (uj,t') G VjnWjo. Clearly, ui and uio are neighbors of uj in G. Since t0 = t1, either (uj ,t') ~ (ui ,t1) or (uj, t') ~ (uio, t0). This is contrary to the fact that there are no edges between Wjo and Wji. Thus, (UkeQ.UA, Vk) n (Uk£Q,o Vk) = 0. Similarly, we have (UkeQ,ouA, o Vk) n (UkeQ, Vk)= 0. Claim 6 Let k G Ai U Ai o. Then Vk n Wjo = 0 and |Vk n S| > n - 1. Assume k G Ai. Then uk G NG(ui). If Vk n Wjo = 0, then take (uk, t) G Vk n W. j Since k G (uk, t) is not adjacent to (ui, t0), and hence t = t0. Consequently, (uk, t) (ui, t1), a contradiction. Thus, Vk n Wjo = 0. By Lemma 2.3, |Vk n Wj | < 1, and hence | Vk n S | > n -1. With a similar argument, we can show that if k G Ai o, then Vk n Wjo = 0 and |Vk n S| > n - 1. Claim 7 (1) n = 3; (2) |Ni| = |Nio | = 0(G)- (3) |Д.| = Д | = 0; (4) |S| = 25(G); (5) S = U fceniun^0 (Vk nS); (6) for each k e Q. UQio, |Vfc n S | = \Vk nW, | = V n Wj | = 1. By the arbitrariness of V and Vio, we may assume that |N | < |Nio1. By Claim 5, |S| > | U (Vk n S)| + |( U (Vk n S))|. (2.1) keo.i0 keQiUAi By Claim 2, if k e Q. U Qio, then |Vk n S| > n - 2, and by Claim 6, if k e Д. U Д.0, then | Vk n S| > n - 1. It follows that |S| > (n - 2)|Qio| + (n - 2)|Qi| + (n - 1)|Д.|. (2.2) By Claim 3, we have |S| > (n - 2)|Nio| + (n - 2)|Ni| + (n - 1)|Д.|. (2.3) Since |Nio | > |Ni | and n > 3, we obtain that |S| > 2(n - 2)|Ni| + (n - 1)|Д.| > (n - 1)dG(ui). (2.4) However, by Lemma 2.1 (4), we have | S| < (n -1) 5( G). So, in the above four inequalities, "=" must hold. By Eq. (2.4) we obtain that n = 3, |Ni| = |Ni01, and 5(G) = dG(ui). Furthermore, for each k e Qi UQio, |Vk nS| = n-1, and for each k e Д., |Vk nS | = n-1. It follows that |S | = 2|Ni| +2|Д<| = 25(G). (2.5) To show that | Д. | = | Д.0 | = 0, we shall first show that ( U Vk) n ( U Vk) = 0. keAi keo.i0 uAio Suppose on the contrary that for some k e Д. , Vk n(U keQ. uA Vk ) = 0. Since |Vk nS| = n - 1, from Claim 6 it follows that |Vk n W, | = 1. Take (uk ,t) e Vk n W,. Then are neighbors of uk in G. Since t0 = ti, either (uk,t) ~ («.,^1) or (uk,t) ~ (uio, t0). This is contrary to the fact that there are no edges between W0 and W1. Thus, (UkeAi Vk) n (Uk£Qi0uAi0 Vk) = 0. It follows that |S| > | UkeQi0uAi0 (Vk n S)| + |(Uke^iUAi(Vk n S))| = | Uke^i0 (Vk n S)| + |(Uke,,uAi(Vk n S))| + |(JkeA,0 (Vk n S)| > 2|Ni| + 21 Д. | +2Д.01 = 25(G) + 2Д |. Combining this with Eq. (2.5) we obtain that |Д.01 = 0. Since ^(u.) = 5(G), we have dc(uio) > dG(ui). Recall that |N.| + |Д.| = dG(ui) and N | + |Д.01 = dG(uio). Since |Ni| = |Nio |, one has Д1 > Д.|, implying |Д.| = 0. At last, from Eq. (2.1) it can be deduced that S = ( U (Vk n S)) U ( U (Vk nS)). (2.6) keQio keQi Claim 8 Wj0 = Nj U Nj0 . Suppose that Wj0 = Nj U Nio. Since Г0 is a component of Г - S, we can take a vertex, say (vkl, t), in Wj1 - (Ni U Nio ) such that (vkl, t) is adjacent to some vertex, say (vk2, t'), in Nj U Nj0. Since (vk2 ,t') G Nj U Nj0, by Claim 7 |Vk2 П Wjl | = 1. By Lemma 2.3, we have |Vkl n Wj01 = 1. By Claim 1, we have Vkl П S = 0. From Eq. (2.6) we see that ki G Oio U Qi, and hence (vkl, t) G Ni U Nio, a contradiction. Thus, Wj0 = Ni U Nio. Wjo S Wjl Figure 3: Explanation of Claims 8,9 Claim 9 m = |G| = 25(G). By Claim 8, |Wj01 = 25(G). So, m > |Wj01 = 25(G). Suppose that m > 25(G). By Claim 7, for each k G Qi U Qio, |Vk n S| = \Vk n Wjo | = |Vk n Wjl | = 1, and S = Ukefi,. Ufi, (Vk n S). This implies that Ukefi,. Ufi, (Vk n Wjl ) is a proper subset of Wjl. By the connectedness of Г1, take an edge e in Г1 such that one end, say (ukl, t), of e is in U kefii Ufi, (Vk n Wjl ) and the other end, say (uk2 , t'), is in Wjl \U kefii Ufi, (Vk n Wjl ). By Claim 7, | Vkl n Wj01 = 1, and by Lemma 2.3, we have | Vk2 n Wjl | = 1. By Claim 1, Sk2 n S| > 1. It follows from Eq. (2.6) that k2 G Oj0 U Qj. This forces that (uk2 ,t') G Ukefii0ufi, (Vk n Wjl), a contradiction. Claim 10 G = Ke,e, where i = 5(G). Clearly, {uk | k g Qi U Qio} С V (G). By Claim 9, m = |G| = 25(G). It follows that V(G) = {uk | k G Oj U Qj0}. Set Bo = {uk | k G Oj} and Bi = {uk | k G Oj0}. Take any two vertices, say ukl and uk2, in B0. Suppose ukl ~ uk2. By Claim 7, we may assume that Vk, n Wj0 = {(uk,, di)} with i = 1 or 2. From Claim 4 we obtain that (ukl, d1) is not adjacent to (uk2, d2), and hence d1 = d2. Since ukl ~ uk2, (ukl, d1) is adjacent to all the remaining vertices in Vk2. Again, by Claim 7, we get that |Vk2 n Wjl | = 1. This implies that there is an edge between Wj0 and Wjl, a contradiction. Therefore, ukl and uk2 are nonadjacent. By the arbitrariness of ukl and uk2, we get that B0 is an independent subset of V(G). Similarly, B1 is also an independent subset of V(G). It follows that G must be a bipartite graph with two partition sets B0 and B1. By Claims 3,7, we know that |Bo| = |B11 = 5(G). This means that G ^ K^, where I = 5(G). □ Lemma 2.6. Let i be a positive integer. Then x K3 is not super-к. Proof. Let B0 = {vi | i G Z^} and B1 = {ui | i G Z^} be the two partition sets of Ke^. Set V(K3) = Z3. Let S = V(K£j£) x {1}. Clearly, |S| = 2i. By [9], «(K^ x K3) = 2i. Set W0 = (B0 x {0}) U (b1 x {2}) and W1 = (B0 x {2}) U (B1 x'{0}). Clearly, V(Ke,e x K3) = S U W0 U W1. It is also easy to see that r[Wi] = K£j£ for i = 0,1. Furthermore, in K^ ( K3 there are no edges between W0 and Wi. It follows that K^ x K3 - S is disconnected with no isolated vertices. Therefore, K^ x K3 is not super-к. □ Proof of Theorem 1.1. By Lemmas 2.4 and 2.5, we can get the necessity. For the sufficiency, by Lemma 2.6, K^ x K3 is not super-к. Now assume that к(Г) = nn(G). Suppose to the contrary that Г is super-к. Then к(Г) = S(Г) = (n - 1 )S(G), and hence (n - 1)S(G) = nn(G). So, k(G) < 0(G). Let T be a minimum separating set of G. Then G - T has no isolated vertices. By Lemma 2.1 (4), T x V(Kn) is a separating set of Г. Clearly, |T x V(Kn)| = nn(G). So, T x V(Kn) is also a minimum separating set of G. Since Г is super-к, T x V(Kn) must be the neighborhood of some vertex, say (ui,j). Let uk e T. Then (uk,j) G T x V(Kn), and hence (u, j ) ~ (uk, j ). This is clearly impossible by the definition of the direct product of graphs. Thus, Г is not super-к. □ Acknowledgements: The author is indebted to the anonymous referee for many valuable comments and constructive suggestions. References [1] N. Alon and E. Lubetzky, Independent sets in tensor graph powers, J. Graph Theory 54 (2007), 73-87. [2] D. Bauer, F. Boesch, C. Suffel and R. Tindell, Connectivity extremal problems and the design of reliable probabilistic networks, in: The Theory and Application of Graphs, Wiley, New York, 1981, 45-54. [3] F.T. Boesch, Synthesis of reliable networks: A survey, IEEE Trans. Reliability 35 (1986), 240246. [4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Elsevier North Holland, New York, 1976. [5] A. Bottreou and Y. Metivier, Some remarks on the Kronecker product of graphs, Inform. Process. Lett. 68 (1998), 55-61. [6] B. Bresar, W. Imrich, S. KlavZar and B. Zmazek, Hypercubes as direct products, SIAM J. Discrete Math. 18 (2005), 778-786. [7] B. Bresar and S. Spacapan, On the connectivity of the direct product of graphs, Austral. J. Combin. 41 (2008), 45-56. [8] S.A. Ghozati, A finite automata approach to modeling the cross product of interconnection networks, Math. Comput. Model. 30 (1999), 185-200. [9] R. Guji and E. Vumar, A note on the connectivity of Kronecker product of graphs, Appl. Math. Lett. 22 (2009), 1360-3163. [10] L. Guo, C. Qin and X. Guo, Super connectivity of Kronecker product of graphs, Inform. Process. Lett. 110 (2010), 659-661. [11] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, Second Edition, CRC Press 2011. [12] R.H. Lammprey and B.H. Barnes, Products of graphs and applications, Model. Simul. 5 (1974), 1119-1123. [13] J. Leskovec, D. Chakrabarti, J. Kleinberg, C. Faloutsos and Z. Gharamani, Kronecker graphs: an approach to modeling networks, J. Mach. Learn. Res. 11 (2010), 985-1042. [14] A. Mamut and E. Vumar, Vertex vulnerability parameters of Kronecker product of complete graphs, Inform. Process. Lett. 106 (2008), 258-262. [15] G. Mekis, Lower bounds for the domination number and the total domination number of direct product graphs, Discrete Math. 310 (2010), 3310-3317. [16] J. Nesetril, Representations of graphs by means of products and their complexity, in: Mathematical Foundations of Computer Science, in: Lecture Notes in Comput. Sci. 118, Springer, Berlin, 1981,94-02. [17] Y. Wang and B. Wu, Proof of a conjecture on connectivity of Kronecker product of graphs, Discrete Math 311 (2011), 2563-2565. [18] W. Wang and N.-N. Xue, Connectivity of direct products of graphs, arXiv: 1102.5180v1 [math.CO] 25 Feb 2011. [19] H. Wang, E. Shan and W. Wang, On the super connectivity of Kronecker product of graphs, Inform. Process. Lett. 112 (2012), 402-405. [20] P.M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 8 (1962), 4-52. [21] A.N. Whitehead and B. Russell, Principia Mathematica, 2, Cambridge University Press. Cambridge, 1912, p.384. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 245-258 Mixed fault diameter of Cartesian graph bundles II Rija Erveš * FCE, University of Maribor, Smetanova 17, Maribor 2000, Slovenia Janez Žerovnik FME, University of Ljubljana, Aškerčeva 6, Ljubljana 1000, Slovenia Received 2 October 2012, accepted 13 December 2012, published online 23 January 2015 The mixed fault diameter D(pq)(G) is the maximum diameter among all subgraphs obtained from graph G by deleting p vertices and q edges. A graph is (p, q)+connected if it remains connected after removal of any p vertices and any q edges. Let F be a connected graph with the diameter D(F) > 1, and B be (p, q)+connected graph. Upper bounds for the mixed fault diameter of Cartesian graph bundle G with fibre F over the base graph B are given. We prove that if q > 0, then D(p+1q)(G) < D(F) + D(pq)(B), and if q = 0 andp> 0, then V^+^G) < D(F) + max{D(p,o)(B),D(p_M)(B')}. Keywords: Mixed fault diameter, Cartesian graph bundle, interconnection network, fault tolerance. Math. Subj. Class.: 05C12, 05C40, 68M10, 68M15, 68R10 1 Introduction Graph products and bundles belong to a class of frequently studied interconnection network topologies. For example meshes, tori, hypercubes and some of their generalizations are Cartesian products. It is less known that some other well-known interconnection network topologies are Cartesian graph bundles, for example twisted hypercubes [9, 12] and multiplicative circulant graphs [23]. *Both authors are also researchers at Institute of mathematics, physics and mechanics. The research supported in part by the Research agency of Slovenia ARRS. E-mail addresses: rija.erves@um.si (Rija Erves), janez.zerovnik@fs.uni-lj.si, janez.zerovnik@imfm.si (Janez Žerovnik) Abstract In the design of large interconnection networks several factors have to be taken into account. A usual constraint is that each processor can be connected to a limited number of other processors and that the delays in communication must not be too long. Furthermore, an interconnection network should be fault tolerant, because practical communication networks are exposed to failures of network components. Both failures of nodes and failures of connections between them happen and it is desirable that a network is robust in the sense that a limited number of failures does not break down the whole system. A lot of work has been done on various aspects of network fault tolerance, see for example the survey [8] and the more recent papers [16, 24, 26]. In particular the fault diameter with faulty vertices, which was first studied in [18], and the edge fault diameter have been determined for many important networks recently [2, 3, 4, 5, 10, 11, 19, 25]. Usually either only edge faults or only vertex faults are considered, while the case when both edges and vertices may be faulty is studied rarely. For example, [16, 24] consider Hamiltonian properties assuming a combination of vertex and edge faults. In recent work on fault diameter of Cartesian graph products and bundles [2, 3, 4, 5], analogous results were found for both fault diameter and edge fault diameter. However, the proofs for vertex and edge faults are independent, and our effort to see how results in one case may imply the others was not successful. A natural question is whether it is possible to design a uniform theory that covers simultaneous faults of vertices and edges. Some basic results on edge, vertex and mixed fault diameters for general graphs appear in [6]. In order to study the fault diameters of graph products and bundles under mixed faults, it is important to understand generalized connectivities. Mixed connectivity which generalizes both vertex and edge connectivity, and some basic observations for any connected graph are given in [13]. We are not aware of any earlier work on mixed connectivity. A closely related notion is the connectivity pairs of a graph [7], but after Mader [20] showed the claimed proof of generalized Menger's theorem is not valid, work on connectivity pairs seems to be very rare. An upper bound for the mixed fault diameter of Cartesian graph bundles is given in [14] that in some case also improves previously known results on vertex and edge fault diameters on these classes of Cartesian graph bundles [2, 5]. However these results address only the number of faults given by the connectivity of the fibre (plus one vertex), while the connectivity of the graph bundle can be much higher when the connectivity of B is substantial. It seems obvious that the upper bound from [14] can be improved. In this paper we provide an upper bound that takes into account the mixed connectivity of the base graph B, i.e. the number of faults allowed is given by the connectivity of the base graph (plus one vertex), thus complementing the result of [14]. We show by examples that the bounds of the new result are tight. In addition, in some cases Theorem 4.6 also improves previously known results on vertex and edge fault diameters on these classes of Cartesian graph bundles [2, 5]. The rest of the paper is organized as follows. General definitions, in particular of the connectivities, are given in section Preliminaries. The third section introduces graph bundles and recalls relevant previous results. In Section 4, the proof of the main theorem is given, followed by a short discussion. 2 Preliminaries A simple graph G = (V, E) is determined by a vertex set V = V (G) and a set E = E (G) of (unordered) pairs of vertices, called edges. As usual, we will use the short notation uv for edge {u, v}. For an edge e = uv we call u and v its endpoints. It is sometimes convenient to consider the union of elements of a graph, S(G) = V (G) U E (G). Given X C S (G) then S (G) \ X is a subset of elements of G. However, note that in general S (G) \ X may not induce a graph. As we need notation for subgraphs with some missing (faulty) elements, we formally define G \ X, the subgraph of G after deletion of X, as follows: Definition 2.1. Let X C S(G), and X = XE UXV, where XE C E (G) and XV C V (G). Then G \ X is the subgraph of (V (G), E (G) \ XE ) induced on vertex set V (G) \ XV. A walk between vertices x and y is a sequence of vertices and edges v0, e1, v1, e2, v2, ..., vk-1, ek, vk where x = v0, y = vk, and ež = vj_ivj for each i. A walk with all vertices distinct is called a path, and the vertices v0 and vk are called the endpoints of the path. The length of a path P, denoted by £(P), is the number of edges in P. The distance between vertices x and y, denoted by dG(x, y), is the length of a shortest path between x and y in G. If there is no path between x and y we write dG(x, y) = то. The diameter of a connected graph G, D(G), is the maximum distance between any two vertices in G. A path P in G, defined by a sequence x = v0, e1, v1, e2, v2,..., vk-1,ek, vk = y can alternatively be seen as a subgraph of G with V (P ) = {v0, v1,v2,... ,vk } and E(P ) = {e1, e2,..., ek}. Note that the reverse sequence gives rise to the same subgraph. Hence we use P for a path either from x to y or from y to x. A graph is connected if there is a path between each pair of vertices, and is disconnected otherwise. In particular, K1 is by definition disconnected. The connectivity (or vertex connectivity) k(G) of a connected graph G, other than a complete graph, is the smallest number of vertices whose removal disconnects G. For complete graphs is к(Кп) = n — 1. We say that G is k-connected (or k-vertex connected) for any 0 < k < k(G). The edge connectivity Л (G) of a connected graph G, is the smallest number of edges whose removal disconnects G. A graph G is said to be k-edge connected for any 0 < k < Л^). It is well known that (see, for example, [1], page 224) k(G) < Л(^) < SG, where SG is smallest vertex degree of G. Thus if a graph G is k-connected, then it is also k-edge connected. The reverse does not hold in general. Here we are interested in mixed connectivity that generalizes both vertex and edge connectivity. Note that the definition used here slightly differs from the definition used in a previous work [13]. Definition 2.2. Let G be any connected graph. A graph G is (p, q)+connected, if G remains connected after removal of any p vertices and any q edges. We wish to remark that the mixed connectivity studied here is closely related to connectivity pairs as defined in [7]. Briefly speaking, a connectivity pair of a graph is an ordered pair (k, £) of two integers such that there is some set of k vertices and £ edges whose removal disconnects the graph and there is no set of k — 1 vertices and £ edges or of k vertices and £ — 1 edges with this property. Clearly (k, £) is a connectivity pair of G exactly when: (1) G is (k — 1, ^)+connected, (2) G is (k, I — 1)+connected, and (3) G is not (k, ^)+connected. In fact, as shown in [13], (2) implies (1), so (k, £) is a connectivity pair exactly when (2) and (3) hold. From the definition we easily observe that any connected graph G is (0,0)+connected, (p, 0)+connectedforany p < k(G) and (0, q)+connectedforany q < Л^). In our notation (i, 0)+connected is the same as (i + 1)-connected, i.e. the graph remains connected after removal of any i vertices. Similarly, (0, j)+connected means (j + 1)-edge connected, i.e. the graph remains connected after removal of any j edges. Clearly, if G is a (p, q)+connected graph, then G is (p', q')+connected for any p' < p and any q' < q. Furthermore, for any connected graph G with k < к (G) faulty vertices, at least k edges are not working. Roughly speaking, graph G remains connected if any faulty vertex in G is replaced with a faulty edge. It is known [13] that if a graph G is (p, q)+connected and p > 0, then G is (p - 1, q + 1)+connected. Hence for p > 0 we have a chain of implications: (p, q)+connected (p - 1, q + 1)+connected ... (1,p + q - 1)+connected (0,p + q)+connected, which generalizes the well-known proposition that any k-connected graph is also k-edge connected. Therefore, a graph G is (p, q)+connected if and only if p < k(G) and p + q < Л (G). Note that by our definition the complete graph Kn, n > 2, is (n - 2,0)+connected, and hence (i, j)+connected for any i + j < n - 2. Graph K2 is (0,0)+connected, and mixed connectivity of K1 is not defined. If for a graph G k(G) = Л^) = k, then G is (i, j)+connected exactly when i + j < k. However, if 2 < k(G) < Л^), the question whether G is (i, j)+connected for 1 < i < k(G) < i + j < Л^) is not trivial. The example below shows that in general the knowledge of k(G) and Л^) is not enough to decide whether G is (i, j)+connected. Example 2.3. For graphs on Fig. 1 we have k(Gx) = k(G2) = 2 and = Л^2) = 3. Both graphs are (1,0)+connected (0,1)+connected, and (0,2)+ connected. Graph Gi is not (1,1)+connected, while graph G2 is. Definition 2.4. Let G be a k-edge connected graph and 0 < a < k. The a-edge fault diameter of G is Df (G) = max {D(G \ X) | X С E(G), |X| = a}. Definition 2.5. Let G be a k-connected graph and 0 < a < k. The a-fault diameter (or a-vertex fault diameter) of G is DV (G) = max {D(G \ X ) | X С V (G), |X | = a}. Note that Df (G) is the largest diameter among the diameters of subgraphs of G with a edges deleted, and DV (G) is the largest diameter over all subgraphs of G with a vertices deleted. In particular, Df (G) = Df (G) = D(G), the diameter of G. For p > k(G) and for q > Л^) we set Df (G) = то, Df (G) = то, as some of the subgraphs are not vertex connected or edge connected, respectively. It is known [6] that for any connected graph G the inequalities below hold. 1. D(G) = D0f (G) < Df (G) < Df (G) < ... < DE(g)_1(G) < . 2. D (G) = DV (G) < DV (G) < DV (G) < ... < dV(g)_I(G) < Definition 2.6. Let G be a (p, q)+connected graph. The (p, q)-mixed fault diameter of G is D(p,q) (G) = max {D(G \ (X U Y )) | X С V (G), Y C E (G), \X | = p, \Y | = q}. Note that by Definition 2.6 the endpoints of edges of set Y can be in X. In this case we may get the same subgraph of G by deleting p vertices and fewer than q edges. It is however not difficult to see that the diameter of such subgraph is smaller than or equal to the diameter of some subgraph of G where exactly p vertices and exactly q edges are deleted. So the condition that the endpoints of edges of set Y are not in X need not to be included in Definition 2.6. The mixed fault diameter D(p,q)(G) is the largest diameter among the diameters of all subgraphs obtained from G by deleting p vertices and q edges, hence D(o,o)(G) = D (G), D«,,a)(G) = Df (G) and D(o,0)(G) = Dva (G). Let HV = {G \ X \ X С V (G), \X \ = a} and Hf = {G \ X \ X С E (G), \X \ = b}. It is easy to see that 1. max {Df (H ) \ H eHVa } = D(0,b)(G), 2. max{DV(H) \ H G Hf} = D(0,b)(G). In previous work [6] on vertex, edge and mixed fault diameters of connected graphs the following theorem has been proved. Theorem 2.7. Let G be (p, q)+connected graph and p > 0. • If q > 0, then Df+q(G) < D(i,p+q_i)(G) < • • • < D^,q)(G). • If q = 0, then Df (G) 0, and B is a connected graph with diameter D(B) > 1. Then we have: • If q > 0, then D(p+1 ,q) (G) < D(p,q)(F) + D(B). • If q = 0, then Dp+1(G) < max{DV(F), D(p-M)(F)} + D(B). Here we prove a similar result for an upper bound for the mixed fault diameter of Cartesian graph bundles, D(p+1q)(G), in terms of diameter of the fibre and mixed fault diameter of the base graph. We consider mixed fault diameter of Cartesian graph bundle G with connected fibre F. If the graph B is (p, q)+connected then Cartesian graph bundle with connected fibre F over the base graph B is at least (p + 1, q)+connected. Theorem 4.6 improves results 3.2 and 3.3 for a = 0. 4 Proof of the main theorem Before stating and proving the main theorem, we prove several lemmas and introduce some notation used in this section. Let G be a Cartesian graph bundle with fibre F over the base graph B. The fibre of vertex x G V (G) is denoted by Fx, formally, Fx = p-1({p(x)}). We will also use notation F (u) for the fibre of the vertex u g V (B ), i.e. F (u) = p-1 ({u}). Note that Fx = F (p(x)). We will also use shorter notation x g F (u) for x G V (F (u)). Let u, v g V (B) be distinct vertices, and Q be a path from u to v in B, and x g F (u). Then the lift of the path Q to the vertex x G V (G), Q x, is the path from x G F (u) to a vertex in F (v), suchthat p(Q x) = Q and i(Q x) = l(Q). Let x,x' G F (u). Then Qx and Q xx have different endpoints in F (v) and are disjoint paths if and only if x = x'. In fact, two lifts Qx and Qx are either disjoint Qx П Qx> = 0 or equal, Qx = Qx>. We will also use notation Q for lifts of the path Q to any vertex in F (u). Let Q be a path from u to v and e = uw G E(Q). We will use notation Q \ e for the subpath from w to v, i.e. Q \ e = Q \ {u, e} = Q \ {u}. Let G be a graph and X C S(G) be a set of elements of G. A path P from a vertex x to a vertex y avoids X in G, if S (P ) П X = 0, and it internally avoids X, if (S(P) \{x,y}) П X = 0. We will use Lemma 4.1 in following proofs. Lemma 4.1. Let F and B be connected graphs, D(F) > 1, and let G be a Cartesian graph bundle with fibre F over the base graph B. Let x, y G V (G) be two vertices, such that p(x) = p(y), and let Q be a path from p(x) to p(y) in B. Then there are (at least) two internally vertex-disjoint paths from x to y in p-1(Q) = FDQ C G of lengths at most D(F )+ *(Q). Proof. Let G be a Cartesian graph bundle with connected fibre F, D(F) > 1, over the connected base graph B. Let x, y G V (G), p(x) = p(y), and Q be a path from p(x) to p(y) in B. Let x' g Fy be the endpoint of Qx. • If x' = y, then there are two paths Pi : x -— y, P2 : x — s -— s' — y, where s g Fx and s' G Fy are neighbors of x and y, respectively. Paths P1, P2 are internally vertex-disjoint paths from x to y in p-1 (Q) and ^(P^ = ^(Q), ^(P2) = 1+ *(Q) + 1 1, and the graph B be (p, 0)+con-nected, p > 0. Then D(p+i,o)(G) = Dp+(G) < D(F) + max{DpV(B),D^_1,1)(B)}. Proof. Let F be a connected graph, D(F) > 1, the graph B be (p, 0)+connected, p > 0, and let G be a Cartesian graph bundle with fibre F over the base graph B. By Theorem 3.4, the Cartesian graph bundle G is (p +1,0)+connected. Let X С V (G) be a set of faulty vertices, |X | = p + 1, and let x, y G V (G) \ X be two distinct nonfaulty vertices in G. We shall consider the distance dG\X (x, y). Note that as graph B is (p, 0)+connected and p > 0, it is also (p - 1,1)+connected and D(p-11)(B) > 2. • Suppose first that x and y are in the same fibre, i.e. p(x) = p(y). If |X n V(Fx)| = 0, then dG\X(x, y) < D(F). If |X П V(Fx) | > 0, then outside of fibre Fx there are at most p faulty vertices. As a graph B is (p, 0)+connected, there are at least p +1 neighbors of vertex p(x) in B. Therefore there exist a neighbor v of p vertex p(x) in B, such that |X П F (v) | = 0, and there is a path x — x' — y' — y, which avoids X, where x',y' G F (v) and ^(P ) < D(F ). Thus dG\X (x,y) < 1+ D(F ) + 1 0}. We distinguish two cases. 1. If p < |XB | < p +1, then let X'B С XB be an arbitrary subset of XB with |XB | = p. The subgraph B \ XB is a connected graph and there exist a path Q from p(x) to p(y) with ^(Q) < DV(B). In p-1(Q) there is at most one faulty vertex. By Lemma 4.1 there are two internally vertex-disjoint paths from x to y in p-1 (Q) and at least one of them avoids the faulty element, thus dG\x(x, y) < *(Q) + D(F) < DV(B) + D(F). 2. If |XB | < p, then the subgraph B \ XB is (at least) (1,0)+connected, thus also (0,1)+connected. If the vertex p(y) is not a neighbor of p(x), then there is a path Q from p(x) to p(y) in B with 2 < ^(Q) < Dp—1(B) < DV(B) that internally avoids XB. Let v G V(Q) be a neighbor of p(x), e' = p(x)v. Then there is a path P Q\e' x — x' -— y' — y, which avoids X, where x', y' G F (v) and ^(P ) < D(F ). Thus dG\x(x, y) < 1 + D(F) + DV(B) - 1 = D(F) + DV(B). If e = p(x)p(y) G E(B), then B \ (Xb U {e}) is a connected graph and there is a path Q' from p(x) to p(y) with 2 < ^(Q') < D(p-11)(B) that internally avoids XB. Similar as before dG\X (x, y) < 1 + D(F) + D(p-1 1)(B) - 1 = D(F ) + D(p-1,1)(B). □ Example 4.3. Lemma 4.2 for p = 1 reads: DV(G) < D(F) + max{DV(B), Df (B)}. 1. Let B = K4 \ {e}. Then D(B) = Df (B) = Df (B) = 2. The 2-vertex fault diameter of Cartesian graph product Р3ПB is Df (P3QB) = D(P3) + Df (B) = 2 + 2 = 4. 2. The 2-vertex fault diameter of Cartesian graph product P3DK3 is DV(P3DK3) = D(P3)+ Df (K3) = 2 + 2 = 4. In both examples the bound of Lemma 4.2 is tight. Lemma 4.4. Let G be a Cartesian graph bundle with fibre F over the base graph B, the graph F be a connected graph with diameter D(F) > 1, and the graph B be (0, q)+con-nected, q > 0. Then D(i,q)(G) < D(F) + Df (B) = D(F) + D(o,q)(B). Proof. Let F be a connected graph, D(F) > 1, and B be (0, q)+connected graph, q > 0. Then Df (B) > 2 and by Theorem 3.4, the Cartesian graph bundle G with fibre F over the base graph B is (1, q)+connected. Let a g V (G) be the faulty vertex and Y С E (G) be the set of faulty edges, | Y| = q. Denote the set of degenerate edges in Y by YD, and the set of nondegenerate edges by YN, Y = YN U YD, p(YB ) С V (B), p(Yw ) С E (B ). Denote the set of faulty elements by X = {a} U Y. Let x, y G V (G) \ {a} be two arbitrary distinct nonfaulty vertices in G. We shall find an upper bound for the distance dG\X (x, y). • Suppose first that x and y are in the same fibre, i.e. p(x) = p(y). If |Fx n X| = 0, then dG\X(x, y) < D(F). If |Fx П X| > 0, then outside of fibre Fx there are at most q faulty elements. As the graph B is (0, q)+connected, there are at least q + 1 neighbors of vertex p(x) in B. Therefore there exist a neighbor v of vertex p(x) in B, such that p(x)v G p(YN) and |F(v) П ({a} U YD)| = 0, p and there is a path x — x' — y' — y which avoids X, where x', y' G F (v) and *(P) < D(F). Thus dG\x(x, y) < 1 + D(F) + 1 < D(F) + Df (B). • Now assume that x and y are in distinct fibres, i.e. p(x) = p(y). Let B' = B \ p(YN). As |p(YN)| < q - |YD |, the subgraph B' is at least (0, |YD |)+ connected and p-1(B') does not contain nondegenerate faulty edges, |p-1(B') n Yn| = 0. Let Y ' = {p(x)v G E (B'); |F (v) П ({a} U Yd )| > 0}. We distinguish two cases. 1. Let a G V(Fx) U V(Fy ), and without of loss of generality assume a G V(Fx). Then |Y'| < |Yd | and the subgraph B' \ Y ' = B \ (Y' U p(YN )) is aconnected graph. Therefore there exists a path Q from p(x) to p(y) in B of length 1 < ^(Q) < Df (B), that avoids p(Yn), and for the neighbor v g V(Q) of the vertex p(x), e = p(x)v, there is no faulty elements in the fibre F (v). Note that the path Q \ e avoids p(a). p If v = p(y) there is a path x — x' — y, where x' G Fy and ^(P) < D(F), that avoids faulty elements, thus dG\X (x, y) < 1 + D(F) < D(F) + Df (b). P Q\e If v = p(y) there is a path x — x' — y' — y, where x',y' G F (v) and ^(P) < D(F), which avoids faulty elements, thus dG\X (x, y) < 1 + D(F) + Df (B) - 1 = D(F) + Df (B). 2. If a G V(Fx) U V(Fy), we distinguish three cases. (a) Suppose | (Fx U Fy) n YD | = 0. There exist a path Q from p(x) to p(y) in B' C B of length £(Q) < Vf (B), that avoids p(YN). By Lemma 4.1 there are two internally vertex-disjoint paths from x to y in p-1(Q), that avoid Y and at least one of them avoids faulty vertex a, thus dG\X (x, y) < £(Q) + D(F) < Vf (B)+ D(F). (b) Suppose, that exactly one of fibres Fx, Fy contains faulty edges, without of loss of generality let |Fx n Yd | > 0 and |Fy n Yd | =0. Then |Y'| < |Yd | and the subgraph B' \ Y' = B \ (Y' Up(YN)) is a connected graph. There exist a path Q from p(x) to p(y) in B of length 1 < t(Q) < Vf (B), that avoids p(YN) and for the neighbor v e Q of vertex p(x), e = p(x)v, there is no faulty elements in the fibre F (v). If v = p(y) then dG\x (x,y) < 1+ V(F ) < V(F ) + Vf (B). If v = p(y), let v' e F (v) beaneighborof x. As |(F (v) U Fy ) n YD | = 0, similar as in (a) there is a path from v' to y in p-1(Q \ e) of length at most V(F) + Vf (B) - 1, that avoids faulty elements, thus dG\X (x, y) < 1 + V(F) + Vf (B) - 1 = V(F) + Vf (B). (c) At last, suppose |Fx n Yd | > 0 and |Fy n Yd | > 0. i. Assume dB(p(x),p(y)) = 1. In this casep(x)p(y) e Y', and |Y'| < |Yd| as |Fx n Yd | > 0. Thus the subgraph B' \ Y' is connected, and there exists a path Q from p(x) to p(y) in B of length 2 < ^(Q) < Vf (B), that avoids p(YN), and for the neighbor v e Q of vertex p(x), e = p(x)v, there is no faulty elements in the fibre F (v). p If ^(Q) = 2, then there is a path x — x' — y' — y, where x', y' e F (v) and ^(P ) < V(F ), which avoids faulty elements, thus dG\x(x, y) < 1 + V(F) + 1 < V(F) + Vf (B). If 3 < ^(Q) < Vf (B), then the path x — x' — y' — s — y where x',y' e F (v), ^(P ') < V(F ), and s e V (Fx), avoids faulty elements, thus dG\X(x, y) < 1 + V(F) + 1 + 1 < V(F) + Vf (B). ii. Assume, dB/(p(x),p(y)) = 2. Then there is at least one common neighbor of vertices p(x) and p(y) in B'. If there exist a common neighbor v of vertices p(x) and p(y) in B' for which there is no faulty elements in the fibre F (v), then as before dG\X (x, y) < 1 + V(F ) + 1 < V(F) + Vf (B). Otherwise suppose, there is some common neighbor w of vertices p(x) and p(y) in B' for which a e F(w) and |F(w) n Yd| > 0. As |Y'| < |Yd| - 1 the subgraph B' \ Y' is (at least) (0,1)+connected graph. If vertex p(a) is a neighbor of p(y) in B', e' = p(y)p(a) C E(B'), then also B' \ (Y' U {e'}) is a connected graph. Therefore there exist a path Q from p(x) to p(y) in B of length 3 < ^(Q) < Vf (B), that avoids p(YN), and for the neighbor u e V (Q) of vertex p(x), there is no faulty elements in the fibre F (u), and for the neighbor v e V(Q) of vertex p(y), v = p(a). If ^(Q) = 3 < Vf (B), then there is a path x — x' y' — s — y, where x',y' e F (u), ^(P ) < V(F ), s e F (v), which avoids faulty elements, thus dG\X(x, y) < 1 + V(F) + 1 + 1 < V(F) + Vf (B). If 4 < ^(Q) < Vf (B), then the path x — x' — y' — s — s' — y, where x', y' e F(u), ^(P') < V(F), s e V(Fx), s' e F(w), avoids faulty elements, thus dG\X (x, y) < 1 + D(F) + 1 + 2 < D(F) + Df (B). The last case to consider is when p(a) is the only common neighbor of vertices p(x) and p(y) in B'. Let Y '' = {p(y)v e E (B '); |F (v) П ({a} U YD )| > 0}. As p (a) is the only common neighbor of vertices p(x) and p(y), |Y' U Y''| < | Yd | and the subgraph B' \ (Y' U Y'') is a connected graph. Therefore there exist a path Q from p(x) to p(y) in B of length 3 < £(Q) < Df (B), that avoids p(YN), and for neighbors u e V(Q) of vertex p(x) and v e V(Q) of vertex p(y), there is no faulty elements in fibres F (u) and F (v). Let xX e F (u) be a neighbor of x and y' e F (v) be a neighbor of y. As in (a) there is a path from x' to y' in p-1(Q\{p(x),p(y)} of length at most D (F ) +Df (B) - 2, that avoids faulty elements, thus dG\X (x, y) < 1 + D (F ) + Df (B) - 2 + 1 = D(F ) + Df (B). iii. Finally, suppose dB>(p(x),p(y)) > 3. As there is no common neighbor of vertices p(x) and p(y) in B', |Y' U Y''| < |Yd | - 2+1 = |Yd | - 1 and as before there exist a path Q from p(x) to p(y) in B of length 3 < £(Q) < Df (B), that avoids p(YN), and for both neighbors u e V(Q) of vertex p(x) and v e V(Q) of vertex p(y), there is no faulty elements in fibres F (u) U F (v), thus dG\X (x, y) < 1 + D(F) + Df (B) - 2 + 1 = D(F) + Df (B). □ Lemma 4.5. Let G be a Cartesian graph bundle with fibre F over the base graph B, the graph F be a connected graph with diameter D(F) > 1, and the graph B be (p, q)+con-nected, q > 0. Then D(p+1 ,q)(G) 0. Let the graph F be a connected graph, D(F) > 1, and the graph B be (p, q)+connected, p, q > 0. Then D(p q)(B) > 2 and by Theorem 3.4, the Cartesian graph bundle G with fibre F over the base graph B is (p + 1, q)+connected. Let X С V (G) be the set of faulty vertices, |X | = p +1, and Y С E (G) be the set of faulty edges, | Y | = q. Denote the set of degenerate edges in Y by YD, and the set of nondegenerate edges by YN, Y = YN U YD, p(Yd ) С V (B), p(Yn ) С E (B). Let x, y e V (G) \ X be two distinct nonfaulty vertices in G. We shall determine an upper bound for the distance dG\(XUY) (x, y). • Suppose first that x and y are in the same fibre, i.e. p(x) = p(y). If |Fx n (X U Yd)| = 0, then dG\(xuY)(x, y) < D(F). If F П (X U Yd)| > 0, then there are at most p + q faulty elements outside of the fibre Fx. As the graph B is (p, q)+connected, there are at least p + q + 1 neighbors of vertex p(x) in B. Therefore there exists a neighbor v of vertex p(x) in B, suchthat p(x)v e p(YN ) and p |F (v) П (X U Yd )| = 0, and there is a path x ^ x' ^ y' ^ y, where x', y' e F (v) and ^(P) < D(F), which avoids X U Y. Thus dG\(XUY) (x, y) < 1 + D(F) + 1 < D(F )+ D(p,q) (B). • Now assume that x and y are in distinct fibres, i.e. p(x) = p(y). Let XB = {v e V (B) \ {p(x),p(y)}; | F (v) П X | > 0}. We distinguish two cases. 1. If p < |Xb | < p +1, then let X'B С Xb , |XB | = p. The subgraph B \ X'B is (0, q)+connected and there is at most one faulty vertex and q faulty edges in p-1 (B \ XB). By Lemma 4.4 there is a path from x to y in p-1 (B \ XB) with length at most D(F) + (B \ XB), that avoids faulty elements, thus do\(xuY )(x,y) < D (F ) + D E (B \ XB ) = D (F ) + D^B). 2. Suppose |Xb | < p. Let Yb = {p(x)v e E(B); |F(v) П YD| > 0} and B' = B \ (Xb U Yb U p(Yw)). Then the subgraph B' is (at least) (1,0)+connected, thus also (0,1)+connected. If dB>(p(x),p(y)) > 2, then there is a path Q from p(x) to p(y) in B' С B with 2 < £(Q) < D(p-1,q) (B) < D(p,q)(B) that internally avoids Xb, it avoids p(YN), and for the neighbor v e V(Q) of vertex p(x), e' = p(x)v, there is no faulty elements in the fibre F (v). Therefore there is a path x — x' — y' Q—e' y, where x', y' e F (v) and ^(P ) < D(F ), which avoids X U Y, thus dG\xuy(x,y) < 1 + D(F) + D(p,q)(B) - 1 = D(F)+ D^B). If dB'(p(x),p(y)) = 1, e = p(x)p(y) e E(B'), then the subgraph B' \ {e} is a connected graph and there is a path Q' from p(x) to p(y) with 2 < ^(Q') < D(p-1,q+1)(B) < D(p q)(B) that internally avoids XB, it avoids p(YN), and for the neighbor v e V(Q) of vertex p(x), there is no faulty elements in the fibre F (v), and as before dG\XUY (x, y) < 1 + D(F ) + D(pq)(B) - 1 = D(F ) + D(p , q) (B). □ Theorem 4.6. Let G be a Cartesian graph bundle with fibre F over the base graph B, the graph F be a connected graph with diameter D(F) > 1, and the graph B be (p, q) +connected, p + q > 0. Then we have: If q > 0, then D(p+i,q)(G) < D(F) + D(p,q)(B). • If q = 0, then D(p+i,o)(G) = Dp+i(G) < D(F) +max{D^(B), D(p_M)(B)}. Proof. The statement of Theorem 4.6 follows from Lemma 4.2 (case q = 0), Lemma 4.4 (case p = 0), and Lemma 4.5 (for positive p and q). □ Remark 4.7. Let G be a Cartesian graph bundle with fibre F over the base graph B, the graph F be a connected graph with diameter D(F) > 1, and the graph B be (p, 0)+con-nected, p > 0. By Theorem 4.6 we have an upper bound for the (vertex) fault diameter Dp+1(G) < D(F)+ DpV (B) + 1 for any graph B. Similarly, Dp++1(G) < D(F )+ D^ (B) if D(p_1,1)(B) 0, D(F) > 1, D(B) > 1, and let G be a Cartesian graph bundle with fibre F over the base graph B. Then we have: If q > 0, then D(p+1,q) (G) < max{D(F) + D^B), D^F) + D(B)}, • If q = 0, then Dp+1(G) < max{D(F) + Df (B), Df (F) + D(B)} + 1, andDf+i(G) < max{D(F) + Df (B),Df (F) + D(B)}, ifD(p_M)(F) < Df (F) andD(p_1,1)(B) < Df (B) hold. We conclude with a conjecture. We know that a Cartesian graph bundle with fibre F over the base graph B, where graph F is (pF, qF )+connected, pF + qF > 0, and where graph B is (pB, qB )+connected, pB + qB > 0, is (pB + pF + 1, qB + qF)+connected [13]. An upper bound for the mixed fault diameter where the number of allowed faulty elements would be the maximal possible may be the following: Conjecture 4.9. Let G be a Cartesian graph bundle with fibre F over the base graph B, where the graph F is (pF, qF)+connected, pF + qF > 0, and where the graph B is (pB, qB)+connected, pB + qB > 0. Then D(PB + 1,qB )(G) < D(pf ,qF )(F ) + D(PB ,qB )(B) + 1. References [1] J. M. Aldous, R. J. Wilson, Graphs and Applications: An introductory Approach, Springer, Berlin, 2000. [2] I. Banic, J. Zerovnik, Fault-diameter of Cartesian graph bundles, Inform. Process. Lett. 100 (2006), 47-51. [3] I. Banic, J. Zerovnik, Edge fault-diameter of Cartesian product of graphs, Lect. Notes Comput. Sc. 4474 (2007), 234-245. [4] I. Banic, J. Zerovnik, Fault-diameter of Cartesian product of graphs, Adv. Appl. Math. 40 (2008), 98-106. [5] I. Banic, R. Erveš, J. Zerovnik, The edge fault-diameter of Cartesian graph bundles, Eur. J. Combin. 30 (2009), 1054-1061. [6] I. Banic, R. Erveš, J. Zerovnik, Edge, vertex and mixed fault diameters, Adv. Appl. Math. 43 (2009), 231-238. [7] L. W. Beineke, F. Harary, The connectivity function of a graph, Mathematika 14 (1967), 197202. [8] J.-C. Bermond, N. Honobono, C. Peyrat, Large Fault-tolerant Interconnection Networks, Graph. Combinator. 5 (1989), 107-123. [9] P. Cull, S. M. Larson, On generalized twisted cubes, Inform. Process. Lett. 55 (1995), 53-55. [10] K. Day, A. Al-Ayyoub, Minimal fault diameter for highly resilient product networks, IEEE T. Parallel. Distr. 11 (2000), 926-930. [11] D. Z. Du, D. F. Hsu, Y. D. Lyuu, On the diameter vulnerability of kautz digraphs, Discrete Math. 151 (2000), 81-85. [12] K. Efe, A variation on the hypercube with lower diameter, IEEE T. Comput. 40 (1991), 13121316. [13] R. Erveš, J. Zerovnik, Mixed connectivity of Cartesian graph products and bundles, to appear in Ars Combinatoria, arXiv:1002.2508v1 [math.CO] (2010). [14] R. Erveš, J. Zerovnik, Mixed fault diameter of Cartesian graph bundles, Discrete Appl. Math. 161 (2013), 1726-1733, doi:10.1016/j.dam.2011.11.020. [15] R. Hammack, W. Imrich, S. KlavZar, Handbook of Product Graphs, second ed., CRC Press, 2011. [16] C. H. Hung, L. H. Hsu, T. Y. Sung, On the Construction of Combined k-Fault-Tolerant Hamil-tonian Graphs, Networks 37 (2001), 165-170. [17] W. Imrich, T. Pisanski, J. Žerovnik, Recognizing Cartesian graph bundles, Discrete Math. 167168 (1997), 393-403. [18] M. Krishnamoorthy, B. Krishnamurty, Fault diameter of interconnection networks, Comput. Math. Appl. 13 (1987), 577-582. [19] S. C. Liaw, G. J. Chang, F. Cao, D. F. Hsu, Fault-tolerant routing in circulant networks and cycle prefix networks, Ann. Comb. 2 (1998), 165-172. [20] W. Mader, Connectivity and edge-connectivity infinite graphs, Surveys in Combinatorics. Lond. Math. S. 38 (1979), 66-95. [21] T. Pisanski, J. Vrabec, Graph bundles, Preprint Series Department of Mathematics, vol. 20, no. 079, p. 213-298, Ljubljana, 1982. [22] T. Pisanski, J. Shawe-Taylor, J. Vrabec, Edge-colorability of graph bundles, J. Comb. Theory B 35 (1983), 12-19. [23] I. Stojmenovic, Multiplicative circulant networks: Topological properties and communication algorithms, Discrete Appl. Math. 77 (1997), 281-305. [24] C. M. Sun, C. N. Hung, H. M. Huang, L. H. Hsu, Y. D. Jou, Hamiltonian Laceability of Faulty Hypercubes, Journal ofInterconnection Networks 8 (2007), 133-145. [25] M. Xu, J.-M. Xu, X.-M. Hou, Fault diameter of Cartesian product graphs, Inform. Process. Lett. 93 (2005), 245-248. [26] J. H. Yin, J. S. Li, G. L. Chen, C. Žhong, On the Fault-Tolerant Diameter and Wide diameter of w-Connected Graphs, Networks 45 (2005), 88-94. [27] B. Žmazek, J. Žerovnik, Recognizing weighted directed Cartesian graph bundles, Discussiones Mathematicae Graph Theory 20 (2000), 39-56. [28] B. Žmazek, J. Žerovnik, On recognizing Cartesian graph bundles, Discrete Math. 233 (2001), 381-391. [29] B. Žmazek, J. Žerovnik, Algorithm for recognizing Cartesian graph bundles, Discrete Appl. Math. 120 (2002), 275-302. [30] B. Žmazek, J. Žerovnik, Unique square property and fundamental factorizations of graph bundles, Discrete Math. 244 (2002), 551-561. [31] J. Žerovnik, On recognizing of strong graph bundles, Math. Slovaca 50 (2000), 289-301. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 259-266 An atlas of subgroup lattices of finite almost simple groups Thomas Connor * Università libre de Bruxelles, Boulevard du Triomphe CP216, Brussels, Belgium Dimitri Leemans University of Auckland, Private Bag 92019, Auckland, New Zealand Received 11 March 2013, accepted 11 March 2014, published online 23 January 2015 We provide algorithms to compute and produce subgroup lattices of finite permutation groups. We discuss the problem of naming groups and we propose an algorithm that automatizes the naming of groups, together with possible ways of refinement. Finally we announce an atlas of subgroup lattices for a large collection of finite almost simple groups made available online. Keywords: Computational methods in group theory, lattices of subgroups Math. Subj. Class.: 20B40, 20E15 1 Introduction The Classification of the Finite Simple Groups (CFSG) emphasizes the importance of the finite simple groups in Group Theory. It is one of the most impressive achievements in the history of Mathematics. We refer to [26] and the references provided there for a broad literature on this wonderful theorem. Among the amazing achievements in this branch of Mathematics, we find the Atlas of Finite Groups [15] as well as the online version of the Atlas of Finite Group Representations [1]. Over the years, the finite simple groups have received a lot of attention with respect to the study of geometry. The Theory of Buildings due to Jacques Tits, who was awarded the Abel Prize in 2008, illustrates this perfectly. We refer for instance to [2] and references *BoursierF.R.I.A. E-mail addresses: tconnor@ulb.ac.be (Thomas Connor), d.leemans@auckland.ac.nz (Dimitri Leemans) Abstract provided there. Much work has been done in this respect with the study of incidence geometries associated to finite almost simple groups (we refer to [7, 9, 20] and references cited there for a large documentation on this aspect). The computations of subgroup lattices and tables of marks of permutation groups, and in particular sporadic simple groups, have been a subject of interest for many decades, linked among others to the search for a unified geometric interpretation of all finite simple groups. Joachim Neubuser gave in [23] the first algorithm that was implemented later on in the computational software Cayley and its successor Magma. Francis Buekenhout computed in 1984 the lattices of Мц and Ji [5]. Then, Herbert Pahlings did the lattice of J2 [24] in 1987. In 1988, Buekenhout and Sarah Rees produced the lattice of M12 (see [6] and [16] for a few corrections). In 1991, Pfeiffer computed the table of marks of J3 and in 1997, those of M22, M23 M24, McL [25]. Also in 1997, Merkwitz got the tables of marks of He and Co3. In 1998, Derek Holt computed all conjugacy classes of subgroups of O'N (personal communication). In a more general setting again, John Cannon, Bruce Cox and Derek Holt described in [11] a new algorithm to compute the conjugacy classes of subgroups of a given group that was used in Magma until 2005. Progresses on the computation of maximal subgroups of a given group by Cannon and Holt [12] led Leemans to a much faster algorithm to compute the subgroup lattice of a given group that is now available in MAGMA. In 2007, Leemans computed the full subgroup lattices of HS, Ru, Suz, O'N, Co2 and Fi22 using permutation degree reduction at each step of the computation. Recently, Naughton and Pfeiffer produced a new algorithm to compute the table of marks of a cyclic extension of a group [22]. The knowledge of the subgroup lattice of a group G is a powerful tool to study the symmetrical objects on which G acts. For instance, in [10], [13] and [21], the authors build flag-transitive coset geometries of ranks 2, 3 and 5 for O'N by identifying boolean lattices in the subgroup lattice of O'N. In [14], the authors develop an algorithm in order to count the number of regular maps on which a finite group G acts regularly. This algorithm makes an intensive use of the knowledge of the subgroup lattice of G. Then they illustrate the algorithm on the group O'N. In a more general approach, [17] discusses the problem of enumerating regular objects with a given automorphism group. The authors introduce, among other things, the Möbius function for a group G as a tool to enumerate regular objects. The knowledge of the Mobius function of G relies on the knowledge of the full subgroup lattice of G. In the spirit of contributing to the study of the finite simple groups, we present here an algorithm that determines the subgroup lattice of a given permutation group. This algorithm, designed by Leemans in 2007, proves itself to be lighter in memory and sometimes even faster than the already implemented function SubgroupLattice in the software Magma [4]. We also present an algorithm to determine structures of groups with a computer. Those two algorithms allow us to produce an atlas of subgroup lattices for a large number of finite almost simple groups. The atlas is made available online at http://homepages.ulb.ac.be/~tconnor/atlaslat Our paper is organised as follows. In section 2, we present the algorithm that computes the subgroup lattice of a permutation group G. It features a systematic reduction of the permutation degree of the subgroups of G and a possibility to start the computation of the subgroup lattice of G with partial information of the lattice already available. This permits to compute the subgroup lattice of very large groups, like the O'Nan sporadic group O'N or even its automorphism group, currently out of reach with the SubgroupLattice function of Magma, but also Co2 and Fi22. In section 3 we discuss the problem of describing the structure of a group in an efficient way. We present an algorithm that provides a structure for a group, based on a choice of suitable normal subgroups. In section 4 we present our atlas of subgroup lattices as an application of the algorithms presented and discussed in this paper. 2 The subgroup lattice of a permutation group We refer to [18] as a reference on subgroup lattices. A lattice is a partially ordered set, or poset, any two of whose elements a, b have a least upper bound a U b and a greatest lower bound a n b. The subgroups of a group G may be taken as the elements of a lattice L(G) under the operations of union and intersection. The poset of conjugacy classes of subgroups forms also a lattice: two conjugacy classes A and B are such that A D B provided that any subgroup of B is contained in some subgroup of A. We call this lattice the subgroup lattice of G, rather than the lattice of conjugacy classes of subgroups of G for the sake of brevity, and we denote it with Л(^). Our terminology is also the one used in Magma. This lattice can be refined with the length of each conjugacy class of subgroups. Moreover, given two conjugacy classes of subgroups A D B, we define nAB to be the number of subgroups of class B contained in any subgroup of class A; alike we define nBA to be the number of subgroups of class A containing a subgroup of class B. Consider the set N of numbers nXY for every couple of classes {X, Y} such that X с Y or X D Y and there does not exist Z such that X с Z с Y or X D Z D Y. The subgroup lattice Л(^) together with the length of each conjugacy class and the set N is called the weighted subgroup lattice of G. We describe in this section a powerful and natural algorithm to compute the weighted subgroup lattice of a given group G. The correctness of this algorithm is obvious. Start with a set classes which is empty and a set sgr containing just one element, namely the group G for which we want to compute the subgroup lattice. While sgr is nonempty, pick one element H out of sgr and put it in classes. Obviously, it is G the first time. Reduce the permutation degree of H and let ф : H ^ H be an isomorphism between H and H where H has a reduced permutation degree. Compute the maximal subgroups of H and for each maximal M, add M := ф-1(М) to sgr provided there is no subgroup in sgr conjugate to M in G. During that process, keep track of inclusions of respective subgroups considered. At the end of this process, in classes there is one representative of each conjugacy class of subgroups of G. Moreover, we also have the maximal inclusions between classes. So the subgroup lattice is determined. The weighted subgroup lattice can be determined in the process by computing weighted inclusions at each step. A Magma implementation of the algorithm described above to compute the subgroup lattice of a given group is available on the webpage of the atlas. Observe that we use the DegreeReduction function in Magma to get ф and H for every subgroup H above. This improvement can save a lot of time and memory. For instance, consider L3(7) : 2, one of the maximal subgroups of the O'Nan sporadic group O'N, acting on 122760 points (the smallest permutation representation of O'N). Then Magma v.2.19 needs 13 seconds and more than 200 Mb of memory to compute its maximal subgroups on a computer running at 2.9 GHz. If we reduce the degree of L3( 7) : 2 on 5586 points by using the DegreeReduction function, then Magma computes them in less than half a second and takes about 20 Mb of memory. Our implementation has three main advantages. 1. For permutation groups of large degree, say at least 1000, our algorithm will perform faster; 2. Our algorithm will also need less memory for these groups; 3. This algorithm permits to compute the subgroup lattices of a group G unreachable for the SubgroupLattice function of Magma when Magma does not know the maximal subgroups of G, as for instance the O'Nan group, Fi22, Co2, etc. Indeed, feeding the function with the maximal subgroups, or even part of the subgroup lattice for groups like Aut(ON), of the group permits to proceed further. On the other end, of course, if the permutation degree of the group is small, Magma will tend to work faster as it is based on our algorithm without the degree reduction and the degree reduction step will slow down the process instead of speeding it up. 3 The structures of a group 3.1 Preliminary remarks Given any finite group G, it is always desirable to identify G in some sense. This identification can be done for instance in a geometrical way by determining the action of G on some set or by algebraic means. In particular, most finite simple groups can be named after their action on some structured set or after the mathematician that discovered them (like the Suzuki groups or most of the sporadic groups). However some groups carry very different names, depending on the incarnation of the group that the context requires to emphasise. This is the case for instance of U4(2). Indeed, S4(3) = U4 (2) = 05(3) = O-( 3) = W (E6). Each of the names of this group emphasizes one of its actions on a structured set of particular interest. Therefore, when speaking about this group, one has to choose carefully the name that should be used depending on the context. This observation means that one has to be aware of possible isomorphisms between different incarnations of a group. In Magma, there exists a database of finite simple groups. Given a simple group G, one can thus ask Magma to name G by using the function NameSimple. This function returns a triple of integers that permits to identify G as a group of one of the infinite families of finite simple groups, or as one of the sporadic groups. Many non simple groups can also be identified in a canonical way. This is the case of most of the almost simple groups for instance, but also the case of the dihedral groups, or the groups AGL(n, q). Abelian groups are also identified easily by a name thanks to the classification theorem of abelian groups. However, most of the finite groups are not almost simple, and identifying them in an efficient way by a name can be tricky. For instance, Leemans exhibited two non iso-morphic primitive groups in [19] that satisfy the following property: they have isomorphic posets of conjugacy classes of subgroups and for each normal subgroup N of the first, there is a normal subgroup isomorphic to N in the second group such that the quotients by N are isomorphic. In other words, it is not possible to make a difference between those two groups by giving them names based on any quotient by a normal subgroup. This shows that the taxonomy of groups is a difficult and possibly not solvable problem. Hence we should not look for a deterministic algorithm that gives names to groups since it is readily impossible. The case of p-groups is also particularly difficult to handle. For instance, there are roughly 50 billions pairwise non-isomorphic groups of order 1024 and hence, finding a way to give distinct names to each of them is hopeless, unless we decide to assign a number to each of them, as is done for instance in the SmallGroups database provided by [3]. 3.2 Algorithmic approach Let G be a group and let N be a normal subgroup of G. Denote by Q the quotient group G/N. Then G can be written as N.Q where the dot " . " denotes an extension that can be split (that is, a direct or a semi direct product) or non split. We denote a direct product by " X ", a semi direct product by " : " and a non split extension by " • ". We recall that a composition series for G is a sequence of subgroups Hi, i G {0,..., n+ 1} such that 1 = Ho < Hi < H2 < ... < Hn < Hn+i = G where all inclusions are strict, i.e. Hi is a maximal normal subgroup of Hi+i. This is equivalent to require that the composition factors Qi = Hi+i/Hi are simple groups, i = 0,..., n. Clearly the group G can be written Hn.Qn. Alike, Hn can be written Hn_i.Qn_i and thus G can be written (Hn_i.Qn_i).Qn. Proceeding inductively, we can finally write G = (... (Q0.QO.Q2) ...).Qn. However in order to reduce the notations, we always suppose that the products are left associate and we can thus avoid to write parentheses whenever there is no possible confusion. Therefore by G = A.B.C we mean G = (A.B).C. The Jordan-Holder theorem states that every finite group has a unique composition series up to the order of the terms [26]. Obviously, two non isomorphic groups can have the same composition series. This is the case for instance of S5 and A5 x 2. In this particular example, it is not enough to use the composition series of those two groups to distinguish them. However S5 = A5 : 2 but S5 £ A5 x 2. On basis of the previous observations, we detail an algorithm that produces a name for a group G in terms of a product of its composition factors. We detail afterwards an improved algorithm that we actually used in order to produce the lattices of our atlas. First of all, given N 0. These were found by Youngs [6] and index 1 current graphs corresponding to these solutions are readily accessible. They can be found in [3]. The biembeddings of the TTS(n)s which we obtain from these biembeddings of Youngs however are not cyclic; they have an automorphism group Zi2m+7 x K4 where K4 is the Klein 4-group. Further we extend our method to find new biembeddings of TTS(n)s for n = 16 (mod 48); these have an automorphism group Zi2m+4 x K4. Finally iterating this latter process we obtain biembeddings of twofold triple systems of order 4s (12m + 4) with an automorphism group Zi2m+4 x (K4)s, s > 1, m > 0. We will also construct our biembeddings from index 1 current graphs lifted under the appropriate current group G of order g. These will satisfy the following properties, which are sufficient to construct a biembedding of a pair of TTS(n)s in an orientable surface, [5], [4], [3]. (i) Each vertex has degree 3. (ii) Each edge is assigned a current from the set G \ {0} so that each current appears exactly once. Note that a current of i in one direction is equivalent to a current of —i in the opposite direction. (iii) At each vertex, the sum of the directed currents is the identity (Kirchoff's current law, KCL). (iv) The directions (clockwise or anticlockwise) assigned to each vertex are such that a complete circuit is formed, that is, one in which every edge of the graph is traversed in each direction exactly once. (v) The graph is bipartite. Hence, such a current graph has 2(g — 1)/3 vertices and g — 1 edges. We use a Möbius ladder graph with (g — 1)/3 "rungs". Set u := (g — 1)/3. The formal definition of such a graph is as follows. The vertex set is {xH,yi | 0 < i < u — 1} and the edge set is {{x^yj, {xi,xi+i}, {yi,yi+i} | 0 < i < u — 2} U {{x„_b y„_i}, {xo,y„_i}, A B Xo X1 x u—2 x u—1 • • • • • • B A У o y 1 Уи—2 Уи—1 Figure 1: A Möbius ladder graph {xu-1, y0}}. In order to obtain a complete circuit, vertices xit 0 < i < u - 1, and yu-1 are assigned a clockwise direction and the vertices yž, 0 < i < u - 2, an anticlockwise direction- In this paper we represent these graphs as shown in Figure 1, where the directions of rotation are not indicated but implicit as defined above. We build the Mobius ladder graphs with currents assigned to the edges, so that Conditions (ii) and (iii) are met, from gadgets, i.e. edge labelled subgraphs which we link together by concatenation; we will define the linking of two gadgets D1 followed by D2 by D1 : D2 and the sequential linking of k gadgets by [Dj :]1 0. From [6] there exists a Mobius ladder graph that lifts under Zv to yield a biembedding of a pair of STS(v)s. Let L be such a graph. We begin by labelling Figure 1 as follows, xž = vj, yž = vi+2m+1, 0 < i < 2m. Thus the bipartition of L consists of the sets of even-indexed and odd-indexed vertices respectively. в V2m + 1 V2m+2 V2m+3 V2i+1 V2i+2 V2i+2m + 1 C2i+2 V2i+2m+2 V2i+2m+3 A V2m-2 V2m-1 V2m v4m-1 V4m v4m+1 в A Figure 2: Vertex and edge labels of L. Noting that by replacing a directed edge label by its inverse in the opposite direction, we can arrange the directed edge labels so that they point in to even-indexed vertices and out of odd-indexed vertices. Thus the directed edge labels will be taken to be as shown in Figure 2. As L satisfies Kirchoff's current law we have the following equations. ai + bi + ci = 0 (mod 12m + 7), (2.1) ai+2 + bi + Ci+2m+2 = 0 (mod 12m + 7). (2.2) We will require the following gadgets where indices are taken modulo 4m + 2: X (ao, 0) (bo, 0) (a0,x) (a0,z) (co,0) (bo,0) (0,x) (co,z) (bo,x) (a.2,y) (co, 0) Д A(c2m+2,z) • > < * < * > < * < * > < • ,y) (ao,z) (ao, x) (co,y) (bo,y) Yi (0, z) (co,x) (bo,z) (bi,y) • > (ai,z) (bi, x) (ai,y) (ci, о) у • > < f > < * > < • < * > bi, y) > (ai,z) (bi, x) (ai+2,y) < f > < * > < • Y^ ><(Ci+2m+2,z) ci (< * (ai, 0) (bi, 0) (ai, x) (bi,z) (ai, 0) (bi, 0) (ai,x) (bi,z) Equations (1.1), (2.1) and (2.2) verify that the gadgets X and, for 1 < j < 2m, Y2j satisfy KCL. Now consider the Möbius ladder graph X : [Y2j :]i 0, with Z12m+7 x K4 as an automorphism group. We conclude this section by giving two examples. Example 2.1. Let m = 0, v = 7 and n = 28. A Mobius ladder graph L, which yields the well known toroidal biembedding of a pair of STS(7)s, has a0 = 1, b0 = 2 and c0 = -3, i.e. A 1 > vo B ■ < * > 2 < B Л-3 A 2 vi 1 In this case our construction gives just the gadget X, labelled as follows. (1,0) > (2, 0) < (1.x) > (1 ,z) (-3,0) > » < (2,0) > (0, x) (-3,z) > » < (2,x) > (i,y) < (-3,0) A —< 1 > < * < * > (1,y) (2, y) (1,z) (1,x) (—3, y) (2, y) (0, z) (—3, x) (2,z) (1,0) 2 < è< Y1? A(—3,z) < 1 > A B B A Example 2.2. Let m =1, v = 19 and n = 76. A Möbius ladder graph, L, yielding a biembedding of a pair of STS(19), is as follows. A ■ B 1 > 4 V2 —9 > f < B A -9 V3 Thus (a0,b0,c0) = (1,7, -8), (a2,b2,c2) = (-4, -9, -6) and (a4,b4,c4) (-2,5, -3). Our construction gives the following Möbius ladder graph. (1,0) (7,0) (1,x) (1,z) (-8,0) (7,0) (0, x) (-8,z) (7,x) (-4,y) B - (-8, 0) Д < 1 > < * < * > 7( < è< T'5, AHM (< 1 > (1,y) (7, y) (1,z) (1,x) (-8, y) (7, y) (0, z) (-8,x) (7,z) (-4,0) (-4,y) (-9, y) (-4, z) (-9, x) (-4, y) (-9,y) (-4,z) (-9,x) (-2,y) —< (-6,0) > > f > < - \ > - > < ' - > > - > < ' - > > - > < k(-8,z) (-4, 0) (-2, y) (-9,0) (5,y) (-4, x) (-2, z) ■ ^ (-9, z) (5,x) (-4, 0) (-2,y) ■ ^ (-9, 0) (5,y) (-4, x) (-2, z) ■ (-9, z) (5,x) (-2,0) (i,y) (-3,0) > > < У < < - СО У - i ^ N - СО У - i - ^o-y ) - < a-6-z) -- (-2,0) (5,0) (-2,x) (5, z) (-2,0) (5,0) (-2,x) (5,z) (1,0) 3 The case n = 16 (mod 48) Let v = n/4 = 12m + 4, m > 0. In [2] a Mobius ladder graph L based on the Colbourn and Colbourn difference triples, [1], on the set Z12m+4 \ {0} was constructed. In that paper the ladders were lifted under the cyclic group Z12m+4 to yield a biembedding of a pair of TTS(v)s. Similarly to Section 2 label the vertices and edges of L as follows (taking indices modulo 8m + 2): B ao b4m bo Aco a4ro+2 a2 C4m+2 b4m+2 V2i -Щ- V2i+1 AC2 b2i AC2i a2i+4m+2 V2i+2 a2i+2 C2i+4m+2 b2i+4m+2 >^c2i+2 V4m + 1 V4m+2 V4m+3 v4m-2 V2i+4m + 1 V2i+4m+2 V2i+4m+3 v4m-1 b4m-2 c4m-2 ®8m V4m c8m a4m г b8m < B b4m У \ c4m. «о v8m-1 V8m v8m+1 A A where v0 corresponds to the difference triple 3m +1, 3m +1, 6m + 2. Note that this means that the vertices with even indices correspond to the Colbourn and Colbourn difference triples. Without loss of generality, in L, either a0 = c0 = 3m + 1 and b0 = 6m + 2 or a0 = b0 = 3m + 1 and c0 = 6m + 2. Both of these cases occur in the Möbius ladder graphs constructed in [2], depending on the residue class of v modulo 72 and we consider them separately in Subsections 3.1 and 3.2, respectively. We will make use of the following gadget where 2 < i < 8m: Vi У (Ci+4m+2 Note that in this case, because the initial Mobius ladder L yields a biembedding of a pair of twofold triple systems Vi, is simpler than the gadget used in Section 2. In fact it is "half" that gadget. As L satisfies Kirchoff's current law we have the following two equations. ai + bi + ci = 0 (mod 12m + 4), (3.1) ai+2 + bi + Ci+4m+2 = 0 (mod 12m + 4). (3.2) These two equations together with Equation (1.1) verify that Vi also satisfies KCL. 3.1 ao = co = 3m + 1 and bo = 6m + 2 In this case it follows, from [2], that L yields the following two equations (6m + 2) + c4m+2 + a2 = 0 (mod 12m + 4), bsm + c4m + (3m + 1) = 0 (mod 12m + 4). (3.3) (3.4) In this case we will require the following gadget: (0,y) V V0 (ao,y) < (0,z) л (ao,x) < (bo, 0) > V(ao,x) A(0,x) (bo, x) -< V (c4m (ao, y) (ao,0) (ao,z) (bo, y) (bo, z) (a2, 0) Equations (1.1) and (3.3) verify that V0 satisfies KCL. As a0 = 3m + 1, Equations (1.1), (3.1), (3.2), (3.3) and (3.4) verify that the Mobius ladder graph V0 : [Vi :]i=2j, i->-« »-<-« »->-< »->-« ^(co,x) у »-<—< »-<-о ^(ao,z) V »->-» 4m+2 z) (ao, y) (ao, 0) (0, x) (co, 0) (ao,z) (a2,0) Equations (1.1) and (3.5) verify that V0' satisfies KCL. As ao = 3m + 1, Equations (1.1), (3.1), (3.2), (3.5) and (3.6) verify that the Möbius ladder graph V0' : [V :]i=2j-, 1 0. Finally note that the gadget V0 contains a vertex with currents (3m +1, x), (3m +1, x) and (6m + 2,0) pointing outwards. Similarly the gadget V0' contains a vertex with currents (3m +1, z), (3m + 1, z) and (6m + 2,0) also pointing outwards. We call this vertex a. Our constructed Mobius ladder graphs L' contain just one of these two gadgets. By reversing the directions on all of the edges of L', labelling the vertex a as v0 and the edge with label (6m + 2,0) as b0, and extending in the same manner as above, the construction, using the gadget V0, can be reapplied. Repeated application of this process yields (Z12m+4 x (K4)s)-biembeddings of a pair of TTS(4s (12m + 4))s, s > 1, m > 0. References [1] M. J. Colbourn and C. J. Colbourn, Cyclic block designs with block size 3, European J. Combin. 2 (1981), 21-26. [2] D. M. Donovan, T. S. Griggs, J. G. Lefevre and T. A. McCourt, Cyclic biembeddings of twofold triple systems, Ann. Comb. 18 (2014), 57-74. [3] M. J. Grannell and T. S. Griggs, Designs and Topology, in Surveys in Combinatorics 2007, A. J. W. Hilton and J. Talbot (Editors), London Math. Soc. Lecture Note Series 346, Cambridge Univ. Press, Cambridge (2007), 121-174. [4] J. L. Gross and T. W. Tucker, Topological Graph Theory, John Wiley, New York (1987). [5] G. Ringel, Map Color Theorem, Springer-Verlag, New York (1974). [6] J. W. T. Youngs, The mystery of the Heawood conjecture, in Graph Theory and its Applications, Academic Press, New York (1970), 17-50. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 275-289 Edmonds maps on the Fricke-Macbeath curve Ruben A. Hidalgo * Departamento de Matemàtica y Estadistica, Universidad de La Frontera, Casilla 54-D, 4780000 Temuco, Chile Received 4 June 2013, accepted 27 December 2014, published online 4 February 2015 In 1985, L. D. James and G. A. Jones proved that the complete graph Kn defines a clean dessin d'enfant (the bipartite graph is given by taking as the black vertices the vertices of Kn and the white vertices as middle points of edges) if and only if n = pe, where p is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart computed the minimal field of definition of them. The minimal genus g > 1 of these types of clean dessins d'enfant is g = 7, obtained for p = 2 and e = 3. In that case, there are exactly two such clean dessins d'enfant (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over Q, but both Edmonds maps cannot be defined over Q; in fact they have as minimal field of definition the quadratic field Q(V—7). It seems that no explicit models for the Edmonds maps over Q( V—7) are written in the literature. In this paper we start with an explicit model X for the Fricke-Macbeath curve provided by Macbeath, which is defined over Q(e2ni/7), and we construct an explicit birational isomorphism L : X ^ Z, where Z is defined over Q( V—7), so that both Edmonds maps are also defined over that field. Keywords: Riemann surface, algebraic curve, dessin d'enfant. Math. Subj. Class.: 30F20, 30F10, 14Q05, 14H45, 14E05 1 Introduction A dessin d'enfant D on a closed orientable surface is given by a bipartite map on it (vertices will be colored black and white). The dessin d'enfant is called clean if the white vertices have all valence 2. A Belyi curve is an irreducible non-singular complex projective algebraic curve (i.e. a closed Riemann surface) S admitting a non-constant meromorphic map ß : S ^ C with * Partially supported by Project Fondecyt 1150003. E-mail address: ruben.hidalgo@ufrontera.cl (Rubén A. Hidalgo) Abstract at most three branch values; which we assume to be inside the set {то, 0,1}; we say that (S, ß) is a Belyi pair. Two Belyi pairs (Si, ßi) and (S2, ß2) are called equivalent, denoted this by the symbol (S1, ß1) = (S2, ß2), if there is an isomorphism f : S1 ^ S2 so that ß2 ◦ f = ßi. _ A subfield K of Q is called a field of definition of a Belyi pair (S, ß) if there an equivalent Belyi pair (S, ß) where S and ß are both defined over K. As a consequence of Belyi's theorem [1], the field of algebraic numbers Q is a field of definition of every Belyi pair. Each Belyi pair (S, ß) defines a dessin d'enfant on S by taking the edges as the components of ß-1((0,1)), the black vertices as the points in ß-1(0) and the white vertices as the points in ß-1(1). Conversely, as a consequence of the uniformization theorem, every dessin d'enfant on a closed orientable surface induces a (unique up to isomorphisms) Riemann surface structure (being a Belyi curve) and a Belyi map so that the original dessin d'enfant is homotopic to the one associated to the Belyi pair [11, 15]. A field of definition of a dessin d'enfant is a field of definition of the corresponding Belyi pair. As there is a natural (faithful) action of the absolute Galois group Gal(Q/Q) on the collection of Belyi pairs [13], it also provides a (faithful) action on dessins d'enfant. This action may help in the study of the internal structure of the group Gal(Q/Q) in terms of combinatorial data. Let us consider a dessin d'enfant D, which is defined by the Belyi pair (S, ß). By Belyi's theorem, we may assume that both S and ß are defined over Q. The field of moduli of D is then defined as the fixed field of the subgroup {a g Gal(Q/Q) : (S, ß) = (S, ßa )}. The field of moduli of D is always contained in any field of definition of it, but it may be that the field of moduli is not a field of definition of it. Both, the computation of the field of moduli of a dessin d'enfant and to decide if the dessin d'enfant can be defined over it, are in general difficult problems. If the dessin d'enfant is regular, that is, the Belyi map ß is a Galois branched cover, then J. Wolfart [19] proved that D can be defined over its field of moduli. Also, if the dessin d'enfant has no non-trivial automorphisms, then it is definable over its field of moduli as a consequence of Weil's descent theorem [16]. So, the problem to decide if the field of moduli is a field of definition appears when it has non-trivial automorphisms but it is non-regular. In 1985, L. D. James and G. A. Jones [10] proved that the complete graph Kn defines a clean dessin d'enfant (the bipartite graph is given by taking as the black vertices the vertices of Kn and the white vertices as middle points of edges) if and only if n = pe, where p is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart [12] computed the minimal field of definition of such clean dessins d'enfant. The minimal genus g > 1 of these types of clean dessins d'enfants is g = 7, obtained for p = 2 and e = 3. In that case, there are exactly two (non-equivalent) such dessins (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over Q, but each of the two Edmonds maps cannot be defined over Q; they have as minimal field of definition the quadratic field Q( V—7) [12]. No explicit models for the Edmonds maps over Q( V—7) seems to be written in the literature. In Section 2 we recall an explicit model X for the Fricke-Macbeath curve provided by Macbeath, which is defined over Q(e2ni/7), and describe both Edmonds maps. We also provide (as matter of interest for specialists) two different equations, over Q, for the Fricke-Macbeath curve which were independently obtained by Bradley Brock (personal communication) and by Maxim Hendriks in his Ph.D. Thesis [7]. In Section 3 we provide an explicit birational isomorphism L : X ^ Z, where Z is defined over Q(V—7). In this model we obtain that the two Belyi maps defining the two Edmonds maps are defined over Q; in particular, this provides explicit models for both Edmonds maps over Q(V-7) as desired. In Section 4 we provide an explicit birational isomorphism L : X ^ W, where W is defined over Q. Unfortunately, the explicit equations over Q are not very simple (they are long ones) and they can be found in [9]. In Section 5 we construct a generalized Fermat curve S of type (2,6) [5] that covers the Fricke-Macbeath curve and we provide a description of the three elliptic curves appearing in the equations of X given by Macbeath. Another model of the Fricke-Macbeath curve is also described. 2 Macbeath's equations of the Fricke-Macbeath curve and the two Edmonds maps In this section we recall equations of the Fricke-Macbeath curve, obtained by Macbeath in [14], and we describe both Edmonds maps discovered in [12]. As a matter of interest to specialists, we also describe two different models over Q, one obtained by Bradley Brock (personal communication) and the other by Maxim Hendriks in his Ph.D. Thesis [7]. 2.1 Hurwitz curves It is well known that | Aut(S) | < 84 (g -1) (Hurwitz upper bound) if S is a closed Riemann surface of genus g > 2. In the case that | Aut(S) | = 84(g -1), one says that S is a Hurwitz curve. In this last situation, the quotient orbifold S/Aut(S) has signature (0; 2,3,7), that is, S = H2/Г, where Г is a torsion free normal subgroup of finite index in the triangular Fuchsian group Д = (x, y : x2 = y3 = (xy)7 = 1) acting as isometries of the hyperbolic plane H2. Wiman [17] noticed that there is no Hurwitz curve in each genera g G {2,4, 5, 6} and there is exactly one Hurwitz curve (up to isomorphisms) of genus three, this being Klein's quartic x3y + y3z + z3x = 0; whose automorphisms group is the simple group PSL(2, 7) (of order 168). 2.2 Macbeath's equations of the Fricke-Macbeath curve In [14] Macbeath observed that in genus seven there is exactly one (up to isomorphisms) Hurwitz curve, called the Fricke-Macbeath curve; its automorphisms group is the simple group PSL(2, 8), consisting of 504 symmetries. In the same paper, Macbeath computed the following explicit equations over Q(p), where p = e2ni/7, for the Fricke-Macbeath curve (involving three particular elliptic curves): X y2 = (x — 1)(x — p3)(x — p5)(x — p6) y2 = (x — p2)(x — p4)(x — p5)(x — p6) U C4. y4 = (x — p)(x — p3)(x — p4)(x — p5) (2.1) In Section 5 we provide a rough explanation about the elliptic curves in the above equations (different from the approach given in [14]) in geometric terms of the highest regular branched Abelian cover of the orbifold X/G of signature (0; 2,2,2,2, 2, 2,2). Another interesting fact on the Fricke-Macbeath curve is that its jacobian variety is isogenous to E7 where E is the elliptic curve with j-invariant j(E) = 1792 (E does not have complex multiplication); see, for instance, [2]. There are not to many explicit examples of Riemann surfaces whose jacobian variety is isogenous to the product of elliptic curves (see [6]). 2.3 Equations over Q of the Fricke-Macbeath curve The uniqueness up to isomorphisms of the Fricke-Macbeath curve asserts that its field of moduli is the field of rational numbers Q. As quasiplatonic curves can be defined over their fields of moduli [19] and Hurwitz curve are quasiplatonic curves, it follows that the Fricke-Macbeath curve can be defined over Q. When the author put a first version of this paper in Arxiv [9] we didn't know of explicit equations of the Fricke-Macbeath curve over Q. Later, Bradley Brock sent me an e-mail in which he told me that, using some suitable change of coordinates on the above equations for X, he was able to compute a plane equation for X over Q, with some simple nodes as singularities, given as 1 + 7 xy + 21x2y2 + 35 x3y3 + 28x4y4 + 2x7 + 2 y7 = 0. An automorphism of order 7 is given by b(x, y) = (px, p-1y) and one of order two is given by ai(x,y) = (y,x). The following model over Q, for the Fricke-Macbeath curve, was recently computed by Maxim Hendriks in his Ph.D. Thesis [7] xix2 + xix0 + Ж2Ж6 + Ж3Ж4 - Ж3Ж5 - Ж3Ж0 - Ж4Ж6 - Ж5Ж6 = 0, 3 + x4x5 - x4x0 - x'5 Ж1Ж3 + Ж1Ж6 - xQ + 2x2x5 + x*2xo - x3 + Ж4Ж5 - Ж4Ж0 - xQ = 0 Ж1Ж4 - 2x1x5 + 2xixo - x^2x6 - x3x4 - x3x5 + x5x6 + x6xo = 0, x2 - 2x1x3 - x2 - x2x4 - x2x5 + 2x2xo + x3 + x3x6 + x4x5 + x5 - x5xo - xQ = 0, xix2 - x1x5 - 2x1x0 + 2x2x3 - x3xo - x5x6 + 2x6xo = 0, -2x1x2 - xix4 - x 1x5 + 2xixo + 2x2x3 - 2x3x0 + 2x5x6 - x6xo = 0, 2x2 + xix3 - xix6 + 3x2x0 + x4x5 - x4xo - x2 + xQ - x0 = 0, 2x2 - xix3 + xix6 + x2 + x2xo + xQ - 2x3x6 + x4x5 - x4xo + x5 - 2x5xo + + x0 = 0, x2 + xix3 - xix6 + 2x2x5 - 3x2x0 + 2x3x6 + xl + x4x5 - x4xo + x6 + 3x0 = 0 с J x - xix3 + x2 - x2x4 - x2x5 - x2xo - x3 + x3x6 + 2x5x0 - xq = 0 In Section 4 we provide an explicit birational isomorphism L : X ^ W, where W is defined over Q. The explicit form of L may be used to compute explicit equation for W; this can be done with MAGMA [3]. 2.4 A description of the two Edmonds maps In the above model X of the Fricke-Macbeath curve it is easy to see a group ZQ = G = (Ai, A2, A3) < Aut(X) where Ai(x,yi,y2,y4) = (x, -yi,y2,y4), A2(x,yi,y2,y4) = (x,yi, -y2,y4), Аэ(ж,У1,У2,У4) = (x,yi,y2, -y4). The quotient Riemann orbifold X/G has signature (0; 2, 2, 2,2,2,2, 2), that is, is the Riemann sphere with exactly 7 cone points of order 2. An automorphism of order 7 of the Fricke-Macbeath curve is given in such a model by Ut \ ( 2 2 2 У1У2 \ в{х,У1,У2,Уа) = px,p У2,р У4,р T-^-^ . V (x — p )(x — p ) J The automorphism B normalizes G and it induces, on the orbifold X/G = C, the rotation T(x) = px. Moreover, X/(G, B) has signature (0; 2, 7,7), that is, the group (G, B) defines aregular dessin d'enfant (X, ß ), where ß(x, y1,y2 ,y4) = x7 (this is one of the two Edmonds maps, but is defined over Q(p)). We may also see that X admits the following anticonformal involution Л \ f 1 y1 P5y2 Р3Ш\ J(x,y1,y2,y4)^x,x2^,^2-J . It can be seen that JB J = B and JAj J = A j, for j = 1,2,3. In this way, one gets another regular dessin d'enfant (X, S), where S(x,y1,y2,y4) = 1/x7 (this is the other Edmonds map, again defined over Q(p)). As S = C о ß o J, where C(x) = x, we have that the two regular dessins d'enfant described above are chirals. 3 An explicit model of the Edmonds maps over Q(V-7) In this section we will construct an explicit biregular isomorphism L : X ^ Z, where Z is defined over Q( V—"7), so that both Edmonds maps are defined over such a field. Note that Q(V—7) = Q(p + p2 + p4 ) since p + p2 + p4 = 2 (V—7 - 1). Most of the computations have been carried out with MAGMA [3] or with MATHEMATICA [20]. 3.1 Let N = Gal(Q(p)/Q(V—7)) = (т) = Z3, where т(p) = p2. If we set x = (x1,x2,x3,x4) = (x, У1,У2,У4), then it is not difficult to check that {fe = I, fT, fT2} is a Weil datum (i.e., they satisfies the Weil co-cycle condition in Weil's descent theorem [16]) with respect to the Galois extension Q(p)/Q(V—7), where I denotes the identity and r f^ ( У2У4 fT(x) = x, y1,y4, ■ (x — p4)(x — p5) f I У2У4 fT2 (x)= (x,y1, (x — p4)(x — p5) ,y2 Ф1 : X -p C12 3.2 Let us consider the rational map 1 (x, У1,У2,Уа) ^ (x,w,v), where W = (W1,W2,W3, W4) = fT (x), V = (V1;V2,V3,V4 ) = fT 2 (x). We may see that Ф1 produces a birational isomorphism between X and Ф1^ ) (its inverse is just the projection on the x-coordinate). Equations defining the algebraic curve Ф1 (X) are the following ones xl = (xi - l)(xi - p3)(x 1 - p5)(xi - p6) x3 = (xi - p2)(xi - p4)(xi - p5)(x 1 - p6) x2 = (xi - p)(xi - p3)(x 1 - p4)(xi - p5) Ф1(Х ) = W1 = xi, W2 = x2, w3 = x4, w4 x^x4 (xi - p4)(xi - p5)' Vi = xi, V2 = x2, V3 = x3x4 (xi - p4)(xi - p5), V4 x3 (3.1) 3.3 We consider the linear permutation action of N on the coordinates of C12 defined by ©i(r )(x,Ww,V) = (W,V,x). Let us notice that the stabilizer of Ф1 (X), with respect to the above permutation action, is trivial since {n G N : в1(п)(Ф1(Х)) = Ф1(Х)} = {n G N : Xn = X} = {e}. 3.4 Each в g Gal(C) induces a natural bijection 0 : C12 ^ C12 : (yi,..., yi2) ^ (e(yi),..., в(Ш)). It is not hard to see that 0(X ) = Xв. 3.5 If в G Gal(C/Q(V-7)), then we denote by eN is projection in N. With this notation, we see that the following diagram commutes (see also [8]) X i-feN Ф1 (X ) 4-Gi(0n ) X °N Л e1(eN )(Ф1(X )) = Ф?N (X °N )=Ф1(X )6N X Ф1 (X) and, for every n, в g Gal(C/Q(V-7))), that (*) ei(nN) ◦ 0 = 0o ei(nN). (3.2) в 3.6 A generating set of invariant polynomials for the linear action 01 (N) can be obtained with MAGMA as t1 = X1 + W1 + V1, t2 = X2 + W2 + V2 t3 = X3 + W3 + V3, t4 = X4 + W4 + V 4 t5 = X2 + w2 + v2, t6 = X2 + + v| tr = Xg + Wg + V3, t8 = x| + w2 + v4 t9 = x3 + w3 + v3, t10 = x3 + w3 + v| t11 = x3 + W3 + V3, t12 = X3 + w| + v4 The map ф1 : C12 ^ C12 (x,w,v) ^ (t1, ..., 112) clearly satisfies the following properties: Щ3 =*1, j =0,1, 2; Ф1 o e1(rj ) = Ф1, j = 0,1, 2. (3.3) Also (as we have chosen a set of generators of the invariant polynomials for the action of 01(N)), it holds that Ф1 is a branched regular cover with Galois group N. It turns out that, if we set Z1 = Ф1(Ф1(Х)) and L1 = Ф1 o Ф1, then L1 : X ^ Z1 is a birational isomorphism (since the stabilizer of Ф1 (X ) is trivial). 3.7 If n € N, then L?(X1) = ФП o фп(Xп) = o 01(п)(Ф1(Х)) = Zi L1(X )п Ф1 o Ф1(X) = L1(X) = Z1, so Z1 can be defined by polynomials with coefficient over Q(V-7). 3.8 Next, we proceed to compute explicit equations for Z1 and the inverse L- Z1 X. The following equalities hold: t1 X1 = "3 , X2 t2 ¥, t4 =t3 (*) X4 = (t3 - X3)(tj - p4)(tj - p5) X3 + (% - P4)(f" - P5) t5 =3", t6 =3", = tr 1 2 2 (**) x2 _(t7 - x3)(t1 - p4)2(t1 - p5)2 4 x2 + ( - p4)2(h. - p5)2 ^ТЦ ri Ц t1 ^2 t9 _-g, ti0 , ti2 _t11 (tli - x3)(% - p4)3(% - p5)3 / \ 3 _ ril x-3)( 3_ (***) X4 _ x3 + (t1 - p4)3(t1 - p5)3 Equality (*) permits to obtain x4 uniquely in terms of t1 and x3 and the equation X2 _ (xi - 1)(xi - p3)(xi - p5)(xi - p6) provides a polynomial equation (relating t1 and t2) given by P1(t1,t2,t3,t7,t11) _ 0, where Pl(tl,t2,t3,t7,tii) II ti -^i + 9^2 Equation -81+27(1 + (p+p2 +p4))ti +9t2-3(p+p2 +p4)t3-ti+9t2 G Q(V-7)[tbt2,t3,t7,tnj. x3 _ (xi - p )(xi - p )(xi - p )(xi - p ) permits to obtain the new equation (1) x2 _ (ti - 3p2)(ti - 3p4)(ti - 3p5)(ti - 3p6)/81, and the equation x2 _ (xi - p)(xi - p3)(xi - p4)(xi - p5) provides the equation (2) x4 _ (ti - 3p)(ti - 3p3)(ti - 3p4)(ti - 3p5)/81. In this way, by replacing the above values for x3 and x2 (obtained in (1) and (2)) in the equality ( ** ), we obtain the polynomial equation P2 (t i, t2, t3, t7, t ii ) _ 0, where P2(ti,t2,t3,t7,tii) II 27 + 27(p + p2 + p4) - 18ti - 3(1 + (p + p2 + p4))t2 - 2t3 - t4 + 27tr G Q(V=7)[ti,t2,t3,t7,tii]. Now, if we replace, inequality (***), x|| by x3(x2 - p2)(x2-p4)(x2 -p5)(x2 -p6)/81 and x4 by x4(ti - 3p)(ti - 3p3)(ti - 3p4)(ti - 3p5)/81, where x4 is given in (*), then we obtain a polynomial which is of degree one in the variable x3. x3 _ (-9p2(-162ti - 18ti + 4t5 - 243(1 + tii) +1?(27 - 54t3) + 6t4t3) + 3(729 + 18t4 -6t5 -27t3(-6+13) -ti(-2+t3) + 243ti(3+13) + 81ti(2+tn +13))+ p3(2187- t\ + 27tl(-6 + tg) + 9t5(-3 + ts) + 486t?ts + 81tf (1+ t3) + 729t1(1 + 2t3))+ p5(2187 + 27tf + 12t1 + t1 - 729ti(-1 + tu - tg) + 729t?tg + 81tf (5 + tg) + 9*5(1 + 2tg)) + p(2916ti + 3tf - t7 - 81tf(—6 + ts) - 2187(-2 + tg) - 27tf(-2 + tg) + 9tf(2 + tg) + 243t?(5 + 2ts)) + p4(2187 +t7 - 729tf(-3 + tn - 2tg) - 81tg(-1 + tg) + 27tf(1+ tg) + 9t5(-1 + 2ts) + 243t?(1 + 3ts)))/(9(tf - 243tu + 27tf(-1 + tg) + 81titg + 9tgtg + 3t4ts + p(3 + ti)(—81 + 18t? - 9tf + 2t4 + 27tftg) + 27p?ti(3 + t? + ti(3 + tg)) + p4ti (243 + 3tg + t4 + 9t i ( — 1 + tg) + 27ti(3 + tg)) + p5(-6ti + t5 + 243(1 + tg ) + 81ti(2 + tg) + 9tg(2 + tg ) + 27t?(3 + tg))+ pgti(162 + 36t? + 6ti + 2ti + 27ti(4 + tg)))) Then, using (*), we obtain x4 = -((3p4 - ti)(3p5 - ti)(-pg(-2187 - 729ti +1} + 243titg(2 + tg)+9ti(3 + tg) + 27t4(6 + tg)+81ti(-1 + 3tg)) + p4(2187 + 27ti + ti + 9ti(-1+ tg) -729ti(-3 + tii + tg) - 243ti(-1+ tg) - 81ti(-1+1?))+ p(4374 + 486tg + 54ti + 3tf - ti - 9t5(-2 + tg) - 243t ?(-5 +1?) - 729ti(-4 - tg + tg)) - 3(ti(-2 + tg) + 3t5(2 + tg ) - 729(1 + tiitg) - 81t?(2 + tii + 2tg - t?) + 9t4(-2 + tg) + 243ti(-3 - tg + t?) + 27tg(-6 + tg + tg)) - 9p?(4t5 - 243(1 + tii) + 81ti(-2 + tg) + 6t4tg + 9tg(-2 + 3tg) + 27ti(1 + tg + t?)) + p5(12tf + t} - 243tit? + 9t5(1 + tg) + 27ti(1 + 2tg) - 81tg(-5 + tg + tg) - 2187(-1 + tg +1?) - 729ti(-1 + tii + tg + t?))))/(9(567ti + 6tf +1} + p(-3 + ti)(-54tg + tf + 9ti(-4 + tg) + 729(-2 + tg) + 243ti(-2 + tg) - 81ti(-2 + tg)) + 27t4(-7 + tg) + 9t5 (—5 + tg) + 2187(2 + tg) + 243ti(-1 + 2tg) + 729ti(1 + 2tg) + p5(2187 + 216t4 + 3tf + 2t i + 729titg + 729ti(1+ tg) + 18ti (2 + tg) + 81ti(16 + tg)) + pgti (9t5 + tf + 27ti(-4 + tg) + 9t4 (3 + tg) + 81t i (5 + tg) + 729(-5 + 2tg) + 243ti(-3 + 2tg)) + p4(2187 + 6ti + 2t7 - 81tg( —14 + tg) + 18t5(-2 + tg) + 1458titg + 27t4(5 + tg) + 243t?(1 + 3tg)) - 9p?(—243 + 243ti - 27ti +15 -54t?(-5 + tg) +14(- 9 + 6tg)))). Now, using such values for xg and x4, and replacing them in (1) and (2) above, we obtain another two polynomials identities Pg (t i, tg, t7, t ii ) = 0 and P4 (t i, tg, t7, t ii ) = 0, where these two new polynomials are defined over Q(p) (see [9] for these long polynomials). In this way, we have obtained the following equations over Q(p) for Zi: t4 = tg, 3t5 = ti, 3t6 = t?, ts = tg 9tg = ti, 9tio = tg, ti? = tii Pi (ti, t?^,^, tii) = 0 P? (ti, t?,tg,t7,tii) = 0 Pg(ti,t?,tg,t7,tii) = 0 P^(ti,t?,tg,t7,tii) = 0 Notice that, by the above computations, we have explicitly the inverse of Li given as L-i : Zi ^ X (ti,..., ti?) ^ (xi,x?,xs,X4), where xi, x?, xg and x4 are in terms of ti, t?, tg, t7 and tii. As the variables ti,..., t^ are uniquely determined only by the variables ti, t?, tg, t7 and tii, if we consider the projection n : с1? ^ C5 (ti,..., ti?) ^ (ti, t?, tg, t7, tii), Zi = с с i? then L = п о L1 : X ^ Z LI (x, yi, У2, У4) \\ ( 2 2 y2y2 3 3 y3y3 \ 3yi, y2 + y4 + (x-p/^X-p5 ), y2 + y4 + (x-p4)2(X-p5)2 , y2 + y4 + (x-p4)3(X-p5)3 J is a birational isomorphism, where ( Pi(ti,t2,t3,tr ,tii)=0 Z = 1 P2(ti,t2,t3,t7 ,tll)=0 I Z Ì P3(tl,t2,t3,t7,tll)=0 >C C [ P4(ti,t2,*3,*7 ,tll) =0 The inverse L-i : Z ^ X is given as L-i(tl,t2,t3,t7,tii) = (X1,X2,X3,X4). We have obtained equations for Z over Q(p). But, as Z^ = Zi, for every n € N, and п is defined over Q, we may see that Zn = Z, for every n € N, that is, Z can be defined by polynomials over Q(V—7). To obtain such equations over Q(V—7), we replace each polynomial Pj (j = 3,4) by the new polynomials (with coefficients in Q( V—7)) Qj,i = Tr(Pj), Qj,2 = Tr(pPj), Qj,3 = Tr(p2Pj) that is Z = Pl(tl,t2,t3,t7,tii) =0 P2(ti,t2,t3,t7,tii) =0 P3(tl,t2,t3,t7, tii) + P3(tl, t2,t3,t7,tii)T + P3(ti, t2, t3,t7,tii)T2 = 0 pP3(tl, t2, t3, t7, tii) + p2P3(tl, t2, t3, t7, tii)T + p4P3(ti, t2, t3, t7, tii)T2 = 0 p2P3(tl, t2, t3, t7, til) + p4P3(tl, t2, t3, t7, tii)T + pP3(ti, t2, t3, t7, tn)T2 = 0 P4(ti, t2, t3, t7, tii) + P4(tl, t2, t3, t7, tn)T + P^ti, t2, t3, t7, tn)T2 = 0 pP4(tl, t2, t3, t7, til) + p2P4(ti, t2, t3, t7, tn)T + /P4 (ti, t2, t3, t7, tn)T 2 = 0 p2P4(tl, t2, t3, t7, til) + p4P4(ti, t2, t3, t7, tn)T + pP4 (ti, t2, t3, t7, tn)T 2 = 0 C C5 We have obtained an explicit model Z for the Fricke-Macbeath curve over Q(V—7) together explicit birational isomorphisms L : X ^ Z and L-i : Z ^ X. 3.9 Finally, notice that the regular dessin d'enfant (X, в), given before, is isomorphic to that provided by the pair (Z, в*), where e*(ti, t2, t3, t7, tii) = в о L i(ti,t2,t3, t7, til) — (ti/3)7; that is, the dessin d'enfant is defined over Q( л/-7). 4 An explicit isomorphism L : X ^ W where W is defined over Q Next we explain how to construct an explicit birational isomorphism L : X ^ W, where W is known to be defined over Q. Let us consider the explicit model Z с C5 over Q(V—7) constructed above. Let M = Gal(Q(V—7)/Q) = {n) = Z2, where n is the complex conjugation. As already noticed, since X admits the anticonformal involution J (defined previously), the curve Z admits the anticonformal involution T = L o J o L-1. It is not difficult to see that by setting ge = I and = S o T, where S(t1, t2, t3, t7, t11) = (t1,t2,t3, t7, t11), we obtain a Weil datum for the Galois extension Q(V—7)/Q. Now, identically as done above, we consider the rational map Ф2 : Z ^ C10 (t1,t2,t3,tr,tn) ^ (t1,t2,t3,tr,tu, S1, S2, S3, S7, S11) where (t1, t2, t3, t7,t11) = (s1, s2, s3, s7, s11). We may see that Ф2 induces abirational isomorphism between Z and Ф2^). In this case, Ф (Z) = J Q1,1(t1,t2,t3,tr,tn) = ••• = Q4,3(t1,t2,t3,t7,t11 ) = 0 1 c10 2( ) \ gn(t1,t2,t3,t7,tn) = (S1, S2, S3, S7, S11) J ' The Galois group M induces the permutation action 02(M) defined as @2 (n)(t1,t2,t3,t7,t11, S1, S2, S3, S7, S11) = (S1, S2, S3, S7, S11, t1, t2, t3, t7, tu) A set of generators for the invariant polynomials (with respect to the previous permutation action) is given by q1 = t1 + S1, q2 = t2 + S2, q3 = t3 + S3, 22 q4 = t7 + S7, q5 = t11 + S11, qe = ti + Sb q7 = t2 + s2, qg = t3 + s3, qg = ^ + S2, 22 q10 = ti1 + S11 Then the rational map Ф2 : C10 C10 (t1, t2, t3, t7, t11, S1, S2, S3, S7, S11) ^ (q1, q2, q3, q4, q5, qe, q7, q8, qg, q10) satisfies the following properties: w2 = ^ (4 1) Ф2 o ©2(n) = ^2' There are two possibilities: 1. Ф2(Z) = 62(n)^2(Z)); in which case Zn = Z and Z will be already defined over Q (which seems not to be the case); and 2. the stabilizer of Ф2 (Z) under 02 (M) is trivial. Under the assumption (2) above, we have that Ф2 : Ф2(^) ^ W = Ф2(Ф2(^)) is a biregular isomorphism and that, as before, W is defined over Q. That is, the map L1 = Ф2 о Ф2 : Z ^ W is an explicit biregular isomorphism and W is defined over Q. In this way, L = Li о L : X ^ W is an explicit birational isomorphism as desired. As R2 and Z are explicitly given, one may compute explicit equations for W over Q(V-7), say by the polynomials A1,..., Am e Q(v/-7)[q1,..., q10] (this may be done with MAGMA [3] or by hands, but computations are heavy and very long). To obtain equations over Q we replace each Aj (which is not already defined over Q) by the traces Aj + An and iAj - iAn. 5 A remark on the elliptic curves in the model X 5.1 A connection to homology covers Let us set A1 = 1, A2 = p, A3 = p2, A4 = p3, A5 = p4, A6 = p5 and A7 = p6, where p = e2ni/7. If S is the Fricke-Macbeath curve, then there is a regular branched cover Q : S ^ C having deck group G = Z' and whose branch locus is the set {A1, A2, A3, A4, A5, A6, A7}. Let us consider a Fuchsian group Г = («1,..., «7 : a' ^7 = «1«2 • • • ®7 = 1} acting on the hyperbolic plane H2 uniformizing the orbifold S/G. If Г' denotes the derived subgroup of Г, then Г' acts freely and S = H2/Г' is a closed Riemann surface. Let H = Г/Г' = Z"; a group of conformal automorphisms of S. Then there exists a set of generators of H, say a1,..., a6, so that the only elements of H acting with fixed points are these and a7 = a1a2a3a4a5a6. In [4, 5] it was noted that S corresponds to the generalized Fermat curve of type (2,6) (also called the homology cover of S/H ) x2 + x2 + X2 = 0 A3 - 1 1 2 3 Г*2 I УУ»2 I yy»2 -'1 + x2 + x4 A4-( A4 - S = A5 -( A5 - A6 -A6 - A7 - 0 Ж2 I /~У* 2 I /-V» 2 1 + X2 + X5 Ж2 I /~У* 2 I ,-y»2 1 + x2 + x6 Ж 2 I yv»2 I /-V»2 1 +г X 2 +г X 7 с PC, that a j is just multiplication by -1 at the coordinate Xj and that the regular branched cover P : S ^ C given by P([x1 : X2 : X3 : X4 : X5 : X6 : X7]) = Ж 2 I /Т»2 2 X1 x2 + a7x1 0 0 0 = z has H has its deck group and branch locus given by the set of the 7th-roots of unity {A1,...,A7}. By classical covering theory, there should be a subgroup K < H, K = Z3, acting freely on S so that there is an isomorphism ф : S ^ S>/K with ^G^-1 = H/K. Let us also observe that the rotation R(z) = pz lifts under P to an automorphism T of S of order 7 of the form T([xi : • • • : x7]) = [cix7 : C2xi : 03x2 : 04x3 : 05x4 : C6x5 : 07x6] for suitable comples numbers cj. One has that Taj T-1 = aj+1, for j = 1,..., 6 and Ta7T-1 = a1. The subgroup K above must satisfy that TKT-1 = K as R also lifts to the Fricke-Macbeath curve (as noticed in the Introduction). 5.2 About the elliptic curves in the Fricke-Macbeath curve The subgroup K * = (a1a3a7, a2a3a5, a1a2a4) = Z| acts freely on S and it is normalized by T. In particular, S * = S/K * is a closed Riemann surface of genus 7 admitting the group L = H/K * = {e, a*,..., a* } = (where a* is the involution induced by a j ) as a group of automorphisms and it also has an automorphism T * of order 7 (induced by T ) permuting cyclically the involutions a*. As S*/(L, T *) = S/(H, T ) has signature (0; 2,7,7), we may see that S = S * and K = K *. We may see that L = (a*,a*,a3) and a4 = a1a2, a5 = a*a3, a* = a*a*a* and a* = a*a*. In this way, we may see that every involution of H/K is induced by one of the involutions (and only one) with fixed points; so every involution in L acts with 4 fixed points on S. Let a*, a* G H/K be any two different involutions, so (a*, a*) = Z2. Then, by the Riemann-Hurwitz formula, the quotient surface S/(a*, a*) is a closed Riemann surface of genus 1 with six cone points of order 2. These six cone points are projected onto three of the cone points of S/H. These points are Aj, Aj and Ar, where a* = a* a*. In this way, the corresponding genus one surface is given by the elliptic curve y2 = П (x — Ak) kQ) will provide a model for the Fricke-Macbeath (affine) curve S. Equations for such an affine model of S are t6tlQ = t5t8 = t4t7tl3 t3t6t7 t3t4t7 t2t5tg = tl + t2 + t3 = 0 A3- 1 A4- l A4- l A5- l A5- l A6- l A6 -l A7-1 tgtl3, t6t7tl2 = t7tl3, t5t6tl2 = tlQtll, t4t6t7 = tgtl3, t3t5tg = tlQtl2, t2tlQ = = t7tll, t2t4tl3 = tl + t2 + 14 = 0 tl + t2 + t5 =0 tl + t2 + t6 = 0 tl + t2 + 1 = 0 t8t11, t5t9t12 tlltl3, t4t8 = = tgtll, t3tll = tlQtl3, t3t5t6 t7tl2, t2tgtl3 tlltl2, t2t4t5t6 2 tlQtll tgtl2 tl2tl3 = t2 = tl3 t8tll t t2t3tg = t8t12, t2t3t4 = t12, t1t12t13 = t8t1Q tltll = t7tg, tlt6tl2 = t8tg, tltstl2 = t7tlQ tlt4tl3 = tgtlQ, tlt4t6 = tg, tlt3t4t5 = t2Q tlt2tl3 = t7t8, tlt2t5 = t7, tlt2t3t6 = t8 C l3 ll Of course, one may see that the variables t2, t3, t4, t5 and t6 are uniquely determined by the variable t1. Other variables can also be determined in order to get a lower dimensional model. 2 Acknowledgments The author is grateful to the referee whose suggestions, comments and corrections done to the preliminary versions helped to improve the presentation of the paper. I also want to thanks J. Wolfart for many early discussions about the results obtained in here. References [1] G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267-276, 479. [2] K. Berry and M. Tretkoff, The period matrix of Macbeath's curve of genus seven, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990), 31-40, Contemp. Math., 136, Amer. Math. Soc., Providence, RI, 1992. [3] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, I. The user language, Computational algebra and number theory (London, 1993), J. Symbolic Comput. 24 (1997), 235-265,http://dx.doi.org/10.1006/jsco.1996.0125 [4] A. Carocca, V. Gonzalez-Aguilera, R. A. Hidalgo and R. E. Rodriguez, Generalized Humbert curves, Israel J. Math. 164 (2008), 165-192. [5] G. Gonzalez-Diez, R.A. Hidalgo and M. Leyton, Generalized Fermat curves, Journal of Algebra 321 (2009), 1643-1660. [6] T. Ekedahl and J.-P. Serre, Exemples de courbes algebriques a jacobienne completement decomposable, C. R. Acad. Sci. Pari Sér. I Math. 317 (1993), 509-513. [7] M. Hendriks, Platonic maps of low genus, Ph. D. Thesis, Technische Universiteit Eindhoven, 2013 [8] R. A. Hidalgo and S. Reyes-Carocca, Towards a constructive proof of Weil's descent theorem, http://arxiv.org/abs/120 3.62 94 [9] R. A. Hidalgo. A computational note about Fricke-Macbeath's curve. http://arxiv.org/ abs/1203.6314 [10] L. D. James and G. A. Jones, Regular orientable imbeddings of complete graphs, J. Combin. Theory Ser. B 39 (1985), 353-367. [11] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (1978), 273-307. [12] G. A. Jones, M. Streit and J. Wolfart, Wilson's map operations on regular dessins and cyclotomic fields of definition, Proc. Lond. Math. Soc. 100 (2010), 510-532. [13] S. K. Lando and A. Zvonkin, Graphs on surfaces and their applications, with an appendix by Don B. Zagier, Encyclopaedia of Mathematical Sciences 141, Low-Dimensional Topology, II. Springer-Verlag, Berlin, 2004. [14] A. Macbeath, On a curve of genus 7, Proc. London Math. Soc. 15 (1965), 527-542. [15] D. Singerman. Automorphisms of maps, permutation groups and Riemann surfaces. Bull. London Math. Soc. 8 (1976), 65-68. [16] A. Weil, The field of definition of a variety, Amer. J. Math. 78 (1956), 509-524. [17] A. Wiman, "Ueber die hyperelliptischen Curven und diejenigen vom Geschlechte p = 3, welche eindeutigen Transformationen in sich zulassen" and "Ueber die algebraischen Curven von den Geschlechtern p = 4, 5 und 6 welche eindeutigen Transformationen in sich besitzen", Bihang Till Kongl. Svenska Vetenskaps-Akademiens Hadlingar, Stockholm, 1895-96. [18] J. Wolfart (joint work with G. Jones and M. Streit), Wilson's operations on regular dessins and cyclotomic fields of definition, Talk in Ankara, March 2011. [19] J. Wolfart, ABC for polynomials, dessins d'enfants and uniformization - a survey. Elementare und analytische Zahlentheorie, 313-345, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006. [20] Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, IL (2010). /^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 8 (2015) 291-295 Counterexamples to a conjecture on injective colorings* Borut Lužar, Riste Skrekovski Faculty of Information Studies, Novo mesto, Slovenia, Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia Received 23 July 2013, accepted 5 May 2014, published online 5 February 2015 Abstract An injective coloring of a graph is a vertex coloring where two vertices receive distinct colors if they have a common neighbor. Chen, Hahn, Raspaud, and Wang [3] conjectured that every planar graph with maximum degree Д > 3 admits an injective coloring with at most \3Д/2] colors. We present an infinite family of planar graphs showing that the conjecture is false for graphs with small or even maximum degree. We conclude this note with an alternative conjecture, which sheds some light on the well-known Wegner's conjecture for the mentioned degrees. Keywords: Injective coloring, planar graph, square graph. Math. Subj. Class.: 05C10, 05C15 1 Introduction An injective k-coloring of a graph G is a mapping c : V (G) ^ {1,2,... ,k} such that c(u) = c(v) for every pair of distinct vertices u,v e V (G) having a common neighbor. The least k such that G is injectively k-colorable is the injective chromatic number of G, denoted by xi(G). Note that this type of coloring is not necessarily proper. The injective coloring of graphs was first introduced by Hahn, Kratochvll, Siran and Sotteau [7] in 2002. The first results on the injective chromatic number of planar graphs were presented by Doyon, Hahn and Raspaud [6] in 2005. As a corollary of the main theorem they obtained that if G is a planar graph of maximum degree Д and girth g(G) > 7, then the injective chromatic number is at most Д + 3. Moreover, if g(G) > 6 then Xi(G) < Д+4, and if g(G) > 5 then Xi(G) < Д+8. Chen, Hahn, Raspaud, and Wang [3] investigated K4 -minor-free graphs and showed that if G is such a graph of maximum degree Д, then Xi (G) < \2 Д]. Moreover, they conjectured that the same bound holds for all planar graphs: * Partially supported by ARRS Program P1-0383 and Creative Core - FISNM - 3330-13-500033. E-mail addresses: borut.luzar@gmail.com (Borut Lužar), skrekovski@gmail.com (Riste Skrekovski) Figure 1: A planar graph with maximum degree Д and girth 4 that needs § Д colors for an injective coloring. Conjecture 1.1 (Chen, Hahn, Raspaud, and Wang). For every planar graph G with maximum degree Д > 1, it holds that Xi(G) < 2Д The conjecture holds for planar graphs with large girth. There is a number of results (see e.g. [1, 2, 4, 5, 8]) showing that if the girth of a planar graph is at least 5, the injective chromatic number of a graph is at most Д + C, where C is some small constant. When considering graphs of girth smaller than 5 the injective chromatic number considerably increases. In Fig. 1 a planar graph of girth 4 and injective chromatic number 2 Д is depicted. In this note we present examples of planar graphs with maximum degree Д > 4 and injective chromatic number Д + 5, for 4 < Д < 7, and |_§ Д + 1, for Д > 8. Thus providing counterexamples to Conjecture 1.1 for planar graphs with maximum degree at most 7 or even. The central problem regarding the chromatic number of squares of planar graphs is the well-known Wegner's conjecture [9] proposed in 1977. Note that for larger Д, the bound of this conjecture is one more than that of Conjecture 1.1. Conjecture 1.2 (Wegner). Let G be a planar graph with maximum degree Д. The chromatic number of square graph G2 is at most 7, if Д = 3, at most Д + 5, if 4 < Д < 7, and at most |_2 Д + 1, otherwise. 2 Planar graphs with largest injective chromatic numbers Using the following result we easily disprove Conjecture 1.1 for every Д g {4, 5,6,7} and for every even Д > 8. We believe that the bound for graphs with maximum degree 3 is correct, however. Theorem 2.1. There exist planar graphs G of maximum degree Д > 3 satisfying the following: (a) Xi (G) = 5, if Д = 3; (b) Xi(G) = A+ 5, if 4 < Д < 7; (c) Xi(G) = L 2 Aj + 1, if A > 8. Proof. For A = 3, i.e., the case (a) of the theorem, a cubic planar graph with the injective chromatic number equal to 5 was presented in [3] (see Fig. 2). For A > 4, the following simple characterization will be used: (*) A graph G has injective chromatic number equal to its order, if and only if 1. G has diameter 2; and 2. every edge of G belongs to a triangle. Figure 3: Planar graphs with diameter 2 and maximum degree A G {4,..., 9}. In Fig. 3 planar graphs with maximum degree A G {4,..., 9} are presented. Note that these graphs are of diameter 2 and of orders 9, 10, 11, 12, 13, and 14, respectively. Figure 4: Constructions of diameter 2 planar graphs with maximum degree Д >8 (even on the left and odd on the right) and the injective chromatic number equal to |_f Д + 1. Moreover, they have the property that each of their edges belongs to a 3-cycle. By (*), it follows that each of them has chromatic index equal to its order. As these graphs have maximum degree 4, 5, 6, 7, 8, and 9, respectively, we conclude that each of them satisfies the identity of the case (b) of the theorem. Finally, we consider the case (c) of the theorem. We give constructions for planar graphs with diameter 2, maximum degree Д > 8, and order [| Д + 1, where each edge belongs to a 3-cycle. Then by (*), the claim (c) of the theorem immediately follows. We distinguish two cases regarding whether Д is even or odd (see Fig. 4 for an illustration). In both cases we start with a path uvw. Then we insert the paths Pa = a1a2 • • • ak, Pb = b1b2 • • • bk, and Pc = cic2 • • • ck+1 (if Д is odd, we introduce also the edge ck+1ck+2). These additional edges are providing paths of length two between the vertices u, v, w and the vertices ai, bi, and ci. The left graph depicted in Fig. 4 has maximum degree Д = 2k + 2 and the right one has Д = 2k + 3, for k > 3. It is easy to see that there is a path of length two between every pair of vertices, thus every vertex in the graph should receive a different color in an injective coloring. □ Let us remark that the presented graphs from the above theorem are not the only ones with such properties. For example in the last construction all the edges of the paths Pa, Pb, and Pc are not really needed to obtain a graph with the desired properties. In fact, it is enough that each vertex of these paths is incident with one edge of the path, so roughly every second is redundant. We conclude the paper with an attempt to correct Conjecture 1.1 by proposing the following Wegner type conjecture for the injective chromatic number of planar graphs: Conjecture 2.2. Let G be a planar graph with maximum degree Д. Then (a) Xi (G) < 5, if Д = 3; (b) Xi (G) < Д + 5, if 4 < Д < 7; (c) Xi (G) < L2 Д] + 1, if Д > 8. Since the injective chromatic number is at most equal to the chromatic number of the square of a graph, proving Wegner's conjecture would imply the truth of Conjecture 2.2. If Wegner's conjecture holds, then the extremal graphs (i.e. the graphs that attain the upper bound) of both conjectures coincide for A's from Theorem 2.2. As the injective coloring is a relaxed version of coloring the square one may reason that colorability of the square is mainly conducted by the injective incidence of the vertices. References [1] O. V. Borodin and A. O. Ivanova, Injective (Д + 1)-coloring of planar graphs with girth 6, Sib. Math. J. 52 (2011), 23-29. [2] Y. Bu, D. Chen, A. Raspaud and W Wang, Injective coloring of planar graphs, Discrete App. Math. 157 (2009), 663-672. [3] M. Chen, G. Hahn, A. Raspaud and W. Wang, Some results on the injective chromatic number of graphs, J. Comb. Optim. 24 (2012), 299-318. [4] D. W. Cranston, S-J. Kim and G. Yu, Injective Colorings of Graphs with Low Average Degree, Algorithmica 60 (2011), 553-568. [5] W. Dong and W. Lin, Injective coloring of planar graphs with girth 6, Discrete Math. 313 (2013), 1302-1311. [6] A. Doyon, G. Hahn and A. Raspaud, Some bounds on the injective chromatic number of graphs, Discrete Math 310 (2010), 585-590. [7] G. Hahn, J. Kratochvll, J. Siran and D. Sotteau, On the injective chromatic number of graphs, Discrete Math. 256 (2002), 179-192. [8] B. Luzar, R. Skrekovski and M. Tancer, Injective coloring of planar graphs with few colors, Discrete Math 309 (2009), 5636-5649. [9] G. Wegner, Graphs with given diameter and a coloring problem, Technical report, University of Dortmund, Germany (1977). ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 297-318 Spherical tilings by congruent quadrangles: Forbidden cases and substructures Yohji Akama * Mathematical Institue, Graduate School of Science, Tohoku University, Sendai, Miyagi 980-0845, Japan Nico Van Cleemputf NTIS - New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia, Univerzitni 22, 30614 Plzen Received 10 December 2013, accepted 18 April 2014, published online 6 February 2015 In this article we show the non-existence of a class of spherical tilings by congruent quadrangles. We also prove several forbidden substructures for spherical tilings by congruent quadrangles. These are results that will help to complete of the classification of spherical tilings by congruent quadrangles. Keywords: Spherical tiling by congruent quadrangles, monohedral tiling, quadrangulation. Math. Subj. Class.: 05B45, 05C10, 51M20, 52C20 1 Introduction In this paper we prove the non-existence of a subclass of spherical tilings by congruent quadrangles which have three equal sides and one side different. We also list several forbidden substructures for this type of spherical tilings. It follows from Euler's formula that spherical tilings by congruent polygons can only exist for triangles, quadrangles and pentagons. *http://www.math.tohoku.ac.jp/akama/stcq/ t Corresponding author, This action is realized by the project NEXLIZ - CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech Republic. E-mail addresses: akama@m.tohoku.ac.jp (Yohji Akama), nico.vancleemput@gmail.com (Nico Van Abstract Cleemput) In [5], Davies completed the classification of spherical tilings by congruent triangles. He however only gave an outline of the proof and his classification contained several duplicates. Ueno and Agaoka [9] worked out the details of the proof, thus completely solving the classification of spherical tilings by congruent triangles. Ueno and Agaoka [8] gave several examples of spherical tilings by congruent quadrangles and showed that the classification of these would be considerably harder than the classification of spherical tilings by congruent triangles. Akama and Sakano [7] completed the classification of spherical tilings by congruent kites, darts and rhombi. Since these quadrangles can be subdivided into congruent triangles, they could rely on the classification by Ueno and Agaoka to solve this classification. The spherical quadrangles can be subdivided into classes based on the cyclic list of edge lengths. Only four of these classes admit a spherical tiling by congruent quadrangles[8]: 1. aaaa 3. aabb 2. aaab 4. aabc The cases handled by Akama and Sakano cover type 1 and type 3. The remaining two cases are spherical tilings by congruent quadrangles which have three equal sides and one side different (type 2), and spherical tilings by congruent quadrangles which have three different sides and an adjacent pair of sides of the same length (type 4). Akama, Nakumara and Sakano [1, 2, 7] showed that if concave quadrangles are allowed, there exist several tilings which have non-congruent tiles but for which the inner angles and the underlying graph are the same. In this paper we focus on convex quadrangles of type 2. We show that there exists no spherical tiling by congruent quadrangles of type 2 if the quadrangles are isosceles. Furthermore we show several forbidden substructures for the underlying graph of spherical tilings by congruent quadrangles of type 2. This paper is organised as follows. We start by giving some general definitions and notations. Then we show the non-existence of spherical tilings by congruent quadrangles of type 2 with isosceles quadrangles. Next we look at the different possible configurations of angles around each vertex and finally we use this to show some forbidden substructures for the underlying graph. 2 Definitions To simplify the notation we will always express angles in n radians. A spherical tiling is a subdivision of the unit sphere into spherical polygons. Edges are always assumed to be parts of great circles. All tilings are edge-to-edge tilings. A spherical quadrangle is of type 2 if the cyclic list of edge-lengths is aaab (with a = b). We use the naming convention shown in Figure 1. We only consider convex spherical quadrangles. This means that we always assume 0 < a, ß, y, S < 1. Throughout this paper G will always refer to a 2-connected, simple graph on the 2-sphere in which all faces are quadrangles. Let A(G) be the set of ordered pairs (f, v) such that f is a face of G, v is a vertex of G and v e f .A chart (G, ф), is an ordered pair consisting of a graph G and a function ф : A(G) ^ {a, ß, y, S} such that for each face A b D Figure 1: Naming conventions in a spherical quadrangle of type 2. of the graph, the cyclic list of the inner angles is (a, ß, y, S) or the reverse. These four parameters, a, ß, 7, S, will take on the role of angles of tiles, so a chart can be seen as a combinatorial spherical tiling by congruent quadrangles. We say a vertex of the tiling has vertex type n1a + n2ß + n3j + n4S, if there are n1 pairs containing v that are mapped to a, n2 pairs containing v that are mapped to ß, n3 pairs containing v that are mapped to 7, and n4 pairs containing v that are mapped to S. It is clear how a chart can be obtained from a spherical tiling by congruent quadrangles. Vice versa, a chart (G, f ) is solvable, if there exist values for the four angles such that there is a spherical tiling realising that graph and those values. There are several conditions that need to be satisfied in order for a spherical tiling by congruent quadrangles to exist. If F is the number of tiles, then the following condition follows from the fact that the area of the tiles need to sum up to the area of the sphere. 4 a + ß + y + S - 2 = — (2.1) F Lemma 2.1. In a convex spherical quadrangle of type 2, we have that a + S < 1 + ß, (2.2) a + ß < 1 + S, (2.3) a + S < 1 + Y, (2.4) Y + S < 1 + a. (2.5) Proof. Draw the diagonal as is shown in Figure 2. The area of the triangle ABD is given by a + ßi + Si - 1. The area of the spherical lune that is formed by the great circles AB and BD is given by 2ß1. Since the area of the triangle is smaller than that of the lune, we have that a + ßi + Si - 1 < 2ßi which can be rewritten as a + Si < 1 + ßi (2.6) A b D Figure 2: A diagonal in a spherical quadrangle of type 2. The triangle BDC is an isosceles triangle. This implies that S2 = ß2 (2.7) If we combine inequality 2.6 and equation 2.7, we find equation 2.2. Equation 2.3 can be proven using the spherical lune formed by AD and BD. Equation 2.4 and equation 2.5 can be proven by using the other diagonal. □ Lemma 2.2. In a convex spherical quadrangle of type 2, we have that a = y (2.8) and S = ß. (2.9) Proof. Assume that S = ß. Draw the diagonal as is shown in Figure 2. The triangle BDC is an isosceles triangle, so we have that S2 = ß2. This implies that Si = ßi, so AB D is an isosceles triangle and a = b. This is however a contradiction, so we find that S = ß .By using the other diagonal, we can prove that a = y . П Lemma 2.3. In a convex spherical quadrangle of type 2, we have that a = S & ß = y. (2.10) Proof. Assume that a = S. The great circles AB and DC in Figure 1 intersect in two points, N and S. Since a = S, the triangle ADN is an isosceles triangle, but since the distance from A to B and from D to C is a, then also the triangle BCN is an isosceles triangle and ß = y. The other direction is completely analogous. □ Lemma 2.4. Let (G, f ) be a solvable chart. Let v be a vertex of G with vertex type n1 a + n2ß + n3Y + n4S in (G, f ), then n1 + n4 is even. Proof. This follows immediately from the fact that each edge of length b that is incident to v contributes exactly two to n1 + n4 and each angle a and S at the vertex v corresponds to exactly one edge of length b incident to v. □ The following lemma can easily be proved using Euler's formula. b Figure 3: An isosceles quadrangle of type 2 Lemma 2.5. Let G be a quadrangulation of the sphere. Let Vi (with 3 < i < Д, where Д is the largest degree of the G) be the number of vertices in G with degree i, then we have the following equality: д V3 = 8 + ^ (i - 4)Vi. i=5 3 Spherical tilings by congruent isosceles quadrangles of type 2 An isosceles spherical quadrangle of type 2 is a convex spherical quadrangle having the cyclic list of edge-lengths aaab (with a = b) and in which a = S and ß = 7. Therefore the cyclic list of the inner angles in a isosceles quadrangle is (a, ß, ß, a). An example of such a quadrangle is given in Figure 3. We can rewrite several of the conditions for general spherical quadrangles of type 2. Equation 2.1 can be rewritten as 4 2a + 2ß - 2 = —. (3.1) F The corresponding lemma for Lemma 2.1 is Lemma 3.1. In an isosceles spherical quadrangle of type 2, we have that 2a < 1 + ß. (3.2) The corresponding lemma for Lemma 2.2 is Lemma 3.2. In an isosceles spherical quadrangle of type 2, we have that a = ß (3.3) We now have the tools to prove the main theorem of this section. Theorem 3.3. There is no isosceles spherical tiling by congruent quadrangles of type 2. Proof. From Lemma 2.5, we know that each quadrangulation contains at least 8 vertices of degree 3. The possible vertex types for a vertex of degree 3 in a spherical tiling by congruent isosceles spherical quadrangles of type 2 are 2a+ß and 3,0. There is no isosceles spherical tiling by congruent quadrangles of type 2 with two vertices of degree 3 with a different vertex type, because in that case we would have a = ß, which does not correspond to a quadrangle of type 2 (cf. Lemma 3.2). So all vertices of degree 3 have the same type. We will examine both possible vertex types. vertex type 2a + ß We first assume that all vertices of degree 3 have vertex type 2a + ß. As a consequence all vertices of degree d > 3 have vertex type dß or da. Otherwise there would be a vertex of degree d > 3 with vertex type with 0 < i < . If we combine this with the vertex type for the vertices of degree 3, then we find that But since d > 3, this has no solution. It is also not the case that all vertices are of degree 3, since that would mean that there are more a's than ß's. This means that there are only a limited number of possibilities for different degrees in this situation: • the quadrangulation has two types of vertices: vertices of degree 3 with vertex type 2a + ß and vertices of degree d > 3 with vertex type dß, or • the quadrangulation has three types of vertices: vertices of degree 3 with vertex type 2a + ß, vertices of degree d > 3 with vertex type dß, and vertices of even degree de > 3 with vertex type dea (de is even due to Lemma 2.4). Assume first that there are only vertices of degree 3 with vertex type 2a + ß and vertices of degree d > 3 with vertex type dß. In this case we get two equations: 2ia + (d - 2i)ß (2i - 2)a +(d - 2i - 1)ß = 0. Since a > 0 and ß > 0, this is equivalent with 2i - 2 = 0 d 2i 1 = 0 2a + ß = 2 dß = 2 This is equivalent to If we substitute these values for a and ß in inequality 3.2, we find This is equivalent to d < 4, which contradicts d > 3. Next we assume that there are only vertices of degree 3 with vertex type 2a + ß, vertices of degree d > 3 with vertex type dß, and vertices of even degree de > 3 with vertex type dea. In this case we get three equations: 2a + ß = 2 dß = 2 dea = 2 This is equivalent to d + d = 2 de d ß= 2 a d a = ~г~ de The first equation has no solution, since de > 4 and d > 3. vertex type 3ß Next we assume that all vertices of degree 3 have vertex type 3ß. This means that 2 3 ß = 2 and from equation 3.1, we then find that 1 2 F + 6 _ „ч a = з + p = (3.4) As 3ß is equal to 2, any vertex type that contains a ß, has at most 2ß. Since there has to be at least one vertex for which the vertex type contains an a, there is a vertex of degree d with one of the following three types: • da, • (d - 1)a + ß, • (d - 2)a + 2ß. We examine the three possibilities: da Combined with equation 3.4, this gives us dF^=2 3F which can be rewritten as 6d = (6 - d)F. Since d and F are both positive integers, and d is even and larger than 3, we find that this only holds if d = 4 and F = 12. (d - 1)a + ß Combined with equation 3.4, this gives us (d - 1) F3+6 =2 - И Figure 4: The quadrangles around a vertex t of degree 3. which can be rewritten as 6(d - 1) = (5 - d)F. Since d and F are both positive integers, and d is odd and larger than 3, we find that this never holds. (d - 2)a + 2ß Combined with equation 3.4, this gives us (d - 2) F+6 = 2 - И which can be rewritten as 6(d - 2) = (4 - d)F. Since d and F are both positive integers, and d is even and larger than 3, we find that this never holds. So the only possibility is a quadrangulation which 12 faces. Such a quadrangulation has 14 vertices, of which at least 8 have degree 3 and vertex type 3ß. This already accounts for all of the 24ß's, so all remaining 6 vertices have degree 4 and vertex type 4a. Assume we have a vertex t of degree 3 as is shown in Figure 4. This vertex has vertex type 3ß. This means that, in the quadrangle tuzw, the angle at vertex t is ß and either the angle at the vertex w or the angle at the vertex u is a. Without loss of generality, we can assume that the angle at the vertex u is a. This implies that the vertex type of u is 4a, and we find that this means that the vertex type of both the vertices w and v is 3ß. But then the quadrangle twyv has three consecutive angles ß. This is a contradiction, so there is no spherical tiling by congruent isosceles quadrangles of type 2 with vertex types 3ß and 4a. This proves that there is no spherical tiling by congruent isosceles spherical quadrangles of type 2. □ 1 2 3 4 5 6 7 8 9 10 1 4.1a 4.1a 4.1a 4.1b 2 4.1a 4.1a 4.1a 4.1c 4.1b 3 4.1a 4.1c 4.1a 4.1a 4 4.1a 4.1b 4.1a 4.1a 5 4.1a 4.1a 4.1b 4.1a 6 4.1a 4.1a 4.1b 4.1a 7 4.1a 4.1a 4.1c 4.1b 4.1a 8 4.1c 4.1a 4.1a 4.1a 9 4.1b 4.1a 4.1a 4.1a 10 4.1b 4.1a 4.1a 4.1a Table 1: Overview of the combinations of two vertex types for vertices of degree 3. For each impossible combination of vertex type, the corresponding case is given. 4 Vertex types in spherical tilings by arbitrary congruent quadrangles of type 2 Since there are at least 8 vertices of degree 3 (and in most cases even more), it can be interesting to look at the possible vertex types for these vertices, and examine whether certain combination are not possible. Owing to Lemma 2.4, there are ten possible vertex types for vertices of degree 3 in a spherical tiling by congruent quadrangles of type 2: 1) 3ß 6) 3 2) 2ß + Y 7) 2y + ß 3) a + S + ß 8) a + S + y 4) 2a + y 9) 2S + ß 5) 2a + ß 10) 2S + y The last five of these types can be obtained from the first five by interchanging a with S, and ß with y. The following lemma shows that several combinations of vertex types for vertices of degree 3 are not possible in a spherical tiling by congruent quadrangles of type 2. Table 1 gives an overview of all combinations. Lemma 4.1. There is no spherical tiling by congruent quadrangles of type 2 which has any of the following combinations of vertex types: a) 3ß and 2ß + y, 3ß and 3y, 3ß and ß + 2y, 2ß + y and 3y, 2ß + y and ß + 2y, a + S + ß and 2a + ß, a + S + ß and a + S + y, a + S + ß and 2S + ß, 2a + y and 2a + ß, 2a + y and a + S + y, 2a + y and 2S + y, 2a + ß and 2S + ß; b) 3ß and 2S + ß, 2ß + y and 2S + y, 2a + y and 3y, 2a + ß and 2y + ß; c) 2ß + y and a + S + y, 2y + в and a + S + ß. Proof. a) Each of these combinations either implies that a = 0, or that ß = 7. This means that the quadrangle is a isosceles quadrangle of type 2. Due to Theorem 3.3, there are no spherical tilings by congruent quadrangles with such a tile. b) The first two combinations imply that ß = 0, but this contradicts inequality 2.9. The last two combinations imply that a = 7, but this contradicts inequality 2.8. c) We will only give the proof for 27 + ß and a + 0 + ß. The other case can be obtained by interchanging a with 0, and ß with 7. When we combine a+0+ß=2 with equation 2.1, we get When we combine this with we get 4 Y = F 2y + ß = 2, в=2 - F. Since ß < 1, this implies that F < 8. However, if F = 6, we have that ß = 7. This is not possible due to Lemma 2.3 and Theorem 3.3. So we find that this combination is not possible. □ Lemma 4.2. In each spherical tiling by congruent quadrangles of type 2 we have the restrictions on the number of faces that are given in Table 2. Proof. We will examine case by case. First we note that for all quadrangulations, we have that F > 6, so F = 6 is equivalent to F > 6. • Vertex type 1 and vertex type 3 In this case we have the following system of equations: 3ß = 2 a + ß + 0 = 2 a + ß + 7 + 0 = 2+ - The last equation in this system corresponds to equation 2.1. If we subtract the second equation from this last equation, we find that 4 Y = F. Owing to Lemma 2.3 and Theorem 3.3, we have that F = 6, because otherwise ß = y By combining the first two equations in the system, we find that 4 a + 0=3. When we substitute these previous two equalities into inequality 2.4, we find that 44 3 = a + 5< 1 + y = 1 + F, which can be rewritten as F < 12. Vertex type 1 and vertex type 8 In this case we have the following system of equations: 3ß = 2 a + y + 5 = 2 a+ß+y+5=2+F By combining the last two equations, we find that ß = F, but together with the first equation of the system, this implies that F = 6. Vertex type 1 and vertex type 10 In this case we have the following system of equations: 3ß = 2 Y + 25 = 2 a+ß+y+5=2+F By substituting the first two equations in the third, we get: 5 4 2 a — о = —--. F 3 In combination with Lemma 2.3 and Theorem 3.3, this implies that F = 6. Vertex type 2 and vertex type 3 In this case we have the following system of equations: 2ß + y = 2 a + ß + 5 = 2 a+ß+y+5=2+F The last two equations give us that 4 Y = F' and using the first equation from the system, this then implies that ß=1 - F ■ In combination with Lemma 2.3 and Theorem 3.3, these last two equations imply that F = 6. Vertex type 2 and vertex type 8 In this case we have the following system of equations: 2ß + y = 2 a+Y+S=2 a+ß+y+S=2+F The last two equations give us that ß = f and using the first equation from the system, this then implies that Y = 2 - f In combination with Lemma 2.3 and Theorem 3.3, these last two equations imply that F = 6. Vertex type 3 and vertex type 4 In this case we have the following system of equations: 2a + y = 2 a + ß + S = 2 a+ß+Y+S=2+F Once again, the last two equations give us that 4 Y = F' and using the first equation from the system, this then implies that 1 2 a =1 - F. In combination with inequality 2.8, these last two equations imply that F = 6. Vertex type 4 and vertex type 9 In this case we have the following system of equations: 2a + y = 2 2S + ß = 2 a+ß+y+S=2+F This is equivalent to the following system: Y = 2 - 2a ß = 2 - 2S a + S = 2 - F By combining the last equation in this system with inequalities 2.2 and 2.4, we find that 4 ß>1-f , and 4 Y> 1 - F. By combining these inequalities with the first two equations in the system, we find that 12 a < - +--, 2 + F' and 12 S<2 + F. If we add up these two inequalities, we get 4 a + S < 1 + —. F If we then combine this last inequality with the last equation of the system, we find the following inequality: 44 2 - F < 1 + F, which is equivalent to F < 8. • Vertex type 5 and vertex type 10 In this case we have the following system of equations: 2a + ß = 2 2S + Y = 2 a+ß+Y+S=2+I This is equivalent to the following system: ß = 2 - 2a Y = 2 - 2S a + S = 2 - F4 Similar to the previous case, we find that 4 a + S < 1 + —. F Together with the last equation of the system, this implies F < 8. The remaining cases are equivalent to one of these cases by interchanging a with S and ß with Y. □ A question that pops up naturally at this point is which combinations of three vertex types for vertices of degree 3 are possible. There are 12 combinations of three vertex types for vertices of degree 3 which we can not exclude at this point. The remaining combinations can be excluded because they contain one of the combinations of two vertex types for vertices of degree 3 that are not allowed by Table 1. The 12 combinations come in pairs, since interchanging a with S, and ß with 7 gives a different combination with the same properties. From these 12 combinations we can also discard combinations (1,5,10) and (5,6,10), since (1,10), resp. (6,5), implies that 6 < F, and (5,10) implies that 6 = F. The remaining 10 combinations are • (1,3,4) and (6,8,9); • (1,3,10) and (5,6,8); • (2,3,4) and (7,8,9); • (1,5,8) and (3,6,10); • (2,4,9) and (4,7,9). Lemma 4.3. There is no spherical tiling by congruent quadrangles of type 2 on more than 8 vertices that contains 3 vertices of degree 3 with pairwise different vertex types. Proof. We need to examine the remaining 5 cases stated above. • (1,3,4): In this case we have the following system of equations 3ß = 2 a+ß+S=2 2a + y = 2 a+ß+y+S=2+F which is equivalent to a = 1 - Fß = 3 Y = 8 S = F + 4 S 3 + F If we combine this with inequality 2.4, we get 4 F + 8 3 < — which is equivalent to F < 8. This is a contradiction because the combination (1,3) implies that 6 < F. (1,3,10): In this case we have the following system of equations 3ß = 2 a+ß+S=2 2S + y = 2 a+ß+Y+S=2+F which is equivalent to a = 8 + F в Y = FF 4 s = 1- F If we combine this with inequality 2.4, we get 4 F + 8 3 <-+- which is equivalent to F < 8. This is a contradiction because the combination (1,3) implies that 6 < F. (2,3,4): In this case we have the following system of equations 2ß + y = 2 a + ß + S = 2 2a + y = 2 a+ß+y+S=2+F which is equivalent to a = ß= 1 - F y=S=i+F This is a contradiction with inequality 2.5. (1,5,8): In this case we have the following system of equations 3ß = 2 2a + ß = 2 a+Y+S=2 a+ß+y+S=2+F which is equivalent to a = ß = 3 s = 3 - y F=6 So we find that a quadrangulation which has this combination, has 8 vertices. (2,4,9): In this case we have the following system of equations 2ß + y = 2 2a + y = 2 2S + ß = 2 a+ß+y+S=2+F which is equivalent to a = i - f ß=a Y = 2 - 2a S = 2 - 2a If we combine this system with inequality 2.3, we get 44 2 - F < 1 + F, Figure 5: An example of a cubic quadrangle which is equivalent to F < 8. So we find that a quadrangulation which has this combination, has F = 6, which implies that it has 8 vertices. □ Theorem 4.4. In a spherical tiling by congruent quadrangles of type 2 there are at most two different vertex types for cubic vertices. Proof. An enumeration of all possible angle assignments for the cube shows that, up to equivalence, only one angle assignment admits a spherical tiling by congruent quadrangles of type 2, and this angle assignment has two vertex types: 3ß and a + y + S. Together with Lemma 4.3 this proves the theorem. □ 5 Forbidden substructures in spherical tilings by arbitrary congruent quadrangles of type 2 5.1 Cubic quadrangles A cubic quadrangle in a quadrangulation is a quadrangle such that all four vertices have degree 3. Figure 5 shows an example of a cubic quadrangle. In Table 3 an overview of the number of quadrangulations which contain a cubic quadrangle is given. Note that the percentage of quadrangulations which contain a cubic quadrangle increases as the size of the quadrangulations increases. We prove the following theorem. Theorem 5.1. A quadrangulation on more than 8 vertices that contains a cubic quadrangle does not admit a realisation as a spherical tiling by congruent quadrangles of type 2. Proof. There are two ways of assigning the edges of length a and b to the edges of the cubic quadrangle and its neighbouring faces. These two ways are shown in Figure 6. If we take the complete quadrangulation into account, then these two ways will of course be realised in different ways, but this is not important for this proof. In both edge assignments there is at least one cubic vertex that is incident to an edge of length b, and one cubic vertex that is not. Owing to Theorem 4.4, all cubic vertices Figure 6: Possible edge assignments for a cubic quadrangle and its neighbouring faces. The bold edge corresponds to an edge of length b. incident to an edge of length b have the same type, and all cubic vertices not incident to an edge of length b have the same type. This means that we can already fix some angle assignments for both cases in Figure 6. This partial angle assignment is shown in Figure 7. The angles in the cubic quadrangle can be fixed, since interchanging a with S and ß with Y gives the same results. The angles in the face that shares an edge of length b with the cubic quadrangle can be fixed, since the other possible assignment implies that there are two different vertex types for cubic vertices incident to an edge of length b: one containing 2a and one containing 2S. We first consider the edge assignment on the left side in Figure 6. The angle assignment for the quadrangle 1562 fixes all remaining angle assignments for the faces neighbouring the cubic quadrangle: either the vertex type of 1 and 4 is a + S + ß and the vertex type of 2 and 3 is 2y + ß, or the vertex type of 1 and 4 is a + S + y and the vertex type of 2 and 3 is 2ß + y. Owing to Lemma 4.1, these combinations are not possible, so this edge assignment is not possible. Next we consider the edge assignment on the right side in Figure 6. The angle assignment for the quadrangle 1562 fixes all remaining angle assignments for the faces neighbouring the cubic quadrangle: either the vertex type of 1, 2 and 4 is a + S + ß and the vertex type of 3 is 3y, or the vertex type of 1, 2 and 4 is a + S + y and the vertex type of 3 is 3,0. Owing to Lemma 4.2, this implies that the quadrangulation has 6 faces, and thus 8 vertices. □ 5.2 Cubic tristars A cubic tristar in a quadrangulation is a cubic vertex v such that all three neighbouring vertices have degree 3. The vertex v is called the central vertex of the cubic tristar. Figure 8 shows an example of a cubic tristar in a quadrangulation. Theorem 5.2. In a spherical tiling by congruent quadrangles of type 2, there is no cubic tristar for which the central vertex is incident to an edge of length b. Proof. We use the vertex labels as given in Figure 8. Assume that the edge 12 has length b. This implies that either edge 36 or edge 46 has length b. Both cases are completely analogous, so we will assume without loss of generality that edge 36 has length b. Figure 8: An example of a cubic tristar in a quadrangulation. Figure 9: Partial angle assignment for a cubic tristar We can fix the angle assignment in the quadrangle 1274, since interchanging a with S and ß with y gives the same results. This also fixes the angle assignment in the quadrangle 1253, since the vertex 1 and the vertex 2 have the same type owing to Theorem 4.4. Since the vertex 3 is incident to an angle ß, also the vertex 1 has to be incident to an angle ß, and so the angle assignment in the quadrangle 1364 is also fixed. This gives the partial angle assignment shown in Figure 9. The third angle at vertex 4 is either ß or y. Owing to Lemma 4.1, ß is not possible. Owing to Lemma 4.2, y implies that the quadrangulation has 6 faces, and thus 8 vertices. This proves the theorem. □ 6 Conclusion For the classification of spherical tilings by congruent quadranglesthere remain two open cases: spherical tilings by congruent quadrangles of type 2 and those of type 4. We show that the most symmetric of type 2 quadrangles, i.e., the isosceles quadrangles of type 2, cannot be used to tile the sphere. This might seem surprising, since spherical tilings by congruent quadrangles of type 2 do exist, but it can be explained because being isosceles and tiling the sphere forces the quadrangle to be of type 1. Next we gave an overview of which vertex types of degree 3 can be used and showed that at most two different types can be used. We also showed that there is no spherical tiling by congruent quadrangles of type 2 for which the underlying graph contains a cubic quadrangle or a cubic tristar containing an edge of length b. As can be seen from Table 3 and Table 4, this excludes already a reasonable percentage of the quadrangulations that can n Quadrangulations contain cubic quadrangle Percentage 8 1 1 100.00% 10 1 0 0.00% 12 3 1 33.33% 14 12 3 25.00% 16 64 24 37.50% 18 510 210 41.18% 20 5 146 2 208 42.91% 22 58 782 25 792 43.88% 24 716 607 319 553 44.59% 26 9 062 402 4 110 016 45.35% 28 117 498 072 54 277 671 46.19% 30 1 553 048 548 731 637 255 47.11% Table 3: Overview of quadrangulations on n vertices that contain a cubic quadrangle. 10 12 14 16 18 20 22 24 26 0 1 2 7 31 217 2 065 22 869 272 106 3 355 499 1 6 68 747 8 804 108 738 1 383 419 2 2 3 15 119 1 249 15 363 201 586 3 5 66 832 11 619 4 2 2 15 259 5 4 Table 4: Overview of the number of cubic tristars in quadrangulations that do not contain a cubic quadrangle. The top row gives the number of vertices, the first column gives the number of cubic tristars and the remaining numbers give how many quadrangulations have that many vertices and that many cubic tristars. appear as the underlying graph of a spherical tiling by congruent quadrangles of type 2, and also limits the possible charts that can correspond to a spherical tiling by congruent quadrangles of type 2. This is why these results can contribute to the completion of the classification of spherical tilings by congruent quadrangles. Table 3 and Table 4 were constructed using plantri[6, 4]. References [1] Y. Akama, Classification of spherical tilings by congruent quadrangles over pseudo-double wheels (i) - a special tiling by congruent concave quadrangles., Hiroshima Math. J. 43, 285304. [2] Y. Akama and K. Nakamura, Spherical tilings by congruent quadrangles over pseudo-double wheels (ii) the ambiguity of the inner angles, submitted, 2013. [3] Y. Akama and Y. Sakano, Spherical tilings by congruent quadrangles over pseudo-double wheels (iii) the essential uniqueness in case of convex tiles, preprint, 2013. [4] G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas and P. Wollan, Generation of simple quadrangulations of the sphere, Discrete Mathematics 305 (2005), 33-54, http: //dx.doi.org/10.1016/j.disc.2005.10.005. [5] H. Davies, Packings of spherical triangles and tetrahedra, in: Proc. Colloquium on Convexity (Copenhagen, 1965), Kobenhavns Univ. Mat. Inst., 1967, 42-51. [6] plantri website, http://cs.anu.edu.au/~bdm/plantri. [7] Y. Sakano and Y. Akama, Classification of spherical tilings by congruent kites, darts, and rhombi - spherical Hilbert's eighteenth problem, submitted, 2013. [8] Y. Ueno and Y. Agaoka, Examples of spherical tilings by congruent quadrangles, Mem. Fac. Integrated Arts and Sci. Ser. IV (2001), 135-144. [9] Y. Ueno and Y. Agaoka, Classification of tilings of the 2-dimensional sphere by congruent triangles, Hiroshima Math. J. 32 (2002), 463-540. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 319-335 Embedded graphs whose links have the largest possible number of components Stephen Huggett School of Computing and Mathematics, University of Plymouth, Plymouth PL4 8AA, UK Israa Tawfik * Department of Mathematics, College of Education for Girls, Tikrit University, Tikrit, Iraq Received 27 May 2013, accepted 10 March 2015, published online 11 March 2015 Abstract We derive the basic properties of graphs embedded on surfaces of positive genus whose corresponding link diagrams have the largest possible number of components. Keywords: Embedded graphs, medial, components of links. Math. Subj. Class.: 05C10, 57M25 1 Introduction A graph G embedded in a surface determines a link diagram D(G), which has a certain number p of components. The relationship between the graph and the link diagram is through the crossing which replaces each edge of the graph, shown in figure 1. In this article, we are only interested in p. So at each crossing, the choice of the over-crossing strand does not matter, and we are therefore actually considering a link universe rather than a diagram of a particular link. However, we will for simplicity refer to a link diagram. The relationship between a graph and the number of components in the corresponding link diagram has been studied by several people. It is shown in [6] that where T (G; x, y) is the Tutte polynomial and q(G) is the number of edges in G. In [4], equation (1.1) is generalised to the projective plane and the torus, while in [7] T (G; -1, -1) * Supported by the Ministry of Higher Education and Scientific Research, Iraq. E-mail addresses: s.huggett@plymouth.ac.uk (Stephen Huggett), israa.tawfik@plymouth.ac.uk (Israa Tawfik) T (G; -1, -1) = (_1)q(G) (_2)m(d(G))-I, (1.1) Figure 1: A crossing replacing an edge. The curved lines are strands of the link diagram, and the dashed straight line joining the two vertices is the edge of the graph replaced by the crossing. is calculated for fans, wheels, and 2-sums of graphs. The number p is the same as the number of "straight-ahead" walks in medial graphs, as described in [8]. The focus in [3] is to characterize the plane graphs G whose p(D(G)) is as large as possible, which is the cycle rank plus one; these are the "extremal" graphs. Maximising p is also our principal interest here, although we will study graphs embedded on various orientable surfaces. In section 2 we show how p depends on the blocks of the graph, we note that p does not change when the graph undergoes a "graph Reidemeister move" or an "embedded" Y ^ Д move, and we show that p cannot change very much when an edge is added to the graph. In section 3 we study plane graphs. Many of our results replicate those in [3], although our emphasis is different because we are preparing to work on other surfaces. Let g be the genus of a surface. Then section 4 shows how to extend many of the results of section 3 to graphs embedded on surfaces with g > 0. In particular, in Theorem 4.3 we show that p < f + 2g, where f is the number of faces in the embedding, so that the extremal graphs now have p = f + 2g. In Theorem 4.5 we give a list of some graph operations which preserve the extremal property, and in Corollary 4.6 and Theorem 4.7 we give some local consequences of this property. We finish in section 5 with some observations on other possible values of p. For plane graphs, the case when p is equal to the cycle rank is considered in [5], where it is found that this class of graphs is quite severely constrained. We comment on two other cases: the case when p = 1, and the Petersen and Heawood families. It is a pleasure to thank Iain Moffatt for many interesting discussions. 2 Basic results Our first theorem comes from the connected sum operation on links. Theorem 2.1. Let G be a graph with blocks B1,B2,..., Bk. Then k p(D(G))= £p(D(Bi)) - (k - 1). i= 1 Proof. For any two adjacent blocks Bi and Bi+1 of G with common vertex v, the two strands at v must be part of the same component. So splitting G at v into two graphs increases the number of components by one. See figure 2. Therefore, splitting G into its k blocks increases the number of components by k - 1, and hence the result. □ Theorem 2.2. Let G be a graph with a bridge e. Then p(D(G))= p(D(G/e)). Bi Bi+1 Figure 2: The curved lines are two strands of a single component at a cut vertex of the graph, which separates blocks Bi and Bi+1. The dashed straight lines are edges of the graph. Proof. Let G1 and G2 be the two components of G \ {e}, and let B be the block in G containing e. Then by Theorem 2.1 p(D(G)) = M(D(Gi))+ m(D(G2))+ p(D(B)) - 2 = p(D(Gi))+ p(D(G2)) - 1, because p(D(B)) = 1. However, G/{e} consists of blocks G1 and G2, so by Theorem 2.1 again p(D(G/{e})) = MD(Gi)) + m(D(G2)) - 1, and hence the result. □ Theorem 2.3. Let G be a graph with parallel edges e1 and e2 bounding a disc. Then p(D(G)) = p(D(G \{e1,e2})). If, instead, e1 and e2 are not parallel edges, but are incident with a common vertex of degree 2, then p(D(G)) = M(D(G/{ebe 2})). Proof. This follows immediately from the Reidemeister 2 move (see figure 3) on the link diagrams, which evidently preserves p. □ Figure 3: The Reidemeister 2 move on a link diagram. We next consider the Y ^ Д moves. These replace a "Y" by a triangle, or vice versa, as in figure 4. For our purposes, we need the graph to be embedded in a surface and the triangle to bound a disc on that surface. Then we refer to the Y ^ Д move as embedded. Otherwise it is an abstract Y ^ Д move. Theorem 2.4. If G1 and G2 are related by embedded Y ^ Д moves, then p(D(G 1))= p(D(G2)). Figure 4: The effect of a Y ^ Д move. In this figure the curved dotted lines are the strands of the link diagram, while the straight lines are the edges of the graph. Proof. This is evident from figure 4. □ Finally in this section we note that adding an edge cannot change the number of components very much. Theorem 2.5. Let e be a new edge connecting two vertices in the same face of a graph G, this face being a disc. Then v(D(G)) - 1 < D(G + e)) < m(D(G)) + 1. Proof. If e is a loop bounding a disc then m(D(G)) = M(D(G + e)), so the result holds. If e is a loop not bounding a disc, or e is not a loop, then there are two cases: we refer to figure 5, where the face is labelled F. Figure 5: The effect of adding an edge in a face F. The arcs a (joining 1 and 2) and ß (joining 3 and 4) may or may not be in different components. (Again, the curved dotted lines are the strands of the link diagram, while the straight lines are the edges of the graph.) (1) If the arcs a (joining 1 and 2) and ß (joining 3 and 4) are contained in different components of D(G), then p(D(G + e)) = m(D(G)) - 1. (2) If the arcs a and ß are contained in the same component of D(G), then there are two further cases. (a) Along this one component, if the order of the four endpoints of the two arcs a and ß is 1, 2, 3,4 then M(D(G + e)) = m(D(G)). (b) If the order of the four endpoints of the two arcs a and ß is 1, 2,4,3 then M(D(G + e)) = m(D(G)) + 1. □ 3 Extremal plane graphs Theorem 3.1. Let G be a connected plane graph. Then 1 < MD(G)) < f (G). Proof. Let T be a spanning tree of the graph G. Then f (T) = 1 and ^(D(T)) = 1, so the theorem is true for T. Now add, one by one, edges to T in order to obtain G. The intermediate graphs are Gì, G2,..., Gs_i. We obtain a sequence of graphs Tri r< r r r = Go, Gì, . . . , GS_1, G s = G. The insertion of an edge increases the number of faces by exactly one, so for i = 0,..., s -1 we have f (Gi+i) = f (G i) + 1 = f (Go) + i + 1. By Theorem 2.5 MD(Gi+i)) < M(D(Gi)) +1 (3.1) < m(D(Go))+ i + 1. (3.2) Since ^(D(G0)) = f (G0), we must have ^(D(Gi+1)) < f (Gi+1) for each i, which means that m(d(g)) < f (G). □ If G is a connected plane graph then G is called extremal if M(D(G)) = f (G). A face of a plane graph is called even if it has an even number of edges. Theorem 3.2. If G is extremal then each face of G is even. Proof. Let T be a spanning tree of G, and let — Go, Gì, G2, . . . , Gs — G be the sequence of graphs in the proof of Theorem 3.1. Since T and G are extremal then each Gi in the sequence is extremal. Otherwise ^(D(Gi)) < f (Gi), and then from Theorem 2.5 we would have ^(D(Gi+1)) < f (Gi+1) and eventually ^(D(G)) < f (G). T has one even face. Suppose that there is a graph in this sequence with an odd face, and let Gi+1 be the first such graph. Gi+1 = Gì U e where e has been inserted into a necessarily even face in Gi, creating two odd faces f1 and f2 in Gi+1. Because all the graphs in the sequence are extremal, we must be in case 2b of Theorem 2.5. Choose the component of D(Gi+1) which includes the arc 13. This component contains exactly one of the faces f1 or f2, suppose it is f1, and the component defines an even circuit in the edges of Gi+1. But the faces inside this circuit are all even except f1, because all except f1 come from Gi, which is a contradiction. □ Corollary 3.3. If G is extremal then G* is eulerian. □ The converse of this corollary is not true. For example, let G be the dual graph of K2,2. Then G* is eulerian but G is not extremal. Corollary 3.4. If G is extremal then G is bipartite. □ Define 6(G) to be the minimum degree of G. Theorem 3.5. If G is extremal, then 6(G) < 3. all have 6(G) > 3. Then 6(Gi) must be 2. When we add an edge to Gi to get Gi+1 we contradict Theorem 3.1, because in Gi+1 the number of faces has increased but the number of components has not: see figure 6. So GCb . . . ,Gi all have 6(G) < 3, while Gi+b . . . ,G: s s. □ Figure 6: Adding an edge to Gi to obtain Gi+1. Lemma 3.6. If G is extremal then it has no loops, and any parallel edges must be between cut-vertices. Proof. G cannot have a pair eb e2 of parallel edges between vertices which are not cut-vertices, because if it did then f (G) = MD(G)) = p(D(G \{ebe2})) by Theorem 3 < f (G \{ei,e2}) by Theorem 3.1 < f (G) - 2. (This argument fails if the parallel edges are between cut-vertices because Theorem 3.1 needs a connected graph.) Similarly, G cannot have a loop e, because if it did then f (G) = M(D(G)) = M(D(G \ {e})) < f (G \ {e}) = f (G) - 1. □ Lemma 3.7. Let G be connected, simple, and non-trivial. Then G* is not extremal. Proof. If G is connected, simple, and non-trivial then G* has no vertices of degree 1 or 2. So from Theorem 3.5 G* is not extremal. □ Lemma 3.8. Let G be connected, simple, and non-trivial. Then MD(G)) f (G \ e). Hence by Theorem 3.1 G \ e is extremal. □ Lemma 3.11. Let G be extremal. Then each component of D(G) only ever crosses itself on a bridge. Proof. Suppose a component of D(G) crosses itself on the edge e in G, not a bridge. By Theorem 3.10, G \ e is extremal. When we delete e the number of components stays the same but the number of faces drops. This is impossible because G \ e is extremal. □ Note that in the following theorem the graph Gj/jxj, y} is obtained from Gi by identifying the vertices xi and y. Theorem 3.12. Let G be a plane graph. G is extremal if and only if it satisfies one of the following conditions. (1) G = Ki (2) G has a bridge e such that G \ e consists of two extremal graphs. (3) G has edges ei = xy for i = 1, 2 such that G \ {e1, e2} consists of two disjoint graphs G1 and G2 with x, yi G V(Gì) and Gi/{xi, yi} extremal. Proof. Denote by f, f1, and f2 the numbers of faces of G, G1, and G2 respectively. Similarly, denote by m, мь and m2 the numbers of components in their link diagrams. Let G be an extremal graph, so that m = f, and suppose that G has at least one edge. If G has a bridge e, with G \ e = G1 U G2, then f = f1 + f2 - 1. because G1 and G2 share a common face. Now by Theorem 2.1 M = M1 + M2 - 1. Since G is extremal M1 + M2 - 1 = M = f = f1 + f2 - 1. Therefore M1 + M2 - 1 = f1 + f2 - 1, which means that M1 + M2 = f1 + f2. Since Mi < fi for each i, we must now have M1 = f1 and M2 = f2 as required. Next, let G be an extremal graph without a bridge. By Theorem 3.5 it must have a vertex v with degree less than 3. However, if d(v) = 0 then G = K1, and if d(v) = 1 we have a bridge. So d(v) = 2. Now there are two cases. (a) If v is adjacent to two distinct vertices x2 and y2, as in figure 7, then G1 = K1 (the vertex v) and x2 = y2 in G2. Clearly m1 = f1. Suppose m2 is the number of components of the link diagram of G2/{x2, y2}, and f2 is the number of faces of G2/{x2,y2}. Then m2 = M and f2 = f because the identification of x and y does not affect the number of components of the link diagram of G2 or the number of faces of G2, which means that G2/{x2, y2} is extremal. Figure 7: The case G1 = Kb adjacent to two distinct vertices in G2. Figure 8: The case G1 = Kb adjacent twice to a vertex in G2. (b) If v is a vertex adjacent twice to another vertex, as in figure 8, then G1 = K1 as before, and since x2 = y2 then ^ = ^2 + 1 and f = f2 + 1. Since G is extremal then G2 / {x2, y2} is extremal. Conversely, suppose that one of the three conditions holds. Then we will show that G is extremal. (1) If G = K1 then G is extremal because ^(G) = f (G) = 1. (2) If G consists of the two extremal graphs G1, G2 and the bridge e between them, then = f1 and = f2 and since e is a bridge then ^ = + ^2 -1. There is a common face between G1 and G2, so f = f1 + f2 - 1, which gives f = f1 + f2 - 1 = + - 1 = Therefore G is extremal. Figure 9: The construction of G from G1 and G2. (3) Suppose that the plane graph G is constructed from two connected plane graphs G1 and G2 by adding two new edges e1 and e2, where e1 = (x1, x2), e2 = (y1, y2) and xi ,yi € G i, as in figure 9. Let ^ be the number of components of the link diagram of Gj/{xj, yi} and fi the number of faces of Gi/{xi, yi}. Then f = f 1 + f2 - 2 (3.3) because we will get two new faces, one in f 1 and another one in f2, when we identify xi and yi. In order to count the components in the various link diagrams, start with Gj/jxj, y} and then "split" the vertices into xi and yi, obtaining the arrangement shown in figure 10. Hence ^ = + - 2. (3.4) From equations (3.3) and (3.4), ^ = f. □ Figure 10: The two components, a and b, crossing from G1 to G2. We finish this section by describing ways of constructing new extremal graphs using the operations of 2-sum and tensor product. (These are natural operations on graphs, but are perhaps most easily defined on matroids: see [2].) Let G and H be any graphs, with distinguished edges e and f. The 2-sum G ф2 H along e and f is obtained by identifying the edges e and f to form a new edge, which is then deleted. The tensor product G ( H is obtained by taking the 2-sum of G with H along each edge of G and the edge f in H. For example, when H is the triangle graph, constructing G ( H amounts to putting a new vertex of degree two in each edge of G. In this case, the embedding of G ( H is induced from that of G, but this only happens because H is so symmetrical. In general, there may be more than one embedding of G ( H for any given embedding of G. Theorem 3.13. Let G be any connected plane graph and H be an odd cycle. Then G ( H is extremal. □ Proof. This follows from the Reidemeister 2 move on the link diagrams. □ Theorem 3.14. Let G be a tree and H be extremal. Then the tensor product G ( H, in which the distinguished edge in H is not a bridge, is extremal. Proof. This follows from Theorem 3.10, part (d). □ 4 Extremal graphs on surfaces of genus g Here we will often restrict to cellular embeddings, in which the interior of each face of the embedded graph is homeomorphic to an open disc. (For plane graphs this implies connectedness, of course.) Given an embedded graph G, a spanning subgraph ф which is connected, has just one face, and is cellularly embedded, is called a pseudo-tree of G. A pseudo-tree can be obtained from any cellularly embedded graph by iteratively deleting edges that lie on two faces, until no such edge can be found. Firstly, let G be cellularly embedded on the torus. Then each block of G is a connected plane graph except for one, which must be cellularly embedded on the torus. Theorem 4.1. If ф is a pseudo-tree cellularly embedded on the torus, then ^(В(ф)) < 3. Proof. We reduce ф as follows. • Contract all bridges in ф. This leaves ^(^(ф)) unchanged, by Theorem 2.2. • For each vertex of degree two in ф whose edges go to distinct vertices, contract both these edges. By Theorem 3 this also leaves р(Б(ф)) unchanged. ф has one meridian M and one longitude L. M n L must be non-empty, and if it had more than one connected component then ф would have more than one face. Now, up to extra meridians or longitudes, there are only four possibilities for the reduced ф, shown in figure 11, and by inspection the link diagrams for A, B, C, and D have ^ = 2,1,1, and 3 respectively. E is the same as A, with an extra meridian. □ A B C Figure 11: A,B,C, and D are the four possibilities for the reduced ф. E is the same as A, with an extra meridian. Theorem 4.2. Let ф be a pseudo-tree embedded on a surface of genus g. Then м(D(ф)) < 1 + 2g. Proof. The result is clearly true when g = 0. Now suppose that for any pseudo-tree фд embedded on a surface Sg of genus g, we have №(фд)) < 1 + 2g. Let фд+1 be a pseudo-tree embedded on Sg+i, a surface of genus g + 1. We will show that MD^g+i)) < 1 + 2(g +1) = 3 + 2g. In other words we will show that ф^ has at most two more components than фg. Consider one of the handles of Sg+1, and let L and M be the longitude and meridian cycles in ф^1 for this handle. Choose an edge eL in L, but not in M, and then delete this edge. By Theorem 2.5 MD^g+i)) < M(D^g+1 \ eL)) + 1. Repeat this process for M by choosing an edge eM in M, but not in L, to get \ eL)) < 1 \ {ем, eL})) + 1. These two deletions yield a graph denoted фд which is no longer a pseudo-tree on Sg+1. It is, however, a pseudo-tree on the surface of genus д obtained from Sg+1 by removing the handle under consideration. We now have М(^(фд+ l)) < МДфд ))+2, as required. □ Theorem 4.3. Let G be a graph cellularly embedded on a surface of genus д. Then 1 < m(D(G)) < f (G) + 2д. Proof. Let ф be a pseudo-tree of the graph G. Then f (ф) = 1, and by Theorem 4.2 we have р(Б(ф)) < 1 + 2д, which means the theorem is true for ф. Now add edges to ф, one by one, in order to obtain G. We obtain a sequence of graphs ф = GCb G1, . . . , Gs-b Gs = G. The insertion of an edge increases the number of faces by exactly one, so for i = 0,..., s -1 we have f (Gi+1) = f (Gi) + 1 = f (Go)+ i + 1. By Theorem 2.5 M(D(Gm)) < M(D(Gi)) + 1 (4.1) < M(D(Gc))+ i + 1. (4.2) Since ^(D(GC)) < f (GC) + 2д, we must have M(D(Gm)) < M(D(Gc )) + i +1 (4.3) < f (Gc) + 2g + i + 1 (4.4) < f (Gi+1) + 2g. (4.5) So ^(D(Gi+1)) < f (Gi+1) + 2g for each i, and hence the result. □ If G is a graph cellularly embedded on a surface of genus д then G is called extremal if MD(G))= f (G) + 2д. Theorem 4.4. If ф is a spanning pseudo-tree of the extremal graph G, then ф is extremal. Proof. Adding edges to ф one by one we obtain a sequence of graphs ф = GC,G1,...,Gs = G. In particular, Gi-1 = Gi \ e, where e is not a bridge. Suppose that Gi is extremal. Then p(D(Gi)) = f (Gi) + 2g. (4.6) Also, f (Gi-i) = f (Gì) - 1. (4.7) By Theorem 2.5 MD(Gi_i)) > M(D(Gi)) -1 = f (Gi)+2g - 1 = f (Gi-i) + 2g. □ With two small modifications, Theorem 3.10 is also true for graphs cellularly embedded on surfaces of genus g: Theorem 4.5. Let G be a graph cellularly embedded on a surface of genus g. Then the following statements are true. (a) Let e be a bridge of G. Then G/e is extremal if and only if G is extremal. (b) Let v be a vertex of degree 2 with exactly one adjacent vertex, the two edges joining these vertices bounding a disc. Then G \ v is extremal if and only if G is extremal. (c) Let v be a vertex of degree 2 with two different adjacent vertices x and y. Then G/{v, x}/{v, y} is extremal if and only if G is extremal. (d) G is extremal if and only if each block of G is extremal. (e) Let G be extremal and e such that G \ e is cellularly embedded. Then G \ e is extremal. Proof. This is exactly as in Theorem 3.10. In part (e) we note that each block of G is a connected plane graph except for one, which must be cellularly embedded on the surface. □ It follows that Lemma 3.11 is true for extremal graphs cellularly embedded on surfaces of genus g: Corollary 4.6. Let G be an extremal graph cellularly embedded on a surface of genus g. Then each component of D(G) only ever crosses itself on a bridge. For any vertex v g V (G) we can define its degree d(v) and we can also count the number of components of the link diagram of G which pass close to v, denoting this by M(v). Theorem 4.7. Let G be an extremal graph cellularly embedded on a surface of genus g, and let v g V (G), not a cut vertex. Then d(v) = p(v). c_ Figure 12: Two arcs, from the same component of the link diagram, passing close to v. There may be many other edges such as a, b, and c incident with v, not shown here. Proof. Suppose that d(v) > p(v). Then there would be two arcs (from the same component of the link diagram) passing close to v, as in figure 12. None of the edges incident with v can be a bridge, or v would be a cut vertex, so by part e of Theorem 4.5 G \ e is also torus extremal. This process can be repeated until our two arcs, passing close to v, cross the same edge incident with v, as in figure 13. But this contradicts Corollary 4.6. Hence the result. □ Figure 13: Two arcs, from the same component of the link diagram, passing close to v and crossing the same edge incident with v. There may be many other edges such as a and b incident with v, not shown here. 5 Concluding remarks It may also be possible to establish results like Theorem 3.2 and Corollary 3.4 for graphs cellularly embedded on surfaces of positive genus. They certainly appear to be true on the torus: Conjecture 5.1. If G is an extremal graph cellularly embedded on a torus then each face of G is even. Conjecture 5.2. If G is an extremal graph cellularly embedded on a torus then G is bipartite. Next, let us make a few observations about plane graphs for which p takes its smallest possible value. We leave the proofs of the results to the reader. Evidently, if G is a tree then p(D(G)) = 1. Similarly, if G is an odd cycle or its dual then p(D(G)) = 1. Theorem 5.3. If G is any cycle and H = Щ, then p(D(G e2 H )) = 1. □ In the next two theorems, the two-sum can be taken at any edge of G, and in fact it can be replaced by the tensor product. Theorem 5.4. Let G be a plane graph with p(D(G)) = 1 and H be an even cycle or its dual. Then p(D(G 02 H)) = 1. □ Theorem 5.5. Let G be a tree and H be any cycle. Then p(D(G ®2 H)) = 1. □ We finish by asking whether there are interesting families of graphs having specific values of p greater than 1 but less than the maximum. Recall that the Petersen family P of graphs are those obtainable from K6 by abstract Y ^ Д moves. P has 7 members, including the Petersen graph itself, and it is of interest partly because of the following intriguing result [9]. (An intrinsically linked graph is one in which all spatial embeddings contain a non-splittable 2-component link.) Theorem 5.6 (Robertson, Seymour, Thomas). P is the minor-minimal family for intrinsically linked graphs. The graphs in P can all be cellularly embedded on the torus, but these embeddings are not unique. Suppose we focus on K6, and suppose we restrict to embedded Y ^ Д moves. Then, for any particular embedding of K6 we will obtain a subfamily of P whose graphs all have the same value of p. Our preliminary results are indicated in the table below, where we have used the graph names given in [1]. (P10 is the Petersen graph.) the choice of embedding p the family obtained using embedded Y ^ Д moves (a) 3 P (b) 3 P\{Pio,Qs} (c) 3 P\ IQs} (d) 5 P\{Qs} (e) 5 P\{Pio,Qs} (f) 5 P\{Pio} (g) 7 P\{Pio} Similarly, the Heawood family H of graphs are those obtainable from K7 by abstract Y ^ Д moves. It has 20 members, which can all be cellularly embedded on the torus. It is shown in [1] that 14 of the graphs in H are intrinsically knotted. Again, choosing specific embeddings and restricting to embedded Y ^ Д moves will yield subfamilies of graphs with constant p values. References [1] R. Hanaki, R. Nikkuni, K. Taniyama and A. Yamazaki, On intrinsically knotted or completely 3-linked graphs, Pacific Journal of Mathematics 252 (2011), 407-425. [2] S. Huggett, On tangles and matroids, Journal of Knot Theory and its Ramifications 14 (2005), 919-929. [3] X. Jin, F. Dong and E. Tay, On graphs determining links with maximal number of components via medial construction, Journal of Discrete Applied Mathematics 157 (2009), 3099-3110. [4] M. Las Vergnas, On Eulerian partitions of graphs, Research Notes in Mathematics 34 (1979), 62-75. [5] Y. Lin, S. D. Noble, X. Jin and W. Cheng, On plane graphs with link component number equal to the nullity, 2011, preprint. [6] P. Martin, Remarkable Valuation of the Dichromatic Polynomial of Planar Multigraphs, Journal of Combinatorial Theory B 24 (1978), 318-324. [7] E. G. Mphako, The component number of links from graphs, Proceedings of the Edinburgh Mathematical Society 45 (2002), 723-730. [8] T. Pisanski, T. Tucker and A. Žitnik, Straight-ahead walks in Eulerian graphs, Discrete Mathematics 281 (2004), 237-246. [9] N. Robertson, P. Seymour and R. Thomas, Sachs' linkless embedding conjecture, Journal of Combinatorial Theory Series B 64 (1995), 185-227. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 337-363 Alternating plane graphs Ingo AlthOfer Department of Mathematics and Computer Science, Friedrich-Schiller Universität, Ernst-Abbe-Platz 2, 07743 Jena - Germany Jan Kristian Haugland Arnstein Arnebergs vei 30, leilighet 308, 1366 Lysaker, Norway Department ofApplied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 - S9 - WE02, 9000 Ghent, Belgium Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society), University of West Bohemia, Univerzitni 8, 30614 Pilsen, Czech Republic Received 17 December 2013, accepted 20 February 2015, published online 29 May 2015 A plane graph is called alternating if all adjacent vertices have different degrees, and all neighboring faces as well. Alternating plane graphs were introduced in 2008. This paper presents the previous research on alternating plane graphs. There are two smallest alternating plane graphs, having 17 vertices and 17 faces each. There is no alternating plane graph with 18 vertices, but alternating plane graphs exist for all cardinalities from 19 on. From a small set of initial building blocks, alternating plane graphs can be constructed for all large cardinalities. Many of the small alternating plane graphs have been found with extensive computer help. Theoretical results on alternating plane graphs are included where all degrees have to be from the set {3,4,5}. In addition, several classes of "weak alternating plane graphs" (with vertices of degree 2) are presented. * Corresponding author. Supported by the project NEXLIZ CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech republic. Karl Scherer 11 Utting Street, Birkdale, 0626 Auckland, New Zealand Frank Schneider Dr.-Bernhard-Klein-Straße 99, 52078 Aachen, Germany Nico Van Cleemput * 4= Abstract 4 3 5 4 1 2 5 1 3 6 2 5 3 2 1 5 2 4 1 4 3 Figure 1: Karl Scherer: Squaring the square with 21 alternating squares. Keywords: Plane graph, alternating degrees, exhaustive search, heuristic search. Math. Subj. Class.: 05C10, 05C75 1 Introduction The concept of alternating plane graphs was introduced by I. Althofer in January 2008. Years before, he had seen K. Scherer's squarings of a square, in particular the nice symmetric one where 21 small squares exactly fill a square of side length 16 in such a way that no two squares with the same side length join an edge or a vertex (see Figure 1). Scherer called such arrangements "alternating tilings". The 21-solution is the second smallest such object. This concept of alternating tilings formed the inspiration for the definition of alternating plane graphs. A large portion of the history of the development of this concept can be found at [8]. The paper is organized as follows. In Section 2 we give the necessary definitions. In Section 3 several theorems about different types of alternating plane graphs are proven. In Section 4 and Section 5 we describe exhaustive and heuristic searches for alternating plane graphs. Section 6 gives an overview of the alternating plane graphs constructed by hand and by these searches. Section 7 and Section 8 deal with several techniques to construct large alternating plane graphs. In Section 9 we describe a relaxation of the definition of alternating plane graphs. E-mail addresses: ingo.althoefer@uni-jena.de (Ingo Althofer), admin@neutreeko.net (Jan Kristian Haugland), karlscherer3@yahoo.co.nz (Karl Scherer), hobblefrank@t-online.de (Frank Schneider), nico.vancleemput@gmail.com (Nico Van Cleemput) 2 Definition Note that a planar graph is a graph that can be embedded in the plane without crossing edges. A plane graph is a particular embedding of a planar graph. Definition 2.1. A plane graph is called an alternating plane graph, when the following conditions are fulfilled: • There are no adjacent vertices with the same degree. • There are no adjacent faces with the same size. • Each vertex has degree at least 3. • Each face has size at least 3. Note that the exterior face is also considered to be a face and also needs to satisfy the conditions above. The following lemma follows immediately from the definition. Lemma 2.2. If G is a 3-edge-connected alternating plane graph, then the dual of G is also an alternating plane graph. For a 2-edge-connected alternating plane graph that is not 3-edge-connected, the dual is not a simple graph, and therefore the dual is not an alternating plane graph. Note that an alternating plane graph is always at least 2-edge-connected, since plane graph with edge connectivity 1 contains a face that is adjacent to itself. 3 Theoretical results Definition 3.1. An alternating plane graph is called an (x\,..., xn)-alternating plane graph if all vertices have degree x1, ..., xn-1 or xn and all faces have x1, ..., xn-1 or xn sides. 3.1 Results for (3,4, 5)-alternating plane graphs Let vi denote the number of vertices of degree i, and let f denote the number of faces with j sides. Theorem 3.2. If G is a (3,4, 5)-alternating plane graph, then v3 = f3, v4 = f4 and V5 = f5. Proof. Suppose G is a (3,4,5)-alternating plane graph. Summing the edges over the vertices shows that the total number of edges equals 3v3+4v4+5v5, while summing over the faces shows that it is 3f3 +4f4+5f5. So we have 22 3v3 + 4v4 + 5v5 = 3fs + 4f4 + 5f5 2 2 Euler's formula gives (3.1) Ol . -U U^f xf xn 3v3 + 4v4 + 5У5 , 3f3 + 4f4 + 5f5 2(v3 + v 4 + v5 ) + 2(f3 + f4 + f5) = -2--1--2--+ 4 which simplifies to v3 + f3 = v5 + f5 + 8 (3.2) The rest of the proof is based on counting (i, j)-combinations, that is, the number of instances of a vertex of degree i incident with a face with j sides. For example, each vertex of degree 3 must be incident with a triangle, a quadrilateral and a pentagon, and each triangle must be incident with one vertex of each degree (3, 4 and 5). So we see that the number of (3, 3)-combinations must be equal to vs, but it must also be equal to fs, so we can deduce that vs = fs (3.3) Note that (3.1) and (3.3) together implies that v5 — f5 must be divisible by 4. Due to parity, each pentagon is incident with at least one vertex of each degree too, while a quadrilateral might have just two values represented. Counting (3, 5)-combinations shows that f5 < V3 (3.4) while a dual argument (counting (5, 3)-combinations) shows that V5 < fs (3.5) Combining (3.1), (3.2), (3.3), (3.4) and (3.5) gives us five possibilities: v5 = vs — 8, f5 = vs v5 = vs — 6, f5 = vs —2 v5 = vs — 4, f5 = vs —4 v5 = vs — 2, f5 = vs —6 v5 = = vs, f5 = vs —8 Let a denote the number of vertices of degree 5 incident with exactly one face with i sides (and two of each of the other two types). Adding these up gives the total number of vertices of degree 5; i.e., as + a4 + a5 = v5. Since a vertex of degree 5 is incident with either 1 or 2 triangles, the number of (5,3)-combinations is as + 2a4 + 2a5, and since each triangle is incident with exactly one vertex of degree 5, we have as + 2a4 + 2a5 = fs = vs. Thus, as = 2v5 — vs and a4 + a5 = vs — v5. It follows that 0 < 0,5 < vs — V5 (3.6) Similarly, with bj denoting the number of pentagons incident with exactly one vertex of degree j, a dual argument shows that bs = 2f5 — vs and b4 + b5 = vs — f5, and it follows that 0 < b5 < vs — f5 (3.7) The number of (5,5)-combinations is 2as + 2a4 + a5 = a5 + 2(v5 — a5) = 2v5 — a5, and a dual argument shows that it is also equal to 2f5 — b5. So we have 05 — b5 = 2(v5 — f5) (3.8) Now let us inspect the possible values of v5 and f5 found earlier in view of (3.6), (3.7) and (3.8): v5 f5 2(v5 - f5) v3 - 8 v3 b5 =0 -16 v3 - 6 v3 - 2 b5 < 2 -8 v3 - 4 v3 - 4 0 v3 - 2 v3 - 6 a5 < 2 8 v3 v3 - 8 a5 =0 16 Since a5 and b5 are both non-negative integers, it is clear that (3.8) is only possible if V5 = f5 = V3 - 4. Of course, (3.1) now also gives v4 = f4 and the proof is complete. □ We can go a little bit further and show that if p(i, j) denotes the number of (i, j)-combinations, then p(i, j) = p(j, i) for all i,j in {3,4, 5}. We have p(3,j) = v3 and p(i, 3) = f3 for each i, j, and we know that v3 = f3. So it remains to prove that p(4, 5) = p(5,4). But p(4,5) = 2b3 + b4 + 2b5 andp(5,4) = 2a3 + a4 + 2a5, so it suffices to verify that aj = bj for every i, and this is immediate from the proof of the theorem. Corollary 3.3. There is no (3,4, 5) -alternating graph with fewer than 17 vertices. Proof. Let r = v3 = f3. So there are r vertices of degree 3, r - 8 vertices incident with a triangle, two quadrilaterals and two pentagons, and 4 other vertices of degree 5, and correspondingly for the dual objects. An immediate consequence is that r is at least 8. The number of edges incident with a triangle is 3r. In addition, each of the r vertices of degree 3 is incident with an edge that is not incident with any triangles. The r edges thus obtained are all distinct, since each one is only incident with one vertex of degree 3. Hence, there are at least 4r > 32 edges, and the result then follows from Euler's formula. □ For larger r, we can get a better estimate for the number of edges than 4r by considering pentagons instead. There are r - 4 pentagons, contributing 5r - 20 edges, and as above, r distinct edges incident with a triangle, a quadrilateral, and a vertex of degree 3. This gives a lower bound of 6r - 20 edges. Any edges not counted so far must be incident with a triangle, a quadrilateral and two vertices of degrees 4 and 5. The number of such edges is bounded by the number of triangles, which gives an upper bound of 7r - 20 edges in total. Furthermore, since we have that the number of edges is 3f3 +4f4 + 5f5 2 , it follows that the number of quadrilaterals is in the interval [r - 5, §r - 5]. 3.2 X, Y-alternating plane graphs Definition 3.4. An alternating plane graph is called an X, Y-alternating plane graph if there are exactly X different vertex degrees and Y different face sizes. Let di,..., dx be the different degrees of vertices sorted in ascending order: 3 < di < d2 < . .. < dx, and let s1,..., sY be the different sizes of faces sorted in ascending order: 3 < S1 < S2 < . . . < SY. Again let vdi, resp. fSi, be the number of vertices with degree di, resp. faces with size si. We denote the total number of vertices, resp. edges and faces, by V, resp. E and F. This means we have X X J^Vdi = V J^diVdi =2 E i=1 i=1 YY E fsi = F E Sifsi =2E I Si 1 i=1 Substituting this in Euler's formula gives X Y X Y ^Y, Vdi fsi = Y; di Vdi + Y; Sifsi + 8 i=1 i=1 i=1 i=1 which simplifies to XY E(4 - di)vdi + E(4 - Si)fsi =8. (3.9) Since these are all positive numbers and di and si are at least 3, this formula gives us that at least one of d1 and S1 is equal to 3. It is immediately clear that X and Y are at least 2. Assume that G is a 2, Y-alternating plane graph. This means that there are only two different vertex degrees and they form a 2-colouring of the vertices. So G is bipartite and thus contains no odd cycles. This also implies that all si are even and thus s1 = 3. This gives us that d1 = 3. Substituting this information in (3.9) gives Y Vd! + (4 - d2)vd2 + E(4 - Si)fSi = 8. (3.10) i=1 <0 Note that d1 = 3 also implies that Y is at least 3. If X = 2, we also have that d1Vdl = d2Vd2 = E, since each vertex is only adjacent to vertices with a different degree. So we have that 3Vdi = d2Vd2. (3.11) Combining (3.10) and (3.11), we find that 2Y (4 - -d2)Vd2 +E(4 - Si)fsi = 8. 3 i=1 <0 From this it follows that 4 - 2 d2 > 0, and so d2 = 4 or d2 = 5. This means that all 2, Y-alternating plane graphs have degrees 3 and 4 or degrees 3 and 5. We can find lower bounds for the number of vertices in 2, Y-alternating plane graphs. We can rewrite (3.10) as Y vdl + (4 - d2)vd2 = 8 + - 4)fSi. (3.12) i= 1 Since d1 = 3, there are at least 3 different face sizes and all face sizes are even, since the graph is bipartite. This means that the left hand side in (3.12) is at least 14 (one face of size 4, one of size 6 and one of size 8). So we find: vdi +(4 - d2)vd2 > 14. (3.13) If d2 = 4, then (3.13) implies that n1 > 14 and, combined with (3.11), this also implies n2 > 11, so we get: V = vdi + vd2 > 25. If d2 = 5, then (3.13) implies that vdl - vd2 > 14. Combined with (3.11), this implies that vdl > 35 and vd2 > 21, so we get: V = vdi + vd2 > 56. This shows that the minimum order of 2, Y-alternating plane graphs lies out of reach of the exhaustive search described in Section 4. However, the restrictions imposed on the relation between the number and size of faces and the number and degree of vertices are quite strong, so we conjecture that there exist no 2, Y-alternating plane graphs. The situation changes if we relax the definition of alternating plane graph to also allow for vertices of degree 2. This is explained in Section 9. 4 Exhaustive search In this section we describe the exhaustive search that was used to verify the minimality of the two alternating plane graphs with 17 vertices. The algorithm described here checks each plane graph with a given number of vertices for being an alternating plane graph. The number of plane graphs however increases too fast with increasing number of vertices to be able to verify all graphs up to 17 vertices in an acceptable time span (see Table 1). Therefore we apply several bounding criteria which prune the graphs so that not all plane graphs need to be verified individually. We use the algorithm described in [2] to generate plane graphs with a given number of vertices. In this algorithm plane graphs are generated by starting from triangulations and removing edges of triangles to obtain the other plane graphs. That each plane graph can be constructed by this algorithm can be realised by looking at the reverse process. We can recursively add an edge between two vertices at distance 2 in a face of size greater than 3. This can be done until we end up with a triangulation. In order to avoid isomorphic graphs the algorithm in [2] uses McKay's canonical construction path method. A plane graph G is the parent of a plane graph G', if in the algorithm described above G' is obtained from G by removing an edge. A plane graph G is an ancestor of a plane graph G' if there exist plane graphs G1,...,Gn such that G is the parent of G1, Gi is the parent of Gi+1 for 1 < i < n and Gn is the parent of G'. n Graphs 4 1 5 2 6 9 7 48 8 429 9 4 794 10 64 968 11 954 362 12 14 791 881 13 237 306 720 14 3 910 739 201 15 65 870 458 907 16 1 130 289 662 773 17 19 709 446 129 094 Table 1: The number of 1-connected plane simple graphs with n vertices. This table was constructed using plantri. The next lemmas follow immediately from the fact that a vertex degree can only decrease during the process above and that the minimum degree is 3. Lemma 4.1. A plane graph with two adjacent vertices of degree 3 is never an ancestor of an alternating plane graph. Lemma 4.2. A plane graph with a triangle with vertices of degree 3, 4 and 4 is never an ancestor of an alternating plane graph. These two lemmas have been used to implement a modification of the program plantri in order to only generate alternating plane graphs. As can be seen in Table 2, Lemma 4.1 gives the most restrictions on the generation process. This motivates our choice to implement the modifications to generate alternating plane graphs as follows. After an edge is removed we check whether we can prune the graph based on Lemma 4.1. During an edge removal only two vertices change their degree. If either of these vertices have degree 3, we need to check whether there are any neighbours with degree 3. Then the algorithm from [2] continues and checks whether the edge removal was canonical. If this test also passes, we then check to see whether the graph can be pruned based on Lemma 4.2. When the algorithm finds a graph that it wants to output, we still need to check it for being an alternating plane graph. However, the number of graphs that remain to be checked, is considerably smaller than when just using such a filter on the unmodified algorithm from [2]. The results of the exhaustive search are shown in Table 3. As can be seen, the running time increases greatly near the end of the table. To obtain these results the jobs were split into several parts. These were run on 2.26 GHz Intel Xeon Nehalem processors and 2.6 GHz Intel Xeon Sandy Bridge processors. The time needed per job is not evenly distributed: some jobs were finished in less than a minute, while other jobs still needed more Lemma 4.1 Lemma 4.2 Lemma 4.1 and Lemma 4.2 n Graphs Ratio Graphs Ratio Graphs Ratio 4 1 100.0% 1 100.0% 1 100.0% 5 1 50.0% 1 50.0% 1 50.0% 6 3 33.3% 6 66.7% 3 33.3% 7 14 29.2% 30 62.5% 13 27.1% 8 105 24.5% 273 63.6% 85 19.8% 9 1 039 21.7% 2 901 60.5% 786 16.4% 10 13 073 20.1% 37 549 57.8% 9 164 14.1% 11 179 961 18.9% 533 883 55.9% 119 395 12.5% 12 2 616 640 17.7% 8 034 607 54.3% 1 664 062 11.2% 13 39 229 044 16.5% 125 435 404 52.9% 24 075 368 10.1% 14 601 955 195 15.4% 2 013 603 025 51.5% 358 017 589 9.2% 15 9 410 493 660 14.3% 33 047 399 191 50.2% 5 438 015 472 8.3% 16 149 488 913 702 13.2% 552 519 039 867 48.9% 84 066 660 749 7.4% 17 2 408 166 869 587 12.2% 9 385 351 956 659 47.6% 1 319 262 418 144 6.7% Table 2: The influence of Lemma 4.1 and Lemma 4.2 on the number of graphs. The ratio shows what percentage of graphs remain to be checked for being an alternating plane graph compared to the total number of plane graphs on n vertices. than a week. This uneven distribution makes it difficult to also obtain the results for 20 vertices even when we split the generation into many jobs. 5 Heuristic searches This section describes the implementation of the algorithm, which was used to find (3,4,5)-alternating plane graphs with 17, 20, 21,..., 41,42 vertices and also some other alternating plane graphs with certain sought properties. The basic idea of the algorithm is to grow a (3,..., x)-alternating plane graph with exactly N vertices by starting with a smallest face (e.g., a triangle) as the current graph and then systematically adding one face at at time "at the border" of the current graph, using backtracking when it becomes obvious that no solution (or no better solution than the best solution found so far) can be found. Note that the border of the faces added so far is also a face of the graph, which we call the exterior face. We call the other faces interior faces. The degree of already created interior faces never changes during the construction and adjacent interior faces always have different degree, i.e., fulfill the face constraint. Vertices adjacent only to interior faces (no longer at the border) are called interior vertices. Their degree also cannot change and so they always have to have different degree (fulfill the vertex constraint). The recursive search takes a graph represented by the faces added so far and the current border and then recursively tries all possible ways to add a new face at the border. Listing 1 gives an overview. The recursive search first checks, if any interior vertices violate the vertex constraint. In that case it backtracks, because the degree of interior vertices does not change by adding faces at the border and so the vertex constraint cannot be fulfilled. Then a lower bound on the number of edges needed to fix vertex constraint violations between border vertices or between border and interior vertices is computed. If the algorithm has already found a recursiveSearch ( currentGraph , remaining number of vertices) { return , if interior vertices violate the vertex — constraint or if no solution with less edges than the best solution found so far is possible . probe hashtable check, if a solution has been found generate all feasible branches s o r t them fo r branches add face to graph recursiveSearch remove added face remember current graph in hashtable }_ Listing 1: Recursive search generateBranches(currentGraph) { for s = start — vertex of the new face for e = end—vertex of the new face for n = 0...N— nVertices new vertices to add g e n e r ate b ranch n—p ath b etween s and e g e n e r ate b ranch n—p ath b etween e and s check if the new branches are feasible } Listing 2: Generating branches solution, it does not search for solutions with more edges. Now, the algorithm probes a hashtable to check if an equivalent graph has been visited previously by the recursive search. The hash table stores the current border (number of vertices, degree of each vertex, degree of each (interior) face on border edges) and the number of vertices and edges used so far. However, it does not store the vertex indices of the border vertices or any interior vertices/faces of the current graph. So on the one hand, the hashing only detects some isomorphisms, but on the other hand, it prevents the algorithm from extending on a current graph, which only differs from a previously searched graph (with identical border vertex degrees and faces) at irrelevant interior vertices/faces. Unlike the exhaustives search in Section 4 the algorithm does not generate all alternating plane graphs but it can be used to find at least one alternating plane graph with sought properties (e.g., a (3,4,5)-alternating plane graph with 17 vertices). If the current graph is a valid alternating plane graph with the sought properties (e.g., a (3,4, 5)-alternating plane graph with 17 vertices), it is returned. Otherwise the search continues by generating all feasible branches. Each feasible branch adds one face at the border of the current graph by adding a single edge or a n-path between two border vertices. The program can be run to search the whole searchspace or (when just trying to find a graph) as a beamsearch. In beamsearch mode, successor nodes of the search are ordered heuristically and only the best X successors are searched. Listing 2 shows how branches are generated. For all pairs of border-vertices s and e and each number n of vertices to be added, generateBranches() tries to generate two branches. The first connects s to e by a n-path and the second connects e to s by a n-path. A branch is feasible if • the degree of the new face is valid (e.g., 3 < degree < 5 for a (3,4, 5)-alternating plane graph ), and • the face-constraint between the new face and interior faces is valid. At the root of the search tree (when adding the second face to the first face), some extra rules are used to prevent generating (too many) isomorphic graphs. For n < 17, the number of nodes needed to fully search (3,4,5)-alternating plane graphs with n vertices is roughly four times that of n — 1. For n =18 and n =19 that factor increases to about 25. This behaviour is caused by the diminishing effect of the hashtable on the number of nodes searched, when the size of the graph increases. 6 List of constructed alternating plane graphs As can be seen in Table 3, the exhaustive search shows that the smallest alternating plane graphs have 17 vertices. Both are (3,4,5)-alternating plane graphs, have 8 triangles, 5 quadrangles and 4 pentagons, and are self-dual. Figure 2 on the left shows the graph found by Frank Schneider using the heuristic search and on the right the graph found in Ghent using the exhaustive search. The minimality of the 17-vertex graphs has been confirmed by an independent implementation. n Graphs Time 4 0 0.0 s 5 0 0.0 s 6 0 0.0 s 7 0 0.0 s 8 0 0.0 s 9 0 0.0 s 10 0 0.0 s 11 0 0.2 s 12 0 2.1 s 13 0 31.4 s 14 0 « 7.9 min 15 0 « 2.0 hours 16 0 « 1.3 days 17 2 « 16.4 days 18 0 « 301.3 days 19 5 « 13.1 years Table 3: The number of 1-connected alternating plane graphs found by the exhaustive search described in Section 4. For the largest orders, the jobs were split into several parts and the cumulated running time is given. These were run on 2.26 GHz Intel Xeon Nehalem processors and 2.6 GHz Intel Xeon Sandy Bridge processors. Š » Tt- ад si -3 си ад U д Its m S u л 11 For weak alternating plane graphs it is clear that 2, Y-weak alternating plane graphs do exist, e.g., take a 3-regular plane graph, substitute each edge by a digon and then subdivide each edge by a vertex. A first result we can prove is that there exists no weak alternating plane graph with degrees 2 and k for k > 11. We do this in two steps. Lemma 9.2. There exists no weak alternating plane graph with vertex degrees 2 and k for k > 12. Proof. Let G be a weak alternating plane graph with degrees 2 and k. Let v,e,f respectively denote the number of vertices, the number of edges and the number of faces. Euler's Formula says: v + f = e + 2. (9.1) Let fj denote the number of faces with j sides. Since the graph is bipartite, all f j for j odd are 0. Due to the definition of weak alternating plane graph, f2 is also equal to 0. Let ers denote the number of edges between r-faces and s-faces. It is rfr _E er,s. (9.2) s>2 We denote this sum by Sr, so (9.2) is equivalent to f _ Sr Jr — . r If we look at all the sums, then we see that er,s occurs two times for each pair (r, s): namely in fr and in fs, so we have: f _ E fr _ E (^ + ef ). (9.3) r> 2 4 vk = k. (9.7) Combining (9.6) and (9.7), we find ч2 k Putting v- and f-values together gives via (9.5) and (9.8) 1 + 1 ) e. (9.8) v+f < G + k)e+12e = ( k+ 12)e. (9.9) If we combine (9.1) and (9.9), we get e+2 < (k + i2)e, which is equivalent to 2+12e < ke. (9.10) For all k > 12, inequality (9.10) does not hold. □ In order to show that there exist no weak alternating plane graph with vertex degrees 2 and 11, we first need the following lemma. Lemma 9.3. A plane multigraph containing no faces of size 2 has a vertex of degree at most 5. Proof. Let G be a plane multigraph containing no faces of size 2. Each face contains at least three edges, and each edge is contained in two faces. If we combine this with Euler's Formula, we get e < 3v - 6. (9.11) Let S be the minimum degree of G. Each vertex is incident to at least S edges. Each edge contains 2 vertices. This gives Sv < 2e. Combining this with (9.11), we find 12 < (6 - S)v. Since v is positive, this means that S is at most 5. □ Lemma 9.4. There exists no weak alternating plane graph with vertex degrees 2 and 11. Proof. Let G be a weak alternating plane graph with vertex degrees 2 and 11. Let G' be the graph obtained from G by smoothing out the vertices of degree 2, i.e., removing each vertex v of degree 2 and connecting the vertices that were neighbours of v. This graph G' can be a multigraph, but since G was a weak alternating plane graph, there are no neighbouring faces of size 2 in G'. All vertices in G' have degree 11. Let G'' be the graph obtained from G' by replacing each face of size 2 by a single edge. This means that G'' is a plane multigraph containing no faces of size 2 and all vertices have degree at least 6. This is a contradiction with Lemma 9.3. □ Theorem 9.5. There exists no weak alternating plane graph with vertex degrees 2 and k for k > 11. Proof. This follows immediately from Lemma 9.2 and Lemma 9.4. □ 9.2 Existence for vertex degrees 2 and k with k < 10 If a 2, Y-weak alternating plane graph exists with degrees 2 and k, then we can show the following result. Lemma 9.6. Let G(V, E) be a 2, Y-weak alternating plane graph with degrees 2 and k. The number of vertices \V \ is a multiple of f+2. So if k is even, then \V \ is a multiple of fY2 and if k is odd, then \ V \ is a multiple of k + 2. Proof. Denote by V2 the set of vertices with degree 2 and by Vf the set of vertices with degree k. Since each edge is incident to exactly one vertex of each degree, we have that \E\ = 2\V2\ = k\Vfc\. So we find that \V\ = \V>\ + \Vfc\ = f \Vfc \ + \Vfc \ = ^ \Vfc\. □ There is a bijection between the weak alternating plane graphs with degrees 2 and k and the vertex-alternating k-angulations with minimum degree 2. Take any weak alternating plane graph with degrees 2 and k. First we smooth out the vertices of degree 2, i.e., we remove the vertex and the two incident edges and connect the two remaining endpoints by a new edge which replaces the two removed edges in the cyclic order for each of the two endpoints. This operations gives a k-regular, plane multigraph that is face-alternating. If we take the dual of this, then we get a vertex-alternating k-angulations with minimum degree 2. The other way around, it is clear to see that applying the inverse of this process to a vertex-alternating k-angulations with minimum degree 2 always leads to a weak alternating plane graph with degrees 2 and k. We used this bijection to generate weak alternating plane graphs with degrees 2 and k. For k = 3 and k = 4, we generated k-angulations using the program plantri and filtered out those k-angulations that are vertex-alternating. For 5 < k < 10, we used the data obtained in [4] and filtered out those k-angulations that are vertex-alternating. For k = 9 and k = 10, the available data was not sufficient to find any weak alternating plane graphs with degrees 2 and k. The results are shown in Table 5. Although no weak alternating plane graphs with degrees 2 and 10 were found using the exhaustive method described in the previous paragraphs, it is clear that they exist due to the following construction for weak alternating plane graphs with degrees 2 and k from 2 -regular plane graphs for k even. Take a f -regular plane graph. Replace each of its edges by a digon. This results in a k-regular, face-alternating, plane multigraph. Finally subdivide Figure 8: An infinite family of 5-regular plane graphs that can be used to construct weak alternating plane graphs with degrees 2 and 10. Figure 9: The first two members of an infinite family of 5-regular plane graphs that can be used to construct weak alternating plane graphs with degrees 2 and 9. A face-alternating matching is shown in bold in each graph. each edge with a vertex to obtain a weak alternating plane graph with degrees 2 and k. Since there exist infinite families of 2-regular plane graphs (the cycles), 3-regular plane graphs (e.g., the prisms), 4-regular plane graphs (e.g., the anti-prisms) and 5-regular plane graphs (e.g., the family shown in Figure 8), this implies that there are infinitely many weak alternating plane graphs with degrees 2 and 4 (respectively 2 and 6, 2 and 8, and 2 and 10). A similar construction can also be used to find weak alternating plane graphs with degrees 2 and 9. A face-alternating matching is a matching in a plane graph that has the property that for each edge e in the matching, e is incident with two faces with distinct sizes. Take a -regular plane graph together with a face-alternating matching. Replace each of its edges that is not in the matching by a digon. This results in a k-regular, face-alternating, plane multigraph. Finally subdivide each edge with a vertex to obtain a weak alternating plane graph with degrees 2 and k. Since there exist infinite families of 3-regular plane graphs with face-alternating matchings (e.g., the prisms on 4n vertices with n > 3), 4-regular plane graphs with face-alternating matchings (e.g., the anti-prisms on 4n vertices with n > 2) and 5-regular plane graphs with face-alternating matchings (e.g., the family shown in Figure 9), this implies that there are infinitely many weak alternating plane graphs with degrees 2 and 5, respectively 2 and 7, and 2 and 9. n\k 5 6 7 8 9 12 15 16 18 20 21 24 25 27 28 30 32 33 35 36 39 40 42 45 48 50 51 55 43 316 2 420 19 648 165 724 1 437 049 1 1 2 7 19 43 125 368 1 264 4 744 18 723 78 657 338 945 1 518 480 139 4 731 11 10 1 83 3 4 1 4 1 0 1 6 7 1 1 0 1 1 Table 5: The number of weak alternating plane graphs with degrees 2 and k on n vertices found using the technique described in Section 9. Due to Lemma 9.6 the orders are always integers and multiples of k^2. 10 Conjectures and open problems • As was explained in Section 3, one conjecture which our intuition suggests is the following. Conjecture 10.1. There are no 2, Y-alternating plane graphs and no X, 2-alternat-ing plane graphs. • What the typical parameters are for large alternating plane graphs is still an open problem. E.g., if we let r be the number of vertices of degree 3 in a (3,4,5)-alternating plane graph, then we know from Theorem 3.2 that the number of vertices of degree 4 is in the interval [r - 5, 3r - 5]. The question is, given this interval, how are the alternating plane graphs distributed. Is there a density function on the interval [1,1.5] which gives the asymptotic fractions of (3,4,5)-alternating plane graphs for large vertex numbers n? If so, what does the density function look like? • The exhaustive search showed that there are no (3,4,5)-alternating plane graphs on less than 17 vertices and on 18 and 19 vertices. The heuristic search found (3,4,5)-alternating plane graphs on all numbers of vertices from 20 to 42. In Section 8 we showed that (3,4,5)-alternating plane graphs exist on all numbers of vertices starting from 111, but the same construction can also construct (3,4,5)-alternating plane graphs on n vertices for n g [21,..., 24] U [39,..., 45] U [57,..., 66] U [75,..., 87] U [93,..., 108]. This means that we do not know whether there exists a (3,4,5)-alternating plane graph on n vertices for n g [46,..., 56] U [67,..., 74] U [88,..., 92] U {109,110}. Conjecture 10.2. For all n > 20 there exist (3,4,5)-alternating plane graphs on n vertices. • In Section 7 we proved that there exist alternating plane graphs on n vertices for any n > 19. The alternating plane graphs that were constructed in that section are not 3-connected, and some are not 2-connected. The (3,4,5)-alternating plane graphs constructed in Section 8 and most of the alternating plane graphs mentioned in Section 6 are 3-connected. That is why we also pose the following conjecture. Conjecture 10.3. For any n > 19 there exists a 3-connected alternating plane graph on n vertices. 11 Concluding remarks One central experience of our investigations is that without computer help we would never have come this far. Only the union of machine power and human creativity together let us achieve the findings in this paper. All the graphs in this paper are also available through the website [8] and can be downloaded from House of Graphs [3] by searching for the keyword apg. Acknowledgements Kathrin Nimczick and Lisa Schreiber found the very first alternating plane graph back in February 2008. In those days, a long thread on alternating plane graphs started in the forum of online game server LittleGolem.net. Thanks to the people who contributed there, in particular to: wccanard (UK; he proposed the empty graph as an extremely small example of an alternating plane graph; he also immediately mentioned graphs with vertices of degree 2) and to FatPhil (Finland), Carroll (France), Hjallti (Belgium). The whole thread can be found at [9]. We would like to thank Mohammadreza Jooyandeh for providing us with the data obtained in [4]. The computational resources (Stevin Supercomputer Infrastructure) used to create Table 3 were provided by Ghent University, the Hercules Foundation and the Flemish Government - department EWI. References [1] J. L. R. Alfonsin, The Diophantine Frobenius Problem, Oxford University Press, Oxford Lecture Series in Mathematics and Its Applications, 2005. [2] G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem. 42(4) (2007), 909-924. http://cs.anu.edu.au/$\sim$bdm/index. html. [3] G. Brinkmann, J. Goedgebeur, H. Melot and K. Coolsaet, House of Graphs: a database of interesting graphs, Discrete Applied Mathematics 161 (2013), 311-314. http://hog.grinvin. org [4] M. Jooyandeh and B. D. McKay, Recursive Generation of k-Angulations, (preprint). [5] K. Scherer, A Puzzling Journey to the Reptiles and Related Animals, Published by the author, 1987. [6] K. Scherer, NUTTS And other Crackers, Published by the author, 1994. http:// karlscherer.com/Mybooks/bknintro.html [7] K. Scherer, New Mosaics, Published by the author, 1997. [8] http://www.althofer.de/alternating-plane-graphs.html [9] http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=1\&topic= 1814 ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 365-379 On global location-domination in graphs* Carmen Hernando Dept. of Applied Mathematics I, Universität Politècnica de Catalunya, Barcelona, Spain Merce Mora Dept. of Applied Mathematics II, Universitat Politecnica de Catalunya, Barcelona, Spain Ignacio M. Pelayo f Dept. of Applied Mathematics III, Universitat Politecnica de Catalunya, Barcelona, Spain Received 23 December 2013, accepted 8 February 2015, published online 29 May 2015 A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number A(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G. The global location-domination number Ag (G) is introduced as the minimum cardinality of a global LD-set of G. In this paper, some general relations between LD-codes and the location-domination number in a graph and its complement are presented first. Next, a number of basic properties involving the global location-domination number are showed. Finally, both parameters are studied in-depth for the family of block-cactus graphs. Keywords: Domination, global domination, locating domination, complement graph, block-cactus. Math. Subj. Class.: 05C35, 05C69 * Research partially supported by projects MTM2012-30951/FEDER, Gen. Cat. DGR 2014SGR46, MTM2011-28800-C02-01, Gen. Cat. DGR 2009SGR1387. t Corresponding author. E-mail addresses: carmen.hernando@upc.edu (Carmen Hernando), merce.mora@upc.edu (Merce Mora), ignacio.m.pelayo@upc.edu (Ignacio M. Pelayo) Abstract 1 Introduction Let G = (V, E) be a simple, finite graph. The open neighborhood of a vertex v e V is NG(v) = {u e V : uv e E} and the close neighborhood is NG[v] = {u e V : uv e E} U {v}. The complement of a graph G, denoted by G, is the graph on the same vertices such that two vertices are adjacent in G if and only if they are not adjacent in G. The distance between vertices v, w e V is denoted by dG(v, w). We write N(u) or d(v, w) if the graph G is clear from the context. Assume that G and H is a pair of graphs whose vertex sets are disjoint. The union G + H is the graph with vertex set V (G) U V (H) and edge set E(G) U E (H ). The join G V H has V (G) U V (H ) as vertex set and E (G) U E (H ) U {uv : u e v (G) and v e V (H )} as edge set. For further notation, see [6]. A set D С V is a dominating set if for every vertex v e V \ D, N (v) П D = 0. The domination number of G, denoted by 7(G), is the minimum cardinality of a dominating set of G. A dominating set is global if it is a dominating set of both G and its complement graph, G. The minimum cardinality of a global dominating set of G is the global domination number of G, denoted with Yg (G) [3, 4, 18]. If D is a subset of V and v e V \ D, we say that v dominates D if D С N (v). A set S С V is a locating set if every vertex is uniquely determined by its vector of distances to the vertices in S. The location number of G ß(G) is the minimum cardinality of a locating set of G [10, 12, 20]. A set S С V is a locating-dominating set, LD-set for short, if S is a dominating set such that for every two different vertices u, v e V \ S, N (u) n S = N (v) П S. The location-domination number of G, denoted by A(G), is the minimum cardinality of a locating-dominating set. A locating-dominating set of cardinality A(G) is called an LD-code [21]. Certainly, every LD-set of anon-connected graph G is the union of LD-sets of its connected components and the location-domination number is the sum of the location-domination number of its connected components. Notice also that a locating-dominating set is both a locating set and a dominating set, and thus e(G) < A(G) and 7(G) < A(G). LD-codes and the location-domination parameter have been intensively studied during the last decade; see [1, 2, 5, 8, 13, 15] A complete and regularly updated list of papers on locating dominating codes is to be found in [16]. A block of a graph is a maximal connected subgraph with no cut vertices. A graph is a block graph if it is connected and each of its blocks is complete. A connected graph G is a cactus if all its blocks are cycles or complete graphs of order at most 2. Cactus are characterized as those graphs such that two different cycles share at most one vertex. A block-cactus is a connected graph such that each of its blocks is either a cycle or a complete graph. The family of block-cactus graphs is interesting because, among other reasons, it contains all cycles, trees, complete graphs, block graphs, unicyclic graphs and cactus (see Figure 1). Cactus, block graphs, and block-cactus have been studied extensively in different contexts, including the domination one; see [7, 11, 17, 22, 23]. The remaining part of this paper is organized as follows. In Section 2, we deal with the problem of relating the locating-dominating sets and the location-domination number of a graph and its complement. Also, global LD-sets and global LD-codes are defined. In Section 3, we introduce the so-called global location-domination number, and show some basic properties for this new parameter. In Section 4, we are concerned with the study of the sets and parameters considered in the preceding sections for the family of block-cactus graphs. Finally, the last section is devoted to address some open problems. 2 Relating Л (G) to Л (G) This section is devoted to approach the relationship between A(G) and A(G), for any arbitrary graph G. Some of the results we present were previously shown in [13] and we include them for the sake of completeness. Notice that Ng(x) П S = S \ NG(x) for any set S С V and any vertex x e V \ S .A straightforward consequence of this fact is the following lemma. Lemma 2.1. Let G = (V, E) be a graph and S С V. If x, y e V \ S, then NG(x) n S = NG(y) n S if and only if NG(x) n S = NG(y) П S. As an immediate consequence of this lemma, the following result is derived. Proposition 2.2. If S С V is an LD-set of a graph G = ( V, E), then S is an LD-set of G if and only if S is a dominating set of G. Proposition 2.3 ([13]). If S С V is an LD-set of a graph G = (V, E ), then S is an LD-set of G if and only if there is no vertex in V \ S dominating S in G. Proof. By Proposition 2.2, S is an LD-set of G if and only if S is a dominating set of G. But S is a dominating set of G if and only if N-(u) n S = 0, for any vertex и e V \ S. This condition is equivalent to NG(u) n S = S for any vertex и e V \ S. Therefore, S is an LD-set of G if and only if there is no vertex и e V \ S such that S С NG(u), that is, there is no vertex in V \ S dominating S. □ Proposition 2.4 ([13]). If S С V is an LD-set of a graph G = (V, E) then there is at most one vertex и e V \ S dominating S, and in the case it exists, S U {и} is an LD-set of G. Proof. By definition of LD-set of G, there is at most one vertex adjacent to all vertices of S. Moreover, и is the only vertex not adjacent to any vertex of S in G. Therefore S U {и} is an LD-set of G and a dominating set of G. By Proposition 2.2, it is also an LD-set of G. □ Theorem 2.5 ([13]). For every graph G, |A(G) - A(G)| < 1. Proof. If S has an LD-code of G not containing a vertex dominating S, then S is an LD-set of G by 2.3. Consequently, Л (G) < Л (G). If S is an LD-code of G_with a vertex u e V \ S dominating S, then S U {u} is an LD-set of G by 2.4. Hence, Л^) < Л^) + 1. In any case, Л(й)-Л^) < 1. By symmetry, Л^)-Л(й) < 1,and thus |Л^)-Л(^)| < 1. □ According to the preceding result, for every graph G, Л^) e ^(G) -1, Л(^), Л(G) + 1}, all cases being feasible for some connected graph G. For example, it is easy to check that the star K1,n-1 of order n > 2 satisfies Л(К1,„_1) = Л(К1,„_1), and the bi-star K2(r, s), r, s > 2, obtained by joining the central vertices of two stars K1jr and K1jS , satisfies Л(K2(r~š)) = Л(K2(r, s)) - 1. We intend to obtain either necessary or sufficient conditions for a graph G to satisfy Л^) > Л^), i.e., Л^) = Л^) + 1. After noticing that this fact is closely related to the existence or not of sets that are simultaneously locating-dominating sets in both G and its complement G, the following definition is introduced. Definition 2.6. A set S of vertices of a graph G is a global LD-set if S is an LD-set of both G and its complement G. Certainly, an LD-set is non-global if and only if there exists a (unique) vertex u e V (G) \ S which dominates S, i.e., such that S C N (u). Accordingly, an LD-code S of a graph G is said to be global if it is a global LD-set, i.e. if S is both an LD-code of G and an LD-set of G. In terms of this new definition, a result proved in [13] can be presented as follows. Proposition 2.7 ([13]). If G is a graph with a global LD-code, then Л^) < Л^). Proposition 2.8. If G is a graph with a non-global LD-set S and u is the only vertex dominating S, then the following conditions are satisfied: 1. The eccentricity of u is ecc(u) < 2; 2. the radius of G is rad(G) < 2; 3. the diameter of G is diam(G) < 4; 4. the maximum degree of G is A(G) > Л^). Proof. If x e N (u), then d(u, x) = 1. If x e N (u), since S is a dominating set of G, then there exists a vertex y e S n N (x) C N (u). Hence, ecc(u) < 2. Consequently, rad(G) < 2 and diam(G) < 4. By the other hand, degG(u) = |NG(u)| > |S| = Л^), implying that A(G) > Л^). □ Corollary 2.9. If G is a graph satisfying Л^) = Л^) + 1, then G is a connected graph such that rad(G) < 2, diam(G) < 4 and A(G) > Л^). The above result is tight in the sense that there are graphs of diameter 4 and radius 2 (resp. A(G) = Л^)), verifying Л^) = Л^) + 1. The graph displayed in Figure 2 is an example of graph satisfying rad(G) = 2, diam(G) = 4 and Л^) = Л^) + 1, and the complete graph Kn is an example of a graph such that A(G) = Л^) and Л^) = Л^) + 1, since Л(Ю = n, Л^) = A(Kn) = n - 1. Figure 2: This graph satisfies: rad(G) = 2, diam(G) = 4, A(G) = 3, A(G) = 4 and {x, y, z} is a non-global LD-code. 3 The global location-domination number Definition 3.1. The global location-domination number of a graph G, denoted by Ag (G), is defined as the minimum cardinality of a global LD-set of G. Notice that, for every graph G, Ag(G) = Ag (G), since for every set of vertices S с V (G) = V (G), S is a global LD-set of G if and only if it is a global LD-set of G. Proposition 3.2. For any graph G = (V, E), A (G) < Ag (G) < A(G) + 1. Proof. The first inequality is a consequence of the fact that a global LD-set of G is also an LD-set of G. For the second inequality, suppose that S is an LD-code of G,i.e. |S| = A (G). If S is a global LD-set of G,then Ag (G) = A (G). Otherwise, there exists a vertex u g V\S dominating S and S U {u} is an LD-set of G. Therefore, Ag (G) < A(G) + 1. □ Corollary 3.3. For any graph G = (V, E), max{A(G), A (G)} < Ag (G) < min{A(G) + 1,A(G) + 1}. Corollary 3.4. Let G = (V, E) be a graph. • If A(G) = A(G), then Ag(G) = max{A(G), A(G)}. • If A(G) = A(G), then Ag (G) g {A(G), A(G) + 1}, and both possibilities are feasible. Proof. Both statements are consequence of Proposition 3.2. Next, we give some examples to illustrate all possibilities given. It is easy to check that the complete graph K2 satisfies 1 = A(K2) = A(K2) = 2 and Ag (K2) = A(K2); toe path P3 satisfies A(P3) = A(P3) = Ag(P3) = 2 and the cycle C5, satisfies A(Cs) = A(Gs) = 2 and Ag(C5) = 3. □ Proposition 3.5. For any graph G = (V, E), Ag (G) = A(G) + 1 if and only if every LD-code of G is non-global. Proof. A global LD-code of G is an LD-set of both G and G. Hence, if G contains at least a global LD-code, then Ag(G) = A(G). Conversely, if every LD-code of G is non-global, then there is no global LD-set of G of size A(G). Then, Ag (G) = A(G) + 1. □ As a consequence of Propositions 2.8 and 3.5, the following corollary holds. Corollary 3.6. If G is a graph with diam(G) > 5, then Ag (G) = A(G). We finalize this section by determining the exact values of A(G), A(G) and Ag(G) for some basic graph families. Lemma 3.7. If n > 7, then X(Cn) = A(P„) = A(Pn_i). Proof. Firsty, we prove that A(Cn) < A(Pn_1) and A(Pn) < A(Pn_1 ). Suppose that V(Pn_i) = {1, 2,..., n - 1} and E(Pn_i) = {(i,i + 1) : i = 1, 2, ...,n - 2} are the vertex set and the edge set of Pn_1, respectively. Assume that S is an LD-code of Pn_1 such that S does not contain vertex 1 neither n - 1 (it is easy to construct such an LD-code from those given in [1]). Since n - 1 > 6, S has at least 3 vertices and there is no vertex in V(Pn_1) \ S dominating S in Pn_1. Hence, S is an LD-set of Pn_1. Next, consider the graph G* obtained by adding to the graph Pn_1 a new vertex u adjacent to the vertices 2,3,..., n - 2, and may be to 1 or n - 1. Clearly, by construction, u is adjacent to all vertices of S in G* and there is no vertex in Pn_1 adjacent to all vertices in S. Therefore, S is an LD-set of G* and A(G*) < A(Pn_1). Finally, observe that if u is not adjacent to 1, neither to n - 1, then G* is the graph Cn and if u is adjacent to exactly one of the vertices 1 or n -1, then G* is the graph Pn, which proves the inequalities before stated. Lastly, we prove that A(Pn_1) < A(G), when G g {Pn, Cn}. Consider an LD-code S of G. Let x be the only vertex dominating S in G, if it exists, or any vertex not in S, otherwise. By construction, S is an LD-set of G - x, hence A(G - x) < A(G). To end the proof, we distinguish two cases. - If G is the cycle Cn, then G - x is the path Pn_1, implying that A(Pn_1) < A(Cn). - If G if the path Pn, then G-x is either the path Pn_1 or the graph Pr +Ps, with r, s > 1 and r + s = n -1 > 6. Since, A(Pr + Ps) = A(Pr ) + A(Ps) = |~2r/5~| +_[2s/5] > [2(r + s)/5| = A(Pn_1), we conclude that, in any case, A(Pn_1) < A(Pn). ^ Proposition 3.8. Let G be a graph of order n > 1. IfG belongs to the set {Pn, Cn, Wn, Kn, K1,n_1, Kr,n_r, K2(r,n - r - 2)}, then the values of A(G) and A(G) areknown andthey are displayed in Tables 1 and 2. Proof. The values of the location-domination number of all these families, except the wheels, are already known (see [1,13,21]). Next, let us calculate the values of the location-domination number for the wheels and for the complements of all these families and also, from the results previously proved, the global location-domination number of them. • For paths, cycles and wheels of small order, the values of A(G) and Ag (G) can easily be checked by hand (see Table 1). • If n > 7, then A(Wn) = A(Cn_1) = [2n_1, since (i) Wn = K V Cn_1, (ii) every LD-code S of Cn_1 is an LD-set of Wn, and (iii) every LD-code of Cn_1 is global. • A(Kn) = A(K1 + • • • + K1) = A(K1) + • • • + A(K1) = n. A(K1,n_1) = A(K1 + Kn_1) = A(K1) + A(Kn_1) = 1 + (n - 2) = n - 1. A(Krn_;) = A(Kr + Kn_r ) = A(Kr) + A(Kn_r) = (r - 1) + (n - r - 1) = n - 2, if 2 < r < n - r. The complement of the bi-star K2(r, s), with s = n - r - 2, is the graph obtained by joining a vertex v to exactly r vertices of a complete graph of order r + s and joining a vertex w to the remaining s vertices of the complete graph of order r + s. It is immediate to verify that the set containing all vertices except w, a vertex adjacent to v and a vertex adjacent to w is an LD-code of K2 (r, s) with n - 3 vertices. Thus, A(K2(r,s)) = n - 3. G P1 P2 P3 P4 P5 P6 C4 C5 C6 W5 W6 W7 Л(С) 1 1 2 2 2 3 2 2 3 2 3 3 1 2 2 2 2 3 2 2 3 3 3 4 Лд (G) = Лд (G) 1 2 2 2 3 3 2 3 3 3 3 4 Table 1: The values of Л (G), Л (G) and Лд (G) of small paths, cycles and wheels. For every n > 7, Л(Рп) = Л(Сп) = \Щ-21. This result is a direct consequence of Lemma 3.7 and the fact that Л(РГ1) = Л(Сп) = \2-1. • According to Lemma 3.7, Л(^) = Л(К + Cn-i) = Л(^) + Л(С--х) = 1 + Л(Рп-2) = 1 + \2(n - 2)/5l = \(2n + 1)/51. D Theorem 3.9. Let G be a graph of order n > 1. If G belongs to the set {Pn, Cn, Wn, Kn, K1,n-1, Kr,n-r, K2(r, n — r — 2)}, then Лд(G) is known and it is displayed in Tables 1 and 2. Proof. All the cases follow from Corollary 3.4, except K1n-1 and Kr n-r, which are trivial. □ G Pn Cn Wn Kn Kl,n-1 Kr,n -r K2 (r, n - r - 2) order n n > 7 n>7 n > 8 n > 2 n > 4 2 < r < n-r 2 < r < n -r-2 Л (G) г 2n 1 г 1 Г1 n - 1 n - 1 n- 2 n - 2 Л (G) г ^n- 1 г ^n- 1 г 1 n n - 1 n- 2 n - 3 Л8^) = = Л g (G) Г 1 Г 1 г1 n n - 1 n- 2 n - 2 Table 2: The values of Л(^), Л(^) and Лд (G) for some families of graphs. 4 Global location-domination in block-cactus This section is devoted to characterizing those block-cactus G satisfying Л(^) = Л(^) +1. By Proposition 2.7, this equality is feasible only for graphs without global LD-codes. We will refer in this section to some specific graphs, such as the paw, the bull; the banner P, the complement of the banner, P ; the butterfly and the corner L (see Figure 3). The block-cactus of order at most 2 are K1 and K2. For these graphs we have Л(^1) = ЛЮ = 1 and ЛЮ = 1 < 2 = Л(Ю. In [5], all 16 non-isomorphic graphs with Л(^) = 2 are given. After carefully examining all cases, the following result is obtained (see Figure 4). Proposition 4.1. Let G = (V, E) be a block-cactus such that Л(^) = 2. Then, Л(^) > Л(С1). Moreover, Л(С1) = Л^) + 1 = 3 if and only if G is isomorphic to the cycle of order 3, the paw, the butterfly or the complement of a banner. —<1С O-—<1 XI Paw Bull Banner, P P Butterfly Corner, L Figure 3: Some special graphs. A(G) = A(G) = 2 A(G) = 3 = A(G) + 1 n = 3 •—•—• < n=4 * * ' * n = 5 X Figure 4: All block-cactus with A(G) = 2. Next, we approach the case A(G) > 3. First of all, let us present some lemmas, providing a number of necessary conditions for a given block-cactus to have at least a non-global LD-set. Lemma 4.2. Let G = (V, E) be a block-cactus and S С V a non-global LD-set of G. If u G V \ S dominates S, then G[N(u)] is a disjoint union of cliques. Proof. Let x, y be a pair of vertices belonging to the same component H of G[N(u)]. Suppose that xy G E and take an x - y path P in H. Let z be an inner vertex of P. Notice that the set {u, x, y, z} is contained in the same block B of G. As B is not a clique, it must be a cycle, a contradiction, since degB (u) > 3. □ Lemma 4.3. Let G = (V, E) be a block-cactus and S С V a non-global LD-set of G. If u G V \ S dominates S and W = V \ N [u], then, for every vertex w G W, the following properties hold. i) 1 <| N (u) n N (w) |< 2. ii) If N (u) n N (w) = {x}, then x G S. iii) If N (u) n N (w) = {x, y}, then xy G E. iv) If w' G W and N (u) n N (w) = N (u) n N (w') = {x}, then w' = w. v) If w' G W, w' = w and |N(u)nN(w) | = |N(u)nN(w') | = 2, then N[w]nN[w'] = 0. Proof. i),ii),iii): | N (u) П N (w) |> 1 as S C N (u) and S dominates vertex w. If N (u) n N (w) = {x}, then necessarily x e S. Assume that | N (u) П N (w) |> 1. Observe that the set N[u] n N[w] is contained in the same block B of G. Certainly, B must be a cycle since uw e E. Hence, | N (u) n N (w) |= 2. Moreover, in this case B is isomorphic to the cycle C4, which means that, if V(B) = {u, x, y, w}, then xy e E. iv): If w' = w, then S n N(w) = S n N(w'), as S is an LD-set. v): Suppose that w = w', N (u) n N (w) = {x, y} and N (u) n N (w') = {z,t}. Notice that {x, y} = {z, t}, since S is an LD-set. If y = z, then the set {u, w, w', x, y, t} is contained in the same block B of G, a contradiction, because B is neither a clique, since uw e E, nor a cycle, as degG(u) > 3. Assume thus that {x,y} n {z, t} = 0. If either ww' e E or N(w) n N(w') = 0, then the set {u, w, w', x, y, z, t} is contained in the same block B of G, again a contradiction, because B is neither a clique, since uw e E, nor a cycle, as degG(u) > 4. □ Lemma 4.4. Let G = (V, E) be a block-cactus and S С V a non-global LD-set of G. If u e V \ S dominates S and W = V \ N[u], then • Every component of G[W] is isomorphic either to Ki or to K2. • If w, w' e W and ww' e E, then the set {w, w'} is contained in the same block, which is isomorphic to C5. Proof. Let w, w' such that ww' e E. According to Lemma 4.3, the set {u} U N[w] U N[w'] forms a block B of G, which is isomorphic to the cycle C5. In particular, no vertex of W \{w,w'} is adjacent to w orto w'. □ As a corollary of the previous three lemmas the following proposition is obtained. Proposition 4.5. Let G = (V, E) be a block-cactus and S С V a non-global LD-set of G. If u e V \ S dominates S, then every maximal connected subgraph of G such that u is not a cut-vertex is isomorphic to one of the following graphs (see Figure 5): a) u is adjacent to every vertex of a complete graph Kr, r > 1, and each one of the vertices of Kr is adjacent to at most one new vertex of degree 1; b) u is a vertex of a cycle of order 4, and each neighbor of u is adjacent to at most one new vertex of degree 1; c) u is a vertex of a cycle of order 5. In the next theorem, we characterize those block-cactus not containing any global LD-code of order at least 3. Theorem 4.6. Let G = (V, E) be a block-cactus such that A(G) > 3. Then, every LD-code of G is non-global if and only if G is isomorphic to one of the following graphs (see Figure 6): a) Ki V (Ki + Kr), r > 3; b) the graph obtained by joining one vertex of K2 with a vertex of a complete graph of order r + 1, r > 3; c) Kr+i, r > 3; Bi Bo u 1 Bk G Kr, r> 1 u (b) Figure 5: If Bi;..., Bk are the maximal connected subgraphs of G with vertex u not being a cut-vertex, each subgraph Bj is isomorphic to one of the graphs displayed in (a), (b), (c). Gray vertices are optional. d) the graph obtained by joining a vertex of K2 with one of the vertices of degree 2 of a corner; e) if we consider the graph Ki V (Kri + • • • + Krt ) and t' copies of a corner, with t +1' > 2 and ri,... ,rt > 2, the graph obtained by identifying the vertex u of Ki with one of the vertices of degree 2 of each copy of the corner. x » (a) r > 3 У (b) r > 3 u Kr (c) r > 3 (d) t') K r 1 i t) K rt (e) t +1' > 2, ri,...,rt > 2 Figure 6: Block-cactus with A (G) > 3 not containing any global LD-code. u u Proof. Firstly, let us show that none of these graphs contains a global LD-code. a) Let G be the graph showed in Figure 6(a). Observe that A (G) = r and, for every LD-code S, |S П {x, u}| = 1 and |S П Kr | = r — 1. Let w be the vertex of Kr not in S. If x G S, then S с N (u). Otherwise, if u G S, then S с N (w). b) Let G be the graph showed in Figure 6(b). Notice that A (G) = r and, for every LD-code S, x g S and |S n Kr | = r — 1. Hence , if S is an LD-code of G, then S с N (u). c) If G = Kn (Figure 6(c)), then G contains no global LD-code. d) Let G be the graph showed in Figure 6(d). Clearly, the unique LD-code of G is S = N (и). e) Let G be the graph showed in Figure 6(e). In this graph, every LD-code contains both vertices adjacent to vertex и in each copy of the corner and, for every i € {1,..., t}, fi — 1 vertices of Kri. Thus, for every LD-code S of G, S C N (и). In order to prove that these are the only graphs not containing any global LD-code, we previously need to show the following lemmas. Lemma 4.7. Let G = (V, E) be a block-cactus and S С V a non-global LD-set of G. If и € V \ S dominates S, then, for every component H of G[N(и)] of cardinality r, |V(H) n S)| =max{1,f — 1}. Proof. This result is an immediate consequence of Lemma4.2 ( G[N(и)] is a disjoint union of cliques), along with the fact that S is an LD-set. □ Given a cut vertex и of a connected graph G, let Ли be the set of all maximal connected subgraphs H of G such that (i) и € V (H ) and (ii) и is not a cut vertex of H. Observe that any subgraph of Ли can be obtained from a certain component of the graph G — и, by adding the vertex и according to the structure of G. Lemma 4.8. Let G = (V, E) be a block-cactus with Л (G) > 3 and let S С V be a nonglobal LD-set of G. If и € V \ S dominates S and the set Ли contains a graph isomorphic to one of the graphs displayed in Figure 7, then G has a global LD-code. (a) z (b) v z u -f 3 and let S С V be a nonglobal LD-set of G. If и € V \ S dominates S and the set Ли contains a pair of graphs Hi and H2 such that H1? H2 € {P2, P3}, then G has a global LD-code. Proof. If Hi is isomorphic to P3, with V(Hi) = {и, v, z} and E(H1 ) = {uv, vz}, then, according to Lemma 4.7, v € S and S' = (S \ {v}) U {z} is an LD-set de G having the same cardinality as S. Hence, S' is a global LD-code of G. If both Hi and H2 are isomorphic to P2, and V (H ) = {u,t} and E (Hi) = {ut}, then, according to Lemma 4.7, v e S and S' = (S \ {t}) U {u} is an LD-set de G having the same cardinality as S. Hence, S' is a global LD-code of G. □ Lemma 4.10. Let G = (V, E) be a block-cactus and S С V a non-global LD-set of G whose dominating vertex is u. If A„ contains three graphs Hi, H2 and H3 such that Hi e {P2, P3} and H2,H3 e {Kr, L}, where L denotes the corner graph displayed in Figure 3, then G has a global LD-code. Proof. If Hi is isomorphic to P2, with V(Hi) = {u, t} and E(Hi) = {ut}, then, according to Lemma 4.7, v e S and S' = (S \ {t}) U {u} is an LD-set de G having the same cardinality as S. Hence, S' is a global LD-code of G. If Hi is isomorphic to P2, V(Hi) = {u, v, z} and E(Hi) = {uv, vz}, then, according to Lemma 4.7, v e S and S' = (S \ {v}) U {z} is an LD-set de G having the same cardinality as S. Hence, S' is a global LD-code of G. □ We are now ready to end the proof of the Theorem 4.6. Suppose that G is a block-cactus such that every LD-code of G is non-global. Let S С V be an LD-code of G and let u e V \ S be a vertex dominating S. Notice that, according to Proposition 4.5, every graph of A„ is isomorphic to one of the graphs displayed in Figure 5. Moreover, having into account the results obtained in Lemma 4.8, Lemma 4.9 and Lemma 4.10, the set A„ is one the following sets: • {P2, Kr}. In this case, G is the graph shown in Figure 6(a). • {P3, Kr }. In this case, G is the graph shown in Figure 6(b). • {P2, L}. Let u, t be the vertices of P2. Then, according to Lemma 4.7, t e S, and S' = (S \ {t}) U {u} is a global LD-code of G. • {P3, L}. In this case, G is the graph shown in Figure 6(d). • {Kr}. In this case, G is the graph shown in Figure 6(c). • A set of cardinality at least two, being every graph isomorphic either to a clique or to a corner. In this case, G is a graph as shown in Figure 6(e). This completes the proof of Theorem 4.6. □ As an immediate consequence of Propositions 3.5 and 4.1 and Theorem 4.6, the following corollaries are obtained. Corollary 4.11. A block-cactus G satisfies Ag(G) = A(G)+1 ifandonlyif G is isomorphic either to one of the graphs described in Figure 6 or it belongs to the set {P2, P5, C3, C5, P, paw, bull, butterfly}. Corollary 4.12. Every tree T other than P2 and P5 satisfies A(T) = Ag (T). Corollary 4.13. Every unicyclic graph G different from the one displayed in Figure 6(d) and not belonging to the set {C3, C5, P, paw, bull} satisfies A(G) = Ag (G). If G is a block-cactus of order at least 2, we have obtained the following characterization. Theorem 4.14. If G = ( V, E ) is a block-cactus of order at least 2, then Л (G) = X(G) + 1 if and only if G is isomorphic to one of the following graphs (see Figure 8): (a) Ki V (Ki + Kr), r > 2; (b) the graph obtained by joining one vertex of K2 with a vertex of a complete graph of order r + 1, r > 2; (c) Kr+i, r > 1; (d) Ki V (Kn + • • • + Krt ), t > 2, ri,...,rt > 2. ~<0 <9 (a) r > 2 (b) r > 2 (c) r > 1 Figure 8: Block-cactus satisfying Л^) = Л^) + 1. Proof. Let us see first that all graphs described above satisfy Л^) < Л^). Recall that if W is a set of twin vertices of a graph G, then every LD-set must contain at least all but one of the vertices of W. Consider one of the graphs described in (a), G = K i V (K + Kr ), r > 2. The complement of G is the graph Ki + Kir. It is easy to verify that Л^) = r < r + 1 = Л^). If G is one of the graphs described in b), then Л^) = r < r + 1 = Л^). Finally, if G = Ki V +(Kri + • • • + Krt ) is a graph of order n, with t > 1 and ri,... ,rt > 2, then we have Л^) = n -1 - 1 3, by Proposition 2.7, G does not contain a global LD-code, and therefore it must be one of those graphs described in Theorem 4.6. Hence, it suffices to prove that the graphs described under items d) or e) with t' > 0, in Theorem 4.6, do not satisfy Л^) = Л^) + 1. The graph G described in item d) satisfies Л^) = Л^) = 3, since an LD-code of G is the set containing the three vertices adjacent to the three vertices of degree 1 in G and an LD-code of G is the set containing the three vertices adjacent to the three vertices of degree 3 in G. Finally, if G is one of the graphs described in item e) obtained from t copies of complete graphs and t' copies of corners, t' > 1, then the set of vertices including all but one vertex of each complete graph and the two vertices of degree 3 of each copy of the corner, is an LD-code of G. If we change exactly one of the vertices of degree 3 of a copy of the corner by the vertex of degree 2 in this copy, then we obtain an LD-code of G. Thus, A (G) = A (G) = 2t' + (n - 1) + • • • + (rt - 1). □ Corollary 4.15. Every tree T other than P2 satisfies A(T) < A(T). Corollary 4.16. Every unicyclic graph G not belonging to the set {C3, P, paw} satisfies A(G) < A(G). 5 Further research This work can be continued in several directions. Next, we propose a few of them. • We have completely solved the equality A (G) = A (G) + 1 for the block-cactus family. In [14], a similar study has been done for the family of bipartite graphs. We suggest to approach this problem for other families of graphs, such as outerplanar graphs, chordal graphs and cographs. • Characterizing those trees T satisfying A (T) = A (T) = Ag (T). • We have proved that every tree other than P2 and P5, every cycle other than C3 and C5, and every complete bipartite graph satisfies the equality A (G) = Ag (G). We propose to find other families of graphs with this same behaviour. References [1] N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, Eur. J. Combin., 25 (2004), 969-987. [2] M. Blidia, M. Chellali, F. Maffray, J. Moncel, A. Semri, Locating-domination and identifying codes in trees, Australas. J. Combin., 39 (2007), 219-232. [3] R.C. Brigham, J.R. Carrington, Global domination, in: T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Domination in Graphs, Advanced Topics, Marcel Dekker, New York, 1998, pp. 30-320. [4] R. C. Brigham, R. D. Dutton, Factor domination in graphs, Discrete Math., 86 (1-3) (1990), 127-136. [5] J. Caceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, Locating-dominating codes: Bounds and extremal cardinalities, Appl. Math. Comput., 220 (2013), 38-45. [6] G. Chartrand, L. Lesniak, P. Zhang, Graphs and Digraphs, fifth edition, CRC Press, Boca Raton (FL), 2011. [7] M. Chellali, Bounds on the 2-domination number in cactus graphs, Opuscula Math., 26 (1) (2006), 5-12. [8] C. Chen, R. C. Lu, Z. Miao, Identifying codes and locating-dominating sets on paths and cycles, Discrete Appl. Math., 159 (15) (2011), 1540-1547. [9] M. Ghebleh, L. Niepel, Locating and identifying codes in circulant networks, Discrete Appl. Math., 161 (13-14) (2013), 2001-2007. [10] F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combinatoria, 2 (1976), 191-195. [11] P. Heggernes, S. H. Sther, Broadcast domination on block graphs in linear time, Lecture Notes in Comput. Sci., 7353 (2012), 172-183. [12] C. Hernando, M. Mora, I. M. Pelayo, C. Seara, D. R. Wood, Extremal graph theory for metric dimension and diameter, Electron. J. Comb., 17 (2010), R30. [13] C. Hernando, M. Mora, I. M. Pelayo, Nordhaus-Gaddum bounds for locating domination, Eur. J. Combin., 36 (2014), 1-6. [14] C. Hernando, M. Mora, I. M. Pelayo, On global location-domination in bipartite graphs, submitted. [15] I. Honkala, T. Laihonen, S. Ranto, On locating-dominating codes in binary Hamming spaces, Discrete Math. Theor. Comput. Sci., 6 (2004), 265-282. [16] A. Lobstein, Watching systems, identifying, locating-dominating and discriminating codes in graphs, http://www.infres.enst.fr/~lobstein/debutBIBidetlocdom. pdf. [17] D. Rautenbach, L. Volkmann, The domatic number of block-cactus graphs, Discrete Math., 187 (1-3) (1998), 185-193. [18] E. Sampathkumar, The global domination number of a graph, J. Math. Phys. Sci., 23 (1989), 377-385. [19] J. Seo, P. J. Slater, Open neighborhood locatingdominating in trees, Discrete Appl. Math., 159 (6) (2011), 484-489. [20] P. J. Slater, Leaves of trees, Congressus Numerantium, 14 (1975), 549-559. [21] P. J. Slater, Dominating and reference sets in a graph, J. Math. Phys. Sci., 22 (1988), 445-455. [22] G. Xu, L. Kang, E. Shan, M. Zhao, Power domination in block graphs, Theoret. Comput. Sci., 359 (1-3) (2006), 299-305. [23] V. E. Zverovich, , The ratio of the irredundance number and the domination number for block-cactus graphs, J. Graph Theory, 29 (3) (1998), 139-149. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 381-408 Quartic integral Cayley graphs* Marsha Minchenko, Ian M. Wanless School of Mathematical Sciences, Monash University, Vic, 3800, Australia Received 19 June 2013, accepted 11 February 2015, published online 5 June 2015 We give exhaustive lists of connected 4-regular integral Cayley graphs and connected 4-regular integral arc-transitive graphs. An integral graph is a graph for which all eigenvalues are integers. A Cayley graph Cay(T, S) for a given group Г and connection set S с Г is the graph with vertex set Г and with a connected to b if and only if ba-1 G S. Up to isomorphism, we find that there are 32 connected quartic integral Cayley graphs, 17 of which are bipartite. Many of these can be realized in a number of different ways by using non-isomorphic choices for Г and/or different choices for S. A graph is arc-transitive if its automorphism group acts transitively on the ordered pairs of adjacent vertices. Up to isomorphism, there are 27 quartic integral graphs that are arc-transitive. Of these 27 graphs, 16 are bipartite and 16 are Cayley graphs. By taking quotients of our Cayley or arc-transitive graphs we also find a number of other quartic integral graphs. Overall, we find 9 new spectra that can be realised by bipartite quartic integral graphs. Keywords: Graph spectrum, integral graph, Cayley graph, arc-transitive, vertex-transitive bipartite double cover, voltage assignment, graph homomorphism. Math. Subj. Class.: 05C50, 05C25 1 Introduction We give exhaustive lists of connected 4-regular integral Cayley graphs and connected 4-regular integral arc-transitive graphs. For reasons which will become apparent, we first restrict our attention to the bipartite case. An integral graph is a graph for which all eigenvalues of the adjacency matrix are integers. The spectrum of a graph is the eigenvalues with their multiplicity. Bipartite graphs have eigenvalues that are symmetric with respect to 0 and r-regular graphs have largest eigenvalue r with multiplicity equal to the number of connected components. For * Research supported by ARC grant DP120100197. E-mail addresses: marsha.minchenko@monash.edu (Marsha Minchenko), ian.wanless@monash.edu (Ian M. Wanless) Abstract details we refer to [10]. Therefore for connected 4-regular bipartite integral graphs, the spectrum has the form {4, 3x, 2y, 1z, 02w, -1z, -2y, -3x, -4}; which we abbreviate by simply specifying the quadruple [x, y, z, w]. There are only finitely many connected 4-regular bipartite integral graphs. Cvetko-vic [6] proved that the diameter D of a connected graph satisfies D < s - 1, where s is the number of distinct eigenvalues. For connected r-regular integral graphs, it follows that R < D < 2r where R is the radius of the graph. Cvetkovic et al. [7] showed that the number of vertices in a connected r-regular bipartite graph is bounded above by (2(r - 1)R - 2)/ (r - 2) if r > 3. Therefore, connected 4-regular bipartite integral graphs have at most 6560 vertices. All graphs in this paper are simple, undirected, and have n vertices. Since a 4-regular graph is integral if and only if each of its components is integral, from this point on we will assume that all graphs are connected. We use the acronym QIG as shorthand for a connected quartic integral graph. Cvetkovic et al. [7] found quadruples [x, y, z, w] that are candidates for the spectrum of a bipartite QIG. They called these possible spectra. Research activities regarding the set of possible spectra fall into two streams: eliminate possible spectra based on new information and/or techniques, or find graphs that realize a possible spectrum. Useful tools include an identity by Hoffman [11] and equations relating the spectral moments to the closed walks of length I < 6. All QIGs that avoid eigenvalues of ±3 and realize a possible spectrum are found in [24]. Stevanovic [23] eliminates spectra using equations arising from graph angles. In the same paper he determines that the possible values for n are between 8 and 1260, except for 5 identified spectra. Stevanovic et al. [25] extend the equations for the ^-th spectral moment to an inequality for I = 8. They make use of a correspondence between closed walks in an r-regular graph and walks in an infinite r-regular tree and find recurrence relations for the number of closed walks. The upper bound for n is improved to give 8 < n < 560. Equations for I > 8 are found in [16] by counting a certain type of closed walk in terms of the counts of small subgraphs of the graph. All of the bipartite QIGs with n < 24 that realize one of the possible spectra were found and are listed with drawings in [25]. We give 12 new graphs that realize possible spectra from the set given in [25]. Of these graphs, 3 are co-spectral to an integral graph listed in [7]. Their spectra are [4,6,4, 5] and [6,16,10,3], and [9,16,19,0]. The spectra not previously known to be realized by a graph are [3,4,1, 6], [3, 5, 9,0], [5,4,7,4], [6,12, 2, 9], [8,10,16,1], [10,14,18, 2], [12, 28, 4,15], [22, 28, 34, 5], and [27, 28,49,0]. Of the 12 graphs, 3 appear in the census of Potocnik et al. [17, 18] but were not recognized as integral. We also list 49 new non-bipartite QIGs that, to our knowledge, do not appear anywhere in the literature. Of these graphs, only 3 appear in the census of Potocnik et al. [17, 18] but were not tested for integrality. A Cayley graph Cay(r, S) for a group Г and connection set S с Г is the graph with vertex set Г and with a connected to b if and only if ba-1 G S. Let Zt, Dt, and Qt denote the cyclic, dihedral, and quaternion groups of order t respectively, and St, At the symmetric and alternating groups of degree t. Klotz and Sander [12] showed that if every Cayley graph Сау(Г, S) over a finite Abelian group Г is integral then Г g {Z2, Z3, Z|, Z2 x Z3, Z2 x Z4}, where s > 1, t > 1. The analogous result for non-Abelian Г was determined independently by Abdol-lahi and Jazaeri [1] and Ahmady et al. [4]: if every Cayley graph Сау(Г, S) over a finite non-Abelian group Г is integral then Г g {S3, Z3 x Z4, Q8 x Z2}, where r > 0. Estelyi and Kovacs [8] considered the groups for which all Cayley graphs Сау(Г, S) over a group Г are integral if |S| < k. The authors proved that for k > 6, Г consists only of the groups above: {Z|, Z3, Z\, Zs2 x Z3, Z2 x U {S3, Z3 x Z4, Q8 x Z2}, s > 1, t > 1, r > 0. Moreover, for k G {4,5} there is only one extra possibility, namely that Г is the generalised dicyclic group with Z3q x Z6 as a subgroup of index 2, where q > 1. Abdollahi and Vatandoost [2] showed that there are exactly 7 connected cubic integral Cayley graphs. They found that Сау(Г, S) is integral for some S with |S| =3 if and only if Г is isomorphic to Z2 x Z2, Z4, Z6, S3, Zf, Z2 x Z4, D8, Z2 x Z6, Di2, A4, S4, D8 x Z3, D6 x Z4 or A4 x Z2. A set of possible orders for Cayley QIGs on finite Abelian groups have been determined by Abdollahi et al. [3]. They showed that for an Abelian group, Г, if Сау(Г, S) is a Cayley QIG then |Г| G {5, 6, 8, 9,10,12,16,18, 20, 24, 25, 32, 36, 40, 48, 50, 60, 64, 72, 80, 96,100,120,144}, but they did not establish whether Cayley QIGs of these orders exist. We find that the precise set of orders of Cayley QIGs on Abelian groups is {5, 6,8,9,10,12,16,18,24,36}. More generally, we consider all groups and find that many Cayley QIGs are on non-Abelian groups. Thus, we show that for any group Г, if Сау(Г, S) is a Cayley QIG then |Г| G {5, 6, 8, 9,10,12,16,18, 20, 24, 30, 32, 36,40,48, 60, 72,120}. Furthermore, for each of these orders Cayley QIGs exist. For a given Cayley graph G, there may exist many different pairs (Г, S) of groups Г and connection sets S such that G = Сау(Г, S). We call isomorphic Cayley graphs on the same group Г equivalent if their connection sets are from the same orbit of the automorphism group of Г (see for example [13]): Definition 1.1. Let Г be a group and Aut^) be the automorphism group of Г. If Cayley graph Сау(Г, S) = Сау(Г, T) and Sff = T for some a G Aut^) then Сау(Г, S) and Сау(Г, T) are equivalent. Any other connection sets give non-equivalent Cayley Graphs. Cayley graphs from different groups are non-equivalent. There are, up to isomorphism, only 32 connected quartic integral Cayley graphs; but each graph is realized in up to 18 non-equivalent ways. Of the 32 graphs, 17 are bipartite. A graph is arc-transitive if its automorphism group acts transitively on the ordered pairs of adjacent vertices. There are, up to isomorphism, only 27 connected quartic integral graphs that are arc-transitive. Of the 27 graphs, 16 are bipartite, 5 of which are not Cayley graphs. In Section 2 we find that most of the feasible spectra from [25] cannot be realized by vertex-transitive QIGs. Section 3 summarises the algorithm used for finding all of the bipartite Cayley QIGs. Section 4 gives our main results. It includes tables giving the details of the Cayley QIGs and the bipartite arc-transitive QIGs, some drawings, and some non-bipartite QIGs that result from finding quotients of our bipartite graphs. 2 Vertex-transitive quartic integral graphs A graph is vertex-transitive if its automorphism group acts transitively on its vertices. In this section, our aim is to compile a set S that includes all possible spectra that might be realized by a vertex-transitive QIG, but is otherwise as small as we can make it. Initially we take S to be all possible spectra from [25], and candidates will be progressively removed from the set as we work through this section. We will need some notation for (unlabelled) subgraphs. We let Ci denote the i-cycle, Cil.i2...ih denote j -cycles sharing a single vertex for j = 1,..., h, Cil-i2 an i i-cycle joined to an i2-cycle by an edge, and ©ilii2i...iih two vertices joined by internally disjoint paths of lengths i j for j = 1,..., h. Examples of this notation for subgraphs appear in Figure 1. (a) C4.3.3 (b) C4-3 (c) ©2,2,1 Figure 1: Subgraph notation If at any point we encounter subgraphs that cannot be described by our notation, we draw a picture of the subgraph like those in Figure 1. For any graph H, let [H] denote the number of subgraphs of G that are isomorphic to H, where the parent graph G will be implicitly specified by the context. In [7], Equations (2.1) and (2.2) are used to determine [C4] and [C6] for a given [x, y, z, w]. 2(44 + 34x + 24y + z)=28n + 8[C4], (2.1) 2(46 + 36x + 26y + z) = 232n + 144[C4] + 12[C6]. (2.2) In [16], these equations were extended to higher spectral moments of general regular graphs. By specialising to 4-regular bipartite graphs, we obtain the following equations: 2(48 + 38x + 28y + z) = 2092n + 2024[C4] + 288[C6] + 16[C8] + 32[C4.4] + 96[в2,2,2,2] + 48[в2,2,2] + 16[6s,3,l], 2(410 + 310x + 210y + z) = 19864n + 26160[C4] + 4860[C6] + 480[C8] + 20[Cio ] + 960[C4.4] + 40[C4-4] + 40[C6-4] + 1440[62,2,2 ] + 520[63,3,l] + 2880[62,2,2,2] + 40[64,2,2] + 20[65,3,l] + 120[63,3,3,l] + 120[64,2,2,2 ] + 120[^>] + 80[^ J]. (2.3) The girth of a graph is the length of the shortest cycle contained in the graph. We use Equations (2.3) to determine the girth where [C4] = [C6] =0 for a given [x, y, z, w] and also to determine the values for [C8] and [C10] where possible. Vertex-transitive graphs have the same number of i-cycles incident with each vertex, so the number of vertices divides i[Ci]. We apply this observation for i g {4,6,8,10} to the possible spectra for which the value of [Ci] can be deduced. We eliminate those quadruples that cannot be realized by a vertex-transitive QIG from S. For example, if we consider [5, 6,11,1] with n = 48, [—4] = 24, and [—6] = 140 then 4[—4] 4(24) _ , 6[—6] 6(140) 35 _ = I = 2 G N but ^ = = 35 G N, n 48 n 48 2 where N denotes the set of non-negative integers. Thus [5,6,11,1] is eliminated from S. In contrast, for [12,12, 20, 3] with n = 96, [C4] = 24, [C6] = 128, and [C8] = 528. We are able to find [C8] from (2.3) by deducing that [C4.4] = [©2,2,2,2] = [©2,2,2] = [©3,3,1] = 0 because there is only one 4-cycle incident with each vertex. In fact, 4[—4] 4(24) 6[C6] 6(128) 8[—8] 8(528) 4—1 = = 1 G N, = ^ ) =8 G N, and = - =44 G N. n 96 n 96 n 96 In this case, [—10] cannot be determined from (2.3), so we consider it unknown. Thus [12,12,20,3] remains in S. It is also plausible to eliminate quadruples from S using arguments specific to particular cases. We give one example to demonstrate the possibility. Consider [24,4,40,3] with [—4] = 72 and [—6] = 0. There are 4(72)/144 = 2 copies of —4 incident at each vertex. Since [—6] = 0, we know [©3,3,1] = 0. Also, with only two 4-cycles at each vertex, [©2,2,2,2] = [©2,2,2] = 0. Since two 4-cycles meet at exactly one vertex of a —4.4, [—4.4] = 144. From Equation (2.3) we get that, 2(48 + 38(24) + 28(4) + 40) = 2092(144) + 2024(72) + 16[—8] + 32(144), which gives the contradiction [—8] = -216. Thus we remove [24,4,40,3] from S. This entry is underlined in Table 1. We eliminate two quadruples from S using the following Lemma [5, Prop. 16.6]: Lemma 2.1. Let G be a vertex-transitive graph which has degree r and an even number of vertices. If Л is a simple eigenvalue of G, then Л is one of the integers 2a — r for 0 < a < r. The orders associated with the eliminated quadruples are 36 and 72. Both entries have 1 as a simple eigenvalue. These entries are underlined and highlighted in bold in Table 1. Using the above methods, we reduced the set S from the initial 828 possible spectra to 59 quadruples in the final version. Henceforth S will refer to this final set of 59 quadruples (see Appendix A). n Girth n Girth n Girth n Girth 8 4 36 4,4,4, q7 96 4,q27,h3 240 6,8,q30,h2 10 4 40 4,q10 112 q34,h2 252 q28,h3 12 4,4 42 4,q14 120 4,4,4,6,6,q28 280 8,q23,h2 14 q1 48 4,q12,h2 126 4,6,q38,h1 288 6,q21,h1 16 4,q1 56 q16,h2 140 q40,h2 336 q14,h2 18 4,q1 60 4,4,4,4,6,q15 144 4,4,6, q31, h1 360 6,6,8,q11 20 4,q3 70 6,q23 160 q33,h2 420 8,q5,h1 24 4,4,4,q3 72 4,4,4,4,6,q18 168 6,q35,h2 480 8,q2 28 q8 80 q22,h2 180 4,6,6,6,q38 504 h1 30 4,4,6,q6 84 q23,h7 210 6,q35,h2 560 10 32 6,q8 90 4,4,6,6,q27 224 q32,h3 Table 1: Finding the set S Table 1 summarizes the process of finding S. For every order, we consider each [x, y, z, w] and check whether we get integer counts at each vertex for each Q where [Q] is known and i e {4, 6,8,10}. A 'qj ' in the table denotes that for the given n there were j possible spectra eliminated because 4[C4]/n e N. An 'hj' in the table denotes that for the given n there were j possible spectra which satisfied 4[C4]/n e N that were eliminated because 6[C6]/n e N. If i[Cj]/n e N for all i where [Q] is known for a possible spectra, then the girth is recorded. Thus an entry of 4,4,6,q6 indicates that there are three possible spectra in S associated with that order. If those quadruples are all realized by graphs (where a graph in this case may actually be a set of cospectral graphs) then two graphs will have girth 4 and the other will have girth 6. It also indicates that 6 possible spectra with 4[C4]/n e N were eliminated because 6[C6]/n e N. 3 The algorithm In this section we outline our method for finding bipartite Cayley QIGs, using the set S compiled in Section 2. Define Q to be the set of orders associated with the spectra in S. Cayley graphs are vertex-transitive, so we only consider groups Г of order n e Q. To reduce the number of groups to be considered, we use a result similar to one in [18]. Let Г' denote the commutator subgroup of a group Г. Lemma 3.1. Let Г be a finite group and let Сау(Г, S) be a connected Cayley graph of degree at most 4. Then Г/Г' is isomorphic to one of Z2 x Z2 x Z2 x Z2; Z2 x Z2 x Za with a > 2; Za x Zb with a, b > 2; or Za with a > 1. Proof. Since Сау(Г, S) is connected and has degree at most 4, Г is generated by an inverse-closed set of at most 4 elements. This must also be true of the quotient group Г/Г'. Now since Г/Г' is Abelian, the result follows. □ By Lemma 3.1, we need only consider groups Г with Г/Г' isomorphic to one of Z2 x Z2 x Z2 x Z2, Z2 x Z2 x Za, Za x Zb, or Za. We denote the set of groups that satisfy this property by Ф. To construct connected simple undirected 4-regular Cayley graphs Сау(Г, S), we considered inverse-closed sets S of four non-identity elements of Г that generate Г. The search was pruned by placing additional restrictions on S. Let g denote the girth of the graph Cay(r,S). • Since Сау(Г, S) is bipartite, the order of s is even for each s g S. • If si, s2 g S and si = s-1, then the order of sis2 is at least g/2 (in particular non-involutions have order no smaller than the girth). • For any set of connection sets that result in equivalent Cayley graphs (in the sense of Definition 1.1), only one representative is chosen. We note that the minimum girth possible for Сау(Г, S) is given by Table 1. We summarize the results of our computations in Table 2. The values for n g О appear as the first column and in the second column the number of groups of order n is given. (We reiterate that О does not include orders eliminated by the vertex-transitive tests of Section 2). The number of groups in Ф of order n are listed in column three. Column 4 contains the number of connection sets S among the groups counted by column 3, subject to the restrictions on S given above. The graphs Сау(Г, S) that are bipartite are counted in column 5. The number of isomorphism classes of these graphs appears in column 6. The number of isomorphism classes of integral graphs is recorded in column 7. The last column gives the number of isomorphism classes of arc-transitive integral graphs. A '-' indicates that there are no integral graphs to consider. n #Groups #Г e Ф #Sets #Bipartite #Isomorphism #Integral #Arc- Г S Cay(r,S) Classes Transitive 8 5 5 13 7 1 1 1 10 2 2 2 2 1 1 1 12 5 5 19 11 3 2 1 16 14 14 66 44 5 1 1 18 5 5 12 12 5 1 1 20 5 5 34 20 8 0 24 15 15 151 98 23 3 1 30 4 4 31 31 17 1 1 32 51 48 58 51 16 1 1 36 14 14 149 105 48 1 1 40 14 14 201 146 54 1 0 42 6 6 55 55 36 0 - 48 52 51 840 616 177 1 0 60 13 13 385 281 161 0 - 70 4 4 96 96 73 0 - 72 50 49 1014 765 338 2 1 90 10 10 236 236 175 0 - 96 231 218 4434 3545 1292 0 - 120 47 47 2833 1968 1123 1 1 126 16 16 427 427 346 0 - 144 197 190 6563 5350 2722 0 - 168 57 57 2388 2212 1601 0 - 180 37 37 2927 2497 1883 0 - 210 12 12 1172 1172 1017 0 - 240 208 205 10884 9885 6791 0 - 280 40 40 4080 3929 3223 0 - 288 1045 968 26391 24815 15695 0 - 360 162 160 15928 14703 11524 0 - 420 41 41 10558 10204 9271 0 - 480 1213 1148 68179 63804 48322 0 - 560 180 177 21764 21433 18704 0 - Table 2: Results at each algorithm step 4 Quartic integral graphs In this section we present the graphs that our computations discovered, starting with the bipartite Cayley case. 4.1 Bipartite Cayley integral graphs As a result of the computation described in Section 3, we have: Theorem 4.1. There are precisely 17 isomorphism classes of connected 4-regular bipartite integral Cayley graphs, as detailed in Table 3. For each bipartite Cayley QIG in Table 3 we give n and the spectrum [x,y, z,w]. Graphs appearing in the paper by Cvetkovic et al. [7] are labelled In,index as in that paper. If the graph is in the census of Potocnik et al. [17, 18] then we give the index in their notation: AT4Val[n][index]. In two columns, we give the groups and connection sets that give rise to each Cayley graph. The first column contains the group, Г, with a presentation of that group. We stick as close as possible to the convention of using generators in {a, b, c, d, e} for cyclic groups, {s, t, u, v} for symmetric or alternating groups, and {r, f} for the quaternion group, the dihedral group, or the quasidihedral group. The last column contains the number of involutions in the connection set, S, followed by the connection set itself in terms of the generators from the previous column. Group Connection Sets (#involutions S) Gì : n = 8 [0, 0, 0, 3] I8,ì AT4Val[8][1] Zs < a | a8 > 0 {a,a3,a5,a7} Z4 X Z2 < a | a4 > X < b | b2 > 2 {a,b,a3,a2b} 0 {a, ab, a3, a3b} Ds < r, f | r4,f2, (rf)2 > 4 {f,fr, fr2,rf } 2 {f, r, fr2,frf } Qs < r, f | r4,f4,r2f2,rfrf 1 > 0 {r,f,r3,r2f} Z2 X Z2 X Z2 < a | a2 > X < b | b2 > X < c | c2 > 4 {a, b, c, abc} G2 : n = 10 [0, 0, 4, 0] Iì0,ì AT4Val[10][2] D10 < r, f | r4,f2, (rf)2 > 4 {f,fr, fr2,r2f } Z10 < a | a10 > 0 {a,a3,a7,a9} G3 : n = 12 [0, 2, 0, 3] Iì2,4 AT4Val[12][2] Z3 x Z4 < a,b | a3, b4, abab-1 > 0 {b,b3,ba,b3a} Z12 < a | a12 > 0 {a,a5,a7,a11} D12 < r, f | r6,f2, (rf)2 > 2 {r2f,f,r5,r} 4 {r4f,rf,r2f,r5f} Za X Z2 < a | a6 > X < b | b2 > 0 {a5,a2b,a,a4b} G4 : n = 12 [0,1, 4, 0] Il2,2 D12 < r, f | r6,f2, (rf)2 > 2 {rf,r3,r,r5} 4 {rf,r3,r5f,r3f } 4 {rf, r4f,r5f,r3f} Za X Z2 < a | a6 > X < b | b2 > 2 {a3,b,a5,a} G5 : n = 16 [0, 4, 0, 3] Ii6,i AT4Val[16][1] Z4 X Z4 < a | a4 > X 0 {a,b,a3,b3} (Z4 X Z2) X Z2 < a,b,c | a4,b2 ,c2, aba~1b~1, (aac)2, (bc)2, baca-1c > 2 {ac, a2bc, a3bc, a2c} 2 {bc, a3b, a2c, ab} 0 {a, a3c,a3,abc} Z4 x Z4 0 {a,a3ba,a3,b} Z8 X Z2 < a,b | a8,b2,aba3b > 0 {a,ab,a3b,a7} QDi6 2 {r, r4 f,r6f,r7} Z4 X Z2 X Z2 < a | a4 > X < b | b2 > X < c | c2 > 2 {a,b,c,a3} Z2 X Ü8 < a | a2 > X < r, f | r4,f2, (rf )2 > 4 {a, f, r3f, r2 f} 4 {f, r3f, af, rf} 4 {f, r3f, af, arf } 2 {a, r, f, r3 } 2 {a, r, af, r3} Z2 X Z2 X Z2 X Z2 < a | a2 > X < b | b2 > X < c | c2 > X < d | d2 > 4 {a, b, c, d} G6 : n = 18 [0, 4, 4, 0] Ii8,i AT4Val[18][2] Z3 X S3 < a | a3 > X < s,t | s2, t3, (st)2 > 2 {s, st, ats, a2ts} 0 {sa, sa2, sat, sa2t} (Z3 X Z3) X Z2 < a, b, c | a3,b3 ,c2, aba-1b-1, (ac)2, (bc)2 > 4 {c, ca, cb, cab} Za X Z3 < a | a6 > X < b | b3 > 0 {a, a5, a3b, a3b2} Gr : n = 24 [0, 8, 0, 3] 124,2 AT4Val[24][1] Z4 X S3 < a | a4 > X < s, t | s2, t3, (st)2 > 2 {s, st, at, a3sts} (Za X Z2) X Z2 < a,b,c | a6 ,b2 ,c2, aba-1b-1, (aac)2, a3 (cb)2 > 2 {a3c,a2c,ab,a5b} Z3 X Ü8 < a | a3 > X < r, f | r4, f2, (rf )2 > 0 {ar3 f, a2r3 f, ar3,a2r} S4 4 {st2sts,t2st,stst2s,tst2} 0 {ts,st2,ststs, tst} 2 {st2sts,tst2,ts,st2} Z2 X A4 < a | a2 > X < s,t | s2,t3, (st)3 > 0 {ast, astst, asts, atst} Z2 X Z2 X S3 < a | a2 > X X < s,t | s2,t3, (st)2 > 4 {s, bs, st, ast} Gs : n = 24 [2, 2, 6,1] Ы,з Z4 X S3 < a | a4 > X < s,t | s2,t3, (st)2 > 2 {a,a3,s,st} 2 {s, st, ats, a3ts} D24 4 {f,rf,r5f,r6f } 2 {f,r3,r9,rsf } Z2 X (Z3 X Z4) < a | a2 > X < b,c | b3,c4, bcbc-1 > 0 {c,c3,ab,ac3bc} (Za X Z2) X Z2 4 {c, b, ca, cbc} 2 {c, bcb, ba, bcac} 2 {c, ca, cbcac, bac} 0 {cb, bc, ba, bcac} Z12 X Z2 < a | a12 > X < b | b2 > 0 {a3,a9,a4b,asb} Z3 X Ds < a | a3 > X 2 {r3f, rf, a2 f, af } 0 {r,r3,ar3f,a2r3f } Z2 X Z2 X S3 < a | a2 > X X < s,t | s2,t3, (st)2 > 4 {s, b, a, st} 4 {s, b, st, ats} 4 {s, sb, ast, ats} 2 {s, b, at, asts} 2 {s, sb, at, asts} Za X Z2 X Z2 < a | a6 > X < b | b2 > X < c | c2 > 2 {a3,b, a2c, a4c} G9 : n = 24 [3, 0, 5, 3] Ы,4 S4 4 {s,t2st, st2sts, stst2st} Z2 X A4 < a | a2 > X < s,t | s2,t3, (st)3 > 2 {a, as, at2s, ast} G10 : n = 30 [0,10, 4, 0] Iso,i AT4Val[30][4] Z5 X S3 < a | a5 > X < s,t | s2,t3, (st)2 > 0 {as, a2st, a4s, a3st} D30 4 {f,r2f,r3f,r11 f} Gii : n = 32 [0,12, 0, 3] Is2,i AT4Val[32][4] Zs x Z4 0 {a,a7, ab, a3b3} (Zs x Z2) x Z2 < a,b,c | as,b2 ,c2 ,a2ba6b, (aac)2, (bc)2 ,ba-1cac > 2 {a4c, a2c, a7bc, a5c} Z2.((Z4 X Z2) X Z2) = (Z2 X Z2).(Z4 X Z2) 0 {ba,a3b,a3,a5} (Z4 X Z4) X Z2 Z4.DS = Z4.(Z4 X Z2) 0 {a, a7 ,a7ba, a4b} (Z4 X Z4) X Z2 < a,b,c | a4,b4,c2,aba-1b-1, (ac)2, (bc)2 > 4 {c, cb, ca, abc} (Zs X Z2) X Z2 < a,b,c | as,b2 ,c2, aba-1b, aca-1c, a4bcbc, (bc)4 > 2 {b, c, abc, a3bc} Z2 X QD16 < a | a2 > X < r,f | rs,f2,rfr5f > 2 {ar, r3,r5, r2f } (Zs X Z2) X Z2 < a, b, c | as,b2,c2,aba-1b-1, (ac)2, a4(bc)2 > 4 {a7c, a2c, ac, a4b} (Z2 X Ds) X Z2 < a,r,f,b | a2 ,r4, f2 ,b2, aba-1b-1 ,r(fa)2, r2(bf )2 > 4 {r2f,ar2,rf,rab} (Z2 X Qs) X Z2 2 {r2b, a, ar3fb, ar3b} < a,r,f,b | a2,r4,f4,b2,ara-1r-1,afa-1f-1,r2f2,rfrf-1, (rrb)2 ,r2 (ab)2 ,brbr-1 f > (Z2 X Qs) X Z2 4 {b, a, br, brf} < a,r,f,b | a2,r4,f4,b2,ara-1r-1,afa-1f-1,r2f2,rfrf-1, (rb)2, fbf-1b-1,arbar-1b > G12 : n = 36 [4, 4, 4, 5] 1зв,з AT4Val[36][3] Z3 X (Z3 X Z4) < a | a3 > X < b,c | b3,c4,bcbc-1 > 0 {ac, a2c3,a2cb,ac3b} (Z3 X Z3) X Z4 < a,b,c | a3,b3,c4,aba-1b-1, (acc)2 ,acac-1b-1 > 0 {a2b2c3,b2c,a2bc3,ac} S3 X S3 < s,t | s2,t3, (st)2 > X 4 {u, s, uv, st} 4 {u, uv, svu, stvu} 2 {su, stu, tsv, tsuvu} 0 {tu, stsu, sv, suvu} Za X S3 < a | a6 > X < s,t | s2,t3, (st)2 > 2 {a,a5,a3s,a3st} 2 {a3s, a3st, a2ts, a4ts} 0 {as, a3t, a5s, a3sts} Z2 X ((Z3 X Z3) X Z2) < a | a2 > X < b,c,d | b3,c3,d2,bcb-1c-1, (bd)2, (cd) 4 {d, dc, adb, adcdbd} 2 > 2 {d, dc, ab, adbd} Za X Za < a | a6 > X 0 {a,a5,b,b5} G13 : n = 40 [4, 6, 4, 5] Z2 X (Z5 X Z4) < a | a2 > X < b,c | b5,c4,cbc-1b2,cb2c-1b-1 > 2 {ac2, ac2b, c, c3} Gi4 : n = 48 [6, 4,10, 3] I48,1 Z2 X Z4 X S3 < a | a2 > X < b | b4 > X < s,t | s2,t3, (st)2 > 2 {s, a, bt, b3sts} Ds X S3 < r,f | r4,f2, (rf)2 > X < s,t | s2,t3, (st)2 > 4 {s, rfs, fts, r2fst} 4 {rf, rfs, fts, r2fst} 2 {s, rfs, r3t, rsts} 2 {rf, rfs, r3t, rsts} Z2 X ((Za X Z2) X Z2) < a | a2 > X < b, c, d b6, c2, d2, bcb-1 c-1, (bbd)2, b3 (dc)2 > 4 {a, c, b4d, b3d} | Za X Ds < a | a6 > X < r,f | r4,f2, (rf)2 > 2 {a3,a3r3f,ar,a5r3} Z2 X S4 < a | a2 > X < s,t | s2,t3, (st)4 > 4 {a, s, stst2s, st2sts} 4 {as, at2st, atst2, stst2st} 2 {s, astst2, atst2s, atst2st} Z2 X Z2 X A4 < a | a2 > X < b | b2 > X < s,t | s2,t3, (st)3 > 2 {a,abtst2, abtst, abt2st2} Z2 X Z2 X Z2 X S3 < a | a2 > X < b | b2 > X X 4 {s, b, cst, ats} ,t | G15 : n = 72 [6,16,10, 3] AT4Val[72][12] Z3 X S4 < a | a3 > X < s,t | s2,t3, (st)4 > 0 {ast, a2tst, a2t2s, aststs} (Z3 X A4) X Z2 4 {atb, ab, tsbt, tbs} < a,s,t,b | a3, s2, t3, b2, asa-1s-1, ata-1t-1, stbsbt-1, (ab)2, (tb)2,(st)3 > A4 x S3 0 {tu, t2u, tsuv, st2uv} < s, t | s2 ,t3, (st)3 > X < u, v | u2 ,v3, (uv)2 > Za X A4 < a | a6 > X < s, t | s2, t3, (st)3 > 0 {ast,a3t,a3t2,a5stst} G16 : n = 72 [8,10,16,1] (Z3 X (Z3 X Z4)) X Z2 < a,b,c,d | a3 ,b3, c4, d2, aba-1b-1, aca-1 c-1, bdb-1d-1, adad-1 ,bcbc-1 ,c2d2 > 2 {dc, dacb, ad, d2ad} (Za X S3) X Z2 < a, b, c, d | a2,b4,c3, d3, (ab-1)2,acac-1, (ad)2, cbcb-1, bdb-1d-1, cdc-1d-1 > 4 {a,ab2, abd, abacda} 2 {ab, abcd, cb, b2cb} Za X (Z3 X Z4) < a | a6 > X < b,c | b3, c4,bcbc-1 > 0 {ab2,a5b,a3b2c,a3b2c3} Z3 X ((Za X Z2) X Z2) < a | a3 > X < b,c,d | b6, c2,d2,bcb-1c-1, (bbd)2,b3(dc)2 > 2 {b5d, b2d, a2b4c,ab2c} 0 {b2cd,b5cd,a2b4c, ab2c} (S3 X S3 ) X Z2 < s, t, u, v, a | s2, t3,u2,v3,a2,tvt-1v-1, (uv)2, (av)2, svst-1 ,asasu > 4 {a, sastsat, s, sast} 2 {sas, stsa,atsa,asat2} 2 {asa, atsat2, sastst, asastsat} 0 {sa, as, atsat, asastst2 } Z2 X ((Z3 X Z3) X Z4) < a | a2 > X < b,c,d | b3, c3,d4,bcb-1c-1, (bdd)2, bdbd-1 c-1 > 2 {ad2, ab2c2d2, ab2d3, ab2cd} 0 {ab2cd,ab2d3,abc2,ab2c} Z2 X S3 X S3 < a | a2 > X < s, t | s2, t3, (st)2 > X 4 {u, s, auvst, atsvu} 4 {u, au, suv, stvu} 2 {u, s, atv, astsuvu} Z2 X Za X S3 < a | a2 > X X < s, t | s2, t3, (st)2 > 2 {ts,ab3ts,bsts,b5t} Gir : n = 120 [12, 28, 4,15] AT4Val[120][4] S5 < s,t | s2,t5,(st)4,(st2st3)2 > 0 {t2st3, tst2st2st, st2stst, tst4} 4 {t(st)2tst4, st2(st)2t,(t2s)2ts, (st2)2st} Z2 X A5 < a | a2 > X < s, t | s2, t3, (st)5 > 2 {a(tst2s)2t,ast(ts)2 ,ast2(st)2, a(st)3ts} S3 X (Z5 X Z4) < s, t | s2, t3, (st)2 > X 2 {sb2,stb2a,tb3,stsb} Z5 X S4 < a | a5 > X < s, t | s2, t3, (st)4 > 0 {a2st, a3t2s, aststs, a4tst} (Z5 X A4) x Z2 4 {tba2, bta, btbsb, tabs} < a, s,t,b | a5, s2, t3, b2, asa-1 s-1, ata-11-1, bsbt-1 st, (st)3, (tb)2, (ab)2 > Table 3: Bipartite Cayley QIGs Drawings for all but the three largest bipartite Cayley QIGs appear below. With over 70 vertices, it is difficult to present Gi5, Gi6, and Gi7 clearly. Gì G2 G3 Gi Gii Gl2 Gi Gi Table 4: Drawings of quartic bipartite integral Cayley graphs G1 to G14 4.2 Bipartite arc-transitive integral graphs We considered all arc-transitive 4-regular graphs from the census of PotoCnik et al. [17, 18] and tested them for integrality. The only arc-transitive bipartite QIGs that are not Cayley and thus not accounted for in Table 3 are the five that appear in Table 5. We let [Г : H] = {Ha : a e Г} denote the set of right cosets of H e Г. A Schreier coset graph Sch(r, H, HS H ) for a group Г, subgroup H < Г, and connection set S с Г is the graph with vertex set [Г : H] and with Ha connected to Hb if and only if ba-1 e HSH. We represent these 5 graphs as Schreier coset graphs. We give the order n and the spectrum [x, y, z, w] followed by the graph index from [17, 18]. Graphs appearing in the paper by Cvetkovic et al. are labelled with the notation of [7]: In,inđex. The first line consists of the group Г, with a presentation of that group. The second line consists of the subgroup H and its generators in terms of the generators of Г followed by the connection set S in terms of the generators of Г. Group Subgroup, Subset Fi : n = 60 [4,16, 4, 5] Ieo,i AT4Val[60][4] Z2 X Z2 X S5 : < a | a2 > x < b | b2 > x < s,t | s2,t5, (st)4, (st2st3)2 > D8 : < bstst2st-1,abstst >, {s,bt2, s4,bt-2} F 2 : n = 70 [6,14,14, 0] Iro,i AT4Val[70][4] Z2 X S7 : x < s,t | s2 ,t7, (st)6, (st2st5)2, (stst-1)3 > S3 X S4 : < t2st-2,t-2st-1(st)2t,t2(st)2(ts)3t-1,t(st)2(ts)3,stst-1s >, {ast4, atstst-1, at, at-1} F3 : n = 90 [9,16,19, 0] Igo,i AT4Val[90][1] Z2 X PrL(2, 9): x (Z2 X D8) x Z2 : < yzxy-1x,x-1yxzx-1y,x2zy-1x-1y-1xy >, {ayx-1y-1x, az, ayxy-1x, axy-1x-1y} F4 : n = 180 [22, 28, 34, 5] AT4Val[180][12] Z2 XS3 XS5 : x < s,t | s2,t3, (st)2 > x < u,v | u2,v5, (uv)4, (uv2 uv3)2 > D8 : < v-2uv2 ,vuv2uv2 ,astuvuv2uv-1 >, {at-1v-2 ,atuv2 ,su,atv2} F5 : n = 210 [27, 28, 49, 0] AT4Val[210][10] S7 : S4 : , {t3s,st4, (st)3, (ts)3} Table 5: Bipartite arc-transitive non-Cayley QIGs The census [17, 18] of arc-transitive graphs contains all arc-transitive graphs with at most 640 vertices. Thus, the upper bound of 560 given in [25] for the order of a bipartite QIG, ensures that Table 3 and Table 5 contain all bipartite arc-transitive QIGs. The non-bipartite arc-transitive QIGs will be given in Sections 4.4 and 4.5. However, first we describe our method for finding all Cayley QIGs. 4.3 Integral graphs as quotients Let V (G) denote the vertices of a graph G, and E (G) the unordered pairs of vertices which are edges of G. A homomorphism from a graph G to a graph H is a map V (G) ^ V (H) which preserves adjacency. Each homomorphism induces an edge map E(G) ^ E(H). If the vertex and edge maps of the homomorphism are both surjective then we say that H is a quotient of G. In this section we find new integral graphs that are quotients of the integral graphs found in Table 3 and Table 5. To specify a quotient of a graph G it suffices to know G and the vertex map (the edges of the quotient are implied by the surjectivity of the edge map). We start by considering special classes of possible homomorphisms. A voltage assignment a for a graph G is a function from the arcs of G to a group Г such that a((u, v)) = a((v, u))-1 for all {u, v} G E (G). The derived graph Vol(G, Г, a) is the graph with vertex set the Cartesian product V(G) x Г with (u, x) connected to (v, y) whenever {u, v} g E (G) and y = x * a((u, v)), where * is the group operation of Г. Projection onto the first coordinate, by definition, maps the derived graph of Vol(G, Г, a) onto G, and this map is a surjective homomorphism. Hence G is a quotient of the derived graph. As an interesting example for quartic integral graphs, we found a voltage assignment a for which the derived graph Vol(F1, Z3, a) is isomorphic to F4. Thus, F1 is a quotient of F4. Given two graphs Gi, G2 with vertex sets V(G1), V(G2), let Gì x G2 be the graph with vertex set the Cartesian product V(G1) x V(G2) with (ui, U2) adjacent to (wi, W2) whenever both u1 is adjacent to w1 in G1 and u2 is adjacent to w2 in G2. The bipartite double cover of G is the bipartite graph G x K2 where K2 denotes the complete graph on two vertices. Equivalently, G x K2 is the derived graph Vol(G, Z2, a), where a is the constant function assigning 1 to every arc of G. We give an example for quartic integral graphs that was also noted in [7]. An odd graph Oi is the graph with one vertex for each of the (i - 1)-element subsets of a (2i - 1)-element set and with edges joining disjoint subsets. The graph F2 is the bipartite double cover of the integral graph O4. Similar to the result by Schwenk [20] used in [24] and [25], we have that if G is a QIG, then the bipartite double cover of G is a bipartite QIG. If G is a bipartite QIG then the bipartite double cover consists of two disjoint copies of G. For this reason, we have restricted our search to integral graphs that are bipartite up to this point. However, we now want to find all graphs which have their bipartite double cover among the bipartite graphs that we have discovered. This requires us to find quotients of our bipartite graphs. Since it is computationally easy to do, we will actually consider a more general class of homomorphisms than what is required for the task just described. This will increase the number of quartic integral graphs that we find. However, we make no effort to be exhaustive in finding all possible quotients. A graph automorphism is k-semiregular if all its orbits have the same size, k. Note that if G = H x K2 then the natural homomorphism from G onto H maps orbits of a 2-semiregular automorphism of G to single vertices of H. With this as motivation, the class of homomorphisms that we consider is the following. We identify any k-semiregular automorphism, ß of a target graph G. Our homomorphism is to collapse each orbit of ß to a single point. We wrote a routine in Magma [21] to find such quotients of a target graph G, as follows. For one representative, ß, of each conjugacy class of (nontrivial) semiregular automorphisms of G, we collapsed the orbits of ß to single vertices to obtain a quotient H. If H was a 4-regular graph we checked to see if it was integral. If it was, then we printed it out and called the routine recursively on H. In some cases we were only interested in finding those H for which G is a bipartite double cover. In such instances, it suffices to only consider 2-semiregular automorphisms and we do not need to make recursive calls to the routine. We applied our Magma routine to all target graphs Gi for i g 1,..., 17 and to most of the arc-transitive graphs from the census of Potocnik et al. [17, 18]. There are graphs in the census with extremely large automorphism groups, and they were impractical for our simple routine. So we decided to only include target graphs from the census if their automorphism group had order no more than 220. The results of our Magma routine will be given in the following subsections. 4.4 Non-bipartite Cayley integral graphs In this section we report all quartic Cayley integral graphs that are not bipartite. We rely on this Lemma: Lemma 4.2. If G is a 4-regular Cayley graph then G x K2, the bipartite double cover of G, is isomorphic to a 4-regular Cayley graph. Proof. If G = Cay(r, S) then we define G' = Cay(r x Z2, {(s, 1) : s G S}). This graph G' is an undirected Cayley graph. It is not hard to verify that G' is isomorphic to G x K2 which gives the desired result. □ Hence we can find all the graphs we seek by applying the Magma routine of Section 4.3 to our graphs Gj where i = 1,..., 17. We use the following result by Sabidussi [19] to decide which of the graphs that we find are Cayley graphs: Lemma 4.3. A graph G is a Cayley graph if and only if Aut(G) contains a regular subgroup. Initial Graph #Non-bipartite #Cayley #Vertex-transitive #Arc-transitive Gi 0 0 0 0 G2 1 1 1 1 G3 1 1 1 1 G4 0 0 0 0 G5 1 1 1 0 G6 2 1 1 1 Gr 2 1 1 1 G8 4 2 2 0 G9 0 0 0 0 G10 1 0 1 1 G11 0 0 0 0 G12 2 1 1 0 G13 1 1 1 0 G14 2 2 2 0 G15 5 1 1 1 G16 13 2 2 0 G17 2 1 1 0 Table 6: Non-bipartite graphs found for Gj Table 6 summarizes our results using the Magma routine of Section 4.3 when restricted to the 2-semiregular automorphisms for each given Gì . We give the number of non-bipartite graphs found, followed by the numbers of those that are Cayley, vertex-transitive, and arc-transitive. The non-bipartite graphs counted in column 2 up to row 8 of Table 6 were previously found by Stevanovic et al. [25]. All graphs counted by column 2 from rows 9 to 17 were previously unknown with the exception of the graph with bipartite double cover Gio and one of the two graphs with bipartite double cover G14. In Table 7, we expand upon the counts of non-bipartite Cayley graphs in column three of Table 6 by producing a breakdown of the groups and the connection sets of the underlying graphs. We follow the same conventions as in Table 3 except that we use different notation for the spectrum, since there is no longer symmetry about the origin. Group Connection Sets (#involutions S) Hi : n = 5 - 14, 41 I5,i AT4Val[5][1] Z5 < a | a5 > 0 {a3,a2,a4,a} H2 : n = 6 - 22, 03, 41 Ie,i AT4Val[6][1] S3 2 {st,t,t2,s} Za < a | a6 > 0 {a5,a2,a,a4} H3 : n = 8 - 23, 03, 21, 41 Is,2 Z4 X Z2 < a | a4 > X < b | b2 > 2 {a,a2,a3,b} Ds 2 {r,r2,r3, fr2} 4 {f,r2,rf, fr} Z2 X Z2 X Z2 < a | a2 > X < b | b2 > X 4 {b, a, abc, ac} H4 : n = 9 - 24,14, 41 I9,2 AT4Val[9][1] Z3 X Z3 < a | a3 > X < b | b3 > 0 {a2b, ab2, a2, a} H5 : n = 12 - 25, 03, 23, 41 112,7 AT4Val[12][1] A4 < s,t | s2,t3,(st)3 > 0 {t2s,ts,st2,st} H« : n = 12 -32,-14, 01,12, 22, 41 I 12,5 Z3 x Z4 0 {a,ba2,b3a2,a2} Zl2 < a | a12 > 0 {a3,a4,a8,a9} Di2 2 {r4,fr4,r2,fr} 2 {r3,f,r4,r2} Za X Z2 < a | a6 > X < b | b2 > 2 {a2,b,a3,a4} Hr : n = 12 - 32, -22, 01,16, 41 112,1 Z3 X Z4 < a,b | a3,b4,abab-1,ab2a2b2 > 0 {b3,b,b2a,b2a2} Z12 < a | a12 > 0 {a3, a10, a2, a9} D12 < r, f | r6,f2, (rf)2 > 2 {f, fr3,r,r5} 4 {r3 ,f,fr,fr5} Za X Z2 < a | a6 > X < b | b2 > 2 {a3b,a3,a2b,a4b} Hs : n = 18 - 32, —24, 05,14, 32, 41 I18,4 Z3 X S3 < a | a3 > X < s,t | s2,t3, (st)2 > 2 {a, s, a2, st2} 0 {t, t2, sat2, sa2t2} (Z3 X Z3) X Z2 < a,b,c | a3, b3, c2, cac-1a-2, bc-1b-2c, aba-1b-1 > 2 {a, c, a2, cb2} Za X Z3 < a | a6 > X < b | b3 > 0 {a2b,a5b2,ab,a4b2} Hg : n = 20 - 26, -14, 05, 34, 41 Z5 X Z4 2 {a2b2,ab2,a2b,a4b3} < a,b | a5, b4, ab3a3b,ab2ab2 > H10 : n = 24 - 33,-23,-15, 03,15, 21, 33, 41 I24,5 S4 2 {s,st2st, (tst)2, t(ts)2} < s,t | s2,t3,(st)4 > Z2 X A4 2 {s, st2, a, ts} < a | a2 > X < s,t | s2,t3, (st)3 > Hii : n = 24 - 34, -2s,-12, 0s, 18, 2i, 32, 4i Z4 X S3 < a | a4 > X < s,t | s2 ,t3, (st)2 > 2 {a3t,at2,sa2,a2} (Za X Z2) X Z2 < a,b,c | a6,b2, c2,aba -1b-1, (a3c)2b, cbc-1b-1, a2 ca2 c > 4 {ca2,cb,b,a3} Z3 X Ds < a | a3 > X 2 {f, r2 ,ar, a2r-1} Z2 X Z2 X S3 < a | a2 > X < b | b2 > X < s, t | s2, t3, (st)2 > 4 {sabt, sbt2, ab, sa} Hi2 : n = 36 - 2i3 -1®, 0s, 14, 2s, 3®, 4i AT4Val[36][6] Z3 X A4 < a | a3 > X < s, t | s2 ,t3, (st)3 > 0 {ta, tst, t2a2, t2st2} His : n = 36 - 34, -2i0, 0i, 1i6, 34, 4i Z3 X (Z3 X Z4) < a,b,c | a3,b3,c4, aba -1b-1,aca-1c-1,bc-1b-2c> 0 {ac2b,cb2,a2c2b2 ,c3b2} (Z3 X Z3) X Z4 < a,b,c | a3,b3,c4, aba -1b-1 ,ac-1a-1bc, ac2ac2 ,acbc-1b, 2 {a2bc3,a2c,ac2,ab2c2} c2bc2b > S3 X S3 < s,t | s2,t3, (st)2 > X < u,v | u2,v3, (uv)2 > 4 {suvt, s, sut2,uv2} Za X S3 < a | a6 > X < s, t | s2 ,t3, (st)2 > 2 {a512, at, sa31, st} Hi4 : n = 36 - 34, -24,-1i2, 0i, 14, 2®, 34, 4i Z3 X (Z3 X Z4) < a,b,c | a3,b3,c4, aba -1b-1,aca-1c-1,bc-1b-2c> 0 {c3b,cb,a2b,ab2} (Z3 X Z3) X Z4 < a,b,c | a3,b3,c4, aba -1b-1 ,ac-1 a-1bc, ac2ac2 ,acbc-1b, 0 {a2bc3,a2c,b2,b} c2bc2b > S3 X S3 < s,t | s2,t3, (st)2 > X < u, v | u2, v3, (uv)2 > 2 {v2t,s,uv,vt2} Za X S3 < a | a6 > X < s, t | s2 ,t3, (st)2 > 2 {a2t,sa3t,a4t2,st} Hi5 : n = 60 - 34, -2i7,-14, 0i5, 2ii, 38, 4i A5 2 {st2(st)2,tst2(st)2t, (st)3ts,((st)2t)2st} Table 7: Non-bipartite Cayley QIGs Thus, by Theorem 4.1 and Lemma 4.2 we have that {Gi : 1 < i < 17} U {Hj : 1 < j < 15} is the complete set of Cayley QIGs. 4.5 Non-bipartite arc-transitive integral graphs In Section 4.2, we listed all bipartite arc-transitive QIGs from the census of Potocnik et al. [17, 18]. When searching this census for integral graphs, we also found arc-transitive QIGs that are not bipartite. There are 6 such graphs that are not Cayley and thus not already accounted for in Table 7. By [25], the bipartite double cover of any QIG has order at most 560, so we can be sure that the census contains all the arc-transitive QIGs. In fact, the following folklore result tells us more: Lemma 4.4. The bipartite double cover of an arc-transitive graph is arc-transitive. Proof. Let G be an arc-transitive graph. Then H = G x K2 has vertices (a, x) for all a G G and x G Z2 and arcs ((a, x), (b, x + 1)) and ((b, x), (a, x + 1)) whenever a is adjacent to b in G. It is not hard to show that the following maps are automorphisms of H : • (a, x) ^ (a(a), x + 1) for all a G G and x G Z2 where a G Aut(G). • (a, x) ^ (a, x + 1) for all a G G and x G Z2. Given these automorphisms, it is routine to check that H is arc-transitive. □ This last result provides a useful cross-check of our results and of the Magma routine from Section 4.3. It tells us that by applying the routine (restricted to 2-semiregular automorphisms) to the bipartite arc-transitive QIGs from Tables 3 and 5, we should find all the arc-transitive integral non-bipartite graphs. This list should tally with the list obtained by directly screening the census for integral graphs, which is what happened in practice. We now list the spectrum of the non-bipartite arc-transitive QIGs that are not Cayley and whose bipartite double cover is one of the Gi for i = 1,..., 17 or Fi for i = 1,..., 5. We denote these graphs by Ji for i = 1,..., 6. Graphs appearing in the paper by Cvetkovic et al. are included using the notation of [7]: /„,index. We give the graph index from the census ofPotocnik et al. [17, 18] in their notation: AT4Val[n][index]. • From Gio, J1 = Ii5,2 = AT4Val[15][1] : [-25, -14, 25, 41], • From F1, J2 = AT4Val[30][2] : [-34, -25,-14,05, 211,41], J3 = I30,4 = AT4Val[30][3] : [-211,-14,05, 25, 34, 41], • From F2, J4 = I35,1 = AT4Val[35][2] = O4 : [-36, -114, 214, 41], • From F3, J5 = I45,1 = AT4Val[45][1] : [-216, -19,110, 39,41], • From F4, J6 = AT4Val[90][8] : [-314, -27, -124,05,110, 221, 38, 41]. Of the arc-transitive non-bipartite non-Cayley graphs, only J2 and J6 were not previously known to be integral. Thus, the arc-transitive QIGs from the census are as follows: G1, G2, G3, G5, G6, G7, G10, G11, G12, G15, G17, F1, F2, F3, F4, F5, H1, H2, H4, H5, H12, J1, J2, J3, J4, J5, and J6. We summarize these results by the following Lemma: Lemma 4.5. There are exactly 27 quartic integral graphs that are arc-transitive; 16 of which are bipartite. 4.6 Other quartic integral graphs Finally, we list the spectra of the remaining QIGs which we found using the Magma routine of Section 4.3 in its full generality. These are graphs that are neither Cayley nor arc-transitive, but are quotients of the graphs Gj for i = 1,..., 17 and/or of the graphs AT4Val[n][index] for n < 640 with automorphism groups of order less than 220. We note that many of these graphs were obtained from multiple starting graphs, but we only list each graph once. We list the spectrum of the bipartite QIGs first. We denote these graphs by Mj for i = 1,..., 9 and follow the same conventions as in the list for Jj where i = 1,..., 6 except that we use the quadruple form for the spectrum of a bipartite graph. • From AT4Val[60][4] we have M1 = /30,3 : [1,8, 3, 2], • From G15 = AT4Val[72][12] we have M2 = /36,1 : [2,8,6,1], M3 = /36,2 : [3, 6, 5, 3], • From G1r = AT4Val[120][4] we have M4 : [3,4,1,6], M5 : [6,12, 2, 9], • From AT4Val[180][12] we have M6 : [9,16,19,0], Mr : [10,14,18, 2], • From AT4Val[216][12] we have M8 : [3, 5, 9,0], • From AT4Val[546][48] we have M9 : [5,4, 7,4]. We do not list graphs with at most 24 vertices since all bipartite QIGs on 24 or fewer vertices are known [25]. The 6 graphs M4,..., M9 were not previously known to be bipartite QIGs. We find that M6 is co-spectral to F3, but 5 of the above spectra were not previously known to be realized by any graph. Next, we list the spectrum of the non-bipartite QIGs. We denote these graphs by Lj where i G 1 , . . . , 44. • From AT4Val[30][3], L1 = /15,4 : [—25, —13,02,23,31,41]. • From G12 = AT4Val[36][3], L2 : [—33, —22, —11,05,13,22,31,41]. • From AT4Val[36][6], L3 = /18 5 : [—2r, —12,01,14,21,32,41], L4 = /18,6 :[—26, —13,03, 12, 33, 41]. • From AT4Val[60][4], L5 : [—33, —2r, —13,05,11,29,31,41], and L6 : [—32, —29, —12,05,12, 2r, 32,41]. • From AT4Val[70][4], Lr : [—35, —24, —19,15,210, 31,41], and L8 : [—34, —26, —18,16, 28, 32, 41]. • From G15 = AT4Val[72][12], L9 : [—31, —25, —13,01,13, 23, 31,41], L10 : [—32, —23, —14, 01,12, 25,41 ], L11 : [—32, —25, 01,16, 23,41], L12 : [—32, —24, —12, 01,14, 24,41 ], L13 : [—31, —25, —13, 01,13, 23, 31, 41], L14 : [—32, —24, —11, 03,14, 22, 31, 41], L15 : [—33, —29, —15,03,15, 2r, 33, 41], L16 : [—34, 29, —12, 03,18, 2r, 32, 41], L1r : [—32, —211, —14,03,16, 25, 34, 41], and L18 : [—33, —29, —15, 03,15, 2r, 33, 41]. • From G16, L19 : [—34, —2r, —16, 01,110, 23, 34,41], L20 : [—34, —29, —12, 01,114, 21, 34,41], L21 : [—34, —25, —110,01,16, 25, 34,41], L22 : [—34, —26, —18,01,18, 24, 34,41], L23 : [-34, —28, — 14, 01,112, 22, 34,41], L24 : [—34, —26, — 18, 01,18, 24, 34,41], L25 : [—34, —28, — 14, 01,112, 22, 34,41], L26 : [—33, —27, — 19, 01,17, 23, 35,41], L27 : [—33, —28, — 17, 01,19, 22, 35,41], L28 : [—34, —27, — 16, 01,110, 23, 34, 41], and L29 : [—34, —26, — 18, 01,18, 24, 34, 41]. • From AT4Val[90][1], L30 : [—34, —210, —19,110, 26, 35,41]. • From AT4Val[90][8], L31 : [—35, —26, —114,15,210, 34,41]. • From G17 = AT4Val[120][4], L32 : [—33, —27, —11,09,11,25,33,41], L33 : [—33, —27, —11, 09,11, 25, 33, 41], L34 : [—34, —25, —12,09, 27, 32, 41], L35 : [—37, —213, —13,015,11, 215, 35,41]. • From AT4Val[180][12], L36 : [—34, —28, —112,02,16, 26, 36,41], L37 : [—39, —217, —119, 05,115, 211, 313, 41], L38 : [—311, —213, — 121,05, 113,215, 311,41], and L39 : [—312, —215, —114,05, 120, 213, 310,41]. • From AT4Val[210][10], L40 : [—316, —29, —129,120, 219,311,41]. • From AT4Val[273][4], L41 : [—31, —24, —16,04,11,34,41]. • From AT4Val[546][48], L42 : [—32, —23, —15,04,12,21, 33,41], L43 : [—33, —22, —14, 04,13, 22, 32, 41], L44 : [—33, —22, —14,04,13, 22, 32, 41]. We do not list graphs with at most 12 vertices since all non-bipartite QIGs on 12 or fewer vertices are known [25]. Of the 44 graphs given above, only L1, L3 and L4 previously appear in the literature about integral graphs. The remaining 41 non-bipartite QIGs are new. 5 Concluding remarks There are precisely 32 connected 4-regular integral Cayley graphs up to isomorphism. Table 3 lists the 17 graphs of the 32 which are bipartite and Table 7 gives the details of the 15 non-bipartite graphs. There are exactly 27 quartic integral graphs that are arc-transitive. We found that 16 of the 27 graphs are bipartite; these appear in Table 3 and Table 5. We found that 16 of the 27 graphs are Cayley graphs; these appear in Table 3 and Table 7. There are integral Cayley bipartite graphs that can be decomposed into H x K2 where H is Cayley and arc-transitive, Cayley but not arc-transitive, or arc-transitive but not Cayley. The graph G10 is our only example of this last possibility; refer to Table 6. The new 4-regular integral graphs that we found that are co-spectral to other graphs are as follows: G13 co-spectral to 140,1 and 140,2, G15 to 172,1, H9 to 120,8, H12 to 136,4, and F3 to M6. We also mention the co-spectral graphs among the known integral graphs: G5 is co-spectral to 116,2 and another graph appearing in [25], G6 to 118,2 and 118,3, G7 to 124,1, F1 to /60,2, H5 to /12,6, and J3 to /30,5. We find that some integral Cayley graphs are co-spectral to integral non-Cayley graphs and that some integral arc-transitive graphs are co-spectral to integral graphs that are not arc-transitive. For example, the arc-transitive Cayley graph H5 has a co-spectral mate /12,6, that is neither arc-transitive nor Cayley. As can also be seen in Table 3, there are isomorphic integral graphs that are non-equivalent Cayley graphs Cay(T, S) and Cay(r*, £*) in the sense of Definition 1.1. This can occur for Г = Г* as well as Г = Г* with S = S*. Consider G12, which has 12 non-equivalent Cayley Graphs on 6 different groups. For Г = S3 x S3, there are 4 non-equivalent Cayley graphs with connection sets occurring for each of the three possible numbers of involutions. There is only one Cayley graph up to equivalence for the graph of order 40. For all other orders the bipartite integral Cayley graphs are not unique up to equivalence. In the non-bipartite case; H1,H4,H5, H9,H12, and H15 are all unique up to equivalence. There are non-isomorphic integral Cayley graphs with the same number of vertices. As can be seen in Table 3 for the bipartite case, there are two graphs on 12 vertices, three graphs on 24 vertices, and two graphs on 72 vertices up to isomorphism. For all other orders there is at most one graph up to isomorphism. There are many more examples in the non-bipartite case (refer to Table 7). There exist non-isomorphic integral Cayley graphs for the same group Г. Consider Gi for i = 7,8,9 in Table 3. The following 6 groups are examples of this: Z2 x A4, Z3 x D8, Z2 x Z2 x S3, S4, (Z6 x Z2) X Z2, and Z4 x S3. We began with the 828 possible spectra from [25], and then narrowed our focus to a set S of 59 candidates for vertex transitive graphs; refer to Table 1 and Appendix A. Of these, we found 22 which are realised by Cayley graphs or arc-transitive graphs. In Section 4.6, by taking quotients, we found 6 new bipartite integral graphs that are neither arc-transitive nor Cayley, but realize a possible spectrum. Overall, we found 9 bipartite quartic integral graphs (namely, G16, G17, F4, F5, M4, M5, M7, M8, Mg) that realise spectra not previously known to be achieved. It remains open whether the remaining possible spectra are realized by any 4-regular bipartite integral graphs. All integral graphs discovered in this paper are available in Magma format from: http://users.monash.edu.au/~iwanless/data/graphs/ IntegralGraphs.html. Acknowledgements A special thanks to Marston Condor, Heiko Dietrich, and Csaba Schneider for their helpful advice and for confirming some of our computational results. The authors used a mixture of the computer algebra packages Magma [21] and GAP [9] (including the GRAPE package [22] for GAP), as well as nauty [14]. This research formed part of the first author's PhD thesis [15]. A Feasible vertex-transitive spectra The following is the set S of possible spectra that might be realized by a connected 4-regular bipartite integral graph G that is vertex-transitive. This set was determined in Section 2. The entries are given as nxyzw [C4] [Ca] where |V (G)| = n and Sp(G) = {4, 3x, 2y, lz, 02w,-lz,-2y, -3x, -4}. 8 0 0 0 3 36 96 10 0 0 4 0 30 130 12 0 1 4 0 27 138 12 0 2 0 3 30 112 16 0 4 0 3 24 128 18 0 4 4 0 18 162 20 0 5 4 0 15 170 24 0 8 0 3 12 160 24 2 2 6 1 30 124 24 3 0 5 3 42 80 30 0 10 4 0 0 210 30 3 2 9 0 30 130 30 4 1 4 5 45 60 32 0 12 0 3 0 192 36 4 4 4 5 36 84 36 5 1 7 4 45 66 40 4 6 4 5 30 100 42 6 0 14 0 42 98 48 6 4 10 3 36 96 60 4 16 4 5 0 180 60 6 9 14 0 15 170 60 7 8 9 5 30 100 60 8 7 4 10 45 30 60 9 1 19 0 45 90 70 6 14 14 0 0 210 72 11 4 13 7 54 24 72 6 16 10 3 0 192 72 8 10 16 1 18 156 72 9 10 7 9 36 60 90 12 13 4 15 45 0 90 13 7 19 5 45 60 90 8 22 4 10 0 150 90 9 16 19 0 0 210 96 12 12 20 3 24 128 120 12 28 4 15 0 120 120 13 22 19 5 0 180 120 15 20 9 15 30 40 120 16 14 24 5 30 100 120 19 6 29 5 60 20 126 13 28 7 14 0 126 126 20 7 28 7 63 0 144 16 28 16 11 0 144 144 20 16 28 7 36 72 168 20 28 28 7 0 168 180 20 40 4 25 0 60 180 21 34 19 15 0 120 180 22 28 34 5 0 180 180 26 19 34 10 45 30 210 27 28 49 0 0 210 240 28 52 4 35 0 0 240 30 40 34 15 0 120 280 34 56 14 35 0 0 288 36 52 28 27 0 48 360 46 64 34 35 0 0 360 47 58 49 25 0 60 360 48 52 64 15 0 120 420 55 70 49 35 0 0 480 64 76 64 35 0 0 560 76 84 84 35 0 0 References [1] A. Abdollahi and M. Jazaeri, Groups all of whose undirected Cayley graphs are integral, European J. Combin. 38 (2014), 102-109. [2] A. Abdollahi and E. Vatandoost, Which Cayley graphs are integral?, Electron. J. Combin. 16 (2009), R122, 17pp. [3] A. Abdollahi and E. Vatandoost, Integral quartic Cayley graphs on abelian groups, Electron. J. Combin. 18 (2011), P89, 14pp. [4] A. Ahmady, J. P. Bell and B. Mohar, Integral Cayley graphs and groups, SIAM J. Discrete Math. 28 (2014), 685-701. [5] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974. [6] D. M. Cvetkovic, Graphs and their spectra, Ph.D. thesis, University of Belgrade, 1971. [7] D. M. Cvetkovic, S. K. Simic and D. Stevanovic, 4-regular integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 9 (1998), 89-102. [8] I. Estelyi and I. Kovacs, On groups all of whose undirected Cayley graphs of bounded valency are integral, Electron. J. Combin. 21 (2014), P4.45. [9] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.5.7, 2012, http: //www.gap-system.org. [10] C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001. [11] A. J. Hoffman, On the polynomial of a graph, Amer. Math. Monthly 70 (1963), 30-36. [12] W. Klotz and T. Sander, Integral Cayley graphs over abelian groups, Electron. J. Combin. 17 (2010), R81, 13pp. [13] C. H. Li, On isomorphisms of finite Cayley graphs: A survey, Discrete Math. 256 (2002), 301334. [14] B. D. McKay, nauty user's guide (version 2.4), 2009, http://cs.anu.edu.au/ ~bdm/ nauty. [15] M. Minchenko, Counting subgraphs of regular graphs using spectral moments, Ph.D. thesis, Monash University, 2014. [16] M. Minchenko and I. M. Wanless, Spectral moments of regular graphs in terms of subgraph counts, Linear Algebra Appl. 446 (2014), 166-176. [17] P. Potočnik, P. Spiga and G. Verret, Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs, arXiv:1010.2546v1 (2010). [18] P. Potocnik, P. Spiga and G. Verret, Cubic vertex-transitive graphs on up to 1280 vertices, J. Symbolic Comput. 50 (2013), 465-477. [19] G. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426-438. [20] A. J. Schwenk, Computing the characteristic polynomial of a graph, in: R. Bari and F. Harary (eds.), Graphs and Combinatorics, Springer-Verlag, Berlin, volume 406 of Lecture Notes in Mathematics, pp. 153-172, 1974. [21] B. J. Smith, R package magma: Matrix algebra on GPU and multicore architectures, version 0.2.2, 2010, http://cran.r-project.org/package=magma. [22] L. H. Soicher, Grape-a gap package, version 4.6.1, http://www.maths.qmul.ac.uk/ -leonard/, May 17, 2012. [23] D. Stevanovic, Nonexistence of some 4-regular integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999). [24] D. Stevanovic, 4-regular integral graphs avoiding ±3 in the spectrum, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. (2003), 99-110. [25] D. Stevanovic, N. M. M. de Abreu, M. A. A. de Freitas and R. Del-Vecchio, Walks and regular integral graphs, Linear Algebra Appl. 423 (2007), 119-135. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 409-416 Strongly light subgraphs in the 1-planar graphs with minimum degree 7 A graph is 1-planar if it can be drawn in the plane such that every edge crosses at most one other edge. A connected graph H is strongly light in a family of graphs G, if there exists a constant À, such that every graph G in G contains a subgraph K isomorphic to H with degG (v) < À for all v e V (K). In this paper, we present some strongly light subgraphs in the family of 1-planar graphs with minimum degree 7. Keywords: Strongly light subgraph, 1-planar graph. Math. Subj. Class.: 05C10 1 Introduction All graphs considered are finite, simple and undirected unless otherwise stated. We denote by V(G) and E(G) the vertex set and the edge set of G. We shall denote by F(G) the set of faces of an embedded graph G. The degree of a vertex v in G, denoted by degG(v), is the number of edges of G incident with v. We denote the minimum and maximum degrees of vertices in G by 6(G) and A(G), respectively. A wheel Wn is a graph obtained by taking the join of a cycle Cn and a single vertex. In an embedded graph G, the degree of a face f, denoted by degG(f), is the number of edges with which it is incident, each cut edge being counted twice. A k-vertex, k+-vertex and k--vertex is a vertex of degree k, at least k and at most k, respectively. Similarly, we can define k-face, k+-face and k- -face. A graph is 1-embeddable in a surface S if it can be drawn in S such that every edge crosses at most one other edge. In particular, a graph is 1-planar if it can be drawn in E-mail address: wangtao@henu.edu.cn (Tao Wang) Tao Wang Institute ofApplied Mathematics, Henan University, Kaifeng, 475004, P. R. China and School ofMathematics and Statistics, Henan University, Kaifeng, 475004, P. R. China Received 23 October 2013, accepted 9 October 2014, published online 11 June 2015 Abstract the plane such that every edge crosses at most one other edge. The concept of 1-planar graph was introduced by Ringel [9] in 1965, while he simultaneously colors the vertices and faces of a plane graph such that any pair of adjacent/incident elements receive different colors. Ringel [9] proved that every 1-planar graph is 7-colorable, and conjectured that every 1-planar graph is 6-colorable. In 1984, Borodin [1] confirmed this conjecture, and later Borodin [2] found a better proof for it. Recently, various coloring problems of 1-planar graphs are considered, see [4, 13, 10]. A connected graph H is strongly light in a family of graphs G, if there exists an integer À, such that every graph G in G contains a subgraph K isomorphic to H with degG(v) < À for all v e V (K). A graph H is said to be light in a family G of graphs if at least one member of G contains a copy of H and there is an integer À(H, G) such that each member G of G with a copy of H also has a copy K of H such that degG (v) < À(H, G) for all v e V (K). Note that a light subgraph may be not strongly light, for example, the graph K5 is light in the family of graphs G = {planar graphs} U {K6}, but K5 is not strongly light in G since not every graph in G contains a subgraph K5. The light subgraphs are well studied when G is a subclass of planar graphs, and we refer the reader to a good survey [8]. Fabrici and Madaras [5] studied the structure of 1-planar graphs, mainly on the light subgraphs of 1-planar graphs. They showed that every 3-connected 1-planar graph contains an edge with each end having degree at most 20, and this bound is the best possible. Hudäk and Madaras [6] proved that each 1-planar graph of minimum degree 5 and girth 4 contains (i) a 5-vertex adjacent to a vertex of degree at most 6, (ii) a 4-cycle whose vertices all have degree at most 9 (the upper bound was further improved to 8 by Borodin, Dmitriev and Ivanova [3]), (iii) a star K14 with all vertices having degree at most 11. In 1965, Ringel [9] found that each 1-planar graph has a vertex of degree at most 7 and the bound is tight. Hudäk and Madaras [7] considered strongly light subgraphs in the family of 1-planar graphs with minimum degree 7, and proved the following theorem. Theorem 1.1 (Hudäk and Madaras [7]). Each 1-planar graph with minimum degree 7 contains (a) two adjacent 7-vertices, (b) a K4 whose vertices all have degree at most 13, (c) a K2*3 whose vertices all have degree at most 13, where K23 is a graph K2 3 with an extra edge between two vertices of the smaller bipartition, (d) a W4 whose vertices all have degree at most 11. In this paper, we also consider strongly light subgraphs in the family of 1-planar graphs with minimum degree 7. 2 Strongly light subgraphs Let G be a graph having been drawn in a surface; if we treat all the crossing points as vertices, then we obtain an embedded graph Gt, and call it the associated graph ofG, call the vertices of G true vertices and the crossing points crossing vertices. In the associated graph, a 3-face is called a false 3-face if it is incident with a crossing vertex; otherwise, it is a true 3-face. Clearly, a false 3-face is incident with exactly one crossing vertex. Note that the set of crossing vertices in the associated graph is independent. In the figures of this paper, the solid dots denote true vertices and the hollow dots denote crossing vertices, and some degree restrictions are beside the vertices. Zhang et al. presented two strongly light subgraphs on four vertices in the family of 1-planar graphs with minimum degree 7. Theorem 2.1 (Zhang et al. [14]). Each 1-planar graph with minimum degree 7 contains a K4 with all vertices of degree at most 11. Theorem 2.2 (Zhang et al. [11]). Each 1-planar graph with minimum degree 7 contains a 4-cycle C = [x1 x2x3x4] with a chord x1 x3, where deg(x1) = 7, deg(x2) < 10, deg(x3) < 8 and deg(x4) < 10. We improve the above two results to the following. A K4 is of type (d1, d2, d3, d4) if its degrees are d1, d2, d3 and d4, respectively. Similarly, we can define a K4 of type (d+, d+, d+, d+), etc. Theorem 2.3. If G is a 1-planar graph with minimum degree 7, then it contains a K4 of the type (7,8-, 8-, 10-). Proof. Suppose that G is a connected counterexample to the theorem, which implies that G contains no K4 or every copy of K4 is of the type (8+, 8+, 8+, 8+) or (7,9+, 9+, 9+) or (7,8-, 9+, 9+) or (7,8-, 8-, 11+). Furthermore, we may assume that G has been 1-embedded in the plane. Clearly, every face of its associated graph is homeomorphic to an open disk. Let Kt be the associated graph of G. By Euler's formula, we have X (degKt (v) - 6) + X (2 degKt (f) - 6) = -12. (2.1) veV(Kt) f eF(Kt) We will use the discharging method to complete the proof. The initial charge of every vertex v is degKt (v) - 6, and the initial charge of every face f is 2degKt (f) - 6. By (2.1), the sum of all the elements' charge is -12. We then transfer some charge from the 4+-faces and the 7+-vertices to crossing vertices, such that the final charge of every crossing vertex becomes nonnegative and the final charge of every 4+-face and every 7+-vertex remains nonnegative, and thus the sum of the final charge of vertices and faces is nonnegative, which leads to a contradiction. The Discharging Rules: (R1) every 4+-face donates its redundant charge equally to incident crossing vertices; (R2) every 7+-vertex donates its redundant charge equally to incident false 3-faces; (R3) after applying (R2), every false 3-face donates its redundant charge to the incident crossing vertex. By the discharging rules, the final charge of every face and every 7+-vertex is nonnegative. So it suffices to consider the final charge of crossing vertices in Kt. By the construction of Kt, every face is incident with at most [^gf(v) j crossing vertices. Thus, every 4+-face sends at least 1 to each incident crossing» vertex. Note that every 7+-vertex v is incident with at most 2 (v) j false 3-faces. More formally, every 7-vertex sends at least 6 to each incident false 3-face; every 8-vertex sends (a) at least 1 to each incident false 3-face; every 9-vertex sends at least | to each incident false 3-face; every 10-vertex sends at least 2 to each incident false 3-face; every ll+-vertex 5 sends at least 2 to each incident false 3-face. Let w be an arbitrary crossing vertex in Kt. Notice that the four neighbors of w are 7+-vertices. If w is incident with at least two 4+-faces, then its final charge is greater than 4 -6 + 2 X 1 = 0. If w is incident with exactly one 4+-face, then its final charge is at least 4 - 6 + 1 + 6 X 6 = 0. If there is no crossing vertex which is incident with four 3-faces, then the sum of the final charge is nonnegative, which leads to a contradiction. So we may assume that w is incident with four 3-faces. It is obvious that the four neighbors of w induce a K4 in G. If this K4 is of the type (8+, 8+, 8+, 8+), then the final charge of w is at least 4 - 6 + 8 X 1 = 0; if this K4 is of the type (7,9+, 9+, 9+), then the final charge of w is at least 4 - 6 + 2 X 1 + 6 X 8 > 0; if this K4 is of the type (7,8-, 9+, 9+), then the final charge of w is at least 4 - 6 + 4 X 6 + 4 X 3 > 0; if this K4 is of the type (7,8-, 8-, 11+), then the final charge of w is at least 4 - 6 + 6 X 6 + 2 X 1 = 0. Finally, all the faces and vertices have nonnegative charge, which leads to a contradiction. □ To the author's knowledge, all the known strongly light graphs have at most five vertices. Now, we give a strongly light graph on 8 vertices in the family of 1-planar graphs with minimum degree 7. Theorem 2.4. If G is a 1-planar graph with minimum degree 7, then G contains a subgraph as illustrated in Fig. (a). Moreover, (i) every vertex in {w2, w3,..., w7} has degree at most 23; (ii) at most one vertex in {w2, w3,..., w7} is a 12+-vertex; (iii) if no vertex in {w2, w3, w5, w7} is a 7-vertex, then w2w3, w3w4, w4w5, w5w6, w6w7, w7w1 € E(G). Proof. Suppose that G is a connected 1-planar graph with minimum degree 7, and it has been 1-embedded in the plane. Clearly, every face of its associated graph is homeomorphic to an open disk. Let Kt be the associated graph of G. By Euler's formula, we have 2 (degKt(v) - 4) + 2 (degKt(f) - 4) = -8. (2.2) vsV(Kt) f eF(Kt) We will use the discharging method to complete the proof. The initial charge of every vertex v is degKt (v) - 4, and the initial charge of every face f is degKt (f) - 4. By (2.2), the sum of all the elements' charge is -8. We then transfer some charge from the 7+-vertices to the 3-faces, such that the final charge of every face and every 8+-vertex is nonnegative, thus there exists a 7-vertex such that its final charge is negative and the local structure is desired. The Discharging Rules: (R1) every 7+-vertex sends 1 to each incident false 3-face and sends 3 to each incident true 3-face; (R2) let f be a face with a face angle w1 ww2 and deg(w) = k > 8, (a) if f is a 3-face with deg(w1) = 7 and deg(w2) > 8, then w sends k-4 - 1 to w1 through f; (b) if f is a 3-face with deg(w1) = deg(w2) = 7, then each of w1 and w2 receives kk4 - 6 from w through f ; (c) if f is a false 3-face with crossing vertex w1 and w1 is on the edge uw of G, then w sends k2k4 - 1 to w2 through f, and additionally w sends 1 - 4 to u through f; 2k 4 2k 4 (d) if f is a 4+-face with crossing vertex w1 and w1 is on the edge uw of G, then w sends to u through f. By the discharging rules, the final charge of every face and every 8+-vertex is nonnegative. Hence, there exists a 7-vertex w0 such that its final charge is negative. If w0 is incident with at least one 4+-face, then its final charge is at least 7 - 4 - 6 x 1 = 0. So we may assume that w0 is incident with seven 3-faces. Notice that the number of incident false 3-faces is even. If w0 is incident with at most four false 3-faces, then its final charge is at least 7 - 4 - 4 x 1 - 3 x 1 = 0. Hence, the vertex w0 must be incident with six false 3-faces and one true 3-face. We also notice that w0 receives less than 3 from all the other vertices; otherwise, its final charge is at least 7 - 4 + 3 - 6 x 1 - 1 = 0. Let w1 w0w2 be the true 3-face. If both w1 and w2 are 8+-vertices, then w0 receives at least 1 - 3 = 6 from each of w1 and w2 by (R2-a), thus w0 receives at least 1 from all the other vertices, a contradiction. Hence, at least one of w1 and w2 must be a 7-vertex, so we may assume that w1 is a 7-vertex, see Fig. (a). (i) Suppose that w0 is adjacent to a 24+ -vertex w in G. By the discharging rules, the vertex w0 receives at least 2 x (12 - 4) = 1 from w, which leads to a contradiction. Hence, every vertex in {w2, w3,..., w7} has degree at most 23. (ii) If at least two vertices in {w2, w3,..., w7} are 12+-vertices, then w0 will receive at least 4 x (1 - 1) = 3, which leads to a contradiction. Hence, at most one vertex in {w2, w3,..., w7} is a 12+-vertex. (iii) Suppose, to derive a contradiction, that w2w3, w3w4, w4w5, w5w6, w6w7, w7w1 € E(G) does not hold. Thus, at least one crossing vertex in Fig. (a) is incident with a 4+-face. By (R2-d), the vertex w0 receives at least 1 from a 8+-vertex through a 4+-face. By (R2-b), the vertex w0 receives at least 4 - 1 = ^ from w2 through the true 3-face w0w1w2, thus it receives at least 1 + ^ = 3 from all the other vertices, which derives a contradiction. □ Corollary 2.5 (Hudäk and Madaras [7]). If G is a 1-planar graph with minimum degree 7, then it contains an edge such that each end has degree exactly 7. (b) (c) (d) (e) (f) Corollary 2.6. Every 1-planar graph with minimum degree 7 contains a K1i7 with the center of degree 7 and the other vertices of degree at most 23. By Theorem 2.4, the wheel W4 is strongly light in the family of 1-planar graphs with minimum degree 7. In the next theorem, we further improve the degree restriction on each vertex in W4. Theorem 2.7. If G is a 1-planar graph with minimum degree 7, then G contains at least one subgraph as illustrated in Fig. (b)-(f). Proof. Suppose that G is a connected 1-planar graph with minimum degree 7, and it has been 1-embedded in the plane. Clearly, every face of its associated graph is homeomorphic to an open disk. Let Kt be the associated graph of G. By Euler's formula, we have 2 (degKt(v) - 4) + 2 (degKt(f) - 4) = -8. (2.3) vsV(Kt) f eF(Kt) We will use the discharging method to complete the proof. The initial charge of every vertex v is degKt (v) - 4, and the initial charge of every face f is degKt (f) - 4. By (2.3), the sum of all the elements' charge is -8. We then transfer some charge from the 7+-vertices to the 3-faces, such that the final charge of every face and every 8+-vertex is nonnegative, thus there exists a 7-vertex such that its final charge is negative and the local structure is desired. The Discharging Rules: (R1) every 7+-vertex sends 1 to each incident false 3-face and sends 3 to each incident true 3-face; (R2) let f be a face with a face angle w1 ww2 and deg(w) = k > 8, (a) if f is a 3-face with deg(w1) = 7 and deg(w2) > 8, then w sends - 1 to w1 through f; (b) if f is a 3-face with deg(w1) = deg(w2) = 7, then each of w1 and w2 receives k-4 1 2k 6 - б from w through f ; (c) if f is a false 3-face with crossing vertex w1, then w sends k-4 - 2 to w2 through f. k 2 By the discharging rules, the final charge of every face and every 8+-vertex is nonnegative. Hence, there exists a 7-vertex w0 such that its final charge is negative. If w0 is incident with at least one 4+ -face, then its final charge is at least 7 - 4 - 6 x 1 = 0. So we may assume that w0 is incident with seven 3-faces. Notice that the number of incident false 3-faces is even. If w0 is incident with at most four false 3-faces, then its final charge is at least 7 - 4 - 4 x 1 - 3 x 1 = 0. Hence, the vertex w0 must be incident with six false 3-faces and one true 3-face. We also notice that w0 receives less than 3 from all the other vertices; otherwise, its final charge is at least 7 - 4 + 3 - 6 x 1 - 1 = 0. Let wi w0w2 be the true 3-face. If both w1 and w2 are 8+-vertices, then w0 receives at least i - 3 = 6 from each of wi and w2 by (R2-a), thus w0 receives at least 1 from all the other vertices, a contradiction. Hence, at least one of wi and w2 must be a 7-vertex, so we may assume that wi is a 7-vertex, see Fig. (a). Case 1. Both deg(w4) and deg(w6) belong to {7,8}. Since the vertex w0 receives less than 3 from the vertex w2, it follows that (^ggw)-4 -1) + (fö-T - i) < i and deg(w2) < 12, see Fig. (b). Case 2. Exactly one of deg(w4) and deg(w6) belongs to {7,8}. Note that max{deg(w4), deg(w6)} > 9, if w2 is a 10+ -vertex, then w0 receives at least 2x (9 - |)+(f - 2)+(ш - 6) > 1, a contradiction. So we may assume that w2 is a9--vertex. If w2 is a 7-vertex and max{deg(w4), deg(w6)} > 12, then w0 will receive at least 2x(| -1) = I, which is a contradiction. If w2 is a 8-vertex and max{deg(w4), deg(w6)} > 11, then w0 will receive at least 2 x (Ц - |) + (4- - 6) > 3, a contradiction. If w2 is a 9-vertex and max{deg(w4), deg(w6)} > 10, then w0 receives at least 2 x (3 - 1) + (5 -1) + (158 - 6) = 11 > 3, which leads to a contradiction. In summary, if w2 is a 7-vertex, then max{deg(w4), deg(w6)} € {9,10,11}, and thus G contains a subgraph isomorphic to that in Fig. (b); if w2 is a 8-vertex, then max{deg(w4), deg(w6)} € {9,10}, and thus G contains a subgraph isomorphic to that in Fig. (b) or Fig. (c); if w2 is a 9-vertex, then max{deg(w4), deg(w6)} = 9, and thus G contains a subgraph isomorphic to that in Fig. (d), Fig. (e) or Fig. (f). Case 3. Both deg(w4) and deg(w6) are at least 9. If w2 is a 9+-vertex, then the vertex w0 will receive at least (18 - 1) + 5 x (9 - 1) > 3, a contradiction. So we may assume that w2 is a 7- or 8-vertex. If min{deg(w4), deg(w6)} > 10, then the vertex w0 will receive at least 4 x (3 - 1) = | > 1, a contradiction. Hence, we have that min{deg(w4), deg(w6)} = 9. If w2 is a 7-vertex and max{deg(w4), deg(w6)} > 11, then the vertex w0 will receive at least 2 x (Ц - 2) + 2 x (9 - 2) > 3, a contradiction. If w2 is a 8-vertex and max{deg(w4), deg(w6)} > 10, then w0 will receive at least 2 x (5 - 2) + 2 x (9 - 2) + (4 - 1 ) > 3, which is a contradiction. In summary, if w2 is a 7-vertex, then G contains a subgraph as illustrated in Fig. (e); if w2 is a 8-vertex, then G contains a subgraph as illustrated in Fig. (f). □ Corollary 2.8. If G is a 1-planar graph with minimum degree 7, then G contains a triangle having vertex degree 7, 7 and at most 9, respectively. As an immediate consequence of Theorem 2.7, the following corollary is an improvement of Theorem 2.2. Corollary 2.9. If G is a 1-planar graph with minimum degree 7, then G contains a 4-cycle C = [x1 x2x3x4] with a chord x1 x3, where deg(x1) = 7, deg(x2) < 9, deg(x3) < 8 and deg(x4) < 9. Corollary 2.10 ([12]). If G is a 1-planar graph with minimum degree 7, then G contains a copy of K1 v (K1 U K2) with all the vertices of degree at most 9. Acknowledgments. The author was supported by NSFC (11101125) and partially supported by the Fundamental Research Funds for Universities in Henan. The author would like to thank the anonymous reviewers for their valuable comments and assistance on earlier drafts. References [1] O. V. Borodin, Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs, Metody Diskret. Analiz. 41 (1984), 12-26, 108. [2] O. V. Borodin, A new proof of the 6 color theorem, J. Graph Theory 19(4) (1995), 507-521. [3] O. V. Borodin, I. G. Dmitriev and A. O. Ivanova, The height of a cycle of length 4 in 1-planar graphs with minimal degree 5 without triangles, Diskretn. Anal. Issled. Oper. 15(1) (2008), 11-16. [4] O. V. Borodin, A. V. Kostochka, A. Raspaud and E. Sopena, Acyclic colouring of 1-planar graphs, Discrete Appl. Math. 114(1-3) (2001), 29-41. [5] I. Fabrici and T. Madaras, The structure of 1-planar graphs, Discrete Math. 307(7-8) (2007), 854-865. [6] D. Hudäk and T. Madaras, On local structure of 1-planar graphs of minimum degree 5 and girth 4, Discuss. Math. Graph Theory 29(2) (2009), 385-400. [7] D. Hudäk and T. Madaras, On local properties of 1-planar graphs with high minimum degree, Ars Math. Contemp. 4(2) (2011), 245-254. [8] S. Jendrol and H.-J. Voss, Light subgraphs of graphs embedded in the plane—a survey, Discrete Math. 313(4) (2013), 406-421. [9] G. Ringel, Ein Sechsfarbenproblem auf der Kugel, Abh. Math. Sem. Univ. Hamburg 29(1) (1965), 107-117. [10] X. Zhang and G. Liu, On edge colorings of 1-planar graphs without adjacent triangles, Inform. Process. Lett. 112(4) (2012), 138-142. [11] X. Zhang andG. Liu, On the lightness of chordal 4-cycle in 1-planar graphs with high minimum degree, Ars Math. Contemp. 7 (2014), 281-291. [12] X. Zhang, G. Z. Liu and J. L. Wu, Light subgraphs in the family of 1-planar graphs with high minimum degree, Acta Math. Sin. (Engl. Ser.) 28(6) (2012), 1155-1168. [13] X. Zhang and J.-L. Wu, On edge colorings of 1-planar graphs, Inform. Process. Lett. 111(3) (2011), 124-128. [14] X. Zhang, J.-L. Wu and G. Liu, New upper bounds for the heights of some light subgraphs in 1-planar graphs with high minimum degree, Discrete Math. Theor. Comput. Sci. 13(3) (2011), 9-16. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 417-423 The multisubset sum problem for finite abelian groups Amela MuratoviC-RibiC University of Sarajevo, Department of Mathematics, Zmaja od Bosne 33-35, 71000 Sarajevo, Bosnia and Herzegovina Qiang Wang * School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada Received 28 October 2013, accepted 29 August 2014, published online 11 June 2015 Abstract We use a similar techique as in [2] to derive a formula for the number of multisubsets of a finite abelian group G with any given size and any given multiplicity such that the sum is equal to a given element g G G. This also gives the number of partitions of g into a given number of parts over a finite abelian group. Keywords: Composition, partition, subset sum, polynomials, finite fields, character, finite abelian groups. Math. Subj. Class.: 11B30, 05A15, 20K01, 11T06 1 Introduction Let G be a finite abelian group of size n and D be a subset of G. The well known subset sum problem in combinatorics is to decide whether there exists a subset S of D which sums to a given element in G. This problem is an important problem in complexity theory and cryptography and it is NP-complete (see for example [3]). For any g g G and i a positive integer, we let the number of subsets S of D of size i which sum up to g be denoted by N (D, i, g) = #{S C D : #S = i, Y s = g}■ ses * Research is partially supported by NSERC of Canada. E-mail addresses: amela@pmf.unsa.ba (Amela Muratovic-Ribic), wang@math.carleton.ca (Qiang Wang) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ When D has more structure, Li and Wan made some important progress in counting these subset sums by a sieve technique [3, 4]. Recently Kosters [2] gives a shorter proof of the formula obtained by Li and Wan earlier, using character theory. N (G,i,g) = n £ (-1 )i+i/s(n/") £ M(s/d)#G[d], n i/s s|gcd(exp(G),i) V 1 7 d|gcd(e(g),s) where exp(G) is the exponent of G, e(g) = max{d : d | exp(G),g G dG}, ^ is the Möbius function, and G[d] = {h G G : dh = 0} is the d-torsion of G. More generally, we consider a multisubset M of D. The number of times an element belongs to M is the multiplicity of that member. We define the multiplicity of a multisubset M is the largest multiplicity among all the members in M. We denote M (D, i, j, g) = #{multisubset M of D : multiplicity (M ) < j, #M = i, £ s = g}. sEM It is an interesting question by its own to count M (D, i, j, g), the number of multisubsets of D of cardinality i which sum to g where every element is repeated at most j times. If j = 1, then M (D, i, j, g) = N (D, i, g). If j > i, this problem is also equivalent to counting partitions of g with at most i parts over D, which is M (D, i, i, g). In this case we use a simpler notation M (D, i, g) because the second i does not give any restriction. Another motivation to study the enumeration of multisubset sums is due to a recent study of polynomials of prescribed ranges over a finite field. Indeed, through the study of enumeration of multisubset sums over finite fields [5], we were able to disprove a conjecture of polynomials of prescribed ranges ove a finite field proposed in [1]. Let Fq be a finite field of q elements and F* be the cyclic multiplicative group. When D is Fq (the additive group) or F*, counting the multisubset sum problem is the same as counting partitions over finite fields, which has been studied earlier in [6]. In this note, we use the similar method as in [2] to obtain M (D, i, j, g) when D = G. However, we work in a power series ring instead of a polynomial ring. Theorem 1. Let G be a finite abelian group of size n and let g G G, i, j g Z with i > 0 and j > 1. For any s | n, we define C (n,j)= £ (-1)f/s - 1)( "j k>0,0 i, the formula gives the number of partitions of g with at most i parts over a finite abelian group. To avoid confusion the multiset consisting of ab ..., an is denoted by {{ai,..., an}}, with possibly repeated elements, and by {ab ..., an} the usual sets. We define a partition of the element g G G with exactly i parts in D as a multiset {{a1, a2,..., aj}} such that all ak's are nonzero elements in D and ai + a2 + ... + aj = g. Then the number of these partitions is denoted by PD (i, g), i.e., Pd(i,g) = j {{ai, a2,. .., aj}} C D : ai + a2 + ... + aj = g, ai,. .., aj = 0 j . It turns out M (D, i, g) = J] k=o PD (k, g) is the number of partitions of g G G with at most i parts in D. Corollary 2. Let G be a finite abelian group of size n and let g G G. Then the number of partitions of g over G with at most i parts is n ? (n/S +//S - 1) ? M(s/d)#G[d]. n z—' V i/s / z—' s\gcd(e®p(G),j) V 1 7 d\gcd(s,e(g)) where exp(G) is the exponent of G, e(g) = max{d : d | exp(G),g G dG}, ^ is the Mobius function, and G[d] = {h G G : dh = 0} is the d-torsion of G. Proof. The number is M (G, i, j, g) when j > i > 0. If j > i, then the linear Diophantine equation sk +1 • lcm(s, j + 1) = i reduces to sk = i and t = 0. The rest of proof follows immediately. □ In Section 2, we prove our main theorem and derive Corollary 1 as a consequence. In Section 3, we extend our study to a subset of a finite abelian group and make a few remarks on how to obtain the number of partitions over any subset of a finite abelian group. 2 Proof of Theorem 1 To make this paper self-contained, we recall the following lemmas (see Lemmas 2.1-2.4 in [2]). Let G be a finite abelian group of size n. Let C be the field of complex numbers and G = Hom(G, C*) be the group of characters of G. Let x G G and x be the conjugate character which satisfies x(g) = x(g) = x(-g) for all g G G. We note that a character x can be naturally extended to a C-algebra morphism x : C[G] ^ C on the group ring C[G]. Lemma 1. Let a = £ G ag g G C [G]. Then we have ag = П £ G x(g)x(a). Lemma 2. Let m be a positive integer and g G G. Then £ x(g) = ^9emG#G[mi xeo,xm=1 where SgemG is 1 if g G mG and it is zero otherwise. Lemma 3. Let x G G be a character and m be its order. Then we have П (1 - x(a)Y) = (1 - Ym)n/m. aeo Lemma 4. Let g G G. The number e(g) is equal to lcm{d : d | exp(G), g G dG}. For d | exp(G) we have g G dG if and only if d | e(g). Let us present the proof of Theorem 1. We use the multiplicative notation for the group. Proof. Fix j > 1. Working in the power series ring C[G][[X]] over the group ring, the generating function of ^ ge0 M (G, i, j, g)g is i _ ffj+iyj+1 £ £m (G, i, j, g)gXi = H(1+cX+■ ■ -+aj Xj ) = П 1_aX G C[G][[X]]. i=0 geo aeo aeo Using Lemma 1, we write £ M (G, j)X i = 1 £ x(g) п jj i=0 xe0 aeo AV ' Separating the first sum on the right hand side, we obtain £ m (G ,i ,j,g)Xi = - £ £ x(g) П jj. i=0 slexp(o) xeo,ord(x)=s aeo y For each fixed x of the order s, we know that xj+1 has the order gcd(sSj+1). Therefore by Lemma 3, we simplify the above as follows: n gcd(s,j + 1) ^ 1 ___^ /1 _ Xlcm(s,j+1)\ s £M(G,i ,j,g)xi = - £ £ x(g)^-(1 -Xl/s-. (2.1) i=0 s|exp(o) xeo,ord(x)=s Note that £ хЫ = £ £ x(g). xeo,xs = i dls xeo,ord(x)=d By Lemma 2 and the Möbius inversion formula, we obtain £ x(g) = £ M(s/d) £ x(g) = £ M(s/d)W#G[d]. xeo,ord(x)=s dls xeo,xd=i dls Because d | s | exp(G), by Lemma 4, g G dG if and only if d | e(g). Hence E X(g)= E p(s/d)Sgedo#G[d}= E p(s/d)#G[d]. xe0,0rd(x)=s d|s d|gcd(s,e(g)) Plugging this into Equation (2.1), we get n gcd(s,j + 1) ~ 1 Xlcm(s,j + 1)\ S EM(G,i,j,g)xi = - E E p(s/d)#G[d\-— X > -. i=0 s | exp(G) d\gcd(s,e(g)) ( ) By applying the binomial theorem to the right hand side and comparing coefficients of Xi in both sides, we single out M (G, i, j, g) and obtain M (G, i, j, g) = — E E p(s/d)#G[d]C (n,i,j,s). - s\exp(G) d\gcd(s,e(g)) After bringing C (n, i, j, s) out of the inner sum we complete the proof. □ Finally we remark that we can derive Corollary 1 using N (G, i, g) = M (G, i, 1, g). When j = 1, let us consider sk +t • lcm(s, j + 1) = sk + t • lcm(s, 2) = i. If s is even, we obtain sk + st = i and thus k +1 = i/s. Note that we have the following power series expansions 1 _e (n/s+k - v (1 - x)n/s k 1 v ' k=0 v 7 2n/s , . ч (1 - x)2n/s =^(-1)< 2f) t = 0 \ / and n/s (1 - x)n/s = ее ( js) (-1)j j=0 Now we compare the coefficients of the term xi/s in both sides of 1 )2n/ (1 - xs)2n/s = (1 - xs)n (1 - xs)n/s after expanding these power series. Hence we obtain с1' s) = E 0,0 (1 - xs)n/s K ' ' x. we obtain V (_iW n/s + k - Л (n/s\ = l_-\Y+i/s(n/s c(n,i, M) = (-1)V k ){ t) = (-1) Vi/ fc>0,0 i and D is a subset of G. We recall that in this case we use the notation M (D, i, g) because j does not really put any restriction. First of all, we note that w 1 w SS M (G \{0},i,g)gX i = fl = (1 - X ) EE M (G,i,g)gX \ i=0 geo aeO,a=0 i= 0 geO By Corollary 2, we obtain M (G \{0},i,g) 1 n f e (n/s +ц:-1) E M(«/d)#GM \s|gcd(exp(0),i) d|gcd(s,e(g)) E (n/s +(i(--1j//'-1) E M«™). s|gcd(exp(0),i-i) V V " 7 d|gcd(s,e(g)) / We note M (G \ {0}, i, g) = Po(i, g). Therefore we obtain an explicit formula for the number of partitions of g into i parts over G. More generally, let D = G \ S, where S = {u1, u2,..., U|S|} = 0. Denote by MS(G, i, g) the number of multisubsets of G of sizes i that contain at least one element from S. Then the number of multisubsets of D = G \ S with i parts which sum up to g is equal to M (G \ S, i, g) = M (G, i, g) - Ms (G, i, g). Note that M (G, 0,0) = 1 and M (G, 0, s) =0 for any s G G \ {0}. The principle of inclusion-exclusion immediately implies that MS(G, i, g) is given in the following formula. We note that the formula is quite useful when the size of S is small in order to compute M (G \ S, i, g). Proposition 1. For all i = 1, 2,... and g G G we have Ms (G, i, g) = E M (G, i - 1,g - u) - ... ues +(-1)t-1 E M (G, i - t, g - (ui + U2 + ... + ut)) + ... {ui,u2,...,ut }CS + (-1)i-2 E M (G, 1, g - (ui + U2 + ... + Ui-i))+ {ui,u2,...,ui-i}CS (-1)i-i E M (G, 1, g - (ui + U2 + ... + Ui)). {ui ,u2 ,...,ui }CS Proof. Fix an element g G G. Denote by Au the family of all the multisubsets of G with i parts which sum up to g and each multisubset also contains the element u. The principle of the inclusion-exclusion implies that |Uu£s Au | = ^ |Au|- ^ |Au! nAu21 + ... (3.1) uES {ui,u2}CS It is obvious to see |Aui n Au21 = M (G, i - 2, g - (ui + u2)) etc. by definition and the result follows directly. □ Acknowledgements We thank anonymous referees for their helpful suggestions. References [1] A. Gacs, T. Heger, Z. L. Nagy, D. Palvolgyi, Permutations, hyperplanes and polynomials over finite fields, Finite Field Appl. 16 (2010), 301-314. [2] M. Kosters, The subset problem for finite abelian groups, J. Combin. Theory Ser. A 120 (2013), 527-530. [3] J. Li and D. Wan, On the subset sum problem over finite fields, Finite Field Appl. 14 (2008), 911-929. [4] J. Li and D. Wan, Counting subset sums of finite abelian groups, J. Combin. Theory Ser. A 119 (2012), no. 1, 170-182. [5] A. Muratovic-Ribic and Q. Wang, On a conjecture of polynomials with prescribed range, Finite Field Appl. 18 (2012), no. 4, 728-737. [6] A. Muratovic-Ribic and Q. Wang, Partitions and compositions over finite fields, Electron. J. Combin. 20 (2013), no. 1, P34, 1-14. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 425-431 Chiral covers of hypermaps Gareth A. Jones * School of Mathematics, University of Southampton Southampton SO171BJ, U.K. Received 17 December 2013, accepted 9 April 2015, published online 26 June 2015 Abstract Generalising a conjecture of Singerman, it is shown that there are orientably regular chiral hypermaps (equivalently regular chiral dessins d'enfants) of every non-spherical type. The proof uses the representation theory of automorphism groups of Riemann surfaces acting on homology and on various spaces of differentials. Some examples are given. Keywords: Chiral hypermap, chiral map, harmonic differential, homology group. Math. Subj. Class.: 05C10, 05E18, 14H57, 20B25, 20C15, 30F10, 30F30 1 Introduction In 1992, in an unpublished preprint [17], Singerman conjectured that if m + n - 2 there is an orientably regular chiral map of type {m, n}. Conder, Hucikova, Nedela and Siran have announced a proof of this in [4] (see also [19, §3.4]). The aim of this note is to prove a similar but more general result for hypermaps (equivalently dessins d'enfants in Grothendieck's terminology [10]) of non-spherical type. Whereas most constructions of maps or hypermaps, including those in [4], involve group theory or combinatorics, this construction is mainly based on analysis (specifically, spaces of differentials on Riemann surfaces) and representation theory (the action of automorphism groups on such spaces and on associated homology groups). For background on the first topic see [9, 11], for the second see [7], and for applications of the second topic to the first see [16]. It is hoped that such techniques may find further applications in this area. The author is grateful to David Singerman and to Jozef Siran for very helpful comments on earlier drafts of this paper. * Supported by the project Mobility - enhancing research, science and education at the Matej Bel University, ITMS code: 26110230082, under the Operational Program Education cofinanced by the European Social Fund. E-mail address: G.A.Jones@maths.soton.ac.uk (Gareth A. Jones) 2 Chiral covers Orientably regular hypermaps H of type (l, m, n) correspond to torsion-free normal subgroups N of the triangle group Д = A(l, m, n); here N is an orientable surface group of genus g equal to that of H, and H has spherical type if g = 0, or equivalently \ + m + n > 1. The orientation-preserving automorphism group G := Aut+H of H is isomorphic to A/N. We say that H is reflexible if N is normal in the extended triangle group A* of type (l, m, n), in which case H has full automorphism group A := Aut H = A * /N ; otherwise H and its mirror image H form a chiral pair, corresponding to N and its other conjugate in A*. Theorem 2.1. If H is a finite orientably regular hypermap of non-spherical type, then H is covered by infinitely many finite orientably regular chiral hypermaps ofthat same type. Proof. Let A and A* be the triangle group and extended triangle group of the given type (l, m, n), so H, which has genus g > 0, corresponds to a torsion-free normal subgroup N of finite index in A with G := Aut+H = A/N. Case 1. Suppose that H is chiral, so N is not normal in A*. Choose any prime p not dividing |G|: by a theorem in Euclid's Elements, there are infinitely many such primes. Let Np = N 'Np, the group generated by the commutators and p-th powers in N ; this is a torsion-free subgroup which is characteristic in N and hence normal in A, so it corresponds to a finite orientably regular hypermap Hp of type (l, m, n), which covers H since Np < N. Now |N : Np| = p2g, which is coprirne to |A : N|, so N/Np is a Sylow p-subgroup of A/Np ; being normal, it is the unique Sylow p-subgroup, so it is a characteristic subgroup of A/np. If Np were normal in A* then N/Np would be normal in A*/Np, and hence N would be normal in A*, against our assumption. Thus Np is not normal in A*, so Hp is chiral. As a smooth p2g-sheeted covering of H it has genus p2g (g - 1) + 1. Case 2. Suppose that H is reflexible, so that N is normal in A *. Let S be the compact Riemann surface U/N canonically associated with H, where U = C or H as g = 1 or g> 1, so that the elements of G and of A \ G induce conformal and anticonformal automorphisms of S. The harmonic differentials on S form a complex vector space V = H1 (S, C) of dimension 2g affording a representation p of A. This space admits a G-invariant direct sum decomposition V + ф Vwhere V + and V- are the g-dimensional spaces of holomor-phic and antiholomorphic differentials on S, affording complex conjugate representations p+ and p- of G. The elements of A \ G transpose holomorphic and antiholomorphic differentials, and hence transpose the subspaces V + and V- of V. Integration around closed paths allows one to identify the first homology group H1(S, C) = H1(S, Z) Z C of S with the dual space V* of V, so it affords the dual (or contragredient) representation p* of A, which is equivalent to the complex conjugate representation p since A is finite. Since p |G is the sum of two complex conjugate representations, p |G is equivalent to p |G. Thus H1(S, C) also decomposes as a direct sum H+ ф H- of two g-dimensional G-invariant subspaces affording complex conjugate representations p+ and p- of G, and these are transposed by elements of A \ G. For any prime p, let Np := N'Np; this is a characteristic subgroup of N and hence a normal subgroup of A*. Let Mp := N/Np. Since N = n1S we have N/N' = H1 (S, Z) = Z2g and hence Mp = Hi (S, Z)/pHi (S, Z) = Hi (S, Z) Fp = Hi (S, Fp) = Fpg where Fp is the field of p elements. Indeed, these are isomorphisms of Fp A-modules, with the natural action of A on homology corresponding to the induced action of Д* /N by conjugation on Mp. Let e be the exponent of A. By a theorem of Dirichlet, there are infinitely many primes p = 1 mod (e). Each such prime p is coprirne to | A|, so Maschke's Theorem holds for representations of A over Fp; moreover, since Fp contains a full set of e-th roots of unity, Fp is a splitting field for A [7, Corollary 70.24], so the representation theory of A over Fp is 'the same' as that over C, and this also applies to any subgroup of A. Specifically, the representation pp of A on Mp can be regarded as the reduction mod (p) of its representation p ~ p on V* = H1(S, C) with respect to a suitable basis for V*, with the same decomposition as a direct sum of absolutely irreducible subspaces. In particular, Mp has a G-invariant decomposition M+ ф M-, with M+ and M- affording g-dimensional representations of G, and the elements of A \ G transpose the two direct factors. (See [16, §3] for full details concerning these representations of G; the extension to A is straightforward.) The inverse images N+ and N- of M+ and M- in N are torsion-free normal subgroups of finite index pg in Д, so let H+ and H- be the corresponding finite orientably regular hypermaps of type (l, m, n). These cover H, and are non-isomorphic as oriented hypermaps since N+ = N-. Elements of Д* \ Д transpose N+ and N- by conjugation, so H+ and H- form a chiral pair. As smooth pg-sheeted coverings of H, they have genus (g - 1)pg + 1 . □ Corollary 2.2. There exist infinitely many orientably regular chiral hypermaps of each non-spherical type. Proof. Being residually finite, the triangle group of the given type has a normal subgroup of finite index which contains no non-identity powers of the canonical generators, and is therefore torsion-free. Applying Theorem 2.1 to the corresponding orientably regular hypermap gives the required chiral hypermaps. □ Remark 2.3. In many cases, the condition p = 1 mod (e) in case 2 is unduly restrictive. It guarantees that every representation of A or G over C decomposes in the same way when regarded as a representation over F , whereas we are interested in just one representation, namely p. There are cases, illustrated in the following examples, where p decomposes in the required way, yielding a chiral pair, for certain primes p = 1 mod (e). Remark 2.4. Theorem 2.1 and Corollary 2.2 can be reinterpreted as statements about dessins d'enfants by replacing the phrase 'orientably regular hypermap' with the equivalent 'regular dessin'. However, their proofs make no essential use of the fact that Д and Д* are triangle groups, or equivalently that the Riemann surface S is quasiplatonic, so they yield more general results concerning coverings of compact Riemann surfaces. It is hoped to explore these in a later paper. Remark 2.5. The method used in case 1 of the proof of Theorem 2.1 is sometimes called the 'Macbeath trick', since it was used by Macbeath [15] to produce an infinite sequence of Hurwitz groups of the form Д/N'Nm, with Д of type (2,3,7) and G = Д/N = L2(7). (See §4 for background, and for more details concerning this example.) The resulting hypermaps (maps of type {3,7}) are all regular, whereas here, in cases 1 and 2 of Theorem 2.1, Macbeath's method is modified to produce chiral hypermaps. Remark 2.6. It is not claimed that there are chiral hypermaps of every non-spherical genus. Indeed, Conder's lists of chiral maps and hypermaps [3] show that there are none of genus 2, 5 or 23. It would be interesting to characterise the genera with this property. Conder, Siran and Tucker [5] have shown that there are no chiral maps of genus g = p+1 for primes p such that p — 1 is not divisible by 3,5 or 8, but finding a similar result for hypermaps would seem to be more difficult. 3 An example in genus 2 There is a unique orientably regular hypermap H of genus 2 and type (8, 2,8). This is a map of type {8, 8}, arising from an epimorphism Д = Д(8, 2,8) ^ G = C8 with kernel N = Д'. By its uniqueness H is reflexible, with A = Д*/М = D8; it is R2.6 in Conder's list of regular maps [3], and is the first entry in [6, Table 9]. One can construct H as a map by identifying opposite sides of an octagon, so that H has one vertex, four edges and one face. Going around the boundary of the octagon gives a presentation N = (a, b,c, d | abcda-1b-1c-1d-1 = 1). The images of a, b, c, d in N/N' form a basis for the homology group V* = H1 (S, C) of the underlying surface S. The automorphism group A of H is induced by the isometries of the octagon: thus G = (r) and A = (r, s) where r is a rotation through n/4 and s is a reflection; these are represented on homology by the matrices 1 1 1 —1 and 1 1 1 1 The Riemann surface S underlying H is the hyperelliptic curve w2 = z(z4 — 1), a quasiplatonic curve with Belyi function ß : (w, z) ^ z4. The vertex is at (0,0), the face-centre is at (то, то), and the edge-centres are at the four points (0, z) with z4 = 1, where ß = 0,1, то respectively. The automorphisms r and s send points (w, z) G S to (Zw, iz) and (w, z) respectively, where Z = exp(2ni/8). The differentials dz z dz = — and Ш2 =- ww form a basis for V + [9, §III.7.5], and their conjugates form a basis for V-. The rotation r sends w1 and w2 to iw1/Z = Z^i. and i2u2/Z = Z3^2, so these span 1-dimensional G-invariant subspaces Ex on which r has eigenvalues A = Z and Z3; there is a similar decomposition for V-, except that here the eigenvalues are Z-1 and Z-3. The action of s is to transpose E^ with , and Е^з with E^-з. Taking the dual space and then reducing mod (p), we see that the same applies to the A-module Mp = H1(S, Fp) for any prime p = 1 mod (8), with Z now interpreted as a primitive 8th root of 1 in the splitting field Fp. This gives a decomposition Mp = M+ e M- with G-submodules M+ = Ez e Ezз and M- = Ez-i e Ez-з transposed by s. Lifting these back to N gives subgroups N+ and N- corresponding to a chiral pair of maps H+ and H- of type {8, 8} and genus p2 +1, as in case 2 of the proof of Theorem 2.1. Remark 3.1. In this example, A has exponent 8, so the smallest prime for which this construction applies is p = 17, giving a chiral pair of orientably regular hypermaps of type {8, 8} and genus 290. They correspond to entry C290.4 in Conder's list of chiral maps [3]. Remark 3.2. This particular example also yields two chiral pairs of p-sheeted coverings of H, corresponding to the four maximal submodules of Mp, each omitting one of the four 1-dimensional eigenspaces Ex. These are orientably regular maps of type {8,8} and genus p + 1, and each pair consists of the duals of the other (see also [1, Theorem 1(a)(iii) and §3, Example (iii)] for the associated groups and Riemann surfaces). When p =17 they correspond to entry C18.1 in [3]. Remark 3.3. It is interesting to see what happens if we use primes p ф 1 mod (8) in this example, so that Fp is not a splitting field for A. If p ф 3 mod (8) then Mp is a direct sum of two irreducible G-submodules, with r having eigenvalues Z and Z3 on one, and their inverses on the other, where Z is now a primitive 8th root of 1 in Fp2 ; these submodules are transposed by s, so we obtain a chiral pair of self-dual maps of type {8, 8} and genus p2 + 1; see C10.3 in [3] for the case p = 3, and C122.7 for p = 11. The same applies if p ф -3 mod (8), except that r now has eigenvalues Z and Z-3 on one submodule, and their inverses on the other; again these submodules are transposed by s, so we obtain a chiral pair of maps of type {8,8} and genus p2 + 1, now duals of each other; see C26.1 and C170.7 in [3] for p = 5 and 13. If p ф -1 mod (8) then Mp is a direct sum of two irreducible G-submodules, with r having eigenvalues Z±x on one and Z±3 on the other; these submodules are both invariant under s, so we obtain a dual pair of regular maps of type {8,8}, rather than a chiral pair; see R50.7 in [3] for the case p = 7. Finally, if p = 2 we again obtain no chiral maps, but there is a unique series M2 > (r - 1)M2 > (r - 1)2M2 > (r - 1)3M2 > (r - 1)4M2 = 0 of A-submodules, giving regular maps of type {8,8} and genus 2,3,5,9 and 17. Remark 3.4. Further examples of chiral and regular hypermaps, arising as elementary abelian coverings of regular hypermaps of genus 2, have been found by Kazaz in [12]; this example is based on his methods (see also [13]). 4 Chiral covers of Klein's quartic curve Klein's quartic curve x3y + y3z + z3x = 0 is a compact Riemann surface S of genus 3, which carries a regular map H of type {3,7} with G = L2(7) and A = PGL2(7); see [14] for a comprehensive study of this curve. The method of proof of case 2 of Theorem 2.1, with Д of type (2,3, 7), yields chiral pairs of maps of type {3, 7} and genus 2p3 + 1 for all primes p ф 1 mod (168). These are of interest because their automorphism groups, as finite quotients of Д, are all Hurwitz groups, attaining Hurwitz's upper bound of 84(g - 1) for the number of automorphisms of a compact Riemann surface of a given genus g > 1. In fact, such chiral pairs exist for all primes p ф 1,2 or 4 mod (7). These can be obtained from a classification by Cohen [2] of those Hurwitz groups which arise as abelian coverings of G; his methods of construction are purely algebraic, using 6 x 6 matrices which can be interpreted as representing generators of Д on various homology modules. (These covers were also obtained in an earlier paper of Wohlfahrt [20], using ideals in the ring of integers of Q( V—7).) The case p = 2 is due to Sinkov [18], giving the chiral pair C17.1 of genus 17 in [3]. In this example, V + and V- are irreducible G-submodules of V, affording complex conjugate representations with characters taking the values 3, -1,0 and 1 on elements of orders 1,2,3 and 4, and (—1± V—7)/2 on the two classes of elements of order 7. Elkies has studied these representations, and their reduction modulo various primes, in [8, §1]. They give irreducible representations over Fp for any prime p such that —7 is a quadratic residue, that is, for p = 1, 2 or 4 mod (7); for such primes we have a G-module decomposition Mp = M+ ф M-, giving a chiral pair H+ and H-. However, no other primes give chiral pairs. The module M7 is indecomposable but reducible, with a submodule of dimension 3 yielding a regular map of genus 687, and the zero submodule yielding one of genus 235299. For primes p = 3,5 or 6 mod (7), Mp is irreducible and again no chiral coverings arise. (Sah's statement in [16, §3(b)] is incorrect for such primes: for instance, M3 yields only a regular map of genus 1459, corresponding to its zero submodule, and not the chiral pair of genus 55 claimed there.) References [1] M. Belolipetzky and G. A. Jones, Automorphism groups of Riemann surfaces of genus p + 1, where p is prime, Glasg. Math. J. 47 (2005), 379-393. [2] J. M. Cohen, On Hurwitz extensions by PSL2(7), Math. Proc. Cambridge Philos. Soc. 86 (1979), 395-400. [3] M. D. E. Conder, Regular maps and hypermaps of Euler characteristic —1 to -200, J. Com-bin. Theory Ser. B 99 (2009), 455-459, with associated lists of computational data available at http://www.math.auckland.ac.nz/~conder/hypermaps.html. [4] M. D. E. Conder, V. Hucfkova, R. Nedela and J. Siran, Chiral maps of any given type, submitted, 2014. [5] M. D. E. Conder, J. Siran and T. W. Tucker, The genera, reflexibility and simplicity of regular maps, J. Eur. Math Soc. (JEMS) 12 (2010), 343-364. [6] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, fourth ed., Springer-Verlag, Berlin - Heidelberg - New York, 1980. [7] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York and London, 1962. [8] N. D. Elkies, The Klein quartic in number theory, in: S. Levy (ed.), The Eightfold Way: the Beauty of Klein's Quartic Curve, MSRI Publications, Cambridge Univ. Press, Cambridge, 1999,51-101. [9] H. M. Farkas and I. Kra, Riemann Surfaces, second ed., Springer, 1992. [10] A. Grothendieck, Esquisse d'un Programme, in: P. Lochak, L. Schneps (eds.), Geometric Galois Actions I, Around Grothendieck's Esquisse d'un Programme, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, Cambridge, 1997, 5-48. [11] J. Jost, Compact Riemann Surfaces, Springer, 1997. [12] M. Kazaz, Finite Groups and Surface Coverings, PhD thesis, University of Southampton, 1997. [13] M. Kazaz, Homology action on regular hypermaps of genus 2, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 51 (2002), 1-18. [14] S. Levy (ed.), The Eightfold Way: the Beauty of Klein's Quartic Curve, MSRI Publications, Cambridge University Press, Cambridge, 1999. [15] A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90-96. [16] C-H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13-42. [17] D. Singerman, Reflexibility and symmetry, preprint, University of Southampton, 1992. [18] A. Sinkov, On the group-defining relations (2, 3, 7; p), Ann. of Math. 38 (1937), 577-584. [19] J. Siran, How symmetric can maps on surfaces be?, in: S. Blackburn, S. Gerke, M. Wildon (eds.), Surveys in Combinatorics 2013, London Math. Soc. Lecture Note Ser. 409, Cambridge University Press, Cambridge, 2013, 161-238. [20] K. Wohlfahrt, Zur Struktur der rationalen Modulgruppe, Math. Ann. 174 (1967), 79-99. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 433-444 The Cayley isomorphism property for groups of order 8p Gabor Somlai * EotvOs Lorànd University, Department of Algebra and Number Theory, 1117 Pàzmàny Péter sétany 1/C, Budapest, Hungary Received 31 December 2013, accepted 10 September 2014, published online 26 June 2015 For every prime p > 3 we prove that Q x Zp is a DCI-group, where Q denotes the quaternion group of order 8. Using the same method we reprove the fact that Z3 x Zp is a CI-group for every prime p > 3, which was obtained in [3]. This result completes the description of CI-groups of order 8p. Keywords: Cayley graphs, CI-groups. Math. Subj. Class.: 05C25 1 Introduction Let G be a finite group and S a subset of G. The Cayley graph Cay(G, S) is defined by having the vertex set G and g is adjacent to h if and only if g-1 h g S. The set S is called the connection set of the Cayley graph Cay(G, S). A Cayley graph Cay(G, S) is undirected if and only if S = S-1, where S-1 = {s-1 G G | s g S }. Every left multiplication via elements of G is an automorphism of Cay(G, S), so the automorphism group of every Cayley graph on G contains a regular subgroup isomorphic to G. Moreover, this property characterises the Cayley graphs of G. Similarly to Cayley graphs one can also define ternary Cayley relational structures. (V, E1, E2,..., El ) is a colour ternary relational structure if E с V3 for i = 1,...,/. We say that a colour ternary relational structure (V, E1,..., El) is a Cayley ternary relational structure of the group G if the automorphism group of (V, E1,..., El) contains a regular subgroup isomorphic to G. It is clear that Cay(G, S ) = Cay(G, p(S )) for every ^ G Aut (G). A Cayley graph Cay(G, S) is said to be a CI-graph if, for each T с G, the Cayley graphs Cay(G, S) * Research supported by the Hungarian Scientific Fund (OTKA), grant no. K84233 E-mail address: gsomlai@cs.elte.hu (Gabor Somlai) Abstract and Cay(G, T) are isomorphic if and only if there is an automorphism ^ of G such that ^(S) = T. Furthermore, a group G is called a DCI-group if every Cayley graph of G is a CI-graph and it is called a CI-group if every undirected Cayley graph of G is a CI-graph. Similarly, a group G is called a CI-group with respect to colour ternary relational structures, if for any pair of isomorphic colour ternary relational structures of G there exists an isomorphism induced by an automorphism of G. Let G be a CI-group of order 8p, where p is an odd prime. It is easy to verify that Z2 X Z4 and the dihedral group of order 8 are not CI-groups. It can easily be seen that every subgroup of a CI-group is also a CI-group. Therefore the Sylow 2-subgroup of G can only be Z8, Z2 or the quaternion group Q of order 8. It was proved by Li, Lu and Palfy [5, Theorem 1.2 (b)] that a finite CI-group of order 8p containing an element of order 8 can only be H = (a, z | ap = 1, z8 = 1, z-1az = a-1) . It was also shown in [5, Theorem 1.3] that H is a CI-group, though not a DCI-group. In view of these results, for the rest of the discussion, we assume that the Sylow 2-subgroup of G is isomorphic to Q or Z2. It was proved by Dobson [2] that Z2 xZp is a CI-group with respect to ternary relational structures if p > 11. Moreover, Dobson and Spiga [3] proved that Z2 x Zp is a DCI-group with respect to colour ternary relational structures if and only if p = 3 and 7. As a consequence of this result it was proved in [3] that Z3 x Zp is a DCI-group for all primes p. If p > 8 or p = 5, then by Sylow's Theorem the Sylow p-subgroup of G is a normal subgroup, therefore G is isomorphic to one of the following groups: Zf x Zp, Q x Zp, Z3 к Zp or Q к Zp. It can also be seen from [5, Theorem 1.2] that neither Q к Zp nor Z3 к Zp is a CI-group. If p = 7, then either the Sylow 7-subgroup is normal, in which case G is as before, or G has 8 Sylow 7-subgroups and the Sylow 2-subgroup of G is normal. Then the Sylow 7-subgroup of G acts transitively by conjugation on the the non-identity elements of the Sylow 2-subgroup. Hence G = Z3 x Z7, which is not a CI-group by [5, Theorem 1.2.(b)]. If p = 3, then the order of G is 24. A complete list of CI-groups of order at most 31 was given in the Ph.D. thesis of Royle, see [7]. The CI-groups of order 24 are the following: Q x Z3, Z8 к Z3 and Z3 x Z3. Spiga [6] proved that Q x Z3 is not a CI-group with respect to colour ternary relational structures. Using different methods depending on whether p > 8, or p = 5,7 we show that the other groups are DCI-groups. By extending our result with the fact that Z2 x Z3 is a CI-group we get that Q x Zp is a CI-group for every odd prime p. Theorem 1.1. For every prime p > 3 the group Q x Zp is a DCI-group. We also prove the following result which was first obtained in [3]. Theorem 1.2 (Dobson, Spiga [3]). For every prime p > 3 the group Z2 x Zp is a DCI-group. Our paper is organized as follows. In Section 2 we introduce the notation that will be used throughout this paper. In Section 3 we collect important ideas which are useful in the proof of Theorem 1.1 and 1.2. Section 4 contains the proof of Theorem 1.1 and 1.2 for primes p > 8 and Section 5 contains the proof of Theorem 1.1 and 1.2 for p = 5 and 7. 2 Technical details In this section we introduce some notation. Let G be a group. We use H < G to denote that H is a subgroup of G and by NG(H) and CG(H) we denote the normalizer and the centralizer of H in G, respectively. Let us assume that the group H acts on the set Q and let G be an arbitrary group. Then by G ta H we denote the wreath product of G and H. Every element g G G ta H can be uniquely written as hk, where k G K = Пш£П Gu and h G H. The group K = Пш£П Gu is called the base group of G ta H and the elements of K can be treated as functions from Q to G. If g G G ta H and g = hk we denote k by (g)b. In order to simplify the notation Q will be omitted if it is clear from the definition of H and we will write G i H. The symmetric group on the set Q will be denoted by Sym(Q). Let G be a permutation group on the set Q. For a G-invariant partition B of the set Q we use GB to denote the permutation group on B induced by the action of G and similarly, for every g g G we denote by gB the action of g on the partition B. For a group G, let G denote the subgroup of the symmetric group Sym(G) formed by the elements of G acting by right multiplication on G. For every Cayley graph Г = Cay(G, S) the subgroup G of Sym(G) is contained in Aut(Г). Definition 2.1. Let G < Sym(Q) be a permutation group. Let G(2) = n G Sym(Q) Va, b G Q 3ga,b G G with n(a) = ga,b(a) and n(b) = ga,b(b) We say that G(2) is the 2-closure of the permutation group G. The following lemma is well-known and follows directly from the definition of G(2). Lemma 2.2. Let Г bea graph. If G < Aut(r), then G(2) < Aut(r). 3 Basic ideas In this section we collect some results and some important ideas that we will use in the proof of Theorem 1.1 and Theorem 1.2. We begin with a fundamental lemma that we will use all along this paper. Lemma 3.1 (Babai [1]). The Cayley graph Cay(G, S ) is a CI-graph if and only if for every regular subgroup G of Aut(Cay(G, S)) isomorphic to G there is a p G Aut(Cay(G, S)) such that GM = G. We introduce the following definition. Definition 3.2. (a) We say that a Cayley graph Cay(G, S) is a ci(2) -graph if and only if for every regular subgroup G of Aut(Cay(G, S)) isomorphic to G there is a a G {G, G)(2) suchthat Gff = G. (b) A group G is called a DCI(2)-group if for every S C G the Cayley graph Cay(G, S ) is aCI(2) -graph. Let R be either Q or Z3. Let us assume that A = Aut(Cay(G, S)) < Sym(8p) contains two copies of regular subgroups, R x Zp and R x Zp. By Sylow's theorem we may assume that Zp and Zp are in the same Sylow p-subgroup P of Sym(8p). If p > 8, then P is isomorphic to Zp. Moreover, P is generated by 8 disjoint p-cycles. It follows that both R and R normalize P so we may assume that R and R lie in the same Sylow 2-subgroup of Na(P). Let P2 denote a Sylow 2-subgroup of Sym(8). It is also well known that P2 is isomorphic to the automorphism group of the following graph Д: 1 2 3 4 5 6 7 8 Every automorphism of Д permutes the leaves of the graph and the permutation of the leaves determines the automorphism, therefore ЛиЬ(Д) can naturally be embedded into Sym( 8). Lemma 3.3. (a) There are exactly two regular subgroups of P2 which are isomorphic to Q. (b) There are exactly two regular subgroups of P2 which are isomorphic to Zf. Proof. (a) Let Q be a regular subgroup of ЛиЬ(Д) isomorphic to the quaternion group with generators i and j. Since Q is regular, for every 1 < m < 4 there is a qm e Q (not necessarily distinct) such that qm (2m - 1) = 2m. These are automorphisms of Д so qm(2m) = 2m - 1 and hence since Q is regular the order of qm is 2. There is only one involution in Q so qm = i2 for every 1 < m < 4 and this fact determines completely the action of i2 on Д. Note that the automorphisms qm are all equal. We can assume that i(1) = 3. Such an isomorphism of Д fixes setwise {1,2,3,4} so we have that i(3) = 2, i(2) = 4 and i(4) = 1 since i is of order 4. Using again the fact that Q is regular on Д and i2 (5) = 6, we get that there are two choices for the action of i: i = (1324)(5768) or i = (1324)(5867). We can also assume that j(1) = 5. This implies that j(5) = j2(1) = i2(1) = 2, and j(2) = 6 since j e Ам£(Д) and j(6) = 1. The action of i determines the action of j on Д since iji = j. Applying this to the leaf 3 we get that j(3) = 8 if i = (1324)(5768) and j(3) = 7 if i = (1324)(5867) so there is no more choice for the action of j. Finally, i and j generate Q and this gives the result. (b) One can prove this using an argument similar to the previous case. □ The previous proof also gives the following. Lemma 3.4. (a) The following two pairs of permutations generate the two regular subgroups of Aut(A) < Sym(8) isomorphic to Q: 11 = (1324)(5768), ji = (1526)(3748) and 12 = (1324)(5867), j2 = (1526)(3847). (b) The elements of these regular subgroups of Aut(A) are the following: Qi: Qr • id (12)(34)(56)(78) id (12)(34)(56)(78) (1324)(5768) (1423)(5867) (1324)(5867) (1423)(5768) (1526)(3847) (1625)(3748) (1526)(3748) (1625)(3847) (1728)(3546) (1827)(3645) (1728)(3645) (1827)(3546) Using the identification given in the following table, Qi and Qr act on Q by left-and right-multiplication with the elements of Q, respectively: Q 1 2 3 4 5 6 7 1 — 1 i —i j k 8 -k (c) Ai = (xi,x2,x3) and A2 = isomorphic to Z2, where xi = (12)(34)(56)(78), x2 yi = (12)(34)(56)(78), y2 = (y1,y2,y3) are subgroups of Aut(A) < Sym(8) = (13)(24)(57)(68), x3 = (15)(26)(37)(48) and = (13)(24)(58)(67), уз = (15)(26)(38)(47). Lemma 3.5. Let us assume that Gi < P2 is generated by two different regular subgroups Qa and Qb of Aut(A) which are isomorphic to Q and G2 < P2 is generated by two different regular subgroups Ai and A2 of Aut(A) which are isomorphic to Щ. Then Gi = G2 . Proof. It is clear that |P21 = | Aut(A) | = 27. One can see using Lemma 3.4 (a) and (c) that Gi and G2 are generated by even permutations. Both Gi and G2 induce an action on the set V = {A, B, C, D} which is a set of vertices of A and it is easy to verify that every permutation of V induced by Gi and G2 is even. This shows that Gi and G2 are contained in a subgroup of P2 of cardinality 25. Lemma 3.4 (b) shows that |Qa П Qb| = 2 and one can also check using Lemma 3.4 (c) that |Ai n A2| = 2. This gives |Gi| > 25 and |G21 > 25, finishing the proof of Lemma 3.5. □ Proposition 3.6. (a) The quaternion group Q is a DCI(2)-group. (b) The elementary abelian group Z2 is a DCI(2)-group. Proof. (a) Let Qa and Qb be two regular subgroups of Sym(8) isomorphic to the quaternion group Q. By Sylow's theorem we may assume that Qa and Qb lie in the same Sylow 2-subgroup of H = (Qa, Qb). Since every Sylow 2-subgroup of H is contained in a Sylow 2-subgroup of Sym(8), we may assume that Qa and Qb are subgroups of Aut (A). Our aim is to find an element n g (Qa, Qb}(2) suchthat Qn = Qb- Let us assume that Qa = Qb- Using Lemma 3.4 (a) we may also assume that Qa and Qb are generated by the permutations (1324)(5768), (1526)(3748) and (1324)(5867), (1526)(3847), respectively. Lemma 3.4 (b) shows that H contains the following three permutations: (12)(34) = (1324)(5768)(1324)(5867) (12)(56) = (1526)(3748)(1526)(3847) (12)(78) = (1728)(3546)(1728)(3645). Now one can easily see that the permutation (12) is in Finally, it is also easy to check using Lemma 3.4 (b) that Qai2) = Qb. (b) One can prove this statement using Lemma 3.4 and Lemma 3.5. Definition 3.7. Let Г be an arbitrary graph and A, B с V(Г) such that A n B = 0. We write A ~ B if one of the following four possibilities holds: (a) For every a g A and b g B there is an edge from a to b but there is no edge from b to a. (b) For every a g A and b g B there is an edge from b to a but there is no edge from a to b. (c) For every a g A and b g B the vertices a and b are connected with an undirected edge. (d) There is no edge between A and B. We also write A ^ B if none of the previous four possibilities holds. The following lemma follows easily: Lemma 3.8. Let A, B be two disjoint subsets of cardinality p of a graph. We write Au B = Zp U Zp. Let us assume that a generator д of Zp acts by g(a1,a2) = (a1 + 1, a2 + 1) on A U B and for a generator a of the cyclic group Zp the action of a is defined by a (a1, a2 ) = (a1 + b, a2 + c) for some b,c G Zp. (a) If b = c, then the action of Zp and Zp on A U B are the same. (b) If A ^ B, then b = c. (c) If A ~ B, then every n G Sym(A U B) which fixes A and B setwise is an automorphism of the graph defined on A U B as long as n| A G Aut(A) and n|B G Aut(B). 4 Main result for p > 8 In this section, we will prove that R x Zp is a DCI-group if p > 8, where R is either Q or Z3. Proposition 4.1. For every prime p > 8, the group R x Zp is a DCI-group. Our technique is based on Lemma 3.1 so we have to fix a Cayley graph Г = Cay(R x Zp,S). Let A = Aut(r) and G = R xZp be aregular subgroup of A isomorphic to RxZp. In order to prove Proposition 4.1 we have to find an a G A such that G a = G = R x Zp. We will achieve this in three steps. 4.1 Step 1 Since p > 8, the Sylow p-subgroup of Sym(8p) is generated by 8 disjoint p-cycles. We may assume Zp and Zp lie in the same Sylow p-subgroup P of Sym(8p). Then both R and R are subgroups of NSym(8p) (P) n A so we may assume that R and R lie in the same Sylow 2-subgroup of NSym(8p) (P) n A which is contained in a Sylow 2-subgroup of A. Since p > 8, the Sylow p-subgroup P gives a partition B = {B1, B2,..., B8} of the vertices of Г, where | B41 = p for every i = l,..., 8 and B is P-invariant. It is easy to see that B is invariant under the action of R and R and hence (G, G) < Sym(p) i Sym(8). Moreover, both G and G are regular so R and R induce regular action on B which we denote by Ri and R2, respectively. The assumption that R and R lie in the same Sylow 2-subgroup of A implies that Ri and R2 are in the same Sylow 2-subgroup of Sym(8). 4.2 Step 2 Let us assume that Ri = R2. We intend to find an element a e A such that (Ra) = R2. We define a graph Г0 on B such that Bm is adjacent to Bn if and only if Bm ^ Bn. This is an undirected graph with vertex set B and both Ri and R2 are regular subgroups of Aut(Г0). It follows that Г0 is a Cayley graph of R. Observation 4.1. Since Ri < Aut(r0) acts transitively on B we have that the order of eah connected component of Г0 divides 8. We can also define a coloured graph Г1 on B by colouring the edges of the complete directed graph on 8 vertices. The vertex Bm is adjacent to the vertex Bn with the same coloured edge as Bm> is adjacent to Bn> in Г1 if and only if there exists a graph isomorphism ф from the induced subgraph of Г on Bm U Bn to the induced subgraph of Г on Bm U Bn/ such that ф(Bm) = Bm> and ф(Bn) = Bn>. The graph Г1 is a coloured Cayley graph of R. Moreover, both R1 and R2 act regularly on Г1. Using the fact that R has property DCI(2), it is clear that there exists an a' e (R1, R2)(2) < Aut(r1 ) such that R2 = R1. We would like to lift a' to an automorphism a of Г such that aB = a'. (a) Let us assume first that Г0 is a connected graph. Lemma 4.2. (a) R x Zp < Zp i Sym(8). (b) If R x Zp < Zp i Sym(8), then for every r e R we have (r)b = id. Proof. (a) We first prove that Zp = Zp. Let x and y generate Zp and Zp, respectively. Since x and y lie in the same Sylow p-subgroup and |B1| = p, we can assume that x|Bl = y|Bl. Using Lemma 3.8(b) we get that x|Bm = y|Bn if there exists a path in Г0 from Bm to Bn. This shows that x = y since Г0 is connected. Moreover, R x Zp < Zp I Sym(8) since the elements of Zp and the elements of R commute. (b) Let A' = A n (Zp i Sym(8)^. We have already assumed that R and R lie in the same Sylow 2-subgroup of A'. Let r be an arbitrary element of R . For every (a, u) e R x Zp we have r(a, u) = (b,u + t) for some b e R and t e Zp, where t only depends on r and a since r < Zp i Sym(8). The permutation group G is transitive, hence there exist r1,r2 e R such that r1(l, u) = (a, u) and r2 (b,u +t) = (1 ,u +1). The order of r2rr 1 is a power of 2 since r2,r, r 1 lie in a Sylow 2-subgroup. Therefore t = 0 and hence (r)b = id. □ Lemma 4.2 says that if Г0 is connected, then {R, R) < Zp l Sym(8) and (r)b = id for every r g {R, R). Therefore we can define a = a'id? to be an element of the wreath product Zp l Sym(8) and clearly a'id? is an element of A with aB = a'. (b) Let us assume that Г0 is the empty graph. Then Lemma 3.8(c) shows that every permutation in {R1,R2)(2) lifts to an automorphism of Г. (c) Let us assume that Г0 is neither connected nor the empty graph. Observation 4.2. If R1 = R2, then {R, R) < A contains ß1, ß2, ß2 such that ß? = (B1ß2)(BsB4), в? = (B1B2)(B5B6), ß22 = (B1B2XB7B8). Proof. Recall from Lemma 3.5 that {R, R) is the same group whether R is Q or By Lemma 3.4 the elements ß1, ß2, ß2 can be generated as products of an element of R and R, as in the proof of Proposition 3.6, if R = Q. □ Lemma 4.3. We claim that B2k-1 and B2k are in the same connected component of Г0 for k = 1,2,3,4. Proof. Since Г0 is a Cayley graph and R1 is transitive on the pairs of the form (B2k-1,B2k) it is enough to prove that B1 and B2 are in the same connected component of Г0. If B1 ^ B2, then B1 is adjacent to B2 in Г0, so we can assume that B1 ~ B2. Since Г0 is not the empty graph B1 is adjacent to Bi for some l > 2, so B1 ^ Bi. By Observation (4.2) there exists ß g A such that ß(B1) = B2 and ß(Bi) = Bi. This shows that B2 ^ Bi and hence there is a path from B1 to B2 in Го. □ Г0 is not connected, so the order of the connected components of Г cannot be bigger than 4. Since B1 and B2 are in the same connected component of Г0 there exists a partition H1 U H2 = B such that |H11 = |H21 =4, B1,B2 g H1 and no vertex in H1 is adjacent to any vertex of H2 in Г0. Lemma 4.4. There exists a g A such that a? = a'. Proof. Let us assume first that H1 = {B1, B2, B2, B4}. Then we define a1 to be equal to ß2 on H1 and the identity on H2, where ß2 is defined in Observation 4.2. Using Lemma 3.8(c) we get that a1 is in {R, R)(2). If H1 = {B1,B2,B5,B6} or H1 = {BUB2,B7,B8}, then we define a2 by a2|Hl = ß1 and a2|H2 = id, where ß1 is defined in Observation 4.2. Lemma 3.8(c) shows again that a2 g A. It is easy to see that a2? — a? — (B1B2). Therefore A contains an element a such в that Rf = R2. □ We conclude that we can assume that R1 = R2. 4.3 Step 3 Let us now assume that R1 = R2. We intend to find 7 G A such that RY = R. Let X and X denote the generators of Zp and Zp, respectively. We may assume that X|bi = X|Bl . Lemma 4.5. There exists f G A such that XY = X. Proof. Let us assume first that Г0 is connected. It is clear by Lemma 3.8 (b) that X = X. So, we may take 7 = 1. Let us assume that Г0 is not connected. In this case there are at least two connected components which we denote by C1,..., Cn. We may assume that B1 G C1. The permutations X and X are elements of the base group of Zp l Sym(8) and hence they can be considered as functions on B. We may assume that X(r, u) = (r, u + 1) for every (r, u) g R X Zp. By Lemma 3.8 (b), the function X is constant on each equivalence class. For every 1 < m < n there exists Гт G R such that Гт (C1 ) = Cm and for every rm g R there exists rm g R such that Г^ = Г£. Let 7 be defined as follows: Y |uC = id YluCm = rmfm1 for 2 < m < n. Let (b,v) G rm(Be) with Be G C1 and we denote rm1(b, v) by (a, u). Since X is constant on Cm we have Xs (b, v) = (b, v + cms) for some cm which only depends on Cm. Thus rm(a, u + s) = (b, v + cms) since X and rm commute and X|Be = X|Be. Therefore we have Y(b, w) = rm(a, w) = rm(a, u + (w - u)) = (b, v + cm(w - u)) for every (b, w) G rm(Be). It is easy to verify that y-1(b, w) = (b, w-ve+"Cm ) for every w G Zp which gives Y-1XY(b, w) = Y-1 X(b, wcm + v - ucm) = Y-1(b, wcm + v - ucm + cm) = (b,w + 1). It follows that Y-1XY = f. It remains to show that f G A. Let y and z be two elements of R x Zp. We denote by By and Bz the elements of B containing y and z, respectively. If By and Bz are in the same connected component of Г0, then either f is defined on By and Bz by ГтоГ,-1 which is the element of the group (G , G?} < A or f (y) = y and f(z) = z. If By and Bz are not in the same connected component, then By ~ Bz. The definition of f shows that fB = id. Using Lemma 3.8 (c) we get that f|By uBz is an automorphism of the induced subgraph of Г on the set By U Bz, which proves that f G A, finishing the proof of Lemma 4.5. □ Using Lemma 4.5 we may assume that X = X. Since X and r commute we have R X Zp < Zp l Sym(8). Now we can apply Lemma 4.2 which gives ( Г)ь = id for every r g R. This proves that R = RR since R1 = R2. Therefore G = G, finishing the proof of Proposition 4.1. □ It is straightforward to check that the proof of Proposition 4.1 only uses the fact that p > 8 in the first step of the argument. We can formulate this fact in Proposition 4.6. Proposition 4.6. Let Г bea Cayley graph of G = Q x Zp or G = Z3 x Zp, where p is an odd prime and let G = Q x Zp or G = Zf x Zp be a regular subgroup of Aut(T) isomorphic to G. Let us assume that there exists a (G, G)-invariantpartition B = {Bi, B2,..., B8} of V(Г), where |Bj | = p for every i = {1,..., 8}. In addition, we assume that Zp is a subgroup of the base group of Zp I Sym(B). Then there is an automorphism a of the graph Г such that Ga = G. 5 Main result for p = 5 and 7 In this section we will prove that Q x Z5, Q x Z7, Z\ x Z5 and Zf x Z7 are CI-groups. The whole section is based on the paper [5], so we will only modify the proof of Lemma 5.4 of [5]. Proposition 5.1. Every Cayley graph of Q x Z5, Q x Z7, Z3 x Z5 and Z3 x Z7 is a CI-graph. We let R denote either Q or Z3, and let p = 5 or 7. Let Г be a Cayley graph of R x Zp and let A = Aut^). We denote by P a Sylow p-subgroup of A. Let us assume that A contains two copies of regular subgroups which we denote by G = R x Zp and G = R x Zp. We can assume that Г is neither the empty nor the complete graph and both Zp and Zp are contained in P. If the order of every orbit of P on V(Г) is p, then it is clear from Proposition 4.6 that Г is a CI-graph. Therefore P has an orbit Л с G such that |Л| = p2 since p3 > |G|. The remaining orbits of P have order p since 2p2 > 8p. It was proved in [5] Lemma 5.4 that the action of A on the vertices of the graph Г cannot be primitive so there is a nontrivial A-invariant partition B = {B0, Bi,..., Bt-1} of V(Г) = G. The elements of the partition B have the same cardinality since the action of A is transitive on B so |Bj| < 4p < p2 for every i = 0,1,..., t — 1. The partition B is P-invariant so P acts on B. Since P is a p-group, the order of every orbit of P is a power of p. Let C = {Co, Ci,..., Cs-i} be an orbit of P on B such that Л С и?-1 Cj. We may assume that Bi = Ci for i = 0,1,..., s — 1. It is clear that s is a power of p. If s > p2, then |uS=r01Ci| > 2p2 > 8p which is a contradiction. Since |C0| = |B0| < p2, we cannot have s = 1 . It follows that 1 < s < p2 which implies s = p. For every i < s and every x g P the following eqalities hold for some j < s x(Bi n Л) = x(Bi) n x^) = Bj n Л. This implies that |Bo n Л| = |Bj n Л| for every 0 < i < s. Therefore p2 = |Л| = lus-o1 (Bi n Л) | = s |Bo n Л| = p |Bo n Л|. This gives |B0 n Л| = p so |B0| can only be p or 8 since |B0| t = 8p and both |B0| and t > s are at least p. If |B01 = p, then Л is the union of p elements of the A-invariant partition B and every orbit Л' of P is an element of the partition B if Л' = Л. For every orbit Л' = Л of P and for every y G Zp U Zp we have y(A') = Л'. In particular, y(B7) = B7. By Proposition 4.6 we may assume that there exists an element x' in Zp U Zp such that x'(B0) = Bj for some j = 0, 7 and clearly x'(B7) = B7. Since both G and G are regular there exists a g CA(x') such that a(B0) = B7. Since a and x' commute we have a(Bj) = B7, which contradicts the fact that a(B0) = B7. We must therefore have |B01 = 8. Let x and x generate Zp and Zp, respectively. Since G and G are regular we have that neither xB nor xB is the identity, so both x and x are regular on B. Since both xB and xB generate a transitive subgroup in Sym(B) of prime order p > 2, and every for r g R U R the permutation rB commutes with one of these two elements, we have rB = id. Since x and x are in the same Sylow p-subgroup of P we may assume that x(Bi) = x(Bi) = Bi+1 for i = 0,1,... ,p — 1, where the indices are taken modulo p. By Proposition 4.6 we may also assume that x = x. For every m there exists an l such that the action of xlx-l is nontrivial on Bm since x = x. Therefore ABm \Bm contains a regular subgroup and a cycle of length p such that p > Щ0-. A theorem of Jordan on primitive permutation groups, which can also be found in [8, Theorem 13.1.], says that such a permutation group is 2-transitive and hence the induced subgraph of Г on Bm is the complete or the empty graph for every m. Lemma 5.2. Bm ~ Bn for 0 < m < n < p — 1. Proof. There exists a unique element g g Zp < P such that g(Bm) = Bn. We also have a unique element g g Zp < P with gB = gB. Since Zp is a cyclic group of prime order and x = x we have g = g. Moreover, we may also assume that g\Bm = g\Bm since g = g and the induced subgraphs of Г on Bm+c U Bn+c are all isomorphic, where both m + c and n + c are taken modulo p. Clearly, g = ggisacycle oflength p on Bn. The vertices of V (Г) \ Л are contained in P-orbits of order p that contain the orbit of the vertex under x, so meet each Bi in a single vertex, so g fixes every vertex of the set Bm U Bn \ Л since gB = id. Let u g Bm \ Л. It is enough to show that if u is adjacent to some v g Bn, then u is adjacent to every vertex of Bn. We will prove that A is transitive on the following pairs: {(u,w) \ w G Bn}. A is transitive on {(u,w) \ w G Bn П supp(g)} = {(u,w) \ w G Bn П Л} since g fixes u. Therefore we may assume that v g Bn \ Л and we only have to find an element a G A such that a(u) = u and a(v) G Bn П Л. The restriction of g to Bn is a cycle of length p so g does not commute with r \ Bn, where r is an involution of R. Since r and g commute we have that there is a u' G Bm such that rg(u') = gr(u'). Since the action of R is transitive on Bm there exists r G R such that r(u) = u'. Then ( rr) g(u) = rgr(u) = rg(u') = gr(u') = g ( rr) (u) so there exists a' G A such that a'g(u) = ga'(u). (5.1) Let us suppose that v = g(u). Notice that g(u) is in a P-orbit of order p, so g(u) G Л. Then the inequality (5.1) gives a'(v) = ga'(u). Since R\Bm is regular on Bm there exists s G R such that s(u) = a'(u) and since s and g commute we have s(v) = sg(u) = gs(u) = ga'(u). Therefore s(v) = a'(v) and hence s-1 a' fixes u and s-1a'(v) = v so we may assume that v = s(u). If p = 7, then v e Bn n Л. Let us assume that p = 5. We claim that there exists t e R such that t(u) e Bm \ Л = Bm \ supp(g) while t(v) e Bn n Л = Bn n supp(g). It is clear that g(Bm n supp(g)) = Bn n supp(g) and g commutes with each element of R. Therefore it is enough to show that if u,v e Bm \ supp(g) with u = v, then there exists t e R such that t(u) e Bm \ supp(g) and t(v) e Bm n supp(g). This can easily be seen from the fact that gcd(|R|, 5) = 1. The permutations t-1glt fix the vertex u for every 0 < l < 4 and t-1gl1 t(v) = t-1 gl2 t(v) if l1 ф l2 (modp). At least one of the the four elements t-1gt, t-1g2t, t-1g3t, t-1 g4t of A fixes u and maps v to an element of Bn n supp(g) = Bn n Л since |Bn \ supp(g) | = 3, finishing the proof of the fact that Bm ~ Bn for 0 < m = n < 7. □ Every permutation of V(Г) which fixes Bm setwise for every m is an automorphism of Г so there is an a e A such that xa = x. Applying Proposition 4.6 we get that there exists a e A such that ^R x Zpj = R x Zp, finishing the proof of Proposition 5.1. References [1] L. Babai, Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329-336. [2] E. Dobson, The isomorphism problem for Cayley ternary relational structures for some abelian groups of order 8p, Dicrete Math. 310 (2010), 2895-2909. [3] E. Dobson, P. Spiga, CI-groups with respect to ternary relational structures: new examples, Ars Math. Contemp. 6 (2012), 351-364. [4] C. H. Li, On isomorphisms of finite Cayley graphs- a survey, Discrete Mathematics 256 (2002), 301-334. [5] C. H. Li, Z. P. Lu, P. P. Palfy, Further restrictions on the structure of finite CI-groups, J. Algebr. Comb. (2007), 161-181. [6] P. Spiga, On the Cayley isomorphism problem for a digraph with 24 vertices, Ars. Math. Con-temp. 1 (2008), 38-43. [7] G. Royle, Constructive enumeration of graphs, Thesis submitted to The University of Western Australia, 1987. [8] H. Wielandt, Finite permutation groups, Academic Press, London-New York, 1964. ARS MATHEMATICA CONTEMPORANEA Author Guidelines Papers should be prepared in ШЕХ and submitted as a PDF file. Articles which are accepted for publication have to be prepared in LTeX using class file amcjou.cls and bst file amcjou.bst (if you use BibTEX). 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By mail: Ars Mathematica Contemporanea UP FAMNIT Glagoljaška 8 SI-6000 Koper Slovenia By fax: +386 5 611 75 71 By e-mail: info@famnit.upr.si ARS MATHEMATICA CONTEMPORANEA National Institute of Chemistry Hajdrlhova 19 1000 Ljubljana, Slovenia FAMNIT - University of Primorska Glagoljaška 8 6000 Koper, Slovenia \ ANNOUNCEMENT PhD Fellowship of the National Institute of Chemistry for the studies of Mathematics in Synthetic Biology at the University of Primorska, Slovenia National Institue of Chemistry and the University of Primorska announce a PhD scholarship in Mathematics in Synthetic Biology sponsored by the ERC grant BioOrigami under the supervision of Professor Roman Jerala (Research Advisor and PhD Co-Advisor at the National Institue of Chemistry Professor Tomaž Pisanski (PhD Advisor at the University of Primorska) Applicants should have a BSc or equivalent training in Mathematics (by September 30, 2015) with interests in computer programming and synthetic biology. 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