ARS MATHEMATICA CONTEMPORANEA Volume 16, Number 1, Spring/Summer 2019, Pages 1-276 Covered by: Mathematical Reviews zbMATH (formerly Zentralblatt MATH) COBISS SCOPUS Science Citation Index-Expanded (SCIE) Web of Science ISI Alerting Service Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES) dblp computer science bibliography The University of Primorska The Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The Slovenian Discrete and Applied Mathematics Society The publication is partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications. ARS MATHEMATICA CONTEMPORANEA Plan S - A Golden Addition to the Diamond Open Access Publishing Model We wish to tell readers about 'Plan S', which is a very recent initiative in Europe, begun in September 2018 with the aim of speeding up a transition from the traditional, subscription-based publishing model to the Gold Open Access model. This plan envisages that all research supported by public money in Europe should be published in Gold OA journals from 2020. Although not all details have been decided, it appears that Plan S could involve imposing sanctions against authors whose research was supported by public research agencies but publish their work subscription-based journals, even in leading journals such as Annals of Mathematics, Acta Mathematica, Inventiones Mathematicae, JAMS and JEMS. An obvious possible consequence of this is that some high quality research will be sponsored by private funding (to avoid such sanctions), and that public funding will support research of lesser importance. Accordingly, it seems to us that Plan S is not about research, or about authors or readers, but is driven by financial considerations. Plan S was put up by cOAlition S, which is a consortium of 16 European National Research Agencies. These agencies intend to pay the costs of APC (article processing charges). For example, the Slovenian Research Agency ARRS (which is a founding member of cOAlition S) issued a call in October 2018 and set aside 300000 € for paying back APC for gold OA for articles published in 2018. At the end of November 2018 the results were published and can be found here: http://www.arrs.gov.si/sl/progproj/rproj/ rezultati/18/inc/rezultati-odprti-dostop-18.pdf About 80 Gold OA papers are listed there as winners. The APC for each of them will be paid by the ARRS. The costs of publication vary from about 500 € to over 4000 € per paper, with the majority of APC sitting between 1000 and 2000 €, and one can easily check that most of this public money goes to private publishing houses. There were three criteria for the eligibility of a paper for this payment: (a) acknowledgement of support by the ARRS, (b) payment for the APC in 2018, and (c) ranking of the journal in which the paper appears as Q1 in WoS or Scopus. We note that the third criterion is not compliant with the 2013 Declaration on Research Assessment (DORA), because it involves the use of impact factors. We also note that to date, no authors of mathematical papers have applied for this support by the ARRS. The pertinent question for Ars Mathematica Contemporanea is how should Diamond OA journals adapt to Plan S. First, we plan to ensure that AMC is compliant with Plan S. For any article in which support is to be acknowledged from a Plan S compliant agency (such as the ARRS), we will offer the authors the option of either removing the acknowledgement, or paying APC at some reasonable rate (such as 2000 €). Before we start implementing this policy, we will negotiate with the ARRS (and through the ARRS with other Plan S agencies and other institutions) to make sure they will indeed reimburse the APC. iii ARS MATHEMATICA CONTEMPORANEA Strictly speaking, this means we will change our publishing model from Diamond OA to Gold OA, but on the other hand, we will waive APC for most papers. We believe other Diamond Open Access journals should consider similar adjustments, to deal with Plan S. An interesting question remains. The supporters of Plan S are being encouraged to sign DORA. But when this happens, it is not at all clear how the 'excellence' of eligible journals will be determined. Klavdija Kutnar, Dragan Marušic and Tomaž Pisanski Editors in Chief iv ARS MATHEMATICA CONTEMPORANEA Contents Splittable and unsplittable graphs and configurations Nino Bašic, Jan Grošelj, Branko Grünbaum, Tomaž Pisanski........ 1 A note on the 4-girth-thickness of Kn,n,n Xia Guo, Yan Yang........' .'..................... 19 On the k-metric dimension of metric spaces Alan F. Beardon, Juan A. Rodríguez-Velázquez...............25 Transversals in generalized Latin squares János Barát, Zoltán Lóránt Nagy....................... 39 On the parameters of intertwining codes Stephen P. Glasby, Cheryl E. Praeger.....................49 On the size of maximally non-hamiltonian digraphs Nicolas Lichiardopol, Carol T. Zamfirescu.................. 59 On chromatic indices of finite affine spaces Gabriela Araujo-Pardo, Gyorgy Kiss, Christian Rubio-Montiel, Adrián Vázquez-Ávila............................67 Pentavalent symmetric graphs of order four times an odd square-free integer Bo Ling, Ben Gong Lou, Ci Xuan Wu.................... 81 The pairing strategies of the 9-in-a-row game Lajos Gyorffy, Géza Makay, András Pluhár.................97 Weight choosability of oriented hypergraphs Marcin Anholcer, Bartlomiej Bosek, Jaroslaw Grytczuk...........111 The Doyen-Wilson theorem for 3-sun systems Giovanni Lo Faro, Antoinette Tripodi ....................119 The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace Irene Sciriha, Didar A. Ali, John Baptist Gauci, Khidir R. Sharaf......141 Regular polygonal systems Jurij Kovic...................................157 F-WORM colorings of some 2-trees: partition vectors Julian D. Allagan, Vitaly Voloshin......................173 Relating the total domination number and the annihilation number of cactus graphs and block graphs Csilla Bujtás, Marko Jakovac.........................183 v ARS MATHEMATICA CONTEMPORANEA The validity of Tutte's 3-flow conjecture for some Cayley graphs Milad Ahanjideh, Ali Iranmanesh ......................203 Edge-transitive bi-p-metacirculants of valency p Yan-Li Qin, Jin-Xin Zhou...........................215 Comparing the expected number of random elements from the symmetric and the alternating groups needed to generate a transitive subgroup Andrea Lucchini, Mariapia Moscatiello...................237 On the domination number and the total domination number of Fibonacci cubes ElifSaygi...................................245 Pappus's Theorem in Grassmannian Gr (3, Cn) Sumire Sawada, Simona Settepanella, So Yamagata.............257 Volume 16, Number 1, Spring/Summer 2019, Pages 1-276 vi ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 1-17 https://doi.org/10.26493/1855-3974.1467.04b (Also available at http://amc-journal.eu) Splittable and unsplittable graphs and configurations* * Nino Basic FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia IMFM, Jadranska 19, 1000 Ljubljana, Slovenia Jan Grošelj Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Branko Griinbaum Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA Tomaž Pisanski FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia IMFM, Jadranska 19, 1000 Ljubljana, Slovenia Received 21 August 2017, accepted 18 March 2018, published online 22 August 2018 We prove that there exist infinitely many splittable and also infinitely many unsplittable cyclic (n3) configurations. We also present a complete study of trivalent cyclic Haar graphs on at most 60 vertices with respect to splittability. Finally, we show that all cyclic flag-transitive configurations with the exception of the Fano plane and the Mobius-Kantor configuration are splittable. Keywords: Configuration of points and lines, unsplittable configuration, unsplittable graph, independent set, Levi graph, Griinbaum graph, splitting type, cyclic Haar graph. Math. Subj. Class.: 5IA20, 05B30 * The work was supported in part by a grant from the Picker Institute at Colgate University and the ARRS of Slovenia, grants P1-0294, N1-0011, N1-0032, J1-6720, and J1-7051. We would like to thank Istvan Estelyi and Jaka Kranjc for fruitful discussions during the preparation of this paper and the two anonymous referees for useful remarks and suggestions that have improved the quality of the paper. E-mail addresses: nino.basic@famnit.upr.si (Nino Basic), jan.groselj@fmf.uni-lj.si (Jan Grošelj), grunbaum@math.washington.edu (Branko Grunbaum), tomaz.pisanski@upr.si (Tomaž Pisanski) ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ Abstract 2 Ars Math. Contemp. 16 (2019) 19-24 1 Introduction and preliminaries The idea of unsplittable configuration was conceived in 2004 and formally introduced in the monograph [8] by Grunbaum. Later, it was also used in [19]. In [20], the notion was generalized to graphs. In this paper we present some constructions for splittable and unsplittable cyclic configurations. In [9], the notion of cyclic Haar graph was introduced. It was shown that cyclic Haar graphs are closely related to cyclic configurations. Namely, each cyclic Haar graph of girth 6 is a Levi graph of a cyclic combinatorial configuration; see also [18]. For the definition of the Levi graph (also called incidence graph) of a configuration the reader is referred to [4]. The classification of configurations with respect to splittability is a purely combinatorial problem and can be interpreted purely in terms of Levi graphs. Let n be a positive integer, let Zn be the cyclic group of integers modulo n and let S Ç Zn be a set, called the symbol. The graph H (n, S) with the vertex set {« | i G Zn} U {v | i g Zn} and edges joining ui to vi+k for each i g Zn and each k g S is called a cyclic Haar graph over Zn with symbol S [9]. In practice, we will simplify the notation by denoting ui by i+ and vi by i-. Definition 1.1. A combinatorial (vk ) configuration is an incidence structure C = (P, B, I), where I Ç P x B, P n B = 0 and |P| = |B| = v. The elements of P are called points, the elements of B are called lines and the relation I is called the incidence relation. Furthermore, each line is incident with k points, each point is incident with k lines and two distinct points are incident with at most one common line, i.e., {(pi,bi), (p2,bi), (pi,b2), (P2,b2)} Ç I,P1 = P2 bi = b2. (1.1) If (p, b) G I then we say that the line b passes through point p or that the point p lies on line b. An element of P U B is called an element of configuration C. A combinatorial (vk) configuration C = (P, B, I) is geometrically realisable if the elements of P can be mapped to different points in the Euclidean plane and the elements of B can be mapped to different lines in the Euclidean plane, such that (p, b) g I if and only if the point that corresponds to p lies on the line that corresponds to b. A geometric realisation of a combinatorial (vk ) configuration is called a geometric (vk) configuration. Note that examples in Figures 2, 3 and 4 are all geometric configurations. The Fano plane (73) is an example of a geometrically non-realizable configuration. An isomorphism between configurations (P, B, I) and (P', B', I') is a pair of bijec-tions ^ : P ^ P' and ^ : B ^ B', such that (p, b) G I if and only if (^(p), y>(b)) G I'. (1.2) The configuration C* = (B, P, I*), where I* = {(b,p) G BxP | (p, b) G I}, is called the dual configuration of C. A configuration that is isomorphic to its dual is called a self-dual configuration. The Levi graph of a configuration C is the bipartite graph on the vertex set P U B having an edge between p G P and b G B if and only if the elements p and b are incident in C, i.e., if (p, b) G I. It is denoted L(C). Condition (1.1) in Definition 1.1 implies that the girth of L(C) is at least 6. Moreover, any combinatorial (vk ) configuration is completely determined by a k-regular bipartite graph of girth at least 6 with a given black-and-white vertex coloring, where black vertices correspond to points and white vertices correspond N. Basic et al.: Splittable and unsplittable graphs and configurations 3 to lines. Such a graph will be called a colored Levi graph. Note that the reverse coloring determines the dual configuration C* = (B, P, I*). Also, an isomorphism between configurations corresponds to color-preserving isomorphism between their respective colored Levi graphs. A configuration C is said to be connected if its Levi graph L(C) is connected. Similarly, a configuration C is said to be k-connected if its Levi graph L(C) is k-connected. Definition 1.2 ([19]). A combinatorial (vk ) configuration C is cyclic if it admits an automorphism of order v that cyclically permutes the points and lines, respectively. In [9] the following was proved: Proposition 1.3 ([9]). A configuration C is cyclic if and only if its Levi graph is isomorphic to a cyclic Haar graph of girth 6. It can be shown that each cyclic configuration is self-dual, see for instance [9]. 2 Splittable and unsplittable configurations (and graphs) Let G be any graph. The square of G, denoted G2, is a graph with the same vertex set as G, where two vertices are adjacent if and only if their distance in G is at most 2. In other words, V(G2) = V(G) and E(G2) = {uv | dG(u, v) < 2}. The square of the Levi graph L(C) of a configuration C is called the Grunbaum graph of C in [19] and [20]. In [8], it is called the independence graph. Two elements of a configuration C are said to be independent if they correspond to independent vertices of the Griinbaum graph. Example 2.1. The Grunbaum graph of the Heawood graph is shown in Figure 1. Its complement is the Mobius ladder M14. Figure 1: The Heawood graph H = H(7, {0,1, 3}) = LCF[5, -5]7 (on the left) is the Levi graph of the Fano plane. Its Griinbaum graph G is on the right. Note that there is an orange solid edge between two vertices of G if and only if they are at distance 2 in H. It is easy to see that two elements of C are independent if and only if one of the following hold: 4 Ars Math. Contemp. 16 (2019) 19-24 (i) two points of C that do not lie on a common line of C ; (ii) two lines of C that do not intersect in a common point of C; (iii) a point of C and a line of C that are not incident. The definition of unsplittable configuration was introduced in [8] and is equivalent to the following: Definition 2.2. A configuration C is splittable if there exists an independent set of vertices S in the Griinbaum graph (L(C))2 such that L(C) - E, i.e., the graph obtained by removing the set of vertices E from the Levi graph L(C), is disconnected. In this case the set E is called a splitting set of elements. A configuration that is not splittable is called unsplittable. This definition carries over to graphs: Definition 2.3. A connected graph G is splittable if there exists an independent set E in G2 such that G - E is disconnected. Example 2.4. Every cycle of length at least 6 is splittable (there exists a pair of vertices at distance 3 in G). Every graph of diameter 2 without a cut vertex is unsplittable. The square of such a graph on n vertices is the complete graph Kn. This implies that |S| = 1. Since there are no cut vertices, a splitting set does not exist. The Petersen graph is an example of unsplittable graph. In [8], refinements of the above definition are also considered. Configuration C is point-splittable if it is splittable and there exists a splitting set of elements that consists of points only (i.e., only black vertices in the corresponding colored Levi graph). In a similar way line-splittable configurations are defined. Note that these refinements can be defined for any bipartite graph with a given black-and-white coloring. There are four possibilities, that we call splitting types. Any configuration may be: (T1) point-splittable, line-splittable, (T2) point-splittable, line-unsplittable, (T3) point-unsplittable, line-splittable, (T4) point-unsplittable, line-unsplittable. Any configuration of splitting type T1, T2 or T3 is splittable. A configuration of splitting type T4 may be splittable or unsplittable. For an example of a point-splittable (T2) configuration see Figure 2. The configuration on Figure 2 is isomorphic to a configuration on Figure 5.1.11 from [8]. For an example of a line-splittable (T3) configuration see Figure 3. Note the following: Proposition 2.5. If C is of type T1 then its dual is also of type T1. If it is of type T2 then its dual is of type T3 (and vice versa). If it is of type T4 then its dual is also of type T4. Since types are mutually disjoint, this has a straightforward consequence for cyclic configurations: Corollary 2.6. Any self-dual configuration, in particular any cyclic configuration, is either of type T1 or T4. N. Basic et al.: Splittable and unsplittable graphs and configurations 5 Figure 2: A point-splittable (153) configuration of type T2. Points that belong to a splitting set are colored red. Its dual is of type T3 (see Figure 3). Figure 3: A line-splittable (153) configuration of type T3. Lines that belong to a splitting set are colored orange. Its dual is depicted in Figure 2. Obviously, unsplittable configurations are of type T4. However, the converse is not true: Proposition 2.7. Any unsplittable configuration is point-unsplittable and line-unsplittable. There exist splittable configurations that are both point-unsplittable and line-unsplittable. Proof. The first statement of Proposition 2.7 is obviously true. An example that provides the proof of the second statement is shown in Figure 4. The splitting set is {0, 8,10, (1, 9,11), (6, 7,14)}. □ Note that configuration in Figure 4 is not cyclic, but it is 3-connected. In [8], the following theorem is proven: Theorem 2.8 ([8, Theorem 5.1.5]). Any unsplittable (n3) configuration is 3-connected. Our computational results show that the converse to Theorem 2.8 is not true. There exist 3-connected splittable configurations. See, for instance, the configuration in Figure 4. 6 Ars Math. Contemp. 16 (2019) 19-24 Figure 4: A splittable (153) configuration of type T4. Elements of a splitting set are points 0, 8 and 10 (colored red) and lines (1,9,11) and (6,7,14) (colored orange). 3 Splittable and unsplittable cyclic (n3) configurations We used a computer program to analyse all cyclic (n3) configurations for 7 < n < 30 (see Table 1, Table 2 and Table 3). In [9] it was shown that cyclic Haar graphs contain all information about cyclic combinatorial configurations. In trivalent case combinatorial isomorphisms of cyclic configurations are well-understood; see [11]. Namely, it is known how to obtain all sets of parameters of isomorphic cyclic Haar graphs. We would like to draw the reader's attention to the manuscript [10], where the main result of [11] is extended to cyclic (nk ) configurations for all k > 3. One would expect that large sparse graphs are splittable. In this sense the following result is not a surprise: Theorem 3.1. Let H(n, {0, a, b}) be a cyclic Haar graph, where 0 < a < b. Let W = {0, a, b, 2b, b + a, b — a, 2b — a, 2b — 2a, 3b — a, 3b — 2a, 2b + a, 3b} and B = {0, a, b, 2b, b + a, b — a, 2b — a, 2b — 2a, 3b — a, 3b — 2a, —a, b — 2a} be multisets with elements from Zn. If all elements of W are distinct and all elements of B are distinct (i.e. W and B are ordinary sets, |W| = |B| = 12) then H(n, {0, a, b}) is splittable and S = {0+, 2b+, (2b — 2a)+, (b — a)-, (b + a)-, (3b — a)-} is a splitting set for H (n, {0, a, b}). Proof. See Figure 5. If W and B are ordinary sets then the graph in Figure 5 is a subgraph of H(n, {0, a, b}). It is easy to see that S is a splitting set. The set S is indeed an independent set in the square of the graph H(n, {0, a, b}) since no two vertices of S are adjacent to the same vertex. In order to see that the subgraph obtained by removing the vertices of S is disconnected, note that one of the connected components is the cycle determined by vertices {b+, (b — a)+, (2b — a)+, b-, 2b-, (2b — a)-}. □ Corollary 3.2. Under conditions of Theorem 3.1, the girth of the graph H(n, {0, a, b}) is 6. N. Basic et al.: Splittable and unsplittable graphs and configurations 7 Figure 5: The set S = {0+, 2b+, (2b - 2a)+, (b - a)-, (6 + a)-, (36 - a)-} is a splitting set for H(n, {0, a, b}). Proof. The girth of such a graph is at most 6 because it contains a 6-cycle (see Figure 5). It is easy to see that the girth cannot be 4. Because the graph H(n, {0, a, b}) is bipartite, each 4-cycle must contain a black vertex. Consider vertex b+ in Figure 5. Its neighborhood is {b-, 2b-, (b + a)-}. None of those vertices have a common neighbor, so b+ does not belong to any 4-cycle. Because of symmetry this argument holds for all black vertices. □ Corollary 3.3. There exist infinitely many cyclic (n3) configurations that are splittable. For example, the following three families of cyclic Haar graphs are splittable: (a) H(n, {0,1,4}) for n > 13, (b) H(n, {0,1, 5}) for n > 16, and (c) H(n, {0, 2, 5}) for n > 16. Proof. Corollary 3.2 implies that each graph from any of the three families has girth 6. From Theorem 3.1 it follows that S = {0+, 6+, 8+, 3-, 5-, 11-} is a splitting set for H(n, {0,1,4}) if n > 13 (see Figure 6), {0+, 8+, 10+, 4-, 6-, 14-} is a splitting set for H(n, {0,1, 5}) if n > 16, and {0+, 6+, 10+, 3-, 7-, 13-} is a splitting set for H(n, {0, 2, 5}) if n > 16. □ If n < 13 then conditions of Theorem 3.1 are not fulfilled. If n = 12 then (n — 1)+ = 11+ which means that the vertices of the graph in Figure 6 are not all distinct. If n = 9 then 9- = 0- since we work with Z9. Similar arguments can be made if n < 16 in the case of the other two families from Corollary 3.3. We investigated the first 100 graphs from the H(n, {0,1,4}) family. All but two are zero symmetric, nowadays called graphical regular representation or GRR for short (see [5]). The exceptions are for n =13 and n = 15. 8 Ars Math. Contemp. 16 (2019) 19-24 (n - 1) + 10- 9 12 Figure 6: The set E where n > 13. = {0+, 6+, 8+, 3-, 5-, 11-} is a splitting set for H(n, {0,1,4}) By Corollary 3.3, there are infinitely many splittable (n3) configurations. However, we are also able to show that there is no upper bound on the number of vertices of unsplittable (n3) configurations: Theorem 3.4. There exist infinitely many cyclic (n3) configurations that are unsplittable. Proof. We use the cyclic Haar graphs X = H(n, {0,1,3}), where n > 7. Clearly, each of them has girth 6. The graph can be written as LCF[5, —5]n. (For the LCF notation see [19].) This means that the edges determined by symbols 0 and 1 form a Hamiltonian cycle while the edges arising from the symbol 3 form chords of length 5. See Figure 1 for an example. Let us assume the result does not hold. This means there exists a splitting set E. By removing E from the graph the Hamiltonial cycle breaks into paths. Each path must contain at least two vertices. Let the sequence n = (pi,p2,... ,pk) denote the lengths of the consecutive paths along the Hamiltonial cycle. The rest of the proof is in two steps: Step 1. If there are no two consecutive numbers of n equal to 2, then the corresponding segments are connected in X - E since there is a chord of length 5 joining these two segments. But this means that all paths are connected by chords, so E is not a splitting set. Step 2. We can show that no two consecutive segments are of length 2. In case of two adjacent segments of length 2 we would have vertices {i — 3,i,i + 3} C E. But that is impossible, since i — 3 is adjacent to i + 3 in X2. □ Note that this is not the only such family. Here is another one: Theorem 3.5. Cyclic configurations defined by H(3n, {0,1,n}), where n > 2, are unsplittable. N. Basic et al.: Splittable and unsplittable graphs and configurations 9 Proof. The technique used here is similar to the technique used in proof of Theorem 3.4. Let X = H(3n, {0,1, n}). The graph X can be written as LCF[2n - 1, -(2n - 1)]3n. Suppose that there exists a splitting set E. The edges determined by symbols 0 and 1 form a Hamiltonian cycle which breaks into paths when the splitting set E is removed. We show that any two consecutive paths are connected in X - E. Without loss of generality (because of symmetry), we may assume that 0+ g E is the vertex adjacent to the two paths under consideration. If 0+ g E then 1-, 0-,n-,n+, 1+, 2n+, (2n + 1)+ G E. We show that vertices 1- and 0- (which belong to the two paths under consideration) are connected in X - E. If (2n +1)- G E then 2n+ and (2n +1)+ are connected in X - E. Since 0- is adjacent to 2n+ and 1- is adjacent to (2n + 1)+, vertices 0- and 1- are also connected in X - E. Now, suppose that (2n + 1)- G E. This implies that 2n-, (n + 1)+(n + 1)- G E. Then 2n+ is adjacent to 2n-, 2n- is adjacent to n+, n+ is adjacent to (n + 1)-, (n + 1)- is adjacent to 1+, and 1+ is adjacent to 1- in X - E. Therefore, 1- and 0- are connected in X - E. □ Cubic symmetric bicirculants were classified in [13] and [16]. These results can be summarised as follows: Theorem 3.6 ([13, 16]). A connected cubic symmetric graph is a bicirculant if and only if it is isomorphic to one of the following graphs: (1) the complete graph K4, (2) the complete bipartite graph K33, (3) the seven symmetric generalized Petersen graphs GP(4,1), GP(5,2), GP(8, 3), GP(10, 2), GP(10, 3), GP(12,5) and GP(24,5), (4) the Heawoodgraph H(7, {0,1, 3}), and (5) the cyclic Haar graph H(n, {0,1, r + 1}), where n > 11 is odd and r G Z*n such that r2 + r + 1 = 0 (mod n). It is well known that an (n3) configuration is flag-transitive if and only if its Levi graph is cubic symmetric graph of girth at least 6. From Theorem 3.6 it follows that the girth of any connected cubic symmetric bicirculant is at most 6. If the girth of such a graph is 6 or more then it is a Levi graph of a flag-transitive configuration. This enables us to characterise splittability of such configurations: Theorem 3.7. The Fano plane (73), the Mobius-Kantor configuration (83), and the Desargues configuration (103 ) are unsplittable. Their Levi graphs are H(7, {0,1,3}), H(8, {0,1,3})= GP(8, 3) and GP(10,3), respectively. If n > 9, all flag-transitive (n3 ) configurations, except the Desargues configuration, are splittable. Proof. We start with the classification given in Theorem 3.6. Only bipartite graphs of girth 6 have to be considered. This rules out the complete graph K4, the complete bipartite graph K3 3, and the generalised Petersen graphs GP(5, 2), GP(10,2) and GP(4,1). Note that GP(4,1) is isomorphic to the cube graph Q3. 10 Ars Math. Contemp. 16 (2019) 19-24 Table 1: Overview of splittable and unsplittable connected cyclic Haar graphs. n (a) (b) (c) (d) (e) (f) 3 1 0 0 1 0 0 4 1 0 0 1 0 0 5 1 0 0 1 0 0 6 2 0 0 2 0 0 7 2 1 0 2 0 1 8 3 1 1 2 0 1 9 2 1 0 2 0 1 10 3 1 1 2 0 1 11 2 1 0 2 0 1 12 5 3 1 4 0 13 3 2 1 2 1 1 14 4 2 2 2 1 1 15 5 4 1 4 1 16 5 3 3 2 2 1 17 3 2 1 2 1 1 18 6 4 3 3 2 19 4 3 2 2 2 1 20 7 5 5 2 4 1 21 7 6 3 4 3 22 6 4 4 2 3 1 23 4 3 2 2 2 1 24 11 9 7 4 6 25 5 4 3 2 3 1 26 7 5 5 2 4 1 27 6 5 3 3 3 28 9 7 7 2 6 1 29 5 4 3 2 3 1 30 13 11 9 4 8 3 (a) Number of non-isomorphic connected cubic cyclic Haar graphs on 2n vertices. (b) Those that have girth 6. (c) Those that are splittable. (d) Those that are unsplittable. (e) Those that are splittable of girth 6. (f) Those that are unsplittable of girth 6. N. Basic et al.: Splittable and unsplittable graphs and configurations 11 It is well known, but one may check by computer that GP(8, 3) = H(8, {0,1,3}). See for instance [9, Table 2]. One may also check by computer that GP(8, 3), GP(10,3) and the Heawood graph H(7, {0,1,3}) are unsplittable. V(GP(n, k)) = {0,1,..., n - 1,0', 1',..., (n - 1)'} and E(GP(n, k)) = {{i', ((i +1) mod n)'}, {i, i'}, {i, (i + k) mod n} | i = 0,..., n - 1}. Note that E = {0', 4', 8', 2, 6,10} is a splitting set for GP(12,5) as shown in Figure 7. Also, GP(12,5) - S = 3G6, i.e., a disjoint union of three copies of G6. The splitting Figure 7: The magenta vertices form a splitting set for the Nauru graph GP(12, 5) [6, 21]. set for GP(24,5) is E = {0', 4', 8', 12', 16', 20', 2, 6,10,14,18, 22} as shown in Figure 8. Note that GP(24,5) - S = 3Gi2. Also, note that GP(24,5) is not isomorphic to a cyclic Haar graph since its girth is 8. Using Theorem 3.1, one may verify that all graphs in item (5) of Theorem 3.6 have girth 6 and for each of them the splitting set is {0+, 2r+, (2r + 2)+,r-, (r + 2)-, (3r + 2)-}. We have W = {0,1, r, r + 1, r + 2, 2r, 2r + 1, 2r + 2, 2r + 3, 3r + 1, 3r + 2, 3r + 3}, B = {0,1, n - 1, r - 1, r, r + 1,r + 2, 2r, 2r + 1, 2r + 2, 3r + 1, 3r + 2}. It is easy to verify that all elements of W are distinct and that all elements of B are distinct. For example, suppose that r = 3r + 3 (mod n). This means that Let 3' 9' 6' 2r = -3 (mod n). From condition r2 + r + 1 = 0 (mod n) we obtain 4r2 +4r + 4=(2r)2 +2 • 2r + 4 = 0 (mod n). (3.1) (3.2) 12 Ars Math. Contemp. 16 (2019) 19-24 1 0' 23' 6' 5' 7' 19 17' 18' 11 12' 13' Figure 8: The magenta vertices form a splitting set for GP (24,5) which was recently named the ADAM graph [14]. Equations (3.1) and (3.2) together imply that (-3)2 + 2 • (-3) + 4 = 7 = 0 (mod n), which is a contradiction since n > 11. All other cases can be checked in a similar way. □ From Theorem 3.7 we directly obtain the following corollary. Corollary 3.8. A cyclic flag-transitive (n3) configuration is splittable if and only if n > 8. The only two exceptions are: (1) H(7, {0,1,3}), i.e. the Fanoplane, and (2) H(8, {0,1,3}), i.e. the Mobius-Kantor configuration. 4 Splittable geometric (nk ) configurations We will now show that for any k there exist a geometric, triangle-free, (nk ) configuration which is of type T1, i.e., it is point-splittable and line-splittable. Let us first provide a construction to obtain a geometric (nk ) configuration for any k. We start with an unbalanced (ki, 1k) configuration, denoted G(1), that consists of a single line containing k points. Let G^ be a configuration that is obtained from Gk-1 by the k-foldparallel replication (see [19, p. 245]). The configuration G(k) is a balanced (k kk) configuration, called a generalised Gray configuration; see [17]. Lemma 4.1. Let C be an arbitrary geometric (nk ) configuration. There exists a geometric (knk) configuration D that is point- and line-splittable. Moreover, if C is triangle-free then D is also triangle-free. Proof. Let C be as stated. Select an arbitrary line L of C passing through points p(1) ,p(2), ... ,p(k) of C as shown in Figure 9(a). Remove line L and call the resulting structure N. Basic et al.: Splittable and unsplittable graphs and configurations 13 Table 2: List of non-isomorphic connected trivalent cyclic Haar graphs H(n, S) with n < 25 and some of their properties. n S (a) (b) (c) (d) n S (a) (b) (c) (d) 3 {0 1 2} ± 4 2 T 18 {0 1 6} ± 6 6 ± 4 {0 1 2} 4 3 T 18 {0 1 9} T 4 9 5 {0 1 2} 4 3 19 {0 1 2} 4 10 6 {0 1 2} 4 4 19 {0 1 3} 6 7 6 {0 1 3} 4 3 19 {0 1 4} T 6 6 7 {0 1 2} 4 4 19 {0 1 8} T 6 5 T 7 {0 1 3} 6 3 T 20 {0 1 2} 4 11 8 {0 1 2} 4 5 20 {0 1 3} 6 8 8 {0 1 3} 6 4 T 20 {0 1 4} T 6 6 8 {0 1 4} T 4 4 20 {0 1 5} T 6 6 9 {0 1 2} 4 5 20 {0 1 6} T 6 6 9 {0 1 3} 6 4 20 {0 1 9} T 6 7 10 {0 1 2} 4 6 20 {0, 1 10} T 4 10 10 {0 1 3} 6 4 21 {0 1 2} 4 11 10 {0 1 5} T 4 5 21 {0 1 3} 6 8 11 {0 1 2} 4 6 21 {0 1 4} T 6 6 11 {0 1 3} 6 5 21 {0 1 5} T 6 6 T 12 {0 1 2} 4 7 21 {0 1 7} 6 7 12 {0 1 3} 6 5 21 {0 1 8} 6 7 12 {0 1 4} 6 5 21 {0 1 9} T 6 6 12 {0 1 5} 6 5 22 {0 1 2} 4 12 12 {0 1 6} T 4 6 22 {0 1 3} 6 8 13 {0 1 2} 4 7 22 {0 1 4} T 6 7 13 {0 1 3} 6 5 22 {0 1 5} T 6 6 13 {0 1 4} T 6 5 T 22 {0 1 6} T 6 7 14 {0 1 2} 4 8 22 {0, 1 11} T 4 11 14 {0 1 3} 6 6 23 {0 1 2} 4 12 14 {0 1 4} T 6 5 23 {0 1 3} 6 9 14 {0 1 7} T 4 7 23 {0 1 4} T 6 7 15 {0 1 2} 4 8 23 {0 1 5} T 6 7 15 {0 1 3} 6 6 24 {0 1 2} 4 13 15 {0 1 4} T 6 5 24 {0 1 3} 6 9 15 {0 1 5} 6 5 24 {0 1 4} T 6 7 15 {0 1 6} 6 5 24 {0 1 5} T 6 7 16 {0 1 2} 4 9 24 {0 1 6} T 6 7 16 {0 1 3} 6 6 24 {0 1 7} T 6 7 16 {0 1 4} T 6 5 24 {0 1 8} 6 8 16 {0 1 7} T 6 5 24 {0 1 9} 6 8 16 {0 1 8} T 4 8 24 {0 , 1,10} T 6 6 17 {0 1 2} 4 9 24 {0,1 ,11} T 6 7 17 {0 1 3} 6 7 24 {0 , 1,12} T 4 12 17 {0 1 4} T 6 5 25 {0,1 ,2} 4 13 18 {0 1 2} 4 10 25 {0,1 ,3} 6 9 18 {0 1 3} 6 7 25 {0,1 ,4} T 6 7 18 {0 1 4} T 6 6 25 {0,1 ,5} T 6 7 18 {0 1 5} T 6 6 25 {0 , 1,10} T 6 7 (a) splittable? (b) girth (c) diameter (d) arc-transitive? 14 Ars Math. Contemp. 16 (2019) 19-24 Table 3: List of non-isomorphic connected trivalent cyclic Haar graphs H(n, S) with 26 < n < 30 and some of their properties. n S (a) (b) (c) (d) 26 {0,1, 2} 4 14 26 {0,1, 3} 6 10 26 {0,1,4} T 6 8 26 {0,1, 5} T 6 7 26 {0,1, 7} T 6 8 26 {0,1, 8} T 6 7 26 {0,1,13} T 4 13 27 {0,1, 2} 4 14 27 {0,1, 3} 6 10 27 {0,1,4} T 6 8 27 {0,1, 5} T 6 7 27 {0,1, 6} T 6 7 27 {0,1, 9} 6 9 28 {0,1, 2} 4 15 28 {0,1, 3} 6 10 28 {0,1,4} T 6 8 28 {0,1, 5} T 6 7 28 {0,1, 6} T 6 8 28 {0,1, 7} T 6 7 28 {0,1, 8} T 6 7 28 {0,1,13} T 6 9 28 {0,1,14} T 4 14 29 {0,1, 2} 4 15 29 {0,1, 3} 6 11 29 {0,1,4} T 6 8 29 {0,1, 5} T 6 7 29 {0,1, 9} T 6 7 30 {0,1, 2} 4 16 30 {0,1, 3} 6 11 30 {0,1,4} T 6 9 30 {0,1, 5} T 6 7 30 {0,1, 6} T 6 7 30 {0,1, 7} T 6 7 30 {0,1, 8} T 6 9 30 {0,1, 9} T 6 7 30 {0,1,10} 6 10 30 {0,1,11} 6 10 30 {0,1,12} T 6 8 30 {0,1,15} T 4 15 30 {0, 2, 5} T 6 8 (a) splittable? (b) girth (c) diameter (d) arc-transitive? N. Basic et al.: Splittable and unsplittable graphs and configurations 15 C'. Make k copies of C': C|, C2,..., Ck and place them equally spaced in any direction v that is non-parallel to the direction of any line of C' (see Figure 9(b)). Point of Ci that (a) (b) Figure 9: Construction provided by Lemma 4.1. correspond to p(j) in C' is denoted pj). Now add lines Mi,M2,..., Mk, such that Mi passes through points p^p^,... ,Pk ^ The resulting structure, denoted D, is clearly a (knk) configuration. The set of lines {Mi, M2,..., Mk} is a splitting set of D which proves that D is line-splittable. The set of points {p[i), pi2),..., pik)} is a splitting set for an arbitrary 1 < i < k which proves that D is also point-splittable. It is easy to see that the resulting structure D is triangle-free. □ Now we can state the main result of this section. Theorem 4.2. For any k > 1 and any n0 there exist a number n > n0, such that there exists a splittable (nk ) configuration. Proof. Let C0 = G^, i.e. the generalised Gray (kkk ) configuration. Let Ci be a configuration obtained from Ci-i by an application of Lemma 4.1. Note that the obtained configuration Ci is not uniquely defined - it depends on the choice of the line L. From Lemma 4.1 it follows that each Ci, i > 1, is a point- and line-splittable configuration. Each configuration Ci is balanced and the number of points of Ci+i is strictly greater than the number of points of Ci. Therefore, for increasing values of i, the number of points will eventually exceed any given number n0. □ Since configurations Ci, C2,... constructed in the proof of Theorem 4.2 are all of type T1, their duals are also of type T1. Example 4.3. The generalised Gray (kkk) configuration for k = 3 is simply called the Gray configuration (see Figure 10(a) and [17]). Let C0 be the Gray configuration. By one application of Lemma 4.1 we obtain a configuration Ci (see Figure 10(b)) which is point-and line-splittable. 5 Conclusion Theorems 3.4 and 3.5, Corollary 3.3, and our experimental investigations (see periodic behaviour of the last column of Table 1 past n = 9) of splittability of cyclic Haar graphs led us to the following conjecture. 16 Ars Math. Contemp. 16 (2019) 19-24 Figure 10: The Gray (273) configuration C0 and the corresponding C\. Conjecture 5.1. A cyclic (n3) configuration is unsplittable if and only if its Levi graph belongs to one of the following three infinite families: (!) H(n, {0,1, 3}) for n > 7; (2) H(3n, {0,1, n}) for n > 2; (3) H(3n, {0,1, n + 1}) for n > 4 where n ^ 0 (mod 3). To show that all other cyclic (n3) configurations are splittable, we expect that the method used in the proof of Theorem 3.1, Corollary 3.2 and Corollary 3.3 can be extended. Nedela and Skoviera [15] showed a nice property of cubic graphs with respect to the cyclic connectivity. Their result is likely to have applications in splittablity. In Section 4 we have shown how to construct geometric point- and line-splittable (nk ) configuration for any k. However, we were not able to obtain any splittable cyclic (nk ) configuration for k > 4 so far. Therefore, we pose the following claim. Conjecture 5.2. All cyclic (nk) configurations for k > 4 are unsplittable. Notions of splittable and unsplittable configurations have been defined via associated graphs. Since splittability is a property of combinatorial configurations, it can be extended from bipartite graphs of girth at least 6 to more general graphs. We expect that results concerning cyclic connectivity such as those presented in [15] will play an important role in such investigations. Note that cyclic Haar graphs have girth at most 6 and form a special class of bicirculants [16]. However, there exist other bicirculants with girth greater than 6. The corresponding configurations have been investigated in [3, 1]. One way of extending this study is on the one hand to consider splittability of these more general bicirculants and on the other hand to study tricirculants [12], tetracirculants and beyond [7]. In the language of configurations, they can be described as special classes of polycyclic configurations [2]. References [1] M. Boben, B. Grünbaum, T. Pisanski and A. Žitnik, Small triangle-free configurations of points and lines, Discrete Comput. Geom. 35 (2006), 405-427, doi:10.1007/s00454-005-1224-9. [2] M. Boben and T. Pisanski, Polycyclic configurations, European J. Combin. 24 (2003), 431457, doi:10.1016/s0195-6698(03)00031-3. [3] M. Boben, T. Pisanski and A. Žitnik, I-graphs and the corresponding configurations, J. Combin. Des. 13 (2005), 406-424, doi:10.1002/jcd.20054. N. Basic et al.: Splittable and unsplittable graphs and configurations 17 [4] H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413-455, doi:10.1090/s0002-9904-1950-09407-5. [5] H. S. M. Coxeter, R. Frucht and D. L. Powers, Zero-Symmetric Graphs, Academic Press, London, 1981, doi:10.1016/c2013-0-10543-0. [6] D. Eppstein, The many faces of the Nauru graph, 2007, https://11011110.github. io/blog/2 007/12/12/many-faces-of.html. [7] B. Frelih and K. Kutnar, Classification of cubic symmetric tetracirculants and pentacirculants, European J. Combin. 34 (2013), 169-194, doi:10.1016/j.ejc.2012.08.005. [8] B. Griinbaum, Configurations of Points and Lines, volume 103 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 2009, doi:10.1090/gsm/ 103. [9] M. Hladnik, D. Marusic and T. Pisanski, Cyclic Haar graphs, Discrete Math. 244 (2002), 137152, doi:10.1016/s0012-365x(01)00064-4. [10] H. Koike, I. Kovacs, D. Marusic and M. Muzychuk, Cyclic groups are CI-groups for balanced configurations, Des. Codes Cryptogr. (2018), doi:10.1007/s10623-018-0517-y. [11] H. Koike, I. Kovacs and T. Pisanski, The number of cyclic configurations of type (v3) and the isomorphism problem, J. Combin. Des. 22 (2014), 216-229, doi:10.1002/jcd.21387. [12] I. Kovacs, K. Kutnar, D. Marusic and S. Wilson, Classification of cubic symmetric tricircu-lants, Electron. J. Combin. 19 (2012), #P24, http://www.combinatorics.org/ojs/ index.php/eljc/article/view/v19i2p2 4. [13] D. Marusic and T. Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croat. Chem. Acta 73 (2000), 969-981, http://hrcak.srce.hr/13197 2. [14] D. Marusic and T. Pisanski, The ADAM graph and its configuration, Art Discrete Appl. Math. 1 (2018), #E1.01, doi:10.26493/2590-9770.1215.1e8. [15] R. Nedela and M. Skoviera, Atoms of cyclic connectivity in cubic graphs, Math. Slovaca 45 (1995), 481-499, http://maslo.mat.savba.sk/paper.php?id_paper=157. [16] T. Pisanski, A classification of cubic bicirculants, Discrete Math. 307 (2007), 567-578, doi: 10.1016/j.disc.2005.09.053. [17] T. Pisanski, Yet another look at the Gray graph, New Zealand J. Math. 36 (2007), 85-92, http://nzjm.math.auckland.ac.nz/index.php/Yet_Another_Look_at_ the_Gray_Graph. [18] T. Pisanski and M. Randic, Bridges between geometry and graph theory, in: C. A. Gorini (ed.), Geometry at Work, Math. Assoc. America, Washington, D.C., volume 53 of MAA Notes, pp. 174-194, 2000. [19] T. Pisanski and B. Servatius, Configurations from a Graphical Viewpoint, Birkhauser Advanced Texts, Birkhauser, New York, 2013, doi:10.1007/978-0-8176-8364-1. [20] T. Pisanski and T. W. Tucker, On the maximum number of independent elements in configurations of points and lines, Discrete Comput. Geom. 52 (2014), 361-365, doi:10.1007/ s00454-014-9618-1. [21] A. Zitnik, B. Horvat and T. Pisanski, All generalized Petersen graphs are unit-distance graphs, J. Korean Math. Soc. 49 (2012), 475-491, doi:10.4134/jkms.2012.49.3.475. ¿^creative , ars mathematica ^commons contemporánea ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 19-24 https://doi.org/10.26493/1855-3974.1488.182 (Also available at http://amc-journal.eu) A note on the 4-girth-thickness of K * n,n,n Xia Guo, Yan Yang t School of Mathematics, Tianjin University, Tianjin, P. R. China Received 20 September 2017, accepted 8 January 2018, published online 24 August 2018 Abstract The 4-girth-thickness 6(4, G) of a graph G is the minimum number of planar subgraphs of girth at least four whose union is G. In this paper, we obtain that the 4-girth-thickness of complete tripartite graph Kn,n,n is [n^] except for 6(4, K 1,1,1) = 2. And we also show that the 4-girth-thickness of the complete graph K10 is three which disprove the conjecture posed by Rubio-Montiel concerning to 6(4, K10). Keywords: Thickness, 4-girth-thickness, complete tripartite graph. Math. Subj. Class.: 05C10 1 Introduction The thickness 6(G) of a graph G is the minimum number of planar subgraphs whose union is G. It was defined by W. T. Tutte [10] in 1963. Then, the thicknesses of some graphs have been obtained when the graphs are hypercube [7], complete graph [1, 2, 11], complete bipartite graph [3] and some complete multipartite graphs [6, 12, 13]. In 2017, Rubio-Montiel [9] defined the g-girth-thickness 6(g, G) of a graph G as the minimum number of planar subgraphs whose union is G with the girth of each subgraph is at least g. It is a generalization of the usual thickness in which the 3-girth-thickness 6(3, G) is the usual thickness 6(G). He also determined the 4-girth-thickness of the complete graph Kn except K10 and he conjectured that 6(4, K10) = 4. Let Kn,n,n denote a complete tripartite graph in which each part contains n (n > 1) vertices. In [13], Yang obtained 6(Kn,n,n) = I"] when n = 3 (mod 6). In this paper, we determine 6(4, Kn,n,n) for all values of n and we also give a decomposition of K10 with three planar subgraphs of girth at least four, which shows 6(4, K10) = 3. * Supported by the National Natural Science Foundation of China under Grant No. 11401430. 1 Corresponding author. E-mail addresses: guoxia@tju.edu.cn (Xia Guo), yanyang@tju.edu.cn (Yan Yang) ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 20 Ars Math. Contemp. 16 (2019) 19-24 2 The 4-girth-thickness of Kn,n,n Lemma 2.1 ([4]). A planar graph with n vertices and girth g has at most (n — 2) edges. Theorem 2.2. The 4-girth-thickness of Kn,n,n is 0(4, K„,„,„) = [^ except for 0(4, K^^) = 2. Proof. It is trivial for n =1, 0(4, K1,1,1) = 2. When n > 1, because |E(Kn,n,n)| = 3n2, |V (Kn,n,n)| = 3n, from Lemma 2.1, we have 0(4, Kw) > r 3n2 2(3n — 2) n1 + 77 + |~n + 1 2) 2 In the following, we give a decomposition of Kn,n,n into [ny1] planar subgraphs of girth at least four to complete the proof. Let the vertex partition of Kn,n,n be (U, V, W), where U = {u1,..., un}, V = {v1,..., vn} and W = {w1,..., wn}. In this proof, all the subscripts of vertices are taken modulo 2p. Case 1: When n = 2p (p > 1). Let G1,..., Gp+1 be the graphs whose edge set is empty and vertex set is the same as V(K2p,2p,2p). Step 1: For each Gj (1 < i < p), arrange all the vertices u1, v3_2j, u2, v4_2j, u3, v5_2j, ..., u2p, v2p_2j+2 on a circle and join uj to vj+2_2j and vj+1_2j, 1 < j < 2p. Then we get a cycle of length 4p, denote it by G1 (1 < i < p). Step 2: For each G1 (1 < i < p), place the vertex w2i-1 inside the cycle and join it to u1,..., u2p, place the vertex w2j outside the cycle and join it to v1,..., v2p. Then we get a planar graph G2 (1 < i < p). Step 3: For each G2 (1 < i < p), place vertices w2j for 1 < j < p and j = i, inside of the quadrilateral w2i-1 M2i_1v1M2i and join each of them to vertices M2i_1 and u2j. Place vertices w2j-_1, for 1 < j < p and j = i, inside of the quadrilateral w2iv2i_1Mfcv2j, in which uk is some vertex from U. Join each of them to vertices v2i-1 and v2j. Then we get a planar graph Gj (1 < i < p). Step 4: For Gp+1, join w2i-1 to both v2j_1 and v2j, join w2j to both u2j_1 and u2j, for 1 < i < p, then we get a planar graph Gp+1. For G1 U • • • U Gp+1 = Kn,n,n, and the girth of Gj (1 < i < p +1) is at least four, we obtain a 4-girth planar decomposition of K2p 2p 2p with p +1 planar subgraphs. Figure 1 shows a 4-girth planar decomposition of K4 4 4 with three planar subgraphs. Case 2: When n = 2p + 1 (p > 1). Base on the 4-girth planar decomposition {G1,..., Gp+1} of K2p,2p,2p, by adding vertices and edges to each Gj (1 < i < p + 1) and some other modifications on it, we will get a 4-girth planar decomposition of K2p+1,2p+1,2p+1 with p + 1 subgraphs. Step 1: (Add u to Gj, 1 < i < p.) For each Gj (1 < i < p), we notice that the order of the p — 1 interior vertices w2j, 1 < j < p, and j = i in the quadrilateral 2 n X. Guo and Y. Yang: A note on the 4-girth-thickness of Kn,n,n 21 u2-V 2-u 3-V 3-u 4-xJ 4 (a) The graph G1. (b) The graph G2. (c) The graph G3. Figure 1: A 4-girth planar decomposition of K4 4 4. 4 w2i-iM2i-iviM2i of Gj has no effect on the planarity of Gj. We adjust the order of them, such that w2j_1M2j_1w2p_2j+2M2j is a face of a plane embedding of Gj. Place the vertex u in this face and join it to both w2j-1 and w2p_2j+2. We denote the planar graph we obtain by Gj (1 < i < p). Step 2: (Add v and w to G?1.) Delete the edge v1u2 in G?1, put both v and w in the face wku1v1wtv2u2 in which wk is some vertex from {w2j- | 1 < j < p} and wt is some vertex from {w2j_1 | 1 < j < p}. Join v to w, join v to u1, u2, and join w to v1, v2, we get a planar graph G1. Step 3: (Add v and w to G?j, 2 < i < p.) For each Gj (2 < i < p), place the vertex v in the face wku2j_1v1u2j in which wk is some vertex from {w2j- | 1 < j < p and j = i}, and join it to u2j-1 and u2j. Place the vertex w in the face wkv2j_1uiv2j in which wk is some vertex from {w2j_1 | 1 < j < p and j = i} and ut is some vertex from U. Join w to both v2j-1 and v2j, we get a planar graph Gj (2 < i < p). Step 4: (Add u, v and w to Gp+1.) We add u, v and w to Gp+1. For 1 < i < 2p, join u to each vj, join v to each wj, join w to each uj, join u to both v and w, and join v1 to u2, then we get a planar graph Gp+1. Figure 2 shows a plane embedding of Gp+1. For G1 U • • • U Gp+1 = K„)Bi„, and the girth of Gj (1 < i < p +1) is at least four, we obtain a 4-girth planar decomposition of K2p+1j2p+1j2p+1 with p + 1 planar subgraphs. Figure 3 shows a 4-girth planar decomposition of K5 5 5 with three planar subgraphs. Case 3: When n = 3, Figure 4 shows a 4-girth planar decomposition of K3 3 3 with two planar subgraphs. Summarizing the above, the theorem is obtained. □ 22 Ars Math. Contemp. 16 (2019) 19-24 v Figure 2: The graph Gp+1. Figure 3: A 4-girth planar decomposition of K5j5j5. Figure 4: A 4-girth planar decomposition of K3 3 3. X. Guo and Y. Yang: A note on the 4-girth-thickness of Kn 23 3 The 4-girth-thickness of K10 In [9], the author posed the question whether 6(4, K10) = 3 or 4, and conjectured that it is four. We disprove his conjecture by showing 6(4, Kw) = 3. Theorem 3.1. The 4-girth-thickness of K10 is three. Proof. From [9], we have 0(4, K10) > 3. We draw a 4-girth planar decomposition of Kio with three planar subgraphs in Figure 5, which shows 0(4, K10) < 3. The theorem We would like to state that after submitting this paper for review, we notice that there exist two results regarding the 4-girth-thickness of K2p,2p,2p and K10. Rubio-Montiel [8] obtained the exact value of the 4-girth-thickness of the complete multipartite graph when each part has an even number of vertices. And by computer, Castaneda-Lopez et al. [5] found the other two decompositions of K10 into three planar subgraphs of girth at least four. In this paper, we give these results in a constructive way. References [1] V. B. Alekseev and V. S. Goncakov, The thickness of an arbitrary complete graph, Math. USSRSbornik 30 (1976), 187-202, http://stacks.iop.org/0025-57 34/30/i=2/ a=A0 4. [2] L. W. Beineke and F. Harary, The thickness of the complete graph, Canad. J. Math. 17 (1965), 850-859, doi:10.4153/cjm-1965-084-2. [3] L. W. Beineke, F. Harary and J. W. Moon, On the thickness of the complete bipartite graph, Math. Proc. Cambridge Philos. Soc. 60 (1964), 1-5, doi:10.1017/s0305004100037385. [4] J. A. Bondy and U. S. R. Murty, Graph Theory, volume 244 of Graduate Texts in Mathematics, Springer-Verlag, London, 2008, doi:10.1007/978-1-84628-970-5. [5] H. Castaneda-L6pez, P. C. Palomino, A. B. Ramos-Tort, C. Rubio-Montiel and C. Silva-Ruiz, The 6-girth-thickness of the complete graph, 2017, arXiv:170 9.07 4 66 [math.CO]. [6] Y. C. Chen and Y. Yang, The thickness of the complete multipartite graphs and the join of graphs, J. Comb. Optim. 34 (2017), 194-202, doi:10.1007/s10878-016-0057-1. [7] M. Kleinert, Die dicke des n-dimensionalen Wurfel-graphen, J. Comb. Theory 3 (1967), 10-15, doi:10.1016/s0021-9800(67)80010-3. [8] C. Rubio-Montiel, The 4-girth-thickness of the complete multipartite graph, 2017, arXiv:1709.03932 [math.CO]. Figure 5: A 4-girth planar decomposition of Kw. follows. □ 24 Ars Math. Contemp. 16 (2019) 19-24 [9] C. Rubio-Montiel, The 4-girth-thickness of the complete graph, ArsMath. Contemp. 14 (2018), 319-327, doi:10.26493/1855-3974.1349.b67. [10] W. T. Tutte, The thickness of a graph, Indag. Math. (Proceedings) 66 (1963), 567-577, doi: 10.1016/s1385-7258(63)50055-9. [11] J. M. Vasak, The thickness of the complete graph, Notices Amer. Math. Soc. 23 (1976), A-479. [12] Y. Yang, A note on the thickness of Ki,m,n, Ars Combin. 117 (2014), 349-351. [13] Y. Yang, Remarks on the thickness of Kn>n,n, Ars Math. Contemp. 12 (2017), 135-144, doi: 10.26493/1855-3974.823.068. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 25-38 https://doi.org/10.26493/1855-3974.1281.c7f (Also available at http://amc-journal.eu) On the k-metric dimension of metric spaces Juan A. Rodríguez-Velázquez * Universität Rovira i Virgili, Departament d'Enginyeria Informática i Matematiques, Av. Pai'sos Catalans 26, 43007 Tarragona, Spain Received 10 January 2017, accepted 27 June 2018, published online 26 August 2018 The metric dimension of a general metric space was defined in 1953, applied to the set of vertices of a graph metric in 1975, and developed further for metric spaces in 2013. It was then generalised in 2015 to the k-metric dimension of a graph for each positive integer k, where k =1 corresponds to the original definition. Here, we discuss the k-metric dimension of general metric spaces. Keywords: Metric spaces, metric dimension, k-metric dimension. Math. Subj. Class.: 54E35, 05C12 1 Introduction The metric dimension of a general metric space was introduced in 1953 in [4, p. 95] but attracted little attention until, about twenty years later, it was applied to the distances between vertices of a graph [12, 14, 15, 18]. Since then it has been frequently used in graph theory, chemistry, biology, robotics and many other disciplines. The theory was developed further in 2013 for general metric spaces [1]. More recently, the theory of metric dimension has been generalised, again in the context of graph theory, to the notion of a k-metric dimension, where k is any positive integer, and where the case k = 1 corresponds to the original theory [7, 8, 9, 10, 11]. Here we develop the idea of the k-metric dimension both in graph theory and in metric spaces. As the theory is trivial when the space has at most two points, we shall assume that any space we are considering has at least three points. *This research was supported in part by the Spanish government under the grant MTM2016-78227-C2-1-P. E-mail addresses: afb@dpmms.cam.ac.uk (Alan F. Beardon), juanalberto.rodriguez@urv.cat (Juan A. Rodriguez-Velazquez) Alan F. Beardon University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 26 ArsMath. Contemp. 16(2019)25-38 Finally, whenever we discuss a connected graph G, we shall always consider the metric space (X, d), where X is the vertex set of G, and d is the usual graph metric in which the distance between two vertices is the smallest number of edges that connect them. Let (X, d) be a metric space. If X is a finite set, we denote its cardinality by |X |; if X is an infinite set, we put |X| = In fact, it is possible to develop the theory with |X | any cardinal number, but we shall not do this. The distances from a point x in X to the points a in a subset A of X are given by the function a ^ d(x, a), and the subset A is said to resolve X if each point x is uniquely determined by this function. Thus A resolves X if and only if d(x, a) = d(y, a) for all a in A implies that x = y; informally, if an object in X knows its distance from each point of A, then it knows exactly where it is located in X. The class R(X) of subsets of X that resolve X is non-empty since X resolves X. The metric dimension dim(X) of (X, d) is the minimum value of |S | taken overall S in R(X). The sets in R(X) are called the metric generators, or resolving subsets, of X, and S is a metric basis of X if S G R(X) and |S| = dim(X). A metric generator of a metric space (X, d) is, in effect, a global co-ordinate system on X. For example, if (x^ ..., xm) is an ordered metric generator of X, then the map A: X ^ Rm given by is injective (for this vector determines x), so that A is a bijection from X to a subset of Rm, and X inherits its co-ordinates from this subset. Now let k be a positive integer, and (X, d) a metric space. A subset S of X is a k-metric generator for X (see [8]) if and only if any pair of points in X is distinguished by at least k elements of S: that is, for any pair of distinct points u and v in X, there exist k points wi,w2,...,wk in S such that A k-metric generator of minimum cardinality in X is called a k-metric basis, and its cardinality, which is denoted by dimk(X), is called the k-metric dimension of X. Let (X) be the set of k-metric generators for X. Since R1(X) = R(X), we see that dim1(X) = dim(X). Also, as inf 0 = this means that dimk (X) = if and only if no finite subset of X is a k-metric generator for X. Given a metric space (X, d), we define the dimension sequence of X to be the sequence and we address the following two problems. • Can we find necessary and sufficient conditions for a sequence (di, d2, d3,...) to be the dimension sequence of some metric space? • How does the dimension sequence of (X, d) relate to the properties of (X, d) ? In Sections 2, 3 and 4 we provide some basic results on the k-metric dimension, and in Section 5 we calculate the dimension sequences of some metric spaces. We then apply these ideas to the join of two metric spaces, and to the Cayley graph of a finitely generated group. (1.1) d(u, Wj) = d(v, wi), i = 1, ..., k. ( dimi (X ), dim2 (X ),..., dimfc (X ),...), A. F. Beardon and J. A. Rodriguez-Velazquez: On the k-metric dimension of metric spaces 27 2 Bisectors As shown in [1], the ideas about metric dimension are best described in terms of bisectors. For distinct u and v in X, the bisector B(u|v) of u and v is given by B(u|v) = {x € X : d(x,u) = d(x,v)}. The complement of B(u|v) is denoted by Bc(u|v); thus Bc(u|v) = {x € X : d(x,u) = d(x, v)}, and this contains both u and v. Whenever we speak of a bisector B, we shall assume that it is some bisector B(u|v), where u = v, so that its complement Bc is not empty. Let us now consider the k-metric dimension from the perspective of bisectors. A subset A of X fails to resolve X if and only if there are distinct points u and v in X such that d(u, a) = d(v, a) for all a in A. Thus A resolves X if and only if A is not contained in any bisector or, equivalently, if and only if for every bisector B, we have |Bc n A| > 1. This leads to an alternative (but equivalent) definition of the metric dimension dim(X), namely dim(X) = inf{|A| : A c X and, for all bisectors B, |Bc n A| > 1}. Again, this infimum may be The extension to the k-metric dimension dimk (X) of X is straightforward: dimk(X) = inf{|A| : A c X and, for all bisectors B, |Bc n A| > k}. (2.1) Note that if X is a finite set then dim|X|+1 (X) = Clearly, the values dimk(X) depend only on the class B of bisectors in X; for example, dim1(X) = 1 if and only if there is some point in X that is not in any bisector. More generally, in all cases, dimk (X) > k, and equality holds here if and only if there are k points of X that do not lie in any bisector. For example, if X is the real, closed interval [0,1] with the Euclidean metric, then dimk (X) = k for k = 1, 2. For a more general example of this type, let X = {Vp : p a prime number} with the Euclidean metric d. If p, q and r are primes, with p = q, then V € B(Vp|Vq) implies Vr = 2(VP + V^); hence 4r = p + q + 2vpq. Since Vpq is irrational, this is false; hence every bisector is empty. It follows that dimk (X) = k for k = 1, 2,...; thus the dimension sequence of (X, d) is (1, 2, 3,...). 3 The monotonicity of dimensions Let (X, d) be a metric space. Then, from (2.1), we have dimk(X) < dimk+1(X), but we shall now establish the stronger inequality dimk (X) + 1 < dimk+1(X) (which is dimk (X) < dimk+1 (X) when the dimensions are finite, but not when they are +to). This inequality is known for graphs; see [8, 10]) where it is an important tool. Theorem 3.1. Let (X, d) be a metric space. Then, for k = 1, 2,..., (i) if dimk (X) < then dimk (X) < dimfc+1 (X); (ii) if dimk (X) = then dimk+1(X) = In particular, dimk (X) + 1 > dim1 (X) + k. 28 Ars Math. Contemp. 16 (2019) 19-24 Proof. First, (ii) follows immediately from (2.1). Next, (i) is true if dimk+1(X) = so we may assume that dimk+1(X) = p < Thus there is a subset {x1,..., xp} (with the xi distinct) of X such that for every bisector B, |Bc n {x1,..., xp}| > k +1. As k > 1 we see that p > 2. Clearly, |Bc n {x1,..., xp-1}| > k for every bisector B; hence dimk (X) < p - 1 < dimk+1(X). The last inequality follows by induction. □ 4 The 1-metric dimension Theorem 3.1 shows that if occurs as a term in the dimension sequence of (X, d), then all subsequent terms are also Thus dim1(X) = if and only if (X, d) has dimension sequence (+ro, ...). The next result shows when this is so. Theorem 4.1. Let (X, d) be a metric space. Then dim1(X) = if and only if every finite subset of X lies in some bisector. In particular, if X is the union of an increasing sequence of bisectors, then dim1(X) = Proof. First, the definition of dim(X) implies that dim1(X) = if and only if every finite subset of X lies in some bisector. The second statement holds because if X = UnBn, where B1, B2,... is an increasing sequence of bisectors, then, given any finite subset {x1,..., xm} of X, each xj lies in some Bij, and {x1,..., xm} c Br, where r = max{i1, ...,im}. □ What can be said if dim1 (X) < It seems that we can obtain very little information from the single assumption that dim1 (X) < for example, for each r > 0 choose a point xr in Rn with ||xr || = r, and let X = {xr : r > 0}. Then {0} is a 1-metric basis for X, and dim1 (X) = 1 but we can say almost nothing about the topological structure of X. However, we can say more if we know that X is compact. Theorem 4.2. Let (X, d) be a compact metric space with dim1(X) = m < Then (X, d) is homeomorphic to a compact subset of Rm. Proof. Suppose that X is compact, and that dim1(X) = m < Then there is a 1-metric basis {x1,..., xm}, and the corresponding bijection A in (1.1) that maps X onto some subset of Rm. Now A is continuous on X since m |A(x) - A(y)| < ^ |d(x,xj) - d(y,xj)| < md(x,y). j=1 As A is a continuous, injective map from a compact space to the Hausdorff space Rm it follows (by a well known result in topology) that it is a homeomorphism. □ This result is related to the following result in [1] (see also [16]). Theorem 4.3. If (X, d) is a compact, connected metric space with dim1(X) = 1 then X is homeomorphic to [0,1]. The compactness is essential here as there is an example in [1] of a connected, but not arcwise connected, metric space X with dim1 (X) = 1. As X is not arcwise connected, it is not homeomorphic to [0,1]. It is conjectured in [1] that if X is arcwise connected, and dim1(X) = 1 then X is a Jordan arc (this means that X is homeomorphic to one of the real intervals [0,1] and [0, +ro)), and we can now show that this is so. A. F. Beardon and J. A. Rodriguez-Velazquez: On the k-metric dimension of metric spaces 29 Theorem 4.4. If X is an arcwise connected metric space with dim^X) = 1, then X is a Jordan arc. Proof. As dim1 (X) = 1, there is a metric basis, say {x0} for X, and every point x of X is uniquely determined by its distance d(x, x0) from x0. Consider the map A: x ^ d(x, x0) of X into [0, +to). This map is (uniformly) continuous because |A(x) - A(y)| = |d(x,xo) - d(y,xo)| < d(x,y), and as X is arcwise connected (and therefore connected), so A(X) is connected. This means that A is an interval of the form [0, a], where a > 0, or [0, b), where 0 < b < Let us consider the case when A(X) = [0, a]. As A is injective, we see that for every r in the interval [0, a] there is some unique xr in X with d(xr, x0) = r. Thus X = {xr : 0 < r < a}. However, as X is arcwise connected, there is a curve, say 7 : [0,1] ^ X with 7(0) = x0 and 7(1) = xa. Now as 7 is continuous, the set {d(7(t), x0) : t G [0,1]} must contain every real number in the interval [0, a], and it cannot contain any other numbers; thus X = 7([0,1]). Now 7([0,1]) is compact for it is the continuous image of the compact interval [0,1]; thus X is compact and so, by Theorem 4.3, X is a Jordan arc. The argument in the case when A(X) = [0, b) is similar. Indeed, the argument above holds for every a with 0 < a < b, and it is easy to see that this implies that A is a homeomorphism from X to [0, b). □ 5 Some examples In order to calculate the k-metric dimension of a metric space we need to understand the geometric structure of its bisectors, and we now illustrate this with several examples. In order to maintain the flow of ideas, the details of these examples will be given later. Example 5.1. Let (X, d) be any one of the Euclidean, spherical and hyperbolic spaces Rn, Sn and Hn, respectively, each with the standard metric of constant curvature 0, 1 and -1, respectively. The bisectors are well understood in these spaces, and we shall show that any non-empty open subset of X has k-metric dimension n + k. In particular, each of these spaces has dimension sequence (n + 1, n + 2, n + 3,...). See [1, 13] for the 1-metric dimensions of these spaces. Example 5.2. Let X be any finite set with the discrete metric d (equivalently, X is the vertex set of a complete, finite graph). For distinct u and v in X we have B(u|v) = X\{u, v}, so that for any subset S of X, we have B(u|v)c n S = {u, v} n S. Thus if |S n Bc | > 1 for all bisectors B, then S can omit at most one point of X. We conclude that dim1(X) = |X |-1. If |Bc n S| > 2 for all bisectors B then S = X, and dim2(X) = |X |. We conclude that (X, d) has dimension sequence (|X| - 1, |X|, ...). Example 5.3. Let X be the real interval [0,1], with the Euclidean metric. Then B is a bisectors if and only if B = {x} for some x in (0,1). Thus {0} is a 1-metric basis, and {0,1} is a 2-metric basis, of [0,1]. We leave the reader to show that if k > 3 then {0, k, |,..., ^j1, 1} is a k-metric basis, so that [0,1] has dimension sequence (1,2,4, 5, 6,...). A similar argument shows that [0, has dimension sequence (1,3,4, 5,...), and that (-to, +to), which is R, has dimension sequence (2,3,4,...). Example 5.4. The Petersen graph, which is illustrated in Figure 1, has dimension sequence (3,4,7, 8, 9,10, ...). The (finite) values dimk(X) for k = 1,..., 6 come from a computer search, and as dim6(X) = 10 = |X|, we have dim7(X) = 30 Ars Math. Contemp. 16 (2019) 19-24 Figure 1: The Petersen graph. Example 5.5. Let G be a group with a given set of generators, let V be the vertex set of the associated Cayley graph of G, and let d be its graph metric. (i) If G is an infinite cyclic group then (V, d) has dimension sequence (2, 3,4,...). (ii) If G is a free group on p generators, where p > 2, then (V, d) has dimension sequence (+TO, +TO, + TO, . . .). (iii) Let G be an abelian group on p generators, where p > 2, and where each generator has infinite order. Then (V, d) has dimension sequence (+ro, ...). 6 Three geometries of constant curvature In this section we give the details of Example 5.1. It is shown in [1] that if U is any non-empty, open subset of any one of the three classical geometries Rn, Sn and Hn, then dimi (U) = n + 1. Here we show that if X is any of these spaces then dimk (X) = n + k for k = 1,2,.... The same result holds for non-empty open subsets of these spaces, and we leave the reader to make the appropriate changes to the proofs. The proof that dimk (X) = n+k when X is one of the three geometries Rn, Sn and Hn, is largely independent of the choice of X, and depends only on the nature of the bisectors in these geometries. Each of these three geometries has the following properties: (P1) dim1(X) = n + 1; (P2) there exists x1, x2,... in X such that if j1 < j2 < • • • < jn then {xj1,..., xjn} lies on a unique bisector B, and no other xj lies on B. Now (P1) and (P2) imply that dimfc(X) = n + k for k = 1, 2,.... Indeed, (P2) implies that for any bisector B, |B n {x1,..., xn+k}| < n, so that |Bc n {x1,..., xn+k}| > k. This implies that dimk(X) < n + k. However, (P1) and Theorem 3.1 show that dimk(X) > n + k. Since we know that each of Rn, Sn and Hn has the property (P1), it remains to show that they have the property (P2), and this depends on the nature of the bisectors in these geometries. We consider each in turn. 6.1 Euclidean space Rn Each bisector in Rn is a hyperplane (that is, the translation of an (n - 1)-dimensional subspace of Rn), and each hyperplane is a bisector. Any set of n points lies on a bisector, A. F. Beardon and J. A. Rodriguez-Velazquez: On the k-metric dimension of metric spaces 31 and there exists sets of n +1 points that do not lie on any single bisector. The appropriate geometry here is the affine geometry of R", but we shall take a more informal view. First, we choose n points x^ ..., xn that lie on a unique hyperplane H. Next, we select a point xn+1 not on H. Then any n points chosen from {x1,..., xn+1} lie on some hyperplane H', and the remaining point does not lie on H'. Now suppose that we have constructed the set {x1,..., xn+p} with the property that any set of n points chosen from this lie on a unique hyperplane, say Ha, and that no other x4 lies on Ha. Then we can choose a point xn+p+1 that is not on any of the (n+p) hyperplanes Ha, and it is then easy to check that the sequence x1, x2,... has the property (P2). Although we have not used it, we mention that there is a formula for the n-dimensional volume V of the Euclidean simplex whose vertices are the n +1 points x1,..., xn+1 in R", namely V 2 = A 2"(n !)2 , where A is the Cayley-Menger determinant given by A: 0 1 ••• 1 1 d2 d2 1 "1,1 d1,n+1 1 d2 . . . d2 1 dn+1,1 dn+1,n+1 and djj = ||xj - Xj||. As V = 0 precisely when the points Xj lie on a hyperplane, we see that this condition could be used to provide an algebraic background to the discussion above. For more details, see [3, 4] and [5]. We also mention that there are versions of the Cayley-Menger determinant that are applicable to spherical, and to hyperbolic, spaces. 6.2 Spherical space Sn Spherical space (Sn, d) is the space {x G Rn+1 : ||x|| = 1} with the path metric d induced on Sn by the Euclidean metric on Rn+1. Explicitly, cos d(x, y) = x-y, where x-y is the usual scalar product in Rn+1. If u and v are distinct points of Sn, we let BE(u|v) be the Euclidean bisector (in Rn+1) of u and v, and BS(u|v) the spherical bisector in the space (Sn, d). Then BE (u|v) is a hyperplane that passes through the origin in Rn+1, and BS(u|v) = Sn n Be(u|v). (6.1) The bisectors BS(u|v) are the great circles (of the appropriate dimension) on Sn. The equation (6.1) implies that the k-metric dimension of the spherical spaces is the same as for Euclidean spaces. Indeed, our proof for Euclidean spaces depended on constructing a sequence x1, x2,... with the property (P2), and it is clear that this construction could be carried out in such a way that each xj lies on Sn. 6.3 Hyperbolic space Hn Our model of hyperbolic n-dimensional space is Poincare's half-space model Hn = {(x1,...,xn) G Rn : xn > 0} equipped with the hyperbolic distance d which is derived from Riemannian metric | dx | /xn. For more details, see for example, [2, 17]. Our argument for Hn is essentially the same as 32 Ars Math. Contemp. 16 (2019) 19-24 for Rn and Sn because if u and v are distinct points in Hn, then the hyperbolic bisector B(u|v) is the set S n Hn, where S is some Euclidean sphere whose centre lies on the hyperplane xn = 0. We omit the details. 7 The metric dimensions of graphs The vertex set V of a graph G supports a natural graph metric d, where d(u, v) is the smallest number of edges that can be used to join u to v. Some basic results on the k-metric dimension of a graph have recently been obtained in [7, 8, 9, 10, 11]. Moreover, it was shown in [19] that the problem of computing the k-metric dimension of a graph is NP-hard. A natural problem in the study of the k-metric dimension of a metric space (X, d) consists of finding the largest integer k such that there exists a k-metric generator for X. For instance, for the graph shown in Figure 2 the maximum value of k is four. It was shown in [7, 10] that for any graph of order n this problem has time complexity of order O(n3). If we consider the discrete metric space (X, d0) (equivalently, a compete graph), then dimi(X) = |X | — 1 and dim2(X) = |X |. Furthermore, for k > 3 there are no k-metric generators for X. In general, for any metric space (X, d), the whole space X is a 2-metric generator, as two vertices are distinguished by themselves. As we have already seen, there are metric spaces, like the Euclidean space Rn, where for any positive integer k, there exist at least one k-metric generator. We shall now discuss the dimension sequences of the simplest connected graphs, that is paths and cycles (and we omit the elementary details). A finite path Pn is a graph with vertices vi,..., vn, edges jvi, v2},..., {vn_i, vn}, and bisectors {v2},..., {vn-1}. We leave the reader to show that Pn has dimension sequence (1, 2, +TO,...) if n = 2, 3; (1, 2, 4, 5,. .., n, ...) if n > 4. A semi-infinite path PN is agraph with vertices v1, v2..., edges {v1, v2}, {v2, v3 },..., and bisectors {v2},____ Thus PN has dimension sequence (1, 3,4,5,...). A doubly-infinite path PZ is the graph with vertices ..., v_1, v0, v1,..., edges ..., {v_1, v0}, {v0, v1}, ..., and bisectors..., {v_1}, {v0}, {v1},.... Thus PZ has dimension sequence (2,3,4, 5,...). We note that a graph G has 1-metric dimension 1 if and only it is Pn or PN [6, 14]. This, together with the results just stated, show that if G is a graph of order two or more, and k > 2, then dimk (G) = k if and only if G is Pn and k = 2 (see also [8]). We now consider cycles. A cycle Cn is a graph with vertices v1,..., vn, and edges {v1, v2},..., {vn_ 1, vn}, {vn, v1}. We must distinguish between the cases where n is even, and where n is odd (which is the easier of the two cases) and, as typical examples, we mention that C7 has dimension sequence (2, 3,..., 7, ...), and C8 has dimension sequence (2, 3,4,6,7, 8, ...). Suppose that n is odd; then the bisectors are the singletons {v}. Thus if S is a set of k + 1 vertices, where k +1 < n, then |Bc n S| > k for every bisector B. Thus if n is odd, then dimk (Cn) = k + 1, and Cn has dimension sequence (2, 3,..., n, ...). We now show that C2q has dimension sequence (2, 3, . . . , q, q + 2, q + 3, . . ., q + q, +TO, . . .). To see this, label the vertices as v, where j e Z, and where v4 = vj if and only if i = j (mod n). The vertices v4 and vj are antipodal vertices if and only if i — j = q (mod 2q); A. F. Beardon and J. A. Rodriguez-Velazquez: On the k-metric dimension of metric spaces 33 thus vj and vj+q are antipodal vertices. The class of bisectors is the class of sets {v, v*}, where v is a vertex, and v* is the vertex that is antipodal to v. For k = 1,..., q - 1 we can take a set of k + 1 points, no two of which are antipodal, as a k-metric basis, so that dimk(C2q) = k + 1 for k = 1,..., q - 1. To find dimq(C2q), we need to take (for a q-metric basis) a set S which contains two pairs of antipodal points, and one more point from each pair of the remining antipodal pairs. We leave the details to the reader. vi Figure 2: For k G {1, 2, 3,4}, dimk(G) = k + 1. As an example which joins a path to a cycle, consider the graph G illustrated in Figure 2 which is obtained from the cycle graph C5 and the path Pt, by identifying one of the vertices of the cycle, say ui, and one of the end vertices of Pt. Let Si = {vi, v2}, S2 = {vi, v2, ut}, S3 = {vi, v2, v3, ut} and S4 = {vi, v2, v3, v4, ut}. Then, for k = 1,2, 3,4, the set Sk is k-metric basis of G. The following lemma is useful when discussing examples in graph theory. Lemma 7.1. Suppose that a graph G does not have any cycles of odd length. Then B(u|v) = 0 when d(u, v) is odd. The proof is trivial for if x G B(u|v) then there is a cycle of odd length (from u to x, then to v, and then back to u). This lemma applies, for example, to the usual grid (or graph) in Rn whose vertex set is Zn. A bipartite graph is a graph G whose vertex set V splits into complementary sets V and V2 such that each of the edges of G join a point of V to a point of V2. As a graph is bipartite if and only if it has no cycles of an odd length, this lemma is about bipartite graphs. Example 7.2. Let us now consider a graph G that is an infinite tree in which every vertex has degree at least three. Now let v be any vertex, select three edges from v, say {v, a}, {v, b} and {v, c}. As G is a tree, if we remove one edge the remaining graph is disconnected. Now let Gc be the subgraph of G that would be the component containing c if we were to remove the edge {v, c} from G. It is clear that if u is a vertex in Gc, then d(a, u) = d(b, u) since any path from a (or b) to u must pass through the edge {v, c}. We conclude that Gc c B(a|b). It is now clear from Theorem 4.1 that G has dimension sequence (+ro, ...). For the rest of this section we shall consider the Cayley graph of a group with a given set of generators as a metric space. Let G be a group and let G0 a set of generators of G. We shall always assume that if g G G0 then g-i g G0 also. Then the Cayley graph of the pair (G, G0) is a graph whose vertex set is G, and such that the pair (gi, g2) is an edge if and only if g2 = g0gi for some g0 in G0. Thus, for example, PZ is the Cayley graph of an infinite cyclic group (on one generator), and Cn is the Cayley graph of an finite 34 Ars Math. Contemp. 16 (2019) 19-24 cyclic group (on one generator). We shall always assume that the set G0 of generators of G is finite; then the Cayley graph is locally finite (that is, each vertex is the endpoint of only finitely many edges). Note also that if a generator g0 has order two then g-1 = g0 so this only provides one edge (not two edges) from each vertex. The following result, which characterises Cayley graphs within the class of all graphs, is well known. Theorem 7.3. A graph is a Cayley graph of a group G if and only if it admits a simply transitive action of G by graph automorphisms. Theorem 7.3 suggests that if we use the homogeneity implied by this result there is a reasonable chance of finding the dimension sequence of a Cayley graph. However, for a graph that is not the Cayley graph of a group, it seems that we are reduced to finding its metric dimensions by a case by case analysis. We shall now verify the claims made in Example 5.5. First, suppose that G is a free group on p generators. Then the Cayley graph of G is a tree in which every vertex has degree 2p; thus, using Example 7.2, we see that G has dimension sequence (+ro, Next, we consider an abelian group G on two generators of infinite order (the proof for p generators is entirely similar). The Cayley graph of G has Z2 as its vertex set and (if we identify the lattice point (m, n) with the Gaussian integer m + in) edges {m + in, m +1 + in} and {m + in, m + i(n + 1)}, where m, n G Z. It is (geometrically) clear that for any m G Z we have, with Z = m + im, B(Z + 1|Z + i) ^ {p + iq : P > m + 1,q > m + 1}. It now follows from Theorem 4.1 (by taking |m| large and m negative) that G has dimension sequence (+ro, ...). In contrast to Example 5.5 we have the following result for the infinite dihedral group whose Cayley graph is an infinite ladder; for example we can take the group generated by the two Euclidean isometries which, in complex terms, are z ^ z +1 and z ^ z. Theorem 7.4. The infinite dihedral group has dimension sequence (3,4, 6, 8,...). Proof. We may assume that (in complex terms) the vertices of the ladder graph are the points m + to, where m G Z and n = 0,1. The key to computing the metric dimensions of the ladder graph is the observation that B(0|1 + i) = {1, 2, 3,...} U {i,i - 1,i - 2,...}. Of course, similar bisectors arise at other pairs of similarly located points; equivalently, each automorphism of the graph maps a bisector to a bisector. All other bisectors are either empty or of cardinality two. We claim that {0,1,«} is a 1-metric basis for the graph so that dim1(G) = 3. Next, it is easy to see that {0,1, i, 1 + «} is a 2-metric basis for X so that dim2(X) = 4. The set {0,1, 2, i, 1 + i, 2 + «} is a 3-metric basis so that dim3(X) = 6. We leave the details, and the remainder of the proof to the reader. □ 8 The join of metric spaces The k-metric dimension of the join G1 + G2 of two finite graphs G1 and G2 was studied in [7]. Let us briefly recall the notion of the join of two graphs G1 and G2 with disjoint A. F. Beardon and J. A. Rodriguez-Velazquez: On the k-metric dimension of metric spaces 35 vertex sets V and V2, respectively. The join G1 + G2 of G1 and G2 is the graph whose vertex set is Vi U V2, and whose edges are the edges in G1, the edges in G2, together with all edges obtained by joining each point in V to each point in V2. Let d1, d2 and d be the graph metrics of G1, G2 and G1 + G2, respectively; then {minjd1(M, v), 2} if u,v G V1; min{d2(u, v), 2} if u,v G V2; 1 if u G Vj, v G Vj, where i = j, because if u, v G V1, say, then for w in V2, we have d(u, v) < d(u, w) + d(w, v) = 2. The join of two metric spaces is defined in a similar way, but before we do this we recall that if (X, d) is a metric space, and t > 0, then d4, defined by d4(x, y) = min{d(x, y), 2t}, is a metric on X. If d(x, y) < 2t then d4(x, y) = d(x, y), so that the d4-metric topology coincides with the d-metric topology on X. As the metric d4 will appear in our definition of the join, we first show how the metric dimension of a single metric space varies when we distort the metric from d to d4 as above. From now on, the k-metric dimension of (X, d4) will be denoted by dimk (X). Theorem 8.1. Let (X, d) be a metric space, and k a positive integer, and suppose that 0 < s < t. Then dimk (X) > dim^ (X) > dimk (X). However, it can happen that lim dim|(X) > dimk(X). (8.1) The join of two metric spaces is defined in a similar way to the join of two graphs, and to motivate this, suppose that (X, d) is a metric space, and that X1 and X2 are bounded subsets X whose distance apart is very large compared with their diameters. Then, in some sense, we can approximate the metric space (X1UX2, d) by replacing all values d(x1, x2), where xj G Xj, by t, where t is some sort of average of the values d(x1, x2). We shall now define the join, so suppose that (X1, d1) and (X2, d2) are metric spaces, with X1 nX2 = 0, and t > 0. Then the join of (X1, d1) and (X2, d2) (relative to the parameter t) is the metric space (X1 U X2, d4), where {d1(u, v) if u, v G X1; d2(u, v) if u, v G X2; t if u G Xj and v G Xj, where i = j. As with graphs, X1 + X2 always represents the metric space (X1 U X2, d4), where in this case t will be understood from the context. We might hope that the metric dimension is additive with respect to the join, but unfortunately it is not. Let X1 = {1,3} and X2 = {2,4}, each with the Euclidean metric, and let t = 1. Then X1 U X2 = {1,2,3,4} with the metric d1, where d1(1,3) = d1(2,4) = 2 and, for all other x and y, d1 (x, y) = 1. The bisectors in X1 + X2 are X1, X2 and 0, and from this we conclude that dim1(X1 + X2) = 3. Obviously, dim1(X1) = dim1(X2) = 1, so that in this case, dim1(X1) + dim1(X2) < dim1(X1 + X2). We now give some inequalities which hold for the join of two metric spaces. 36 ArsMath. Contemp. 16(2019)25-38 Theorem 8.2. Let (Xj, dj), j = 1,2, be metric spaces with Xi n X2 = 0, and consider the join (Xi U X2, d4). Then, for any positive integer k, we have dimfc (Xi) + dimfc(X2) < dimk (Xi) + dim4fc(X2) < dimk (Xi + X2). (8.2) We shall now give an example which shows that (8.1) can hold; then we end with the proofs of Theorems 8.1 and 8.2, and stating a consequence of Theorem 8.2. Example 8.3. Let X = R and d(x, y) = |x - y|, so that dimi(X) = 2. We shall now show that if t > 0 then dimi(X) = so that (8.1) can hold. Suppose that a < b, and consider the bisector B4(a|b). If x < a — 2t, then d4(x, a) = d4(x, b) = 2t so that x € B4(a|b). Thus B4(a|b) D (—to, a — 2t]. Now let S be any finite set, and let s be the largest element in S. Then Bt(s + 2t, s + 3t) D (—to, s] D S, so that dimi(X) = +to. This is a convenient place to describe the notation that will be used in the following two proofs. We have metric spaces (Xi, di) and (X2, d2) with Xi n X2 = 0. For j = 1,2 we use Bj(u|v) for the bisectors in Xj, and dimk(Xj) for their metric dimensions. Now consider the join (Xi U X2, d4), and its metric subspaces (Xj, d4). We use B4(u|v) and Bj(u|v) for the bisectors in these spaces, and dim|(Xi + X2) and dim|(Xj) for their metric dimensions. In general, we write [B]c for the complement of a bisector B of any type. We shall need the following lemma in our proof of Theorem 8.1. Lemma 8.4. Let (X, d) be a metric space, and suppose that 0 < s < t. Then B(u|v) C B4(u|v) C Bs(u|v). Proof. First, observe that for all real r, and all real, distinct a and we have min{a, r} = minj^, r} if and only if (i) r < min{a, or (ii) a = Now suppose that x € B4(m|v). Then d4(x,w) = d4(x, v) so that min{d(x,u),t} = min{d(x, v), t}. This implies that t < min{d(x, u), d(x, v)} or d(x, u) = d(x, v), and (since s < t) in both cases we have ds(x, u) = ds(x, v). Thus B4(u|v) C Bs(u|v). The proof that B(u|v) C B4(u|v) is trivial: if x € B(u|v) then d(x,u) = d(x, v) so that d4(x, u) = d4(x, v); hence x € B4(u|v). □ The proof of Theorem 8.1. Let A be any finite subset of X. Then, by Lemma 8.4, for all u and v in X with u = v, we have |A n [B(u|v)]c| > |A n [Bt(u|v)]c| > |A n [Bs(u|v)]c|. It follows that if A is a k-metric generator for (X, ds) (that is, if, for all u and v, |A n [Bs(u|v)]c| > k), then it is also a k-metric generator for (X, d4). Thus the minimum of |S| taken over all k-metric generators S of (X, d4) is less than or equal to the minimum over all k-metric generators of (X, ds); hence dimk (X) > dim^ (X). The proof that dimk (X) > dimk(X) is entirely similar. □ The proof of Theorem 8.2. The first inequality follows from Theorem 8.1. The inequality is trivially true if dim| (Xi + X2) = so we may assume that there is a k-metric basis, say W, of Xi + X2. Thus |W| = dim| (Xi + X2). Now take any u and v in Xi; then B4(u|v) = {x € Xi U X2 : d4(x, u) = d4(x, v)} = B^(u|v) U X2, A. F. Beardon and J. A. Rodriguez-Velazquez: On the k-metric dimension of metric spaces 37 so that, from Lemma 8.4, [Bf (u|v)]c = [Bf (u|v)]c c Xi. Weput Wj = W n Xj, j = 1, 2. Then, if we let u and v vary over Xi, with u = v, we find that k < |[Bf(u|v)]c n W| = |[Bf(u|v)]c n Xi n w| = |[Bf(u|v)]c n Wi|, so that dimk(Xi) < |Wi|. Similarly, dimk(X2) < |W2|, so that dimk(Xi) + dimk(X2) < |Wi| + |W2| = |W| = dim4fc(Xi + X2) as required. □ If (Xj,dj), j = 1, 2, are metric spaces, each with diameter less than t, such that Xi n X2 = 0, the for any k-metric basis A of (Xj, dj), Ai U A2 is a k-metric generator for the join (Xi U X2, df). This shows that dimjk (Xi + X2) < dimk (Xi) + dimk (X2), and so Theorem 8.2 leads to the following corollary. Corollary 8.5. Let (Xj, dj), j = 1, 2, be metric spaces, each with diameter less than t, such that Xi n X2 = 0. Then, for k = 1, 2,..., dimk(Xi + X2) = dimfc(Xi) + dimfc (X2). References [1] S. Bau and A. F. Beardon, The metric dimension of metric spaces, Comput. Methods Funct. Theory 13 (2013), 295-305, doi:10.1007/s40315-013-0024-0. [2] A. F. Beardon, The Geometry of Discrete Groups, volume 91 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1983, doi:10.1007/978-1-4612-1146-4. [3] M. Berger, Geometry I, Universitext, Springer-Verlag, New York, 1987, doi:10.1007/ 978-3-540-93815-6, translated from the French by M. Cole and S. Levy. [4] L. M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, Oxford, 1953. [5] L. M. Blumenthal, Theory and Applications of Distance Geometry, Chelsea Publishing, New York, 2nd edition, 1970. [6] G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99-113, doi:10.1016/ s0166-218x(00)00198-0. [7] A. Estrada-Moreno, Y. Ramirez-Cruz and J. A. Rodriguez-Velazquez, On the adjacency dimension of graphs, Appl. Anal. Discrete Math. 10 (2016), 102-127, doi:10.2298/aadm151109022e. [8] A. Estrada-Moreno, J. A. Rodriguez-Velazquez and I. G. Yero, The k-metric dimension of a graph, Appl. Math. Inf. Sci. 9 (2015), 2829-2840, http://www.naturalspublishing. com/Article.asp?ArtcID=10013. [9] A. Estrada-Moreno, I. G. Yero and J. A. Rodriguez-Velazquez, k-metric resolvability in graphs, in: J. A. Rodriguez-Velazquez and I. G. Yero (eds.), Jornadas de Matemática Discreta y Algorítmica, Elsevier, Amsterdam, volume 46 of Electronic Notes in Discrete Mathematics, 2014 pp. 121-128, doi:10.1016/j.endm.2014.08.017, papers from the 9th Conference held in Tarragona, July 7-9, 2014. [10] A. Estrada-Moreno, I. G. Yero and J. A. Rodriguez-Velazquez, The k-metric dimension of corona product graphs, Bull. Malays. Math. Sci. Soc. 39 (2016), S135-S156, doi:10.1007/ s40840-015-0282-2. 38 Ars Math. Contemp. 16 (2019) 19-24 [11] A. Estrada-Moreno, I. G. Yero and J. A. Rodriguez-Velazquez, The fc-metric dimension of the lexicographic product of graphs, Discrete Math. 339 (2016), 1924-1934, doi: 10.1016/j.disc. 2015.12.024. [12] F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combinatoria 2 (1976), 191-195. [13] M. Heydarpour and S. Maghsoudi, The metric dimension of geometric spaces, Topology Appl. 178 (2014), 230-235, doi:10.1016/j.topol.2014.09.012. [14] S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70 (1996), 217-229, doi:10.1016/0166-218x(95)00106-2. [15] R. A. Melter and I. Tomescu, Metric bases in digital geometry, Comput. Vis. Graph. Image Process. 25 (1984), 113-121, doi:10.1016/0734-189x(84)90051-3. [16] G. P. Murphy, A metric basis characterization of Euclidean space, Pacific J. Math. 60 (1975), 159-163, http://projecteuclid.org/euclid.pjm/11028 684 44. [17] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, volume 149 of Graduate Texts in Mathematics, Springer, New York, 2nd edition, 2006. [18] P. J. Slater, Leaves of trees, in: R. S. D. Thomas (ed.), Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica Publishing, Winnipeg, Manitoba, volume 14 of Congressus Numerantium, 1975 pp. 549-559, held at Florida Atlantic University, Boca Raton, Florida, February 17-20, 1975. [19] I. G. Yero, A. Estrada-Moreno and J. A. Rodriguez-Velazquez, Computing the fc-metric dimension of graphs, Appl. Math. Comput. 300 (2017), 60-69, doi:10.1016/j.amc.2016.12.005. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 39-47 https://doi.org/10.26493/1855-3974.1316.2d2 (Also available at http://amc-journal.eu) Transversals in generalized Latin squares* Janos Baratf University ofPannonia, Department of Mathematics, 8200 Veszprem, Egyetem utca 10., Hungary and MTA-ELTE Geometric and Algebraic Combinatorics Research Group, H-1117 Budapest, Pdzmdny P. sitdny 1/C, Hungary Zoltan Lorant Nagy * MTA-ELTE Geometric and Algebraic Combinatorics Research Group, H-1117 Budapest, Pazmany P. setany 1/C, Hungary and ELTE Eotvos Lorand University, Department of Computer Science, H-1117 Budapest, Pazmany P. setany 1/C, Hungary Received 31 January 2017, accepted 29 July 2018, published online 9 September 2018 We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order n is equivalent to a proper edge-coloring of Kn,n. A transversal corresponds to a multicolored perfect matching. Akbari and Alipour defined l(n) as the least integer such that every properly edge-colored Kn,n, which contains at least l(n) different colors, admits a multicolored perfect matching. They conjectured that l(n) < n2/2 if n is large enough. In this note we prove that l(n) is bounded from above by 0.75n2 if n > 1. We point out a connection to anti-Ramsey problems. We propose a conjecture related to a well-known result by Woolbright and Fu, that every proper edge-coloring of K2n admits a multicolored 1-factor. Keywords: Latin squares, transversals, anti-Ramsey problems, Lovdsz local lemma. Math. Subj. Class.: 05B15, 05C15, 60C05 * We are grateful for an anonymous referee for pointing out numerous inaccuracies and for a valuable observation in the proof of Proposition 3.5, hence thereby improving the presentation of our paper. tSupported by Szechenyi 2020 under the EFOP-3.6.1-16-2016-00015 and OTKA-ARRS Slovenian-Hungarian Joint Research Project, Grant No. NN-114614. i Supported by OTKA Grant No. K 120154 and by the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences. E-mail addresses: barat@cs.elte.hu (Janos Barat), nagyzoli@cs.elte.hu (Zoltan Lorant Nagy) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 40 Ars Math. Contemp. 16 (2019) 19-24 1 Multicolored matchings and generalized Latin squares A subgraph H of an edge-colored host graph G is multicolored if the edges of H have different colors. The study of multicolored (also called rainbow, heterochromatic) subgraphs dates back to the 1960's. However, the special case of finding multicolored perfect matchings in complete bipartite graphs was first studied much earlier by Euler in the language of Latin squares. Since then this branch of combinatorics, especially the mentioned special case, has been flourishing. Several excellent surveys were dedicated to the subject, see [8,9, 10, 13]. In this paper we mainly focus on the case when the host graph is a complete bipartite graph Kn,n, and the multicolored subgraph in view is a perfect matching (1-factor). There is a natural constraint on the coloring: it has to be proper. These conditions can be reformulated in the language of Latin squares. A Latin square of order n is an n x n matrix, which has n different symbols as entries, and each symbol appears exactly once in each row and in each column. A generalized Latin square of order n is an n x n matrix, in which each symbol appears at most once in each row and in each column. A diagonal of a generalized Latin square of order n is a set of entries, which contains exactly one representative from each row and column. If the symbols are all different in a diagonal, then we call it a transversal. Generalized Latin squares correspond to properly edge-colored complete bipartite graphs, while transversals correspond to multicolored 1-factors (perfect matchings). The so called partial transversals correspond to multicolored matchings. This intimate relation allows us to use the concept of symbols and colors interchangeably. It is known that there exist Latin squares without a transversal. One might think that using more symbols should help finding a transversal. Therefore, it is natural to seek the sufficient number of symbols. We recall the following Definition 1.1 (Akbari and Alipour [1]). Let 1(n) be the least number of symbols satisfying 1(n) > n that forces a transversal in any generalized Latin square of order n that contains at least 1(n) symbols. In the terminology of matchings, they asked the threshold for the number 1(n) of colors such that any proper 1-coloring of Kn,n contains a multicolored perfect matching if 1 > l(n). Notice that the function l(n) is not obviously monotone increasing. Akbari and Alipour determined l(n) for small n: 1(1) = 1,1(2) = 1(3) = 3,1(4) = 6. They also proved that 1(n) > n + 3 for n = 2° - 2 (2 < a e N). They posed the following Conjecture 1.2 (Akbari and Alipour [1]). The difference 1(n) -n is not bounded if n ^ to, while 1(n) < n2/2 if n > 2. Our main contribution is the following Theorem 1.3. 1(n) < 0.75n2 if n > 1. Although we conjecture that 1(n) = o(n2), we must mention that if we relax the settings by allowing symbols to appear more than once in the columns, then for all n, there exist n x n transversal-free matrices, which contain n2/2 + O(n) symbols [2]. The paper is built up as follows. In Section 2 we show the connection of the problem to a classical Erdos-Spencer result. We prove an upper bound on 1(n) using a refined variant of the Lovasz local lemma. We present the proof of Theorem 1.3, which is mainly built on Konig's theorem. Finally in Section 3, we propose the study of a function similar to 1(n), and investigate the relation to certain anti-Ramsey problems. J. Barat and Z. L. Nagy: Transversals in generalized Latin squares 41 2 Two approaches to bound the number of symbols 2.1 Lovasz local lemma It is a classical application of the Lovasz local lemma (LLL) that there exists a transversal in an n x n matrix if no color appears more than 1 n times. In fact, Erdos and Spencer [7] weakened the conditions of LLL by introducing the so called lopsided dependency graph G of the events, on which the following holds for every event Ei and every subfamily F of events {Ej : j & NG [i]}: P(Ei | HjeFEj) < P(E), where NG [i] denotes the closed neighborhood of vertex i in graph G. Under this assumption, it is enough to show the existence of an assignment i i—> > 0) which fulfills P(Ei) < =-^--(2.1) Lsc»G[i] 1 Ijes Mj to obtain P(HiE) > 0. Applying the ideas of Scott and Sokal [11]; Bissacot, Fernandez, Procacci and Scop-pola [4] observed that LLL remains valid if the summation in Inequality (2.1) is restricted to those sets S which are independent in G. Let c(aij) denote the number of occurrences of the symbol aij in an n x n array A (n > 1). Let c„ (A) and c* j (A) measure the average occurrence in row i and column j as Ci*(A) = ^^ c(ait)^ - n and c*j(A) = ^^ c(atj- n. It can be viewed as some kind of weight-function on the rows and columns, where the weight is zero only if all entries admit uniquely occurring colors. We follow the proof of the improvement on the Erdos-Spencer result in [4]. We show that P (nv Ev) > 0 holds for the set of events {Ev} that a random diagonal contains a particular pair v of monochromatic entries. Here |NG [v] | in the lopsided dependency graph G depends only on the number of monochromatic pairs (v, v') of entries, which shares (at least) one row or column with an entry from both v and v'. Thus if v consists of aij and aki, then |Ng[v]| < c„(A) + c*j(A) + cfc*(A) + c*;(A). Also if w, w' G Ng[v] covers the same row from {i, k} or column from {j, 1} then w and w' are adjacent in G. If we set mv := M ^v, then it is enough to provide a m such that P(Ev)= . 1 . < Mv — M n(n 1) J2sCNa[v],S indep^jeS Mj J2sCNG[v], S indep. M|S| Consequently, it is enough to set m in such a way that M ^2sCNa [v], S indep. M |S| > M > 1 (1 + c„(A)m)(1 + c*j(A)m)(1 + ck*(A)M)(1 + c*i(A)m) ~ n(n - 1) holds. 42 Ars Math. Contemp. 16 (2019) 19-24 It is easy to see that (1 + Up)(1 + Vp) < (1 + m)2 for all U, V e R, hence M > 1 (1 + cVm)4 > n(n - 1) implies the required condition, where _ c„(A) + cj(A) + ck„(A) + c*(A) 4 Thus if we set ^ := , we obtain the following Proposition 2.1. There always exists a transversal in a generalized Latin square unless 4 f -J (c„(A) + cj (A) + cfc*(A) + c*;(A)) > n(n - 1) (2.2) holds for a pair of monochromatic entries aj and aki. Corollary 2.2. l(n) < (1 - 2=5)n2 + 22=73n ~ 0.895n2 if n > 1. Proof. Observe that n2 - c„ (A) or n2 - c*j (A) bounds from above the number of colors in A for then A for all i, j e [1, n]. Consequently, if the number of colors is at least (1 - )n2 + 253n, ^ c„(A) < 4(n2 - n) and Q) c*j(A) < 4(n2 - n) for every row i and column j, which in turn provides the existence of a transversal according to Proposition 2.1. □ Remark 2.3. Note that while the proof of Erdos and Spencer points out the existence of one frequently occurring symbol, the proof above reveals that in fact many symbols must occur frequently to avoid a transversal. 2.2 Using Konig's theorem We start with a lemma on the structure of partial transversals, which is essentially the consequence of the greedy algorithm. The following easy observation is due to Stein [12]. Result 2.4. Consider r rows in a generalized Latin square A of order n. If n^1 > r, then there exists a partial transversal of order r in A covering the r rows in view. We need the following consequence: Lemma 2.5. Consider p rows and q columns in an n x n generalized Latin square. If q < P < (n + 1)/2, then either Case (a) q < p/2 and there exists a partial transversal of size p covering the p rows and q columns, or Case (b) q > p/2 and there exists a partial transversal of size |_p/2j + q covering the p rows and q columns. J. Barat and Z. L. Nagy: Transversals in generalized Latin squares 43 Proof. Both parts follow from the fact that we can build a partial transversal by choosing first min{q, [p/2]} different symbols in the array formed by the intersection of the p rows and q columns, and then we can extend this greedily by entries in the uncovered rows and columns of the array (essentially using Result 2.4). □ We proceed by recalling a variant of Konig's theorem, see Brualdi, Ryser [5]. Lemma 2.6. There exists an all-1 diagonal in a 0/1 square matrix of order n if and only if there does not exist an all-0 submatrix of size x x y, where x + y > n + 1. Now we prove another upper bound on l(n). Theorem 2.7. If a generalized Latin square of order n contains at least 0.75n2 symbols, then it has a transversal. Proof. First notice that the statement holds for n = 1,2. We proceed by induction. Consider a generalized Latin square A of order n, which contains at least 0.75n2 symbols. A symbol is a singleton if it appears exactly once in A. We refer to the other symbols as repetitions. A submatrix is called a singleton-, resp. repetition-submatrix if every entry of the matrix is a singleton, resp. repetition. Let p be the number of rows consisting only of repetitions and q be the number of columns consisting only of repetitions. We refer to these as full rows and columns, and assume that q < p. Notice that p < n/2, since the number of symbols is at least 0.75n2. Our aim is to choose a partial transversal that covers all full rows and columns, and then we complete this to a transversal by adding only singletons. First we apply Lemma 2.5 to get a partial transversal that covers the full rows and columns. Next, we omit the rows and columns that are covered by the chosen partial transversal. We obtain a generalized Latin square A' of order n - p in Case (a) or of order n - |_p/2j - q in Case (b). Now we are done by Lemma 2.6, if there are not too large repetition-submatrices in A'. Suppose to the contrary that such a repetition-submatrix of size x x y exists in one of the cases, where x + y is larger than the order of A'. Note first that in either case, A' does not contain full rows and columns. Therefore, we can choose a singleton ai in A' such that at least x repetitions appear in its row. Similarly, we can choose a singleton a2 in A' such that at least y repetitions appear in its column. Claim 2.8. There exists a singleton a such that the row of a or the column of a contains more than n/2 repetitions in the original Latin square A. Proof. In Case (a) of Lemma 2.5: q < p/2. The number of repetitions in the row of a1 is at least q + x and number of repetitions in the column of a2 is at least p + y. Thus the statement holds since p + q + x + y>p + q +(n - p) > n. In Case (b) of Lemma 2.5: q > p/2. The number of repetitions in the row of a1 is at least q + x and number of repetitions in the column of a2 is at least p + y. Thus the statement holds since p + q + x + y>p + q +(n -|_p/2j- q) > n. □ In view of Claim 2.8, if we omit the row and column of the singleton a, we obtain a generalized Latin square B of order n - 1, which admits more than 0.75n2 - (2n - 1) + n/2 > 0.75(n - 1)2 symbols. By the induction hypothesis, there exists a transversal in B, hence it can be completed to a transversal of A by adding a. □ 44 Ars Math. Contemp. 16 (2019) 19-24 3 Discussion At the time of submission, we learned that Best, Hendrey, Wanless, Wilson and Wood [3] achieved results similar to ours. As the best upper bound, they proved l(n) < (2 - i/2)n2. Nevertheless, not only the conjecture of Akbari and Alipour remained open, but it is plausible that it can be strengthened in the order of magnitude as well. In fact, the bound 2n2 is intimately related to the number of singletons, which took a crucial part in both our proof and the proof in [3]. If the number of colors does not exceed 2n2, then there might be no singletons at all. However, our first probabilistic proof implies also that either there exists a transversal in a generalized Latin square of order n with Cn2 colors (C > 0.45), or the number of singletons is large. This fact points out that the constant 1/2 in Conjecture 1.2 is highly unlikely to be sharp. More precisely, we show the following Proposition 3.1. If the number of singletons is less than (2C + 0.25 (| )3 — 1 + o(1))n2 in a generalized Latin square of order n with Cn? symbols, then there exists a transversal. Proof. Suppose first that in every row and column, the sum c„(A) and c*¿ (A) are below 0.25 (4)3 (n2 — n). This in turn implies the existence of a transversal by Proposition 2.1. On the other hand, if for example c„(A) exceeds that bound, then consider only the symbols not appearing in row i, and let us denote by nk the number of symbols which occur exactly k times overall, with none of those occurrences being in row i. Clearly Efc nk = Cn2 — n and £k knk = (n(n — 1) — a,(A)) < (1 — 0.25 (4)3)(n2 — n). Consequently, for the number of singletons not appearing in the ith row, ni > 2 ^ nk — ^ knk > (2C + 0.25 — 1 + o(1))n2, kk4 which makes this case impossible. □ A special case, that appears as a bottleneck in some arguments concerns generalised Latin squares, where each repeated symbol has maximum multiplicity. We show that also in this special case, one can find a transversal. Lemma 3.2. If A is a generalised Latin square of order n, where each symbol has multiplicity 1 or n (and both multiplicities occur), then A has a transversal. Proof. We associate an edge-colored complete bipartite graph GA to A such that vertices on one side correspond to rows the other side to columns and the colors of the edges to the symbols. Our goal is to find a multicolored matching. Notice that the Latin property implies that a symbol with multiplicity n corresponds to a perfect matching. Let us remove all edges corresponding to symbols with multiplicity n. If there are r such colors, then the remaining bipartite graph is (n — r)-regular. As an easy corollary of Hall's theorem, any regular bipartite graph contains a perfect matching. In our case there are only singleton colors on the edges, so the perfect matching is multicolored. □ It seems likely that if the number of colors is large, then we not only obtain one transversal, but also a set of disjoint transversals. This motivates the study of the following function. J. Barat and Z. L. Nagy: Transversals in generalized Latin squares 45 Definition 3.3. Let I* (n) be the least integer satisfying I* (n) > n such that for any proper edge-coloring of Kn,n by at least 1*(n) colors, the colored graph can be decomposed into the disjoint union of n multicolored perfect matchings. Conjecture 3.4. 1*(n) < n2/2 if n is large enough. We remark that the difference of l(n) and l* (n) is at least linear in n if l(n) = n. Proposition 3.5. 1*(n) — l(n) > n — 1. Proof. Suppose first that there exists a transversal-free generalised Latin square of order n, i.e., l(n) > n. For n < 2 the claim is straightforward. Suppose n > 3. By definition, there exists a transversal-free generalized Latin square A of order n with l(n) — 1 symbols. Since l(n) < 0.75n2, we can find a set S of n — 1 different repetitions, where n — 1 < 0.25n2. We assign new symbols to the entries of S to create a new generalized Latin square A' of the same order. Since S cannot cover n disjoint transversals, and there were no transversals disjoint to S, matrix A' cannot be decomposed to n transversals, but contains l(n) + n — 2 symbols. Now consider the case when l(n) = n, which implies that n must be odd. Take the cyclic Latin square of order n — 1 (which has no transversal, since n — 1 is even) and add one row and column of singletons. The resulting matrix has n—1+2n —1 = 3n—2 symbols in it. However, it cannot be decomposed into transversals because such a decomposition would need to include a transversal of the embedded cyclic group table. □ Remark 3.6. Observe that the above result implies 1*(n) > 2n — 1 for all n. Notice that there are some orders n, for which l(n) = n, e.g. n € {1,3,7}, see also [3]. The question we studied concerning l(n) clearly has an anti-Ramsey flavor. The anti-Ramsey number AR(n, G) for a graph family G, introduced by Erdos, Simonovits and Sos [6], is the maximum number of colors in an edge coloring of Kn that has no multicolored (rainbow) copy of any graph in G. To emphasize this connection, we propose the following problem. Problem 3.7. What is the least number of colors t(n, 2), which guarantees a rainbow 2-factor subgraph on at least n — 1 vertices in a properly edge-colored complete graph Kn colored by at least t(n, 2) colors? Perhaps the size n — 1 of the 2-factor subgraph seems artificial in some sense at first, or at least it could be generalized to any given function f (n). We recall that for the function t(n, 1) corresponding to 1-factors, Woolbright and Fu provided the following related result. In Problem 3.7, we have to allow two values n — 1 and n to avoid parity issues. Proposition 3.8 ([14]). Every properly colored K2n has a multicolored 1-factor if the number of colors is at least 2n — 1 and n > 2. That is, t(n, 1) = n — 1. In another formulation, the necessary number of colors for a proper edge-coloring is already sufficient to guarantee a multicolored perfect matching. It might happen that it also forces a much larger structure as required in Problem 3.7. We propose the following Conjecture 3.9. Any proper edge-coloring of K2n by at least 2n — 1 colors contains a multicolored 2-factor on 2n — 1 or 2n vertices. 46 ArsMath. Contemp. 16(2019)25-38 If the above conjecture fails, then possibly there are proper edge-colorings of Kn without multicolored 2-factors of size n or n -1. In that case, we can use a connection between t(n, 2) and l(n) to show a lower bound. Proposition 3.10. l(n) > t(n, 2) + 1. Proof. Consider an edge-coloring C of the complete graph Kn on vertex set V without multicolored 2-factors of size n or n - 1. We associate to C a coloring of the complete bipartite graph Kn n on partite classes U and W as follows: let us assign the color of ViVj e E(Kn) (i,j e [1, n]) to the edge uiWj e E(Kn,n) if i = j, and color the set of independent edges uiwi (i e [1,n]) by a separate color. Suppose that we found a multicolored 1-factor M in the complete bipartite graph. We omit at most one edge of M if we delete the edges uiwi and M' remains. Consider the edges vkvi in Kn, for which ukwl is contained in the multicolored M' of edges. This edge set is multicolored too, and each vertex has degree 2. □ References [1] S. Akbari and A. Alipour, Transversals and multicolored matchings, J. Combin. Des. 12 (2004), 325-332, doi:10.1002/jcd.20014. [2] J. Barat and I. M. Wanless, Rainbow matchings and transversals, Australas. J. Combin. 59 (2014), 211-217, https://ajc.maths.uq.edu.au/pdf/5 9/ajc_v59_p211. pdf. [3] D. Best, K. Hendrey, I. M. Wanless, T. E. Wilson and D. R. Wood, Transversals in Latin arrays with many distinct symbols, J. Combin. Des. 26 (2018), 84-96, doi:10.1002/jcd.21566. [4] R. Bissacot, R. Fernandez, A. Procacci and B. Scoppola, An improvement of the Lovasz local lemma via cluster expansion, Combin. Probab. Comput. 20 (2011), 709-719, doi:10.1017/ s0963548311000253. [5] R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, volume 39 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1991, doi:10.1017/ cbo9781107325708. [6] P. Erd6s, M. Simonovits and V. T. S6s, Anti-Ramsey theorems, in: A. Hajnal, R. Rado and V. T. S6s (eds.), Infinite and Finite Sets, Volume II, North-Holland, Amsterdam, volume 10 of Col-loquia Mathematica Societatis Janos Bolyai, pp. 633-643, 1975, proceedings of a Colloquium held at Keszthely, June 25 - July 1, 1973, dedicated to Paul Erd6s on his 60th birthday. [7] P. Erd6s and J. Spencer, Lopsided Lovasz local lemma and Latin transversals, Discrete Appl. Math. 30 (1991), 151-154, doi:10.1016/0166-218x(91)90040-4. [8] S. Fujita, C. Magnant and K. Ozeki, Rainbow generalizations of Ramsey theory: a survey, Graphs Combin. 26 (2010), 1-30, doi:10.1007/s00373-010-0891-3. [9] M. Kano and X. Li, Monochromatic and heterochromatic subgraphs in edge-colored graphs -a survey, Graphs Combin. 24 (2008), 237-263, doi:10.1007/s00373-008-0789-5. [10] C. F. Laywine and G. L. Mullen, Discrete Mathematics Using Latin Squares, volume 49 of Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1998. [11] A. D. Scott and A. D. Sokal, The repulsive lattice gas, the independent-set polynomial, and the Lovasz local lemma, J. Stat. Phys. 118 (2005), 1151-1261, doi:10.1007/s10955-004-2055-4. [12] S. K. Stein, Transversals of Latin squares and their generalizations, Pacific J. Math. 59 (1975), 567-575,http://projecteuclid.org/euclid.pjm/1102905365. J. Barat and Z. L. Nagy: Transversals in generalized Latin squares 47 [13] I. M. Wanless, Transversals in Latin squares: a survey, in: R. Chapman (ed.), Surveys in Combinatorics 2011, Cambridge University Press, Cambridge, volume 392 of London Mathematical Society Lecture Note Series, pp. 403-437, 2011, doi:10.1017/cbo9781139004114.010, papers from the 23rd British Combinatorial Conference held at the University of Exeter, Exeter, July 3-July 8, 2011. [14] D. E. Woolbright and H.-L. Fu, On the existence of rainbows in 1-factorizations of K2n, J. Combin. Des. 6 (1998), 1-20, doi:10.1002/(sici)1520-6610(1998)6:1(1::aid-jcd!)3.0.co;2-j. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 49-58 https://doi.org/10.26493/1855-3974.1547.454 (Also available at http://amc-journal.eu) On the parameters of intertwining codes * Stephen P. Glasby, Cheryl E. Praeger Centre for Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Crawley 6009, Australia Received 8 December 2017, accepted 6 March 2018, published online 13 September 2018 Let F be a field and let Frxs denote the space of r x s matrices over F. Given equinu-merous subsets A = {Ai | i e I} C Frxr and B = {Bi | i e I} C Fsxs we call the subspace C(A, B) := {X e Frxs | AiX = XBi for i e I} an intertwining code. We show that if C(A, B) = {0}, then for each i e I, the characteristic polynomials of Ai and Bi and share a nontrivial factor. We give an exact formula for k = dim(C(A, B)) and give upper and lower bounds. This generalizes previous work. Finally we construct intertwining codes with large minimum distance when the field is not 'too small'. We give examples of codes where d = rs/k = 1/R is large where the minimum distance, dimension, and rate of the linear code C(A, B) are denoted by d, k, and R = k/rs, respectively. Keywords: Linear code, dimension, distance. Math. Subj. Class.: 94B65, 60C05 1 Introduction Let F be a field and let Frxs denote the space of r x s matrices over F. Given equinu-merous subsets A = {Ai | i e I} C Frxr and B = {Bi | i e I} C Fsxs we call the subspace C(A, B) := {X e Frxs | AiX = XBi for i e I} an intertwining code. The parameters of this linear code are denoted [n, k, d] where n = rs, k := dim(C(A, B)) and d is the minimum distance of C(A, B). Given u, v e Fn the Hamming distance d(u, v) = |{i | ui = vi}| is the number of different coordinate entries, and a subspace C < Fn has minimal (Hamming) distance d(C) := min{d(u, v) | u = v} which equals min{d(0,w) | w e V where w = 0}. If |I| = 1 we write C(A, B) instead *The authors acknowledge the contribution of Robin Chapman who emailed us in August 2016 a proof of Theorem 2.8. His formula for dim(C(N^, NM)) involved a double sum which can be reduced to a single sum using Theorem 3.1. The authors gratefully acknowledge the support of the Australian Research Council Discovery Grant DP160102323. E-mail addresses: Stephen.Glasby@uwa.edu.au (Stephen P. Glasby), Cheryl.Praeger@uwa.edu.au (Cheryl E. Praeger) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 50 ArsMath. Contemp. 16(2019)25-38 of C({A}, {B}). Centralizer codes [1] have the form C(A, A) and twisted centralizer codes [2, 3] have the form C(A, aA) where A G Frxs and a G F. Intertwining codes C(A, B) are more general still, so our dimension formula (Theorem 2.8) has particularly wide applicability. Furthermore, the greater abundance of intertwining codes turns out to help us construct intertwining codes with large minimum distance, cf. Theorem 4.3 and [3, Theorem 3.2]. Intertwining codes have the advantage of a short description, and fast matrix multiplication algorithms give rise to efficient syndrome computations which, in turn, may be used for decoding as described in [3, §3]. Given representations g ^ Aj and g ^ Bj a group algebra F (g | i g I}, elements of C(A, B) are homomorphisms between the associated modules. Hence Lemma 2.2 generalizes the fact that irreducible representations with distinct characters are inequivalent. An exact formula for k := dim(C(A, B)) is given in Theorems 2.9 and 2.8 of Section 2. The formula for k is simplified by an identity involving partitions proved in Section 4. Simpler upper and lower bounds for k are given in Section 5. In Theorem 4.3 in Section 4, we give an algorithm to construct A, B for which the minimum distance is d(C(A, B)) = [r/kjs. These examples have dR < 1 where R = rs is the rate of C (A, B). Corollary 4.4 to Theorem 4.3 shows that there exist matrices A G Frxr and B G Fsxs such that the intertwining code C(A, B) has dimension min{r, s} and minimum distance max{r, s}. We wonder how much this result can be improved. 2 A formula for dimF (C(A, B)) Throughout this section A = {Aj | i G I} C Frxr and B = {Bj | i G I} C Fsxs for a field F. The idea underlying this section is to use the Jordan form over the algebraic closure F of F to compute dimF(C(A, B)). To implement this idea we must simultaneously conjugate each Aj G A, and each Bj G B, into Jordan form. This is always possible when |I I = 1. Let GL(r, F) denote the general linear group of r x r invertible matrices over F. An ordered pair (R, S) G GL(r, F) x GL(s, F) acts on Frxs via X= R-1XS. Clearly (X (Rl,Sl))(R2,S2) = X (RlR2,Si S2) (XSi)(R'S) = X (R'S)Sf, and (RiX )(R'S) = RrX (r's). Lemma 2.1. If (R, S) G GL(r, F) x GL(s, F), then C (A, B)(r's) = R-1C (A, B)S = C(ar, bs ) where AR := {R-1AjR | i G I} and BS := {S-1BjS | i G I}. Proof. For each i G I, the equation AjX = XBj is equivalent to Arx (r-s) = (AjX )(R'S) = (XBj)(R'S) = X (R'S)Bf. □ Let cA(t) = det(tI - A) be the characteristic polynomial of A. Lemma 2.2. If C(A, B) = {0}, then gcd(cAi (t),cBi (t)) = 1 for all i G I. S. P. Glasby and C. E. Praeger: On the parameters of intertwining codes 51 Proof. Suppose that for some i G I we have gcd(cAi(t), cBi(t)) = 1. Then there exist polynomials f (t),g(t) such that f (t)cAi(t) + g(t)cBi(t) = 1. Evaluating this equation at t = Bj, and noting that cBi(Bj) = 0, shows f (Bj)cA. (Bj) = I. Hence cAi(Bj) is invertible. For X G C(A, B), \ve have AjX = XBj. Thus* * afc Ak I X = X afcBf I , \k>0 J \^k>0 J for all ak G F, and therefore cAi (Aj)X = XcAi (Bj). Since cAi (Aj) = 0, post-multiplying by cAi(Bj)-1 shows that X =0, and hence C(A, B) = {0}. □ Henceforth when we wish to emphasize the field F, we write CF(A, B). Lemma 3.1 of [2], in essence, says C^(A, B) = CF(A, B) ( F. This immediately yields Lemma 2.3. Lemma 2.3. If K is an extension field of F, then dimF (CF (A, B)) = dimK (CK (A, B)). In particular, dimF(CF(A, B)) = dim^(Cf(A, B)) where F is the algebraic closure of F. Lemma 2.3 allows us to assume that F is algebraically closed, which we shall do for the rest of this section. Given A g Frxr and B g Fsxs define A © B to be the block diagonal matrix (A B ), and define A © B to be {Aj © Bj | i g I} C F(r+s)x(r+s). Lemma 2.4. If A1 C FriXri,..., A™ C Fr™xrm and B1 C FS1XS1,..., Bn C Fs"xsn are subsets, all with the same finite cardinality, then (m n \ m n 0 Aj, 0 Bj I = 00 C(Aj, Bj). j=i j=i / j=i j=i Proof. Write X = (Xjj) as a block matrix where Xjj has size rj x sj. The condition X G C (®m=1 Aj, 0"=1 Bj) is equivalent to Xjj g C(Aj, Bj) for each i, j. □ Corollary 2.5. Suppose that A1,..., A™ and B1,..., Bn are as in Lemma 2.4, and suppose that for i = j, the characteristic polynomials of matrices in Aj are coprime to the characteristic polynomials of matrices in Bj. Then (m n \ min{m,n} 0 Aj, 0 Bj I = 0 C(Aj, Bj). j=1 j=1 / j=1 Proof. Use Lemma 2.4, and note that C(Aj, Bj) = {0} for i = j by Lemma 2.2. □ Remark 2.6. Let F be a finite field. Standard arguments, for example [6, p. 168], can be used to relate dim^ (C^ (A, B)) to data computed over F. This remark and Remark 2.10 explain the details. Let p1(t),p2(t),... enumerate the (monic) irreducible polynomials over F and write cA(t) = nj>1 Pj(t)ki and cB(t) = nj>1 Pj(t)£i, respectively. This gives rise to the A-invariant primary decomposition Fr = 0j>1 ker(pj(A)ki), and the B-invariant decomposition Fs = 0j>1 ker(pj(A)£i). Let Aj be the restriction of A to ker(pj (A)ki) and Bj the restriction of B to ker(pj(A)£i). Corollary 2.5 shows that dim(C(A, B)) = J2j>1 dim(C(Aj, Bj)). The second ingredient involves partitions and is described in Remark 2.10. 52 ArsMath. Contemp. 16(2019)25-38 It is straightforward to see that C(A, B) = p|ieJ C(A^ Bj) where C(A^ Bj) means C({Aj}, {Bj}). Recall that a matrix A g Frxr is nilpotent if and only if Ar = 0. We say that A is a-potent, where a g F, if (A — ai)r = 0. The following lemma and theorem reduce our deliberations from a-potent matrices to nilpotent matrices. For A = {Aj | i g I} C Frxr, let A — air denote the set {Aj — air | i g I}. Lemma 2.7. If AC Frxr, B C Fsxs and a g F, then C (A, B) = C (A — aIr, B — aIs). Proof: For i g I, AjX = XBj holds if and only if (Aj — aIr )X = X (Bj — aIs). □ A partition A of r, written A h r, is a sequence A = (Ai, A2,...) of integers satisfying Ai > A2 > • • • > 0 and Ai + A2 + • • • = r. We call Aj the ith part of A, and we usually omit parts of size zero. Let Nr be the r x r nilpotent matrix with all entries 0 except for an entry 1 in position (i, i + 1) for 1 < i < r. Let NA =0 NAi where A h r. Every nilpotent r x r matrix is conjugate to some NA for a unique A h r. Furthermore, if an r x r matrix R has eigenvalues p1,..., pm and associated generalized eigenspaces of dimensions r1,..., rm where r1 + • • • + rm = r, then R has Jordan form 0™ 1 (pjIri + NA .) where Aj is a partition of rj (not a part of a partition). Theorem 2.8. Suppose A g Frxr, B g Fsxs and gcd(cA(t),cB(t)) has distinct roots Z1,..., Cm in F. Suppose that the sizes of the Jordan blocks of A associated with the generalized Q-eigenspace of A determine a partition aj, and the sizes of the Jordan blocks of B associated with the generalized Q-eigenspace of B determine a partition fa. Then m dim(C(A, B)) = ^dim(C(Nai, Nft)). j=1 Proof. By Lemma 2.3 we may assume that F = F. Let Aj be the restriction of A to its generalized Creigenspace {v | v(A — ZjI)k for some k > 0}. Then Aj is Crpotent, and so determines a partition aj. Similarly, the restriction Bj of B to the Creigenspace determines a partition By Corollary 2.5 and Lemma 2.7, we have mm dim(C (A, B)) = ^ dim(C (Aj,Bj)) = ^ dim(C (Na ,Nft)). □ j=1 j=1 Theorem 2.9. Given partitions A of r and m of s, the dimension of C(NA , NM) equals dim(C (Na,Nm)) = ^^ min{Aj,Mj }. Proof. As A h r and m h s, we have J2Aj = r and Mj = s. Lemma 2.4 shows that C(NA,NM) = 0C(NAi, NMj). Taking dimensions, it suffices to show dim(C(NAi, NMj)) = min{Aj, mj}. This can be shown by solving NAiX = XNMj for X and counting the number of free variables. Alternatively, FAi is a uniserial (NAi}-module with 1-dimensional quotient modules, and similarly for FAj. As their largest common quotient module is Fmin{Ai-Ajwe have dim(C(NAi, NAj)) = min{Aj, Aj}. □ S. P. Glasby and C. E. Praeger: On the parameters of intertwining codes 53 Remark 2.10. Suppose |F | = q is finite. Following on from Remark 2.6 it suffices to consider the case where cA(t) = p(t)r/d, cB (t) = p(t)s/d, where p(t) is irreducible over F of degree d. The field K := F [t]/(p(t)) has order qd. In this case the structure of the modules Fr = Kr/d and Fs = Ks/d is determined by partitions A h r/d and p h s/d. It turns out that A is conjugate (see below) to NXp := diag(NAj,p, Na2,p, ...) G Frxr where /C(p) I \ N„ C (p) \ I C (p)/ G F' dm X dm and C(p) G Fdxd is the companion matrix of p(t). Now C(p) is conjugate in GL(d, K) to diag(Zi,..., Zd) where Zi,..., Zd are the (distinct) roots of p(t) in K. It follows from Theorems 2.9 and 2.8 that dim(C(A, B)) = dim(C(NA,P> NMiP)) (2.1) As an example, suppose A is cyclic and cA(t) = p(t)3 where d = deg(p) = 3. In this case r = 9 and A = (3). Write p(t) = t3 + p2t2 + pit + po = (t - Zi)(t - Ca)(t - Zs). Then A is conjugate in GL(9, F) by [5] to 'C(p) N 0 0 C(p) N 0 0 C (p). where C (p) = 0 0 - p o 1 0 0 1 - p i - p 2 and N 000 0 0 0 ,0 0 1, As p(t) is separable, [5, Theorem 1] implies that A is conjugate in GL(9, F) to 'C (p) 0 0 I C (p) 0 0 I C (p), Hence A is conjugate in GL(9, K) to 'D(Ci) 0 0 0 D(Z2) 0 0 0 D(C3)> where D(Z ) 'C 1 0 Z 00 This explains the factor of d = deg(p(t)) in equation (2.1) and relates the generalized Jordan form of A over F to the Jordan form of A over K. 54 ArsMath. Contemp. 16(2019)25-38 3 Conjugate partitions In this section we simplify the formula in Theorem 2.9 for dim(C(NA, NM)). We prove an identity in Lemma 3.2 involving partitions which replaces multiple sums by a single sum. In order to state the simpler dimension formula we need to define 'conjugate partitions'. The conjugate of A h r is the partition A' = (A1, A2,...) of r whose parts satisfy Aj = |{j | Aj > i}|, for each i. The Young diagram of A', is obtained from that of A by swapping rows and columns as shown in Figure 1. n ] Figure 1: Young diagrams for A = (5, 3, 3,1) and A' = (4,3, 3,1,1). For the following result, note that the number of nonzero Aj is A'x, and r = J2A=i Aj. Theorem 3.1. Given partitions A of r and of s, the dimension of C(NA, NM) equals min{ Ai } dim(C (Na,Nm)) = £ Aj ¡j = £ AjMj. j>1 j=i To prove Theorem 3.1 we need a technical lemma which we have not been able to find in the literature, see [4]. Lemma 3.2 below says J2j>1 Aj = J2j>1 Aj when k =1. We only need the case k = 2 for the proof of Theorem 3.1, however, the argument for k > 2 is not much harder. Lemma 3.2. If A, ..., w are partitions and A', ¡',..., w' are their conjugates, then Ai ^i Wi min{ Ai ,...,wi} min{Aj,Mj ,...,wk} = Aj^j •••w'. (3J) j=1 j=1 k=1 j=1 Proof. By permuting the partitions A, ..., w if necessary, we can assume that A1 ^ ¡1 ^ • • • ^ W1. If A1 = 0, then A h 0 and both sides of (3.1) are zero. If A1 = 1, then Ai ^i w1 min{ Ai ,...,Wi} LHS(1) = £ £ ••• £ 1 = A1m1 ••• w1 = £ AjMj ••• Wj' = RHS(1). j=1 j=1 k=1 j=1 Suppose now that A1 > 1. We use induction on A1. Let A be the partition of (^ j>1 Aj)-A'1 obtained by deleting the first column of the Young diagram of A. Since 1 < < • • • < w1, we define ¡A,..., W similarly. It is clear that Aj = Aj - 1 for 1 < i < A1 and Aj = Aj+1 for i > 1, and similarly for ¡A,..., W. As A1 < A1, induction shows Ai /ii W1 min{ Ai ,^i,...,Wi} min{Aj,/Aj ,...,wfc} = Aj'Aj' j=1 j=1 k=1 j=1 A A wA . S. P. Glasby and C. E. Praeger: On the parameters of intertwining codes 55 Note that Aj = 0 for each i e [A'x + 1, A'J since A'x = A2, so the upper limit A'x of the sum£*= 1 can be replaced by A'x. Similarly, the upper limits /¿i,..., O can be replaced by ^1,..., wj . Hence, since Aj = Aj - 1,..., o = wj - 1, we have Ai m'i "i • • min{Aj - - 1,..., wfc - 1} = j=1 j = 1 k = 1 min{A' — 1,^' — 1,...,"' -1} Ai+1Mj+1 ••• °i+1. j=1 Re-indexing the right sum, and using J2A= 1 J2j=1 • • • 2"i 1(-1) = -A1M1 • • • W1 gives A' ¡jl' "1 min{ A',...,"'} -AiMi ••• + min{Aj,^j ,...,°k} = Kni ••• . j=1 j=1 k=1 j=2 Adding A/1^/1 • • • w1 to both sides completes the inductive proof of (3.1). □ Proof of Theorem 3.1. Apply Theorem 2.9 and Lemma 3.2 with k = 2. □ 4 Minimum distances In Section 2 a formula is given for k := dim(C(A, B)); where we suppress mention of the field F in our notation. In this section we choose A and B to maximize the value of the minimum distance d := d(C(A, B)) as a function of k. We focus on the case when |A| = |B| = 1. The action of GL(r, F) x GL(s, F) of C(A, B) fixes k = dim(C(A, B)) but can change d wildly, e.g. from 1 to rs as setting k = 1 in Theorem 4.3 illustrates. Let Ej denote the r x s matrix with all entries 0, except the (i, j) entry which is 1. Lemma 4.1. Suppose r, s, k e Z where 1 < k < min{r, s}, and suppose F is afield with |F| > k+min{1,r-k}+min{1, s-k}. Fixpairwise distinct scalars Z1,..., Zk, a, 0 e F and set Ao := diag(Z1,..., Zk, a,..., a) e Frxr and Bo := diag(C1,...,Ck,£,...,£) e Fsxs. Then C(A0, B0) = (En,..., Ekk) has dimension k and minimum distance 1. Proof. Note first that if k = min{r, s}, then A0 has no as, or B0 has no ,0s. Thus the assumption |F| > k + min{1,r - k} + min{1, s - k} ensures that distinct scalars C1,...,Ck,a,0 in F exist. Using a direct calculation of C (A0, B0), or Corollary 2.5, shows that C(A0, B0) = (E11,..., Ekk). Since d(0,En) = 1, we have d(C(A0,B0)) = 1. □ We now seek R e GL(r, F) and S e GL(s, F) such that R—1(En,..., Ekk)S has large minimum distance. For brevity, we write T := R—1. Denote the ith row of a matrix A by A„ and its jth column by Aj. Lemma 4.2. Suppose r, s, k e Z where k < min{r, s}. Fix S e Fsxs and T e Frxr and define X(1),..., X(k) e Frxs by X= T^for 1 < ^ < k. Then TE«S = Xfor 1 < I < k. 56 ArsMath. Contemp. 16(2019)25-38 Proof. Suppose Sj is 1 if i = j and 0 otherwise. Then the (i, j) entry of Eff is SifSfj. The (i', j') entry of T*fSf* is t^ sfj-/. This agrees with the (i', j') entry of TEffS, namely SifSfj Sjjl = ti'£S£j' . □ i=1 j = 1 Theorem 4.3. Suppose r, s, k G Z where 1 < k < min{r, s}, and suppose F is afield with |F| > k + 2. Then there exist A G Frxr and B G Fsxs such that the linear code C(A, B) has dimension k and minimum distance d = |_r/kj s. Proof. By Lemma 4.1 there exist diagonal matrices A0 G Frxr and B0 G Fsxs such that C(A0, B0) = (E11,..., Ekk} has dimension k. We seek invertible matrices R G Frxr and S G Fsxs such that A = AR and B = BS give C (A, B) = (En,.. .,Ekk }(R'S) with minimum distance d = |_r/kj s. Let X(f) = R-1EffS for 1 < £ < k. The r x s matrices X (f) , 1 ^ £ ^ k, will have a form which makes it clear that d = |_r/kj s. First, we partition the set {1,..., r} of rows into the following k subsets: I = 1 {1,...,[Ij},l2 = { +1,..., 2 [kj}..... r -1) bd + 1,...,^. ik = Choose the ith row of the matrix X(f) to be zero if i G If, and to be a vector with all s entries nonzero otherwise. Since |_|J = |If | < |Ik | for £ < k, it follows that for 1 < £ < k d(0,X(f)) = ^ s = |If|s > ieit with equality if £ < k. The choice of these matrices is such that for each nonzero X in the span (X(1),..., X(k)} we also have d(0, X) > d(0, X(f)) for some £, and hence (X(1),..., X(k)} has minimum distance d = |_r/kjs. It is well known that if the first few rows of a square matrix are linearly independent, then the remaining rows can be chosen so that the matrix is invertible. A similar remark holds if the first few columns are linearly independent. Our construction uses k linearly independent 1 x s row vectors u1,..., uk which give the first k rows of S G GL(s, F), and k linearly independent r x 1 column vectors v(1),...,v(k) which give the first k columns of R-1 G GL(r, F). The pair (R, S) will be used to construct A and B. Henceforth suppose that 1 ^ £ ^ k. Since |F| ^ 3, we may choose 7 G F \ {1,1 — s}. Let J be the s x s matrix with all entries 1. Then the s x s matrix S' = (7 — 1)I + J is invertible as det(S') = (7 — 1)s-1(y + s — 1) is nonzero. Let uf = (1,..., 1,7,1,..., 1) be the £th row of S'. Since u1,..., uk are linearly independent, there exists an invertible matrix S G GL(s, F) with Sf* = uf. Of course S = S' is one possibility. Similarly, let v(f) be the r x 1 column vector (f) v; 1 if i G If, 0 if i G if. As v(1),..., v(k) are linearly independent, there exists an r x r invertible matrix, which we call R-1, whose first k columns are v(1),..., v(k). Lemma4.2 shows that R-1EffS = X (f) for 1 < £ < k. Hence C (AR, BS) = C (A0 ,B0)(R'S) = (X (1),...,X(k)} has minimum distance |_r/kj s as desired. □ S. P. Glasby and C. E. Praeger: On the parameters of intertwining codes 57 Corollary 4.4. If |F| > min{r, s} + 2, then there exist matrices A e Frxr and B e Fsxs such that C(A, B) has dimension min{r, s} and minimum distance max{r, s}. Proof. Since AX = XB if and only if XTAT = BTXT we see that C(BT, AT) equals C(A, B)t. Because C(A, B) and C(A, B)T have the same dimension and minimum distance, we may assume that r < s. If |F| > r + 2, then applying Theorem 4.3 with k = r gives the desired result. □ Remark 4.5. Suitable matrices A and B in Theorem 4.3 are constructed by first choosing the diagonal matrices A0 and B0 in Lemma 4.1, and then taking A = R-1A0R and B = S-1B0S where R and S are constructed in the proof of Theorem 4.3. It is desirable for a code to have both a high rate, viz. R = k/n, and a high distance d. Can the product Rd be a constant for intertwining codes? By setting r = s = k in Theorem 4.3 we obtain a rate of R = 1/r and a distance of d = r, so the answer is affirmative. It is natural to ask how the maximum value of Rd for an intertwining code depends on (r, s, F)? We wonder whether there is a sequence C1, C2,... of intertwining codes over a field F with parameters [r^, kj, dj] for which Rjdj = ^^ approaches infinity. 5 Upper and lower bounds for dimF (C(A, B)) Denote that rank and nullity of A e Frxr by Rk(A) and Null(A), respectively. Note that Rk(A) + Null(A) = r and Null(NA) = A^. In this section we bound k = dim(C(A, B)) in terms of the rank and nullity of A and B. If A h r and ^ h s, Theorem 2.9 implies that A>1 < £ AjMj = dim(C(Na,Nm)) < I £ Aj I I £I = rs. (5.1) j>1 \j >1 ) \j>1 ) View A e Frxr as acting on an r-dimensional vector space over the algebraic closure F. Let the a-eigenspace, and the generalized a-eigenspace, of A have dimensions kA,a and mA,a, respectively. Then cA(t) = f](t - a)mA,a where mA,a = 0 for finitely many a e F and 0 < kA,a < mA,a. The following result generalizes [2, Theorems 2.8 and 4.7]. Theorem 5.1. If A e Frxr and B e Fsxs, then (a) J2kA,akB,a < dim(C(A, B)) < ]T mA,amB,a, and (b) (r - Rk(A))(s - Rk(B)) < dim(C(A, B)) < (r - Rk(A))(s - Rk(B)) + Rk(A) Rk(B). Proof. Part (a) follows immediately from Theorem 2.8 and (5.1). (b) The lower bound follows from part (a) since r - Rk(A) = Null(A) = kAj0. For the upper bound, note that A is similar to a diagonal direct sum N\ © A' where N\ is nilpotent of size m0jA and A' is invertible of size r - m0jA. Similarly, B is similar to 58 Ars Math. Contemp. 16 (2019) 97-109 NM © B' where NM is nilpotent of size m0,B and B' is invertible of size s - m0,B. It follows from Theorem 2.8 that dim(C(A,B)) = dim(C(NA,NM)) + dim(C(A','b')). Further by Theorem 2.9 dim(C(NA, NM)) = ^i>x Ai^i where, as usual, A' and m' denote conjugate partitions. We use the observation: if 0 < x < a and 0 < y < b, then (a — x)(b — y) + xy < ab (5.2) to show that dim(C(A, B)) = Ai^i + ^ Ai^i + dim(C(A', B')) i> 2 < Ai^i + (mo,A — Ai)(mo,B — m1) + (r — mo,i)(s — mo,B) < A>i + (r — Ai)(s — Mi). The result follows since Ai = Null(NA) = Null(A) = r — Rk(A) and m1 = s — Rk(B). □ The Singleton bound d + k < n +1 implies that if d is close to n = rs, then k is small, and the lower bound of Theorem 5.1(b) implies that A or B has high rank. Setting k =1 in Theorem 4.3, shows that this bound is attained for intertwining codes. The code C(A, B) is the row nullspace of AT ( Is + Ir ( B and the column nullspace of A ( Is + Ir ( Bt where T denotes transpose. References [1] A. Alahmadi, S. Alamoudi, S. Karadeniz, B. Yildiz, C. Praeger and P. Sole, Centraliser codes, Linear Algebra Appl. 463 (2014), 68-77, doi:10.1016/j.laa.2014.08.024. [2] A. Alahmadi, S. P. Glasby and C. E. Praeger, On the dimension of twisted centralizer codes, Finite Fields Appl. 48 (2017), 43-59, doi:10.1016/j.ffa.2017.07.005. [3] A. Alahmadi, S. P. Glasby, C. E. Praeger, P. Sole and B. Yildiz, Twisted centralizer codes, Linear Algebra Appl. 524 (2017), 235-249, doi:10.1016/j.laa.2017.03.011. [4] S. P. Glasby, Lemmas involving two partitions of integers, MathOverflow, 2017, https:// mathoverflow.net/q/25 8 7 22. [5] D. W. Robinson, Classroom notes: the generalized jordan canonical form, Amer. Math. Monthly 77 (1970), 392-395, doi:10.2307/2316152. [6] R. Stong, Some asymptotic results on finite vector spaces, Adv. in Appl. Math. 9 (1988), 167199, doi:10.1016/0196-8858(88)90012-7. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 59-66 https://doi.org/10.26493/1855-3974.1291.ee9 (Also available at http://amc-journal.eu) On the size of maximally non-hamiltonian digraphs* * Nicolas Lichiardopol Lycee A. de Craponne, Salon, France Carol T. Zamfirescuf Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, 9000 Ghent, Belgium and Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania A graph is called maximally non-hamiltonian if it is non-hamiltonian, yet for any two non-adjacent vertices there exists a hamiltonian path between them. In this paper, we naturally extend the concept to directed graphs and bound their size from below and above. Our results on the lower bound constitute our main contribution, while the upper bound can be obtained using a result of Lewin, but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open. Keywords: Maximally non-hamiltonian digraphs. Math. Subj. Class.: 05C45, 05C20 1 Introduction Throughout this paper all graphs will be simple, finite, connected, and will not admit multiple edges or loops. In a digraph, each edge between two adjacent vertices u and v may be oriented from u to v, from v to u, or both ways. We call a digraph an oriented graph if no edge has both orientations. *We are grateful to the referees for their helpful comments. We would also like to thank Kenta Ozeki for fruitful discussions. t Zamfirescu is supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO). E-mail address: czamfirescu@gmail.com (Carol T. Zamfirescu) ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ In memory of Nicolas Lichiardopol. Received 14 January 2017, accepted 12 June 2018, published online 14 September 2018 Abstract 60 Ars Math. Contemp. 16 (2019) 97-109 We were led to the study of the titular subject from a related concept: homogeneous traceability, a notion introduced by Skupien. A digraph is called homogeneously traceable if every vertex is the start-vertex of a hamiltonian path. If additionally every vertex is also the end-vertex of some hamiltonian path, the digraph is called bihomogeneously traceable. A graph or digraph D is called hypohamiltonian if D is non-hamiltonian, but for any vertex v in D, the graph D - v is hamiltonian. Obviously, any digraph that is hamiltonian or hypohamiltonian is bihomogeneously traceable. But not every homogeneously traceable digraph is hamiltonian [4]. Not even bihomogeneous traceability implies hamiltonicity. At a meeting in Kalamazoo (in 1980) Skupien showed that for all n > 7 there exists a 2-diregular bihomogeneously traceable non-hamiltonian oriented digraph of order n, see [16], which appeared in 1981. Moreover, Skupien [17] later constructed exponentially many bihomogeneously traceable non-hamiltonian oriented graphs. Independently, in another paper which also appeared in 1981, Hahn and T. Zamfirescu [12] also constructed an infinite sequence of bihomoge-neously traceable non-hamiltonian oriented graphs, and gave three special examples: the first is arc-minimal (i.e. with the smallest possible number of arcs for a given number of vertices) and order 7, the second is planar and has 8 vertices (it is proven in [12] that there are no smaller examples), and the third is both arc-minimal and planar, and has 9 vertices. Note that arc-minimality amounts in this context to 2-diregularity. Hahn and T. Zamfirescu asked in [12] the natural question whether infinitely many planar bihomogeneously traceable non-hamiltonian oriented graphs exist, as very few were known. Infinite families of planar hypohamiltonian digraphs containing opposite arcs have been found by Fouquet and Jolivet [10]. In [20], Thomassen proved that a planar hypohamiltonian digraph with n vertices (and many edges with both orientations—in fact, all but six) exists for each n > 6. It was shown by the second author [21] that, indeed, there exist infinitely many planar bihomogeneously traceable non-hamiltonian oriented graphs. A stronger result was recently obtained by van Aardt, Burger, and Frick [1], who showed that there exist infinitely many planar hypohamiltonian oriented graphs, thereby solving a problem of Thomassen [20]. If one now asks for an even larger set of spanning paths, one may be led to the case demanding that between any two non-adjacent vertices there exists a hamiltonian path. Such graphs have been studied in the non-oriented case, see for instance [7] and [2], and are called maximally non-hamiltonian (which, in the following, will often be abbreviated to MNH). For a digraph D, we write V(D) and A(D) for its set of vertices and arcs, respectively. Mirroring the non-directed definition, a digraph D is maximally non-hamiltonian if D is non-hamiltonian, but for every x, y G V(D) with yx G A(D) there is a hamiltonian path from x to y. A few words concerning the notation. For a set X, we denote with |X | the cardinality of X. In a graph G, for adjacent vertices x and y in G we denote by xy the edge between x and y. If G is a digraph, xy will be the arc from x to y. A digraph D is called strong if for any pair x, y G V(D) there exists a (directed) path from x to y. Denote with ¿+(D) (¿-(D)) the minimum out-degree (minimum in-degree) and with ¿0(D) the minimum semi-degree of D, which is the minimum of ¿+(D) and ¿-(D). N + (x) (N-(x)) shall be the set of out-neighbours (in-neighbours) of a vertex x. For a set of vertices S of D we denote the digraph induced by S in D as D[S]. Further definitions follow when needed. N. Lichiardopol and C. T. Zamfirescu: On the size of maximally non-hamiltonian digraphs 61 2 Results An important direction of research on non-directed MNH graphs has been determining the smallest size of an MNH graph of order n, which we shall denote by /(n). This study was initiated by Bondy [6], who showed that for n > 7 we have /(n) > |~3f 1. Bollobas [5] conjectured that there exist infinitely many graphs for which this lower bound is in fact attained. This was proven to be correct for all even n > 36 by Clark and Entringer [7]. In the same paper graphs of order n and size [3fl + 1 were constructed for all odd n > 55. Horak and Siran [13] also found such almost extremal graphs by using a construction of Thomassen [19] (which was originally intended to construct hypohamiltonian graphs). They also proved that for every n > 48 there exists a triangle-free MNH graph of order n. Clark and Entringer asked in [7] whether for infinitely many n the MNH graph on n vertices of smallest size is unique or not. Combining the results from [7, 8, 9, 15], one obtains that for infinitely many n there exist two non-isomorphic MNH graphs of order n and smallest size. Still, there were infinitely many orders for which only one MNH graph of smallest size was known. This was investigated by Stacho [18]; he proved that for any n > 88 there exist three pairwise non-isomorphic MNH graphs of order n and smallest size. In the following we extend the study of the size of MNH graphs to directed graphs. First, we characterise the non-strong MNH digraphs. A digraph is symmetric if for every arc in D the corresponding inverted arc also lies in D. For integers m > 1 and p > 1, let Dm,p denote the digraph whose vertices are the vertices of two vertex-disjoint complete symmetric digraphs Km and Kp and whose arcs are those of the digraphs Km, Kp, and additionally the arcs xy with x in Km and y in Kp. We claim: Lemma 2.1. The non-strong MNH digraphs are the digraphs Dm,p. Proof. Clearly, a digraph Dm p is a non-strong MNH digraph. Conversely, let D be a non-strong MNH digraph. There exists a partition V1, V2 of V(D) such that there are no arcs from V2 to V1. Let x and y be two distinct vertices of D. It is easy to see that if x, y are both in Vi or in V2 there is no hamiltonian path from x to y. Furthermore, if x is in V2 and y is in V1, there is also no hamiltonian path from x to y. Then, since D is MNH, it follows that D [V1] and D [V2] are complete symmetric digraphs and that every ordered pair xy with x e V1 and y e V2 is an arc of D. This means that D is a digraph Dm p, and so we are done. □ The upper bound contained in the following theorem can be obtained by using a result of Lewin [14], but we choose to give here a different proof. (In fact, our proof of the upper bound is a new proof of Lewin's [14, Corollary 1].) In upcoming arguments, we require the following. Theorem 2.2 (Ghouila-Houri [11]). A strong digraph D with ¿0(D) > |V(D)|/2 ishamil-tonian. Theorem 2.3. For an MNH digraph D of order n > 4 we have |A(D)| < (n - 1)2 and this upper bound is attained. Proof. There exists a vertex x e V(D) with d+(x) + d-(x) < n (for otherwise, by Theorem 2.2, D would be hamiltonian). Let us put B = N+(x) n N-(x), A = N+(x) \ B, and C = N-(x) \ B. 62 Ars Math. Contemp. 16 (2019) 97-109 Furthermore, let a, b, and c be the respective cardinalities of A, B, and C. Clearly, for a vertex y in A or in C we have d+(y) + d_ (y) < 2n — 3. For a vertex y in B, we have d+ (y) + d_ (y) < 2n — 2, and for a vertex y not adjacent with x, we have d+ (y) + d_ (y) < 2n — 4. By addition, we get 2 x |A(D)| < n — 1 + a(2n — 3) + b(2n — 2) + c(2n — 3) + (n — 1 — a — b — c)(2n — 4), hence 2 x |A(D)| < n — 1 + (n — 1)(2n — 4) + a + c + 2b =(n — 1)(2n — 3) + a + c + 2b. But d+ (x) + d- (x) < n means that we have a + c + 2b < n — 1. It follows that 2 x |A(D)| < (n — 1)(2n — 3) + n — 1 < 2(n — 1)2, and the upper bound is proved; it is attained since D1jn-1 and Dn-1j1 are MNH digraphs of size (n — 1)2. □ For integers r > 1 and n with n > 2r +1 define the digraph Hn,r of order n as follows. The vertices of Hn,r are those of a complete symmetric digraph K^-r-1 and r +1 additional vertices y1,..., yr+1. Let x1,..., xr be r vertices of K^-r-1. The arcs of Hn,r are the arcs of K*-r-1, y4x where 1 < i < r + 1, x G V(K*-r-1), and x^y^ where 1 < i < r and 1 < j < r +1. It is easy to see that a digraph Hn ,r, like its converse, is MNH, of minimum semi-degree r and of strong connectivity r. By Theorem 2.3, the maximum size of a non-strong MNH digraph is at most (n — 1)2 and this bound is reached. It was proved in [3] that a digraph D with minimum semi-degree r and with more than a„,r = n2 — (r + 2)n + (r + 1)2 arcs is hamiltonian. When r > (n — 1)/2, by Theorem 2.2, D is hamiltonian and therefore cannot be MNH. But when r < (n — 1)/2, the result of [3] shows that the maximum size of an MNH digraph D of order n with ¿0(D) = r is at most an,r, and since Hn,r is MNH, of minimum semi-degree r and of size an,r, this upper bound is reached. If 1 < r < s < (n — 1)/2, then an,r > an,s, so the maximum size of an MNH digraph D of order n and of strong connectivity k(D) = r is at most an,r. As K(Hn,r) = r, the bound is sharp. We now establish a lower bound on the size of an MNH digraph. Lemma 2.4. For an MNH digraph D on at least four vertices, either ¿°(D) > 2 or |A(D)| > 3n — 4. Proof. We proceed by induction on n. It is easy to verify that the assertion is true for n = 4. Now let n > 5 and suppose the assertion is true for all k < n — 1. Consider a digraph D of order n having the required property. Let us put V (D) = {x1,...,xn}. Let there exist a vertex of D which is of out-degree at most 1. W.l.o.g., we may assume that this vertex is x1. Suppose first that d+(x1) = 0. N. Lichiardopol and C. T. Zamfirescu: On the size of maximally non-hamiltonian digraphs 63 In this case D is a non-strong MNH digraph, and by Lemma 2.1, D is in fact D„_i i. We then have |A(D)| = (n - 1)2 > 3n - 4, and so we are done. Suppose now that (x1) = 1. W.l.o.g., we may assume that the unique out-neighbour of x1 is x2. We claim that x2x1 G A(D). Suppose the opposite. Then there exists a hamil-tonian path P from x1 to x2. But then x1 has at least two out-neighbours, a contradiction. We also claim that all of the vertices of V(D) \ {x1, x2} are in-neighbours of x2. Suppose the opposite. Then there exists a vertex xj, i > 3, which is not an in-neighbour of x2. Thus, there exists a hamiltonian path from x2 to x^ whence, x1 has an out-neighbour in this hamiltonian path distinct from x2, a contradiction. We claim that D' = D - x1 is MNH, i.e. that for any two vertices y, z of D' such that zy G A(D), there exists a hamiltonian path in D' from y to z. Observe first that y = x2. There exists in D a hamiltonian path P from y to z. P contains the arc x1x2, and since x1 is not the first vertex of P it admits an in-neighbour u in P. Then P' = P — x1 + ux2 (i.e. the path P from which we delete the vertex x1 and add the arc ux2) is a hamiltonian path in D' from y to z. So, the claim is proved. By induction hypothesis, either every vertex of D' is of in-degree at least 2 in D', or |A(D')| > 3(n — 1) — 4 = 3n — 7. In the first case, since x2 is of in-degree n — 2 in D', we have |A(D') | > n — 2 + 2(n — 2), hence |A(D') | > 3n — 6. Since x1x2 and x2x1 are arcs of D but not of D', we get | A(D) | > 3n — 4, and the theorem is proved in this case. Suppose now that |A(D')| > 3n — 7. Assume first that there are no distinct vertices xj, xj, where 3 < i, j < n, such that xjxj is not an arc of D'. Then we have |A(D')| > (n — 2)2 > 3n — 6. Thus | A(D) | > 3n — 4, and again we are done. Suppose now that there exist distinct vertices xj, xj, where 3 < i, j < n, such that xj xj G A(D'). Then there exists a hamiltonian path of D from xj to xj, and then necessarily x1 has an in-neighbour xk with 3 < k < n. It follows that | A(D) | > 3n — 7 + 3 = 3n — 4, and we are done. □ As a corollary, we can state: Theorem 2.5. For an MNH digraph D of order n > 4 we have | A(D) | > 2n. Proof. If | A(D) | > 3n — 4, we are done. If | A(D) | < 3n — 4, by Lemma 2.4, each vertex of D is of out-degree at least 2, and we get | A(D) | > 2n. □ Observe that for n = 3, this lower bound inequality is untrue: consider the digraph D = ({x, y, z}, {xy, yx, xz, yz}). D is an MNH digraph of order 3 and size 4. Also note that for n = 4, the lower bound is tight due to the digraph D2 2, which is not regular. From [6] we know that for n > 7 the size of an MNH graph of order n is at least . For every n > 19, this lower bound is reached; consult [18] and [15] for details. Thus, for n > 19, there exists an MNH graph G of order n and size . It is easy to prove that the symmetric digraph obtained by doubly orienting each edge of G is an MNH digraph of order n and size 2 x [3f a. S° the minimum size of an MNH digraph of order n > 19 is at least 2n and at most 2 x |"3f 1. We now give a lower bound on the size of an MNH non-strong digraph: Theorem 2.6. Let D be an MNH non-strong digraph of order n > 2. Then 3 |A(D)|> - n2 — n. 64 Ars Math. Contemp. 16 (2019) 97-109 Proof. We know that D is of the form Dan,(1-a)n, where 0 < a < 1 and an is an integer. Then we have |A(D)| = an(an — 1) + (1 — a)n((1 — a)n — 1) + an(1 — a)n = n2(a2 — a +1) — n > 4n2 — n, since a2 — a + 1 > 4. □ If the strong connectivity is 1, we have the following result. Theorem 2.7. Let D be an MNH digraph of order n > 4 and of strong connectivity 1. Then |A(D)| > min < 3n — 4, 5 2 n 1 Proof. If S+ (D) < 1 or S-(D) < 1, by Lemma 2.4, we have | A(D)| > 3n — 4, and the result is proved. Suppose now that S+ (D) > 2 and S- (D) > 2, i.e. each vertex of D is of out-degree at least 2 and of in-degree at least 2. There is a vertex x such that the digraph D — x is not strong, so there exists a partition of V(D) \ {x} into two non-empty sets A and B such that there are no arcs from B to A. W.l.o.g., we suppose that |B| < |A|, so |A| > 2. We claim that all the vertices of B are out-neighbours of x. Let us suppose this not to be the case. Then there exists a vertex y of B such that xy G A(D). Since D is MNH, there exists a hamiltonian path P from y to x. Necessarily, P contains an arc uv with u G B and v G A, which is impossible. Similarly, all the vertices of A are in-neighbours of x. Since x has at least one out-neighbour in A and at least one in-neighbour in B, we get d+(x) + d- (x) > n + 1. Suppose now that x has exactly one out-neighbour y in A. It is easy to see that A \ {y}, B U {x} is a partition of V(D) \ {y} into non-empty sets such that there are no arcs from B U {x} to A \ {y}. So, D — y is not strong, and from the preceding arguments, we have d+(y) + d-(y) > n + 1. Since d+(z) + d-(z) > 4 for z distinct from x and y, by addition we get 2 x |A(D)| > 2(n + 1) + 4(n — 2), hence 2 x |A(D)| > 6n — 6, and | A(D) | > 3n — 3 > 3n — 4, and the result is proved. Suppose now that x has at least two out-neighbours in A. We then get d+ (x) + d- (x) > n + 2, and for z = x we have d+(z) + d-(z) > 4. By addition this yields 2 x |A(D)| > n + 2 + 4(n — 1), hence |A(D)| > [§ n] — 1, as stated. With above conclusions in mind, an MNH digraph D with □ |A(D)| < min < 3n — 4 5 2 n 1, 3 2 4 n is necessarily of strong connectivity at least 2, and thus, of semi-degree at least 2. Now it is easy to see that for n > 5, if the lower bound 2n is attained by an MNH digraph D of order n, then necessarily D is a 2-diregular digraph of order n and of strong connectivity 2. We were not able to show the existence of such digraphs, but an advance in this direction is perhaps the following: n Theorem 2.8. Let D be a 2-diregular MNH digraph of order n. Then D is hypohamilto-nian. N. Lichiardopol and C. T. Zamfirescu: On the size of maximally non-hamiltonian digraphs 65 Proof. We know that D is non-hamiltonian. Suppose for the sake of a contradiction that there exists a vertex z such that D — z is non-hamiltonian. Let x be an in-neighbour of z, and let y be the other out-neighbour of x. We claim that zy G A(D). Suppose the opposite. Then there exists a hamiltonian path from y to z, and it is easy to see that the in-neighbour of z in P is x. But then P — z + xy is a hamiltonian cycle of D — z, a contradiction. Assume that y is an in-neighbour of z. Then y and z are of in-degree 2 in the induced sub-digraph D[{x, y, z}]. Since D is 2-strong and n > 5, this is not possible. Therefore y is not an in-neighbour of z. Suppose that x is an out-neighbour of z. Then x and z are of out-degree 2 in the induced sub-digraph D[{x, y, z}]. Since D is 2-strong and n > 5, this is not possible. So x is not an out-neighbour of z. Then z has an in-neighbour u distinct from x, y, and z. The vertex y is not an out-neighbour of u (for otherwise y would be of in-degree at least 3, which is impossible), and x is not an out-neighbour of u (for otherwise, with a previous argument, zx would be an arc of D, which is false). Thus u has an out-neighbour v distinct from x, y, and z. Then, by previous arguments, zv is an arc of D, and v is not an in-neighbour of z. The last statement implies that there exists a hamiltonian path P' from z to v. The second vertex of P' is necessarily y, and then it is easy to see that x has an out-neighbour w distinct from y and z. Since x is of out-degree 2, this is impossible. Thus D — z cannot be non-hamiltonian, and so the theorem is proved. □ The above discussions concerning the size of MNH digraphs led us to the following question. Problem. Does every non-hamiltonian oriented graph of order at least 3 contain an arc xy for which there is no hamiltonian path from x to y? Lastly, we would like to point out the connection between maximally non-hamiltonian graphs and so-called platypus, which are non-hamiltonian graphs in which every vertex-deleted subgraph is traceable. They contain the family of all hypohamiltonian and hy-potraceable graphs. The second author showed [22] that a maximally non-hamiltonian graph G is a platypus if and only if A(G) < |V(G)| — 1, where A(G) denotes the maximum degree of G. Directed platypus have not yet been investigated, but the above results may provide a good starting point. References [1] S. A. van Aardt, A. P. Burger and M. Frick, An Infinite Family of Planar Hypohamiltonian Oriented Graphs, Graphs Combin. 29 (2013), 729-733, doi:10.1007/s00373-012-1165-z. [2] A. Ainouche and N. Christofides, Conditions for the existence of hamiltonian circuits in graphs based on vertex degrees, J. London Math. Soc. 32 (1985), 385-391, doi:10.1112/jlms/s2-32.3. 385. [3] D. Amar, I. Fournier and A. Germa, Some conditions for digraphs to be Hamiltonian, in: M. Rosenfeld and J. Zaks (eds.), Convexity and Graph Theory, North-Holland, Amsterdam, volume 20 of Annals of Discrete Mathematics, 1984 pp. 37-41, doi:10.1016/s0304-0208(08) 72804-4, proceedings of the conference held in Jerusalem, March 16 - 20, 1981. [4] J.-C. Bermond, J. M. S. Simoes-Pereira and C. M. Zamfirescu, On nonhamiltonian homogeneously traceable digraphs, Math. Japonica 24 (1979), 423-426. [5] B. Bollobas, Extremal Graph Theory, volume 11 of London Mathematical Society Monographs, Academic Press, London, 1978. 66 Ars Math. Contemp. 16 (2019) 97-109 [6] J. A. Bondy, Variations on the Hamiltonian theme, Canad. Math. Bull. 15 (1972), 57-62, doi: 10.4153/cmb-1972-012-3. [7] L. Clark and R. Entringer, Smallest maximally nonhamiltonian graphs, Period. Math. Hungar. 14 (1983), 57-68, doi:10.1007/bf02023582. [8] L. H. Clark, R. P. Crane, R. C. Entringer and H. D. Shapiro, On smallest maximally nonhamiltonian graphs, in: F. Hoffman, R. C. Mullin, R. G. Stanton and K. B. Reid (eds.), Proceedings of the Seventeenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica Publishing, Winnipeg, Manitoba, volume 53 of Congressus Numerantium, 1986 pp. 215-220, held at Florida Atlantic University, Boca Raton, Florida, February 10- 14, 1986. [9] L. H. Clark, R. C. Entringer and H. D. Shapiro, Smallest maximally nonhamiltonian graphs II, Graphs Combin. 8 (1992), 225-231, doi:10.1007/bf02349959. [10] J.-L. Fouquet and J.-L. Jolivet, Hypohamiltonian oriented graphs, Cahiers Centre Etudes Rech. Oper. 20 (1978), 171-181. [11] A. Ghouila-Houri, Une condition suffisante d'existence d'un circuit hamiltonien, C. R. Acad. Sci. Paris 251 (1960), 495-497. [12] S. Hahn and T. Zamfirescu, Bihomogeneously traceable oriented graphs, Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), 137-145. [13] P. Horak and J. Siran, On a construction of Thomassen, Graphs Combin. 2 (1986), 347-350, doi:10.1007/bf01788108. [14] M. Lewin, On maximal circuits in directed graphs, J. Comb. Theory Ser. B 18 (1975), 175-179, doi:10.1016/0095-8956(75)90045-3. [15] X. Lin, W. Jiang, C. Zhang and Y. Yang, On Smallest Maximally Non-Hamiltonian Graphs, Ars Combin. 45 (1997), 263-270. [16] Z. Skupien, On homogeneously traceable non-Hamiltonian digraphs and oriented graphs, in: G. Chartrand, Y. Alavi, D. L. Goldsmith, L. Lesniak-Foster and D. R. Lick (eds.), The Theory and Applications of Graphs, John Wiley & Sons, New York, 1981 pp. 517-527, proceedings of the Fourth International Conference held at Western Michigan University, Kalamazoo, Michigan, May 6-9, 1980. [17] Z. Skupien, Exponential constructions of some non-Hamiltonian minima, in: J. Nesetril and M. Fiedler (eds.), Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity, North-Holland, Amsterdam, volume 51 of Annals of Discrete Mathematics, 1992 pp. 321-328, doi:10.1016/s0167-5060(08)70649-6, proceedings of the symposium held in Prachatice, 1990. [18] L. Stacho, Non-Isomorphic Smallest Maximally Non-Hamiltonian Graphs, Ars Combin. 48 (1998), 307-317. [19] C. Thomassen, Hypohamiltonian and hypotraceable graphs, Discrete Math. 9 (1974), 91-96, doi:10.1016/0012-365x(74)90074-0. [20] C. Thomassen, Hypohamiltonian graphs and digraphs, in: Y. Alavi and D. R. Lick (eds.), Theory and Applications of Graphs, Springer-Verlag, Berlin, volume 642 of Lecture Notes in Mathematics, 1978 pp. 557-571, proceedings of the International Conference held at Western Michigan University, Kalamazoo, Michigan, May 11 - 15, 1976. [21] C. T. Zamfirescu, An Infinite Family of Planar Non-Hamiltonian Bihomogeneously Traceable Oriented Graphs, Graphs Combin. 26 (2010), 141-146, doi:10.1007/s00373-010-0900-6. [22] C. T. Zamfirescu, On Non-Hamiltonian Graphs for which every Vertex-Deleted Subgraph Is Traceable, J. Graph Theory 86 (2017), 223-243, doi:10.1002/jgt.22122. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 67-79 https://doi.org/10.26493/1855-3974.1546.c5e (Also available at http://amc-journal.eu) On chromatic indices of finite affine spaces* Gabriela Araujo-Pardo Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510, Mexico City, Mexico Gyorgy Kiss Department of Geometry and MTA-ELTE Geometric and Algebraic Combinatorics Research Group, Eotvos Lomnd University, H-1117 Budapest, Pízmany s. 1/c, Hungary and FAMNIT, University of Primorska, 6000 Koper, Glagoljaska 8, Slovenia Christian Rubio-Montiel Department of Algebra, Comenius University, Mlynska dolina, 84248, Bratislava, Slovakia and Division de Matematicas e Ingeniería, FES AcatMn, Universidad Nacional Autónoma de Mexico, 53150 Naucalpan, Mexico Adrian Vazquez-Avila Subdireccion de Ingeniería y Posgrado, Universidad Aeronautica en Queretaro, Parque Aeroespacial Queretaro, 76270, Queretaro, Mexico Received 6 December 2017, accepted 14 April 2018, published online 17 September 2018 A line-coloring of the finite affine space AG(n, q) is proper if any two lines from the same color class have no point in common, and it is complete if for any two different colors i and j there exist two intersecting lines, one is colored by i and the other is colored by j. The pseudoachromatic index of AG(n, q), denoted by ^'(AG(n, q)), is the maximum number of colors in any complete line-coloring of AG(n, q). When the coloring is also proper, the maximum number of colors is called the achromatic index of AG(n, q). We prove that ^'(AG(n,q)) - qi.Sn-1 for even n, and that qL5("-1) < ^'(AG(n,q)) < q1.5"-1 for odd n. Moreover, we prove that the achromatic index of AG(n, q) is q15"-1 for even n, and we provide the exact values of both indices in the planar case. *The authors gratefully acknowledge funding from the following sources: Gabriela Araujo-Pardo was partially supported by CONACyT-Mexico under Projects 178395, 166306, and by PAPIIT-Mexico under Projects IN104915 and IN107218. Gyorgy Kiss was partially supported by the bilateral Slovenian-Hungarian Joint Research Project, grant no. NN 114614 (in Hungary) and N1-0032 (in Slovenia), and by the Hungarian National Foundation for Scientific Research, grant no. K 124950. Christian Rubio-Montiel was partially supported by a CONACyT-Mexico Postdoctoral fellowship, and by the National scholarship programme of the Slovak Republic. Adrian Vazquez-A'vila was partially supported by SNI of CONACyT-Mexico. ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ Abstract 68 Ars Math. Contemp. 16 (2019) 97-109 Keywords: Achromatic index, complete coloring, finite affine space, pseudoachromatic index. Math. Subj. Class.: 05B25, 51E15, 05C15 1 Introduction This paper is motivated by the well-known combinatorial conjecture about colorings of finite linear spaces stated by Erdos, Faber and Lovasz in 1972. As a starting point, we briefly recall some definitions and state the conjecture. Let S be a finite linear space. A line-coloring of S with k colors is a surjective function ? from the lines of S to the set of colors [k] = {1,..., k}. For short, a line-coloring with k colors is called k-coloring. If ?: S ^ [k] is a k-coloring and i e [k] then the subset of lines ?-1(i) is called the i-th color class of A k-coloring of S is proper if any two lines from the same color class have no point in common. The chromatic index x'(S) of S is the smallest k for which there exists a proper k-coloring of S. The Erdos-Faber-Lovasz conjecture (1972) states that if a finite linear space S contains v points then x'(S) < v, see [12, 13]. Several papers have investigated the conjecture for particular classes of linear spaces. For instance, if each line of S has the same number k of points then S is called a block design or a (v, n)-design. The conjecture is still open for designs even for k = 3, however, it was proved for finite projective spaces by Beutelspacher, Jungnickel and Vanstone [8]. It is not hard to see that the conjecture is also true for the n-dimensional affine space AG(n, q) of order q defined over the Galois field GF(q). Indeed, qn - 1 x'(AG(n,q)) = . q -1 For some related results, see for instance [6, 7]. A natural question is to determine similar, but slightly different color parameters in finite linear spaces. A k-coloring of S is complete if for each pair of different colors i and j there exist two intersecting lines of S, such that one of them belongs to the i-th and the other one to the j-th color class. Observe that any proper coloring of S with x'(S) colors is a complete coloring. The pseudoachromatic index ^'(S) of S is the largest k such that there exists a complete k-coloring (not necessarily proper) of S. When the k-coloring is required to be complete and proper, the parameter is called the achromatic index and it is denoted by a'(S). Therefore, we have that x'(S) < a'(S) < V'(S). Several authors studied the pseudoachromatic index, see [2, 3, 4, 5, 9, 14, 15, 17]. Moreover, in [1, 10, 18] the achromatic indices of some block designs were also estimated. In this paper we study the pseudoachromatic and achromatic indices of finite affine spaces. In the proofs we will often use the notion of the projective closure of AG(n, q). This is the finite projective space PG(n, q) = AG(n, q) U where the points of correspond to the parallel classes of lines in AG(n, q). The space is isomorphic to PG(n - 1, q), and it is called the hyperplane at infinity. We assume that the reader is E-mail addresses: garaujo@matem.unam.mx (Gabriela Araujo-Pardo), kissgy@cs.elte.hu (Gyorgy Kiss), christian.rubio@apolo.acatlan.unam.mx (Christian Rubio-Montiel), adrian.vazquez@unaq.edu.mx (Adrian Vazquez-Avila) G. Araujo-Pardo et al.: On chromatic indices of finite affine spaces 69 familiar with the most important properties of affine and projective geometries. For the detailed description of these spaces we refer to [16]. The main results in the paper are proved in Sections 2 and 3. They are stated in Theorems 1.1, 1.2 and 1.3. In these theorems v = qn always denotes the number of points of the finite affine space AG(n, q). Theorem 1.1. For all n: V>'(AG(n,q)) < v/Vy(V . 1) - Q(qVV/2). q — 1 Theorem 1.2. If n is even: 1 VV(v — 1) 2 ^ q — 1 i VV(v — 1) If n is odd: sjq q — 1 Theorem 1.3. If n is even: 1 yv(v — 1) 3 ^ q — 1 — Q(Vv/2) < V'(AG(n,q)). — ©(^y/v/q5) < ^'(AG(n, q)). + Q(v/q) < a'(AG(n, q)). Note that when n is even Theorems 1.1 and 1.2 show that ^'(AG(n, q)) grows asymptotically as ©(v1-5/q), while Theorems 1.2 and 1.3 show that a'(AG(n, q)) grows asymptotically as ©(w15/q). Let us remark that no similar estimates regarding the asymptotic behavior of these indices have appeared so far in the literature. Finally, in Section 4 we determine the exact values of pseudoachromatic and achromatic indices of arbitrary (not necessarily Desarguesian) finite affine planes and we improve the previous lower bounds in dimension 3. 2 Upper bounds In this section, upper bounds for the pseudoachromatic index of AG(n, q) are presented when n > 2. The following lemma is pivotal in the proof. Lemma 2.1. Let L be a set of s lines in AG(n, q), n > 2. Then the number of lines of AG(n, q) intersecting at least one element of L is at most 2 ( q"-1 - 1 ( 1) q s-;--(s -1) V q -1 Proof. In AG(n, q) there are q ^- l) = q2 (q g--1) lines intersecting any fixed line. The number of lines intersecting two lines, say and l2, is at least q2, because if £1 n l2 = 0 then the q2 lines joining a point of and a point of l2 intersect both and ¿2, while, if l\_ n i2 = {P} then the other qn—r - 2 > q2 lines through P intersect both ^ and . Consequently, the number of lines intersecting at least one element of L is at most "2 (^) - <• -1"2' 70 Ars Math. Contemp. 16 (2019) 97-109 Notice that the previous inequality is tight, since if L consists of s parallel lines in a plane then there are exactly q2 ^s q q_-1 - (s - 1) j lines intersecting at least one element of L. □ Lemma 2.2. Let n > 2 bean integer. Then the colorings of the finite affine space AG(n, q) satisfy the inequality *'(AG(n, q)) < ^4q"(q"- 1)(q" -^q211)2(q-1)2+. (2.1) Proof. Consider a complete coloring which contains ^'(AG(n, q)) color classes. Then the number of lines in the smallest color class is at most qn-1(qn - 1) s — (q - 1)V"(AG(n, q))" Each of the other ^'(AG(n, q)) - 1 color classes must contain at least one line which intersects a line from the smallest color class. Hence, by Lemma 2.1, we obtain V>'(AG(n, q)) - 1 < q2 f sq" 1 - 1 - (s - . q-1 Multiplying it by ^'(AG(n, q)), we get a quadratic inequality on ^'(AG(n, q)), whence the assertion follows. □ We are in a position to prove our first main theorem. Proof of Theorem 1.1. For n > 2 a straightforward computation shows 4qn(qn - 1)(qn - q2) + (q2 + 1)2(q - 1)2 = (2q2 (qn - 1) - q2 (q2 - 1))2 - qn(q2 - 1)2 + (q2 + 1)2(q - 1)2 < (2q2 (qn - 1) - q2 (q2 - 1)) 2 because n > 2 implies that qn(q2 - 1)2 > (q2 + 1)2(q - 1)2. This together with Inequality (2.1) give *'<**,,)) < ,2 (^ 1 - q2 (i+i) + 11 q -1 J V 2 ) 2 ' which proves the theorem for n > 2. For n = 2 the statement is clear. □ 3 Lower bounds In this section complete colorings of AG(n, q) are presented. These constructions give different bounds on ^'(AG(n, q)) depending on the parity of n. First, we prove some geometric properties of affine and projective spaces. G. Araujo-Pardo et al.: On chromatic indices of finite affine spaces 71 Proposition 3.1. Let n > 1 be an integer, ni and n2 be subspaces in PG(n, q) = AG(n, q) U Hœ. Let d denote the dimension of n for i = 1,2. Suppose that n1 n n2 n %œ is an m-dimensional subspace and d1 + d2 = n + 1 + m. Then n1 n n2 n AG(n, q) is an (m + 1)-dimensional subspace in AG(n, q). In particular, n1 n n2 is a single point in AG(n, q) when n1 n n2 n %œ = 0 and di + ¿2 = n. Proof. Since n1 n n2 n %œ is an m-dimensional subspace, dim(n1 n n2) < m +1. On the other hand, the dimension formula yields dim(n1 n n2) = dimn1 + dimnv - dim(n1, n2) > d1 + d2 - n = m + 1. Thus n1 n n2 is an (m + 1)-dimensional subspace in PG(n, q), therefore n1 n n2 n AG(n, q) is an (m + 1)-dimensional subspace in AG(n, q) if m > 0. If m = -1, then n1 n n2 n %œ = 0 and dim(n1 n n2) = 0. Hence n1 n n2 is a single point in AG(n, q). □ In the following proposition we present a partition of the points of PG(2k, q) that we will call a good partition in the rest of the paper. Proposition 3.2. Let k > 1 be an integer and Q G PG(2k, q) be an arbitrary point. The points of PG(2k, q) \ {Q} can be divided into two subsets, say A and B, and one can assign a subspace S (P) to each point P G A U B, such that the following holds true: • P G S (P) for all points; 2k |A| = q2 ( qo2_11 ) and, if A G A then S(A) is a k-dimensional subspace; l 2k_-| \ • |B| = q ( qq2_1 j and, if B G B then S(B) is a (k — l)-dimensional subspace; • S(A) n S(B) = 0 for all A G A and B gB. Proof. We prove the assertion by induction on k. If k = 1 then let ¿4,..., } be the set of lines through Q. Let A and B consist of points PG(2, q) \ {4i} and \ {Q}, respectively. If A g A then let S(A) be the line AQ, if B g B then let S(B) be the point B. These sets clearly fulfill the prescribed conditions, so PG(2, q) admits a good partition. Now, let us suppose that PG(2k, q) admits a good partition. In PG(2k + 2, q) take a 2k-dimensional subspace n which contains the point Q. Then n is isomorphic to PG(2k, q), hence it has a good partition {Q} uA' UB' with assigned subspaces S'(P). Let H0,..., Hq be the pencil of hyperplanes in PG(2k + 2, q) with carrier n. Let B = B' U (H0 \ n) and A = PG(2k + 2, q) \ (B U {Q}). Notice that A' and B' have the required cardinalities, because q2fc+3 _ 1 q2k+3 _ 1 /q2k+2 _ 1 \ |A'| = \ — 1 — (|B| + 1) = (q + 1)q q2 1 1 — q ( q ,2 t M — 1 q — 1 q2 — 1 V q2 — 1 /q2k+2 _ i ^ 2 q 1 q lB'l = ibi + IH\ni = q q2 — 1 q2 - 1 q2k _ 1 \ /q2k+2 _ 1 q_M + q2k+1 = q q_1 72 Ars Math. Contemp. 16 (2019) 97-109 We assign the subspaces in the following way. If A 6 A' then let S(A) be the (k + 1)-dimensional subspace (S'(A),P} where P 6 uf=1H( is an arbitrary point, whereas, if A 6 (uq=1Hi) \ n then let S(A) be the (k + 1)-(dimensional subspace (A, S'(P)} where P 6 A' is an arbitrary point. In both cases S(A) C u|=1H( for all A 6 A. Similarly, if B 6 B' then let s(b) be the k-dimensional subspace (S'(B),P} where P 6 H0 is an arbitrary point, whereas, if B 6 H0 \ n then let S(B) be the k-dimensional subspace (B, S'(P)} where P 6 B' is an arbitrary point. Also here, in both cases, S(B) C H0 for all B 6 B. Moreover, the assigned subspaces satisfy the intersection condition because if A 6 A and B 6 B are arbitrary points then S(A) n S(B) = (S(A) n (u?=1H()) n (S(B) n H0) = S'(A) n S'(B) n n = 0. Hence PG(2k + 2, q) also admits a good partition, and the statement is proved. □ The next theorem proves Theorem 1.2 for even dimensional finite affine spaces. Notice that the lower bound depends on the parity of q, but its magnitude is — ^ in both cases, where v = qn. Theorem 3.3. If k > 1 then the colorings of the even dimensional affine space, AG(2k, q), satisfy the inequalities {^kfr———, if q is odd, qkM) +1 f q iseven 2(q—1) + 1 " q even. Proof. The hyperplane at infinity in the projective closure of AG(2k, q), is isomorphic to PG(2k - 1, q), hence it has a (k - 1)-spread S = {S1, S2,..., Sqk+1}. The elements of S are pairwise disjoint (k - 1)-dimensional subspaces (see [16, Theorem 4.1]). Let {Pj, P2(,..., P(qfc — 1)/(q—1)} be the set of points of S( for i = 1, 2,..., qk + 1. For a point P 6 let S(P) denote the unique element of S that contains P, and A(P) = {nPj1, nP,2,..., nP,qk} denote the set of the qk parallel k-dimensional subspaces of AG(2k, q) whose projective closures intersect in S(P). We define a pairing on the set of points of which depends on the parity of q. On the one hand, if q is odd then let (Pj Pj+1) be the pairs for i = 1, 3, 5,..., qk and _ On flip nthpr honH i q—1 he p i = 4,6,..., qk and j = 1,2,..., q—, and let (P/, Pj2), (P?+1, Pf+1), (P/+1, Pf) and (P-2, Pf) be the pairs for i = 1, 2, 3 and j = 2,4,6,..., 1 - 1. Let (U, V) be any pair of points. Then, by defintion, S(U) = S(V). Let the color class Cu,v,( contain the lines joining either U and a point from nU (, or V and a point from nV,(, for i = 1,2,..., qk. Clearly, (U, V) defines qk color classes, each one consists of the parallel lines of one subspace in A(U) and the parallel lines of one subspace in A(V). Finally, if q is even, then let the color class C1 consist of all lines of AG(2k, q) whose point at infinity is P11 . 2k_1 2k_ We divided the points of into |(q—1) pairs if q is odd, and into |(q—1) pairs if q is 2k_1 , even. Consequently, the number of color classes is equal to 2(—1) qk when q is odd, and it is equal to |(q—qk + 1 when q is even. j = 1,2,..., q—j-. On the other hand, if q is even then %œ has an odd number of points, thus we give the pairing on the set of points \ {P-}: let (Pj, PJ+-) be the pairs for G. Araujo-Pardo et al.: On chromatic indices of finite affine spaces 73 Now, we show that the coloring is complete. The class C obviously intersects any other class. Let CUtv,i and Cw,z,j be two color classes. Then S(U) and S(V) are distinct elements of the spread S and S(W) is also an element of S. Hence we may assume, without loss of generality, that S(U) n S(W) = 0. As dim(S(U) U nU,i) = dim(S(W) U nWj) = k in PG(2k, q), by Proposition 3.1, we have that nU,i n nW,j consists of a single point in AG(2k, q). Notice that the coloring is not proper, because the same argument shows that nU,i n nVi is also a single point in AG(2k, q). □ For odd dimensional spaces we have a slightly weaker estimate. In this case, the magnitude of the lower bound is ^q ■ , where v = qn. Theorem 3.4. If k > 1 then the colorings of the odd dimensional affine space, AG(2k + 1, q), satisfy the inequality qfc+2 ( +1 - +1 q)). Proof. The hyperplane at infinity in the projective closure of AG(2k+1, q), is isomorphic to PG(2k, q). Hence, by Proposition 3.2, admits a good partition = AUBU {Q} with assigned subspaces S(U). Let A = {Pi, P2,..., Pt} and B = {Ri, R2,..., R } where t = q2 (i^) and s = q ($-r) . For a point Pi e A let A(Pi) = {nPiji, nPi,2,..., nP.qk} denote the set of the qk parallel (k+1)-dimensional subspaces of AG(2k+1, q) whose projective closures intersect in S(Pi). Similarly, for a point Rj e B let B(Rj) = {nR. ,i, nR.,2,..., nRj ,,fc+i} denote the set of the qk+1 parallel k-dimensional subspaces of AG(2k + 1, q) whose projective closures intersect in S(Rj). Now, we define the color classes. Let Ci be the color class that contains all lines of AG(2k + 1, q) whose point at infinity is Q. Let the color class Ci,j,m contain the lines joining either P(j-i)q+i and a point from np(._1)f+.,m, or Rj and a point from nR. ,(i-i)qk +m for j = 1,2,..., s, i = 1,2,..., q and m = 1, 2,..., qk. Counting the number of color classes of type Ci,j,m, we obtain s ■ q ■ qk = qk+2 ^qp—1) . Each color class consists of the parallel lines of one subspace in A(P(j-i)q+i) and the parallel lines of one subspace in B(Rj). Clearly, the total number of color classes is 1 + qk+2 ^ qq2 j-1 j . The color class Ci contains q2k lines and each of the classes of type Ci,j,m consists of qk + qk-i lines. To prove that the coloring is complete, notice that the class Ci obviously intersects any other class. Let Ci,j,m and Ci/,j/,m' be two color classes other than Ci. Consider the projective closures of those elements of A(P(j-i)q+i) and B(Rj) whose lines are contained in Ci,j,m and in Ci',j',m', respectively. One of these subspaces is a (k + 1)-dimensional, whereas the other one is a k-dimensional subspace in PG(2k + 1, q), and they have no point in common in Thus, by Proposition 3.1, their intersection is a single point in AG(2k + 1, q). The coloring is not proper, because the same argument shows that nP(j_1)q+.,m n nRj ,(i-i)qfc+m is also a point in AG(2k + 1, q), thus Ci,j,m contains a pair of intersecting lines. □ 74 Ars Math. Contemp. 16 (2019) 97-109 Now, we are ready to prove our second main theorem. Proof of Theorem 1.2. If n is even then Theorem 3.3 gives the result at once. If n is odd then v = q2fc+1, hence \Jv/q = qk. From the estimate of Theorem 3.4 we get / q2k _ i \ q3k+2 _ qk+2 qk+2( ^—r ) +1 = q _2 ! +1 ;) q3k+1 + qk+2 _ qk + 1 _ qk + 1 q2 -1 q2 -1 (q + 1)(q3k+1 - qk) q3fc+1 + qk+2 - qfc+1 - qk q2 - 1 q2 - 1 1 Vv(v - 1) q3k+1 + qk+2 - qk+1 - qk + 1, v^ q - 1 q2 - 1 which proves the statement. □ Next, recall that a lower bound for the achromatic index require a proper and complete line-coloring of AG(n, q). We consider only the even dimensional case. Theorem 3.5. Let k > 1 and e = 0,1 or 2, such that qk + 1 = e (mod 3). Then the achromatic index of the even dimensional finite affine space AG(2k, q) satisfies the inequality q + 1 - ' (qk + 2) + e) q--1 < a'(AG(2k, q)). 3 ^ ' / q - 1 Proof. The hyperplane at infinity in the projective closure of AG(2k, q), is isomorphic to PG(2k - 1, q), hence it admits a (k - 1)-spread L = ... , ^qfc+1}. Let A(£j) = jn£i, 1, n£i, 2,..., , qk} denote the set of the qk parallel k-dimensional subspaces in AG(2k, q) whose projective closures intersect in £j. Then, by Proposition 3.1, the intersection n£.jS nn^.,t is a single affine point for all i = j and 1 < s,t < qk. First, to any triple of (k - 1)-dimensional subspaces, e, f, g G L, we assign qk + 2 color classes as follows. Take a fourth (k - 1)-dimensional subspace d g L, and, for u = (qk - 1)/(q-1), denote the points of the (k-1)-dimensional subspaces d, e, f and g by D1, D2,..., Du, E1, E2,..., Eu, F1, F2,..., Fu and G1, G2,..., Gu, respectively. For any triple (Dj, e, g) there is a unique line through Dj which intersects the skew subspaces e and g. We can choose the numbering of the points Ej and Gj such that the line Ej Gj intersects d in Dj for i = 1, 2,..., u; the numbering of the points Fj, such that the line DjEj+1 intersects d and g for i = 1,2,..., u - 1, and, finally, choose the line D„E1 that intersects d and g. Notice that this construction implies that the line DjEj does not intersect g for i = 1,2,..., u. Let the points of nd1 denote by M1, M2,..., Mqk. We can choose the numbering of the elements of A(e), A(f) and A(g) such that nejj nn ^ nnSjj = {Mj} for i = 1, 2,..., qk. We define three types of color classes for i = 1, 2,..., u and j = 1,2,..., qk. Let f g and Bj' f g be the color classes that contain the lines through Mj whose point at infinity is Ej and Fj, respectively. Let Cj ' f g be the color class that contains the lines in ne, j whose point at infinity is Ej, except the line Ej Mj, the lines in n , j whose point at infinity is Fj, except the line Fj Mj, and the lines in ng , j whose point at infinity is Gj. Hence each of Bj,' f g and Bj,' f g contains qk lines and Cj'f g contains 3qk-1 - 2 lines. Notice that for each i G {1,2,..., u}, the union of the color classes v-j = pi'0 , , Bj '1 , ,qfc Cj 'j Ke , f, g = Be, f ,g U Be , f ,g Uj=1 Ce , f, g G. Araujo-Pardo et al.: On chromatic indices of finite affine spaces 75 contains all lines whose point at infinity is E^ Ej or Gj. Each of the two sets of lines belonging to Bj f g or Bj,' f g, naturally defines a (k +1)-dimensional subspace of PG(2k, q), we denote these subspaces by nE. and nFi, respectively. For t = 0,1,..., L(qfc - 2 - 'e)/3j let'e = 4t+1, f = ¿st+2, g = 4t+s, d = 4t+4, define ^qfc+2-e as and make the qk + 2 color classes Bj,' f g, Bj,' f g and Cjf g. Finally, for each point P in the subspace £qk+1 if e =1, or in £qk if e = 2, define a new color class which contains all lines whose point at infinity is P. Clearly, the coloring is proper and it contains, by definition, the required number of color classes. Now, we prove that it is complete. Notice that each color class of type obviously intersects any other color class. In relation to the other cases we have that: • The color classes Bj ' 3 e e and Bj' 3 ,, e intersect, because both of them contain all points of the k-dimensional subspace n^3m+4 , 1. • If t = m then the color classes Bj '3 „ e and Bj '3 „ e intersect, ' ¿3t + 1 /3t + 2, ¿3t+3 (3m+1, ¿3m+2, (3m+3 because the (k - 1)-dimensional subspaces ^st+4 and ^Sm+4 are skew in hence the 2-dimensional intersection of the (k + 1)-dimensional subspaces nEi or nFi, according as j = 1 or 2, and nE, or nF,, according as j' = 1 or 2, is not a subspace of Thus Proposition 3.1 implies that their intersection contains some affine points. • The color classes Bj'3 „ „ and Gj '3 „ e intersect in both cases (3m+1 ' (3m+2 /3m + 3 ¿3t + 1 , ¿3t+2 , ¿3t+3 m = t and m = t, because the (k - 1)-dimensional subspaces ^Sm+4 and ^st+s are skew in . Again, Proposition 3.1 implies that the intersection of the k-dimensional subspaces n£3m+4,1 (which is a subspace of either the (k + 1)-dimensional subspace nEi or nFi, according as j = 1 or 2) and n^3 is an affine point. • If t = m then each pair of color classes Cj '3 e e and Cj '3 e e , ' 1 ^3t + 1 '^3t + 2 '^3t + 3 t3m + 1 '^3m + 2 '^3m + 3 7 intersects since, as previously, the (k - 1)-dimensional subspaces ^st+s and ^3m+s are skew in thus Proposition 3.1 implies that the projective closures of the k-dimensional subspaces n^3t+3, j and n^3m+3, j/ intersect each other in AG(2k, q). • Finally, we prove that each pair of classes Cj '3 e „ and Cj '3 e „ intersects. It is obvious when i = i'. Suppose that i = i', let Mj = n^3t+1, j n n^3t+2' j n n^3t+3' j and Mj/ = n^3t+1' j, n n^3t+2' j, n n^3t+3' j,. Since the points Mj and Mj, are in n^3t+4 , 1, the line MjMj, intersects in ^st+4. Take the point T = MjMj, n ^st+4 and the lines E3T and E3 T. Clearly, at least one of these lines does not intersect ¿st+s, we may assume without loss of generality, that E3T n ¿st+s = 0. By Proposition 3.1, there exist affine points Nj = n^3t+1 j n n^3t+3, j, and Nj, = n^3t+1 'j, n n£3t+3,j. Suppose that Nj G EjMj, and Nj, G E3MM». Then ¿st+1 n MjMj, = 0, hence (^st+1, MjMj,} is a (k + 1)-dimensional subspace Ek+1, which intersects in a k-dimensional subspace £fc. Obviously, £fc also contains the points Ej and E3-,. Then £fc = (^st+1, T}, and £fc n ^st+s is a single point, say U. As the lines Nj, Mj and NjMj, are in the k-dimensional subspaces n^3t+3, j and n^3t+3, j,, respectively, there exist the points Nj,Mj n ^st+s and NjMj, n ^st+s. Moreover, we have that Nj,Mj n ^st+s = NjMj, n ^st+s = U. Hence the points Nj, Mj, Nj, and Mj, are contained in a 2-dimensional subspace S2, and S2 n contains the points U, E3, Ej, and T. Consequently, S2 n is the line E3 T and it contains the point U, thus E3 T intersects the subspace ^st+s, contradiction. 76 ArsMath. Contemp. 16(2019)97-109 Thus Nj G Ejr Mj/ or N^ G Ej Mj. This implies that Nj or N^ is a common point of the color classes In consequence, the coloring is complete. and C' j , , . □ To conclude this section we prove our third main theorem. Proof of Theorem 1.3. As v = q2k, from Theorem 3.5 we get + 1 - e, k , , ^ qk - 1 _ q3k + (2 - e)q2k + (2e - 1)qk - 2 - e 3 + 2) + e q - 1 3(q - 1) 1 Vv(v - 1) (2 - e)v + 2eVV - 2 - e = 3 q - 1 + 3(q - 1) ' which proves the statement. 4 Small dimensions □ In this section, we improve on our bounds in two and three dimensions. First, we prove the exact values of achromatic and pseudoachromatic indices of finite affine planes. Due to the fact that there exist non-desarguesian affine planes, we use the notation Aq for an arbitrary affine plane of order q. For the axiomatic definition of Aq we refer to [11]. The basic combinatorial properties of Aq are the same as of AG(2, q). Theorem 4.1. Let Aq be any affine plane of order q. Then xX (Aq ) = a'(Aq ) = q +1. Proof. Let Si, S2,..., Sq+1 denote the q +1 parallell classes of lines in Aq. Two lines have a point in common if and only if they belong to distinct parallel classes. Hence, if we define a coloring ^ with q +1 colors such that a line I gets color i if and only if I G Sj then ^ is proper, so q +1 < x'(Aq). Since x'(Aq) < a'(Aq), it is enough to prove that a'(Aq) < q +1. Suppose to the contrary that ^ is a complete and proper coloring with m > q +1 color classes. As ^ is proper, each color class must be a subset of a parallel class. By the pigeonhole principle, m > q+1 implies that there exist at least two color classes that are subsets of the same parallel class. Hence they do not contain intersecting lines, contradicting to the completeness of Thus a'(Aq) < q +1, the theorem is proved. □ Theorem 4.2. Let Aq be any affine plane of order q. Then Proof. First, we prove that ^'(Aq) < (q+1)2 (q+1)2 . Suppose to the contrary that f is a com- plete coloring of Aq with (q+1)2 + 1 color classes. As Aq has q2 + q lines, this implies that f has at most q2 + q - (q+1) J + 1) color classes of cardinality greater than one. Thus, there are at least (q+1)2 + 1 - q2 + q - (q+1)2 + 1 = q + 2, if q is even, q + 3, if q is odd, 2 2 2 2 2 G. Araujo-Pardo et al.: On chromatic indices of finite affine spaces 77 color classes of size one. Hence, again by the pigeonhole principle, there are at least two color classes of size one belonging to the same parallel class. They have empty intersection, so ( is not complete. This contradiction shows that ^'(Aq ) < We go on to give a complete coloring of Aq with a point and e1, e2, Sq+1 be the lines through P. (q+1)2 2 For i (q+1)2 2 color classes. Let P be 1, 2,,..., q +1 let S, be the parallel class containing ei and denote the q — 1 lines in the set Si \ {ei} by ti.i (q+1)+i, X (q-2)(q+1)+i Then: U (Si \{ei}) = {h,l2,...,lq2-1}, and ij and ij+1 are non-parallel lines for all 1 < j < q2 -1. For better clarity, we construct q2 — 1 color classes with odd indices. Let the q +1 color classes with even indices and color class C2k consist of one line, ek, for k = 1,2,..., q + 1. Let the color class C2k—1 a2 — 1 contain the lines i2k-1 and i2k for k = 1,2,..., , finally, if q is even, let the color class Ca2—3 contain the line ia2_1, too. The coloring is complete, because color classes having even indices intersect at P, and each color class with odd index contains two non-parallel lines whose union intersects all lines of the plane. □ Our last construction gives a lower bound for the achromatic index of AG(3, q). As a'(AG(3, q)) < ^'(AG(3, q)), this can be considered as well as a lower estimate on the pseudoachromatic index of AG(3, q) and this bound is better than the general one proved in Theorem 3.4. We use the cyclic model of PG(2, q) to make the coloring. The detailed description of this model can be found in [16, Theorem 4.8 and Corollary 4.9]. We collect the most important properties of the cyclic model in the following proposition. Proposition 4.3. Let q be a prime power. Then the group Zq2+a+1 admits a perfect difference set D = {d0,d1,d2,... ,da}, that is the q2 + q integers di — dj (i = j) are all distinct modulo q2 + q +1. We may assume without loss of generality that d0 =0 and d1 = 1. The plane PG(2, q) can be represented in the following way. The points are the elements of Za2+a+1, the lines are the subsets D + j = {di + j : di e D} for j =0,1,..., q2 + q, and the incidence is the set theoretical inclusion. Theorem 4.4. The achromatic index of AG(3, q) satisfies the inequality: q(q +1)2 2 + 1 < a'(AG(3,q)). Proof. The plane at infinity in the projective closure of AG(3, q), is isomorphic to PG(2, q), hence it has a cyclic representation (described in Proposition 4.3). Let v = q2 + q +1, let the points and the lines of be P1, P2,..., Pv, and i1, i2,..., iv, respectively. We can choose the numbering such that for i = 1,2,3,..., v the line ii contains the points Pi, Pi+1 and Pi—d (where 0 = d =1 is a fixed element of the difference set D, and the subscripts are taken modulo v). 78 Ars Math. Contemp. 16 (2019) 97-109 Let A(Pj) = {nPiii,nPi,2,...,nPi,q} denote the set of the q parallel planes in AG(3, q) whose projective closures intersect in £j, and nPi,j denote the projective closure of nPij- for i = 1,3,..., v, and j = 1,2,..., q. Let Wj be a plane whose projective closure intersects in £j_d. Then the projective closure of each element of A(Pj) U A(Pj+1) intersects Wj in a line whose point at infinity is Pg, so we can choose the numbering of the elements of A(Pg) and A(Pj+1), such that nPi,j n nPi+1ij- c Wj for i = 1, 3,..., v - 2, and j = 1,2,..., q. Let ej denote the line nPij- n nPi+1ij-. We assign q +1 color classes to the pair (Pj, Pj+1) for i = 1, 3,..., v - 2. Let the color class Cg contain the lines el, e2 ..., eq. For j = 1, 2,..., q, let the color class Cj contain those lines of nP. j whose point at infinity is Pg, except the line ej, and the q parallel lines of nPi+1,j whose point at infinity is Pj+1. Finally, let the color class Cv contain all lines whose point at infinity is Pv. In this way we constructed ,v - 1 q(q + 1)2 (q + 1) —+ 1 = q(q + ) +1 color classes and each line belongs to exactly one of them, because C0i contains q lines, Cj contains 2q - 1 lines for each j = 1,2,..., q. and Cv contains q2 lines. The coloring is proper by construction. The color class Cv obviously intersects any other class. For other pairs of color classes, two major cases are distinguished when we prove the completeness. On the one hand, if i = k then we have: • Cg n C0fc = 0, because the planes Wj and W0 intersect each other; • if j > 0 then Cg n Ck = 0, because the planes Wg and nPk+1 ,j intersect each other; • if m > 0 and j > 0 then C^ n CO = 0, because the planes nPi+1jm and nPfc+1ij intersect each other. On the other hand, color classes having the same superscript also have non-empty intersection: • Cg n Cj = 0, because the planes Wg and nPi+1ij- intersect each other; • if j = k then the planes nPi j and nPi+1 0 intersect in a line f and f = ej, hence its points are not removed from nPi,j, so Cj n CO = 0. Hence the coloring is also complete, this proves the theorem. □ References [1] G. Araujo-Pardo, Gy. Kiss, C. Rubio-Montiel and A. Vazquez-Avila, On line colorings of finite projective spaces, 2018, arXiv:1702.06769 [math.CO] . [2] G. Araujo-Pardo, J. J. Montellano-Ballesteros, C. Rubio-Montiel and R. Strausz, On the pseu-doachromatic index of the complete graph II, Bol. Soc. Mat. Mex. 20 (2014), 17-28, doi: 10.1007/s40590-014-0007-9. [3] G. Araujo-Pardo, J. J. Montellano-Ballesteros, C. Rubio-Montiel and R. Strausz, On the pseu-doachromatic index of the complete graph III, Graphs Combin. 34 (2018), 277-287, doi: 10.1007/s00373-017-1872-6. [4] G. Araujo-Pardo, J. J. Montellano-Ballesteros and R. Strausz, On the pseudoachromatic index of the complete graph, J. Graph Theory 66 (2011), 89-97, doi:10.1002/jgt.20491. G. Araujo-Pardo et al.: On chromatic indices of finite affine spaces 79 [5] G. Araujo-Pardo and C. Rubio-Montiel, Pseudoachromatic and connected-pseudoachromatic indices of the complete graph, Discrete Appl. Math. 231 (2017), 60-66, doi:10.1016/j.dam. 2017.03.019. [6] G. Araujo-Pardo, C. Rubio-Montiel and A. Vazquez-Avila, Note on the Erdos-Faber-Lovasz conjecture: quasigroups and complete digraphs, Ars Combin., in press. [7] G. Araujo-Pardo and A. Vazquez-Avila, A note on Erdos-Faber-Lovasz conjecture and edge coloring of complete graphs, Ars Combin. 129 (2016), 287-298. [8] A. Beutelspacher, D. Jungnickel and S. A. Vanstone, On the chromatic index of a finite projective space, Geom. Dedicata 32 (1989), 313-318, doi:10.1007/bf00147923. [9] A. Bouchet, Indice achromatique des graphes multiparti complets et reguliers, Cahiers Centre Etudes Rech. Oper. 20 (1978), 331-340. [10] C. J. Colbourn and M. J. Colbourn, Greedy colourings of Steiner triple systems, in: A. Bar-lotti, P. V. Ceccherini and G. Tallini (eds.), Combinatorics '81, North-Holland, Amsterdam, volume 18 of Annals of Discrete Mathematics, pp. 201-207, 1983, proceedings of the International Conference on Combinatorial Geometries and their Applications held in Rome, June 7 -12, 1981. [11] P. Dembowski, Finite Geometries, Classics in Mathematics, Springer-Verlag, Berlin, 1997, doi:10.1007/978-3-642-62012-6, reprint of the 1968 original. [12] P. Erdos, Problems and results in combinatorial analysis and combinatorial analysis, in: C. St. J. A. Nash-Williams and J. Sheehan (eds.), Proceedings of the Fifth British Combinatorial Conference, Utilitas Mathematica Publishing, Winnipeg, Manitoba, volume 15 of Congressus Numerantium, pp. 169-192, 1976, held at the University of Aberdeen, Aberdeen, July 14-18, 1975. [13] P. Erdos, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25-42, doi:10.1007/bf02579174. [14] R. P. Gupta, Bounds on the chromatic and achromatic numbers of complementary graphs, in: W. T. Tutte (ed.), Recent Progress in Combinatorics, Academic Press, New York, pp. 229-235, 1969, proceedings of the Third Waterloo Conference on Combinatorics, May 1968. [15] F. Harary, S. Hedetniemi and G. Prins, An interpolation theorem for graphical homomorphisms, Portugal. Math. 26 (1967), 453-462, http://eudml.org/doc/1150 4 7. [16] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Mathematical Monographs, The Clarendon Press, New York, 2nd edition, 1998. [17] R. E. Jamison, On the edge achromatic numbers of complete graphs, Discrete Math. 74 (1989), 99-115, doi:10.1016/0012-365x(89)90202-1. [18] A. Rosa and C. J. Colbourn, Colorings of block designs, in: J. H. Dinitz and D. R. Stinson (eds.), Contemporary Design Theory, John Wiley & Sons, New York, Wiley-Interscience Series in Discrete Mathematics and Optimization, pp. 401-430, 1992. /^creative ^commor ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 81-95 https://doi.org/10.26493/1855-3974.1333.68f (Also available at http://amc-journal.eu) Pentavalent symmetric graphs of order four times an odd square-free integer* Bo Ling t School of Mathematics and Computer Sciences, Yunnan Minzu University, Kunming, Yunnan 650504, P. R. China Ben Gong Lou School ofMathematics and Statistics, Yunnan University, Kunming, Yunnan 650031, P. R. China Ci Xuan Wu School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P. R. China Received 24 February 2017, accepted 6 June 2018, published online 18 September 2018 Abstract A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. in 2011 and Pan et al. in 2013 determined all pentavalent symmetric graphs of order 4pq. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each such graph r, either the full automorphism group Aut r is isomorphic to PSL(2,p), PGL(2,p), PSL(2,p) x Z2 or PGL(2,p) x Z2, or r is isomorphic to one of 9 graphs. Keywords: Arc-transitive graph, normal quotient, automorphism group. Math. Subj. Class.: 05C25, 05E18, 20B25 *The authors are very grateful to the referee for the constructive comments and suggestions. The first author was supported by the National Natural Science Foundation of China (11701503, 11861076, 11761079), Yunnan Applied Basic Research Projects (2018FB003) and the Scientific Research Foundation Project of Yunnan Education Department (2017ZZX086). The second author was supported by the National Natural Science Foundation of China (11861076, 11231008, 11461004, 11301468). 1 Corresponding author. E-mail addresses: bolinggxu@163.com (Bo Ling), bengong188@163.com (Ben Gong Lou), wucixuan@gmail.com (Ci Xuan Wu) ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 82 Ars Math. Contemp. 16 (2019) 97-109 1 Introduction All graphs in this paper are assumed to be finite, simple and undirected. Let r be a graph and denote Vr and Ar the vertex set and arc set of r, respectively. Let G be a subgroup of the full automorphism group Aut r of r. Then r is called G-vertex-transitive and G-arc-transitive if G is transitive on Vr and Ar, respectively. An arc-transitive graph is also called a symmetric graph. It is well known that r is G-arc-transitive if and only if G is transitive on Vr and the stabilizer Ga := {g G G | a9 = a} is transitive on the neighbor set r (a) of the vertex a of r. The cubic and tetravalent graphs have been studied extensively in the literature. In recent years, attention has moved on to pentavalent symmetric graphs and a series of results have been obtained. For example, all the possibilities of vertex stabilizers of pentavalent symmetric graphs are determined in [7, 20]. Also, for distinct primes p, q and r, the classifications of pentavalent symmetric graphs of order 2pq and 2pqr are presented in [9, 19], respectively. A classification of 1-regular pentavalent graph (that is, the full automorphism group acts regularly on its arc set) of square-free order is presented in [13]. Recently, pentavalent symmetric graphs of square-free order have been completely classified in [11]. Furthermore, some classifications of pentavalent symmetric graphs of cube-free order also have been obtained in recent years. For example, the classifications of pentavalent symmetric graphs of order 12p, 4pq and 2p2 are presented in [8, 16, 5]. More recently, symmetric graphs of any prime valency which admit a soluble arc-transitive group have been classified in [14]. The main purpose of this paper is to extend the results in [8, 16] to four times an odd square-free integer case. The main result of this paper is the following theorem. Theorem 1.1. Let n be an odd square-free integer and let r be a pentavalent symmetric graph of order 4n. If n has at least three prime factors, then one of the following statements holds. (1) Aut r = PSL(2,p), PGL(2,p), PSL(2,p) x Z2 or PGL(2,p) x Z2, where p > 29 is a prime. Furthermore, the stabilizer (Aut r)a and the prime p appear in Table 5 or Table 6. (2) The triple (r, n, Aut r) lies in the following Table 1. Remark 1.2 (Remarks on Theorem 1.1). (a) The graphs in Table 1 are introduced in Example 3.2. (b) The graphs C5852 and C380 in Table 1, and the graphs in part (1) with automorphism group PSL(2,p) x Z2 or PGL(2,p) x Z2 can also be constructed from the bipartite double cover (the definition of bipartite double cover see Section 3) of a pentavalent symmetric graph of square-free order (see [11, Example 4.3 and Example 4.5] and [19, Example 3.9 and Example 3.11] for details on these graphs). 2 Preliminaries We now give some necessary preliminary results. The first one is a property of the Fitting subgroup, see [18, p. 30, Corollary]. Lemma 2.1. Let F be the Fitting subgroup of a group G. If G is soluble, then F = 1 and the centralizer CG(F) < F. B. Ling et al.: Pentavalent symmetric graphs of order four times an odd square-free integer 83 Table 1: Nine 'sporadic' pentavalent symmetric graphs of order four times an odd squarefree integer. Row r Aut r (Aut r )a Transitivity Bipartite? 1 r1 C17556 3 • 7 • 11•19 J1 D10 1-transitive No 2 r2 C17556 3 • 7 • 11•19 J1 D10 1-transitive No 3 r3 C17556 3 •7 • 11•19 J1 D10 1-transitive No 4 r4 C17556 3 •7 • 11•19 J1 D10 1-transitive No 5 5 C17556 3 •7 • 11•19 J1 D10 1-transitive No 6 C5852 7 • 11 • 19 J1 x Z2 A5 2-transitive Yes 7 1 C780 3 • 5 • 13 PSL(2,25) x Z2 F20 2-transitive No 8 2 C780 3 • 5 • 13 PSL(2, 25) x Z2 F20 2-transitive No 9 C380 3 • 5 • 13 PSL(2, 25) x Z2 F20 2-transitive Yes The maximal subgroups of PSL(2,p) are known, see [4, Section 239]. Lemma 2.2. Let T = PSL(2,p), where p > 5 is a prime. Then a maximal subgroup of T is isomorphic to one of the following groups: (1) Dp-!, where p = 5,7, 9,11; (2) Dp+i, where p = 7, 9; (3) Zp : Z(p-1)/2; (4) A4, where p = 5 or p = 3,13, 27, 37 (mod 40); (5) S4, where p = ±1 (mod 8) (6) A5, where p = ±1 (mod 5). By [2, Theorem 2], we may easily derive the maximal subgroups of PGL(2,p). Lemma 2.3. Let T = PGL(2,p) with p > 5 a prime. Then a maximal subgroup of T is isomorphic to one of the following groups: (1) Zp : Zp-i; (2) D2(p+1); (3) D2(p-i), wherep > 7; (4) S4, where p = ±3 (mod 8); (5) PSL(2,p). From [6, pp. 134-136], we can obtain the following lemma by checking the orders of nonabelian simple groups. Lemma 2.4. Let n be an odd square-free integer such that n has at least three prime factors. Let T be a nonabelian simple group of order 2® • 3j • 5 • n, where 1 < i < 11 and 0 < j < 2. Let p be the largest prime factor of n. Then T is listed in Table 2. n 84 Ars Math. Contemp. 16 (2019) 97-109 Table 2: Nonabelian simple groups of order 2® • 3j • 5 • n with 1 < i < 11 and 0 < j < 2. T |T| n M22 27 • 32 • 5 • 7 • 11 3 7 • 11 M23 27 32 • 5 • 7 • 11 • 23 7 11 • 23 J1 23 • 3 • 5 • 7•11 • 19 7 11 • 19 J2 27 • 33 • 52 • 7 3 5 • 7 Sz(32) 210 • 52 • 31 • 41 5 31 • 41 PSU(3, 4) 26 3 • 52 • 13 3 5 • 13 PSp(4, 4) 28 32 • 52 • 17 3 5 • 17 PSL(2, 25) 23 3 • 52 • 13 3 5 • 13 PSL(2, 28) 28 • 3 • 5 • 17 • 257 3 17 • 257 PSL(5, 2) 210 • 32 • 5 • 7 • 31 3 7 • 31 PSL(2, 26) 26 • 32 • 5 • 7 • 13 3 7 • 13 M23 27 • 32 • 5 • 7 • 11 • 23 3 7 • 11 • 23 M24 210 • 33 • 5 • 7 • 11 • 23 3 7 • 11 • 23 J1 23 • 3 • 5 • 7•11 • 19 3 7 • 11 • 19 PSL(2,p) p(p+12(p-1) (p > 29) Proof. If T is a sporadic simple group, by [6, p. 135-136], T = M22, M23, M24, J1 or J2. If T = An is an alternating group, since 34 does not divide |T |, we have n < 8, it then easily exclude that T = A5, A6, A7 or A8. Hence no T exists for this case. Suppose now T = X (q) is a simple group of Lie type, where X is one type of Lie groups, and q = rd is a prime power. If r > 5, as |T| has at most three 3-factors, two 5-factors and one p-factor, it easily follows from [6, p. 135] that the only possibility is T = PSL(2,p) with p > 29 (note that PSL(2,p) with 5 < p < 23 does not satisfy the condition of the lemma) or PSL(2,25), where p is the largest prime factor of n. If r < 3, as 212 and 34 do not divide |T|, then we have T = Sz(32), PSU(3,4), PSp(4,4), PSL(2,26), PSL(2, 28) or PSL(5,2). □ For a graph r and a positive integer s, an s-arc of r is a sequence a0, a1;..., as of vertices such that a® are adjacent for 1 < i < s and ai-1 = ai+1 for 1 < i < s - 1. In particular, a 1-arc is just an arc. Then r is called (G, s)-arc-transitive with G < Aut r if G is transitive on the set of s-arcs of r. A (G, s)-arc-transitive graph is called (G, s)-transitive if it is not (G, s + 1)-arc-transitive. In particular, a graph r is simply called s-transitive if it is (Aut r, s)-transitive. Let F20 denote the Frobenius group of order 20. The following lemma determines the stabilizers of pentavalent symmetric graphs, refer to [7, 20]. Lemma 2.5. Let r be a pentavalent (G, s)-transitive graph, where G < Aut r and s > 1. Let a G Vr. Then one of the following holds. (a) If Ga is soluble, then s < 3 and |Ga| I 80. Further, the pair (s,Ga) lies in the B. Ling et al.: Pentavalent symmetric graphs of order four times an odd square-free integer 85 following table. s Ga 1 Z5, D10, D2o 2 F2o, F20 x Z2 3 F20 x Z4 (b) If Ga is insoluble, then 2 < s < 5, and \Ga\ | 29 • 32 • 5. Further, the pair (s, Ga) lies in the following table. s Ga | Ga | 2 3 4 5 A5, S5 A4 x A5, (A4 x A5) : Z2, S4 x S5 ASL(2,4), AGL(2,4), ASL(2,4), ArL(2,4) Z6 : rL(2,4) 60, 120 720, 1440, 2880 960, 1920, 2880, 5760 23040 A typical method for studying vertex-transitive graphs is taking normal quotients. Let r be a G-vertex-transitive graph, where G < Aut r. Suppose that G has a normal subgroup N which is intransitive on V r. Let V rN be the set of N-orbits on Vr. The normal quotient graph rN of r induced by N is defined as the graph with vertex set VrN, and B is adjacent to C in rN if and only if there exist vertices ft G B and 7 G C such that ft is adjacent to 7 in r. In particular, if val(r) = val(rN), then r is called a normal cover of rN. A graph r is called G-locally primitive if, for each a G Vr, the stabilizer Ga acts primitively on r(a). Obviously, a pentavalent symmetric graph is locally primitive. The following theorem gives a basic method for studying vertex-transitive locally primitive graphs, see [17, Theorem 4.1] and [12, Lemma 2.5]. Theorem 2.6. Let r be a G-vertex-transitive locally primitive graph, where G < Aut r, and let N < G have at least three orbits on Vr. Then the following statements hold. (i) N is semi-regular on Vr, G/N < Aut rN, and r is a normal cover of rN; (ii) Ga = (G/N)Y, where a G Vr and 7 G VrN; (iii) r is (G, s)-transitive if and only if rN is (G/N, s)-transitive, where 1 < s < 5 or s = 7. For reduction, we need some information of pentavalent symmetric graphs of order 4pq, stated in the following lemma, see [8, Theorem 4.1] and [16, Theorem 3.1]. Lemma 2.7. Let r be a pentavalent symmetric graph of order 4pq, where q > p > 3 are primes. Then the pair (Aut r, (Aut r )a) lies in the following Table 3, where a G V r. Remark 2.8 (Remarks on Lemma 2.7). (a) Suppose that r is one of the graphs in Lemma 2.7 and M is an arc-transitive subgroup of Aut r. Then M is insoluble (for convenience, we prove this conclusion in Lemma 4.4 and we remark that Lemma 4.4 is independent where it is used). (b) By Magma [1], the graphs C^ and Cf32 in [8, Theorem 4.1] are isomorphic, Aut(Cf32) = PGL(2,11) x Z2. 86 Ars Math. Contemp. 16 (2019) 97-109 Table 3: Pentavalent symmetric graphs of order 4pq. r C60 r1 C132 2 < i < 4 C132, 2 — ' — C132 (2) C574 C4IO8 (p, q) Aut r (Aut r) (3, 5) A5 X D10 D10 (3,11) PSL(2,11) x Z2 D10 (3,11) PGL(2,11) D10 (3,11) PGL(2,11) x Z2 D20 (7,41) PSL(2,41) x Z2 A5 (13, 79) PSL(2, 79) A5 a The final lemma of this section gives some information about the pentavalent symmetric graphs of square-free order, refer to [19, Theorem 1.1] and [11, Theorem 1.1]. Lemma 2.9. Let r be a pentavalent symmetric graph of order 2n, where n is an odd square-free integer and has at least three prime factors. Then one of the following statements holds. (1) Aut r is soluble and Aut r = D2n : Z5. (2) Aut r = PSL(2,p) or PGL(2,p), where p > 5 is a prime. (3) The triple (r, 2n, Aut r) lies in the following Table 4. Table 4: Two 'sporadic' pentavalent symmetric graphs. r 2n Aut r (Aut r)a ~c3io 390 PSL(2,25) F20 C2926 2926 Ji A5 3 Some examples In this section, we give some examples of pentavalent symmetric graphs of order 4n with n an odd square-free integer. In order to construct our graphs we first introduce the definition of a coset graph. Let G be a finite group and let H be a core-free subgroup of G. Let t g G and t2 G H. Define the coset graph Cos(G, H, t) of G with respect to H as the graph with vertex set [G : H] such that Hx, Hy are adjacent if and only if yx-1 G HtH. The following lemma about coset graphs is well known and the proof of the lemma follows from the definition of coset graphs. Lemma 3.1. Using the notation as above, the coset graph r = Cos(G, H, t) is G-arc-transitive graph and (1) val r = |H : H n HT (2) r is connected if and only if (H, t) = G. B. Ling et al.: Pentavalent symmetric graphs of order four times an odd square-free integer 87 Conversely, each G-arc-transitive graph E is isomorphic to the coset graph Cos(G, Gv,t), where t g NG(Gvw) is a 2-element such that t2 G Gv, and v G VE, w G E(v). We next introduce the definition of the bipartite double cover of a graph. Let r be a graph with vertex set Vr. The standard double cover of r is defined as the undirected bipartite graph r with biparts V0 and V, where V = {(v, i) | v G Vr}, such that two vertices (x, 0) and (y, 1) are adjacent if and only if x, y are adjacent in r. It is easily shown that thestandard double cover can be represented as a direct product: r = fx K2. Furthermore, r is connected if and only if r is connected and non-bipartite. For a given small permutation group X, we may determine all graphs which admit X as an arc-transitive automorphism group by using Magma [1]. It is then easy to have the following result. Example 3.2. (1) There is a unique pentavalent symmetric graph of order 5852 which admits Ji x Z2 as an arc-transitive automorphism group; and its full automorphism group is J1 x Z2. This graph is denoted by C5832 which satisfies the conditions in Row 6 of Table 1. (2) There are five pentavalent symmetric graphs of order 17556 admitting J1 as an arc-transitive automorphism group; and their full automorphism group are all isomorphic to J1. These five graphs are denoted by Cj7556 which satisfy the conditions in Row 1 to Row 5 of Table 1, where 1 < i < 5. (3) There are three pentavalent symmetric graphs of order 780 which admit PSL(2, 25) x Z2 as an arc-transitive automorphism group; and their full automorphism group are all isomorphic to PSL(2,25) x Z2. These three graphs are denoted by C78o which satisfy the conditions in Row 7 to Row 9 of Table 1, where 1 < j < 3. Remark 3.3 (Remarks on Example 3.2). (a) Let r be a pentavalent symmetric graph of order 4n with n an odd square-free integer and having at least three prime factors. Then the graphs appearing in Example 3.2 are the only sporadic graphs of such r. In fact, let A = Aut r. If A is insoluble and has no nontrivial soluble normal subgroup, then Lemma 4.2 shows that Cj7556 with 1 < i < 5 are the only sporadic graphs. If A is insoluble and has a soluble minimal normal subgroup N = Z2, then Lemma 4.3 shows that C5832 and Cj80 with 1 < j < 3 are the only sporadic graphs. If A is soluble or has a soluble minimal normal subgroup N = Zr with r > 2, then Lemma 4.1 and Lemma 4.6 show that no such exists. (b) Since both C2926 and C390 are non-bipartite, the bipartite double cover of both C2926 and C390 is connected pentavalent symmetric graph of order 4n. In fact, the graph C5832 is isomorphic to the bipartite double cover of C2926 and the graph C380 is isomorphic to the bipartite double cover of C390. Example 3.4. Let p be a prime such that p = 49, 79, 81,111 (mod 160) and let A = PSL(2,p). Then by Lemma 2.2, A has a subgroup H = A5. Let K < H with K = A4. Then NA(K) = K : (t} = S4, where t g A — H is an involution. Let r = Cos(A, H, HtH). Then r is a connected pentavalent symmetric graph. 88 Ars Math. Contemp. 16 (2019) 97-109 Example 3.5. Let p be a prime such that p = 9, 39, 41, 71 (mod 80) and let A = PGL(2,p). Then by Lemma 2.2 and Lemma 2.3, A has a subgroup H = A5. Let K < H with K = A4. Then NA(K) = K : (t} = S4 is a maximal subgroup of A, where t g A — H is an involution, and so (H, t} = A. Let r = Cos(A, H, HtH). Then r is a connected pentavalent symmetric graph. Example 3.6. Let p be a prime such that p = 9, 39, 41, 71 (mod 80) and let A = PSL(2,p) x Z2 = T x (z}, where T = PSL(2,p) and (z} = Z2. Then T has a subgroup H = A5. Let K < H with K = A4. Then Na(K) = K : (t} x (z} = S4 x Z2, where t g T — H is an involution. Let r = Cos(A, H, HtzH). Then r is a connected pentavalent symmetric graph. Example 3.7. Let p be a prime such that p = 11,19, 21, 29 (mod 40) and let A = PGL(2,p) x Z2 = T x (z}, where T = PGL(2,p) and (z} = Z2. Then T has a subgroup H = A5. Let K < H with K = A4. Then Na(K) = K : (t} x (z} = S4 x Z2, where t g T — H is an involution. Let r = Cos(A, H, HtzH). Then r is a connected pentavalent symmetric graph. 4 Proof of Theorem 1.1 Let n be an odd square-free integer and n has at least three prime factors. Let r be a pentavalent symmetric graph of order 4n. Set A = Aut r. By Lemma 2.5, | Aa | | 29 • 32 • 5, and hence |A| | 211 • 32 • 5 • n. Assume that n = p1p2 • • • ps, where s > 3 and pj's are distinct primes. Lemma 4.1. The group A is insoluble. Proof. Suppose to the contrary that A is soluble. Let F be the Fitting subgroup of A. By Lemma 2.1, F = 1 and CA(F) < F. Further, F = O2(A) x OPl (A) x OP2 (A) x • • • x Ops (A), where 02(A), Op1 (A), Op2 (A), ..., Ops (A) denote the largest normal 2-, pi-, p2-,..., ps-subgroups of A, respectively. For each p4 g {p1,p2,... ,ps}, OPi(A) has at least three orbits on Vr, by Theorem 2.6, OPi(A) is semi-regular on Vr. Therefore, F is semi-regular on Vr and so |F| divides |Vr| = 4n. Since n = p1p2 • • • ps, we have OPi (A) < ZPi. This argument also proves O2(A) < Z4 or Z2. If O2(A) = Z4 or Z2, then by Theorem 2.6, the normal quotient graph rO2(A) is a pentavalent symmetric graph of odd order, which is a contradiction. Thus, O2(A) < Z2, F = Zm, where m | 2n. It implies that CA(F) > F, and so Ca(F ) = F. If F has at least three orbits on Vr, then, by Theorem 2.6, rF is A/F-arc-transitive. Since A/F = A/Ca(f) < Aut(F) is abelian, we have (A/F)s = 1, where 6 g VrF, which is a contradiction. B. Ling et al.: Pentavalent symmetric graphs of order four times an odd square-free integer 89 Thus, F has at most two orbits on Vr. If F is transitive on Vr, then F is regular on Vr, a contradiction with F = Zm, where m | 2n. Hence F has two orbits on Vr and F = Z2n. Let K = OP3 (A) x OP4 (A) x • • • x OPs (A). Then K = ZP3P4...Ps. Since K< A has 4pip2 orbits on Vr, by Theorem 2.6(i), rK is an A/K-arc-transitive pentavalent graph of order 4p1p2, and hence rK satisfies the conditions in Table 3. Since A/K is soluble, by Remark 2.8, a contradiction occurs. Hence A is insoluble. This completes the proof of the Lemma. □ We now consider the case where A is insoluble and has no nontrivial soluble normal subgroup. Lemma 4.2. Assume that A is insoluble and has no nontrivial soluble normal subgroup. Then Aut r = J1, PSL(2,p) or PGL(2,p) with p > 29. Further, if Aut r = J1, then r = q7556 satisfies the conditions in Row 1 to Row 5 of Table 1 of Theorem 1.1, where 1 < i < 5. If Aut r = PSL(2,p) or PGL(2,p), then r satisfies the conditions in Table 5. Table 5: Aut r is almost simple. Aut r (Aut r )a r Remark PSL(2, p) A5 Example 3.4 p = 49, 79, 81,111 (mod 160) PGL(2,p) A5 Example 3.5 p = 9, 39,41, 71 (mod 80) PSL(2, p) Dio p = 9, 39,41, 71 (mod 80) PGL(2,p) Dio p = 11,19, 21, 29 (mod 40) PSL(2, p) D20 p = 49, 79, 81,111 (mod 160) PGL(2,p) D20 p = 9, 39,41, 71 (mod 80) Proof. Let N be the socle of A. Then N is insoluble and 4 divides |NIf N has more than three orbits on Vr, then by Theorem 2.6, rN is a pentavalent symmetric graph of odd order, a contradiction. Hence, N has at most two orbits on Vr, so 2n divides |N|. Assume that A has at least two minimal normal subgroups Ni and N2. Then by a similar argument as above, we have that 2n divides both | N11 and | N21. Hence 4n2 divides |A| = 211 • 32 • 5 • n, and so n divides 29 • 32 • 5. It implies that n = 3 • 5, a contradiction with n having at least three prime factors. So A has a unique minimal normal subgroup and we may write N = Sd, where S is a nonabelian simple group and d > 1. Sinceps > 5, ps divides |N| andp?s does not divide |N| as |A| | 211 • 32 • 5 • p1p2 • • • ps, we conclude that d = 1 and N = S is a nonabelian simple group. Hence A is almost simple with socle S. If Sa = 1, then S acts regularly on Vr. Hence S is a non-abelian simple group such that |S| = 4n. By checking the orders of nonabelian simple groups (see [6, pp. 135-136] for example), we have that S = PSL(2,p) and so A < Aut(S) = PGL(2,p), which is impossible as A is transitive on Ar, |A|< 2|S| and ^r| = 5|S|. Hence Sa = 1. Since r is connected and S < A, we have 1 = S^(a) < A^(a), it follows that 5 | |Sa|, we thus have 10 • p1p2 • • • ps divides |S|. 90 Ars Math. Contemp. 16 (2019) 97-109 Thus, soc(A) = S is a nonabelian simple group such that |S| | 211 • 32 • 5 • n and 10 • n | |S|. Hence the triple (S, |S|, n) lies in Table 2 of Lemma 2.4. We will analyse all the candidates one by one in the following. Assume (S,n) = (J1, 3 • 7 • 11 • 19). Then |Vr| = 17556 and A = J1 as Out(J1) = 1. It then follows from Example 3.2 that r = Ci7556 satisfies the conditions in Row 1 to Row 5 of Table 1 of Theorem 1.1, where 1 < i < 5. Assume (S,n) = (Sz(32), 5 • 31 • 41). Since Out(Sz(32)) = Z5 (see Atlas [3] for example), A = Sz(32) or Sz(32).Z5, so |Aa| = -n = 1280 or 6400, which is not possible by Lemma 2.5. Similarly, for the case (S, n) = (PSL(5,2), 3 • 7 • 31), then A = PSL(5,2) or PSL(5, 2).Z2 as Out(PSL(5, 2)) = Z2. Thus, |Aa| = JA = 3840 or 7680, which is impossible by Lemma 2.5. For the case where (S, n) = (PSL(2, 28), 3 • 17 • 257), since A = PSL(2,28).O, where O < Out(PSL(2, 28)) = Z8, we have |Aa| = Jn = 2k • 5, where 6 < k < 9, which is also impossible by Lemma 2.5. For the case where (S, n) = (PSU(3,4), 3 • 5 • 13), since A = PSU(3,4).O, where O < Out(PSU(3,4)) = Z4, we have |Aa| = jn = 2k • 5, where 4 < k < 6, which is impossible by Lemma 2.5. Assume (S, n) = (PSp(4,4), 3 • 5 • 17). Since S < A < Aut(S) = PSp(4,4).Z4, we have |Aa| = -Jn = 960, 1920 or 3840. If |Aa| = 960 or 1920, then by Lemma 2.5, Aa ^ ASL(2,4) or A£L(2,4). However, by Atlas [3], PSp(4,4) has no subgroup isomorphic to ASL(2,4) and PSp(4,4)Z has no subgroup isomorphic to A£L(2,4). If |Aa| = 3840, then also by Lemma 2.5, a contradiction occurs. Assume (S,n) = (PSL(2,26), 3 • 7 • 13). Recall that S has at most two orbits on Vr, |Sa| = 4J = 240 or 2n = 480. However, by Lemma 2.2, PSL(2, 26) has no maximal subgroup with order a multiple of 240, a contradiction occurs. Similarly, for the case (S, n) = (J2,3 • 5 • 7), then |Sa| = 4§ = 2880 or 2n = 5760. By Atlas [3], J2 has no maximal subgroup with order a multiple of 2880, a contradiction also occurs. Assume S = M23. Then n = 3 • 7 • 11 • 23 or 7 • 11 • 23, and as Out(M23) = 1, we have A = S and |Aa| = ^r = 480 or 1440. By Lemma 2.5, it is impossible for the case |Aa| = 480. For the latter case, by a direct computation using Magma [1], no graph r exists. If (S, n) = (M22,7 • 11 • 23), as Out(M22) = Z2, we have A = M22 or M22.Z2, so | Aa | = = 480 or 960, a computation by MAGMA [1] shows that no graph r exists. Similarly, we can exclude the case where (S, n) = (PSL(2, 25), 3 • 5 • 13) by MAGMA [1]. Assume (S, n) = (M24, 3 • 7 • 11 • 23) or (J1, 3 • 7 • 11 • 19). Since Out(M24) = Out(J1) = 1, we always have A = S. Hence |Aa | = -Jn = 11520 or 10. A computation by MAGMA [1] also shows that no graph r exists. Finally, assume S = PSL(2,p) with p > 29 a prime. Then A = PSL(2,p) or PGL(2,p). By Lemma 2.2, Lemma 2.3 and Lemma 2.5, we have Aa = Z5, D1o, D2o or A5. If Aa = Z5, then r is an arc-regular pentavalent graph of order four times an odd square-free integer. However, by [15, Theorem 1.1], no such r exists. Hence Aa = D10, D20 or A5. If Aa = A5, then by Lemma 2.2 and Lemma 2.3, we have p = ±1 (mod 5). Since |A : Aa| = 4n, we have |A| is divisible by 16, but not by 32. Since |A| = | PSL(2,p)| = p(p-12(p+1) or | PGL(2,p)| = p(p - 1)(p + 1), we havep = ±15 (mod 32) for A = PSL(2,p) or p = ±7 (mod 16) for A = PGL(2,p). Since p = ±1 (mod 5), we have p = 49,79,81,111 (mod 160) for A = PSL(2,p) or p = 9, 39,41,71 (mod 80) for A = PGL(2,p). These graphs are constructed in Example 3.4 and Example 3.5. Similarly, if Aa = D10 or D20, then p satisfies the condition in Table 5. This completes the proof of the Lemma. □ B. Ling et al.: Pentavalent symmetric graphs of order four times an odd square-free integer 91 We next assume that A has a nontrivial soluble normal subgroup. Let N be a soluble minimal normal subgroup of A. Then there exists a prime r | 4n such that N = ZJ?. Further, N has at least three orbits on Vr. It follows from Theorem 2.6 that N is semi-regular on Vr, and so |N| = |Zr |d | |Vr| = 4n. If d > 2, then (r, d) = (2,2). It follows that rN is an arc-transitive graph of odd order, a contradiction. Hence d = 1, N = Zr. The next lemma consider the case where r = 2. Lemma 4.3. Assume that A is insoluble and has a soluble minimal normal subgroup N = Z2. Then one of the following statements holds: (1) Aut r = PSL(2,p) x Z2 or PGL(2,p) x Z2, where p > 29 is a prime. Furthermore, r satisfies the conditions in Table 6. (2) Aut r = PSL(2,25) x Z2 and r is isomorphic to Qso in Table 1, where 1 < i < 3. (3) Aut r = Ji x Z2 and r is isomorphic to C5852 in Table 1. Table 6: Aut r has a normal subgroup isomorphic to Z2. Aut r (Aut r )a r Remark PSL(2,p) x Z2 A5 Example 3.6 p = 9, 39,41, 71 (mod 80) PGL(2, p) x Z2 A5 Example 3.7 p = 11,19, 21, 29 (mod 40) PSL(2,p) x Z2 D10 p = 11,19, 21, 29 (mod 40) PSL(2,p) x Z2 D20 p = 9, 39,41, 71 (mod 80) PGL(2, p) x Z2 D20 p = 11,19, 21, 29 (mod 40) Proof. Since N has more than three orbits on Vr, then by Theorem 2.6, rN is an A/N-arc-transitive pentavalent graph of order n = 2n. It follows that rN is isomorphic to one of the graphs in Lemma 2.9. Since A/N < Aut rN and A/N is insoluble, we have that Aut rN is insoluble and so Aut rN = PSL(2,p), PGL(2,p), PSL(2, 25) or J1. Let A := Aut r. Suppose that A = PSL(2,p) or PGL(2,p). Since A/N is insoluble, by Lemma 2.2 and Lemma 2.3, A/N is isomorphic to A5, PSL(2,p) or PGL(2,p). If A/N = A5, then since rN is an A/N-arc-transitive pentavalent graph of order n = 2n, we have 2n • 5 | | A51. It implies that n divides 6, a contradiction with n having at least three odd prime factors. Thus, A/N is isomorphic to PSL(2,p) or PGL(2,p). Therefore, A = N. PSL(2,p) or N. PGL(2,p), that is, A = PSL(2,p) x Z2, SL(2,p), PGL(2,p) x Z2 or SL(2,p).Z2. Assume first that A = SL(2,p). Note that SL(2,p) has a unique central involution. Then by Lemma 2.5, Aa = Z5. It follows that |Vr| = |A : Aa| is divisible by 8 as | SL(2,p)| is divisible by 8, a contradiction. Assume next that A = SL(2,p).Z2. Then A contains a normal subgroup H isomorphic to SL(2,p). Since 8 | |H|, we have Ha = 1. By Theorem 2.6, H has at most two orbits on Vr and so 144 1 2. If H is transitive on Vr, then H \Ha\ I is arc-transitive. A similar argument with the case A = SL(2,p), a contradiction occurs. Therefore, H has two orbits on Vr and so Ha = Aa. Since H has a unique central involution, by Lemma 2.5, Aa = Z5, it follows that |Vr| = |A : Aa| is divisible by 16, a contradiction. Therefore, A = PSL(2,p) x Z2 or PGL(2,p) x Z2 in this case. By a similar 92 Ars Math. Contemp. 16 (2019) 97-109 argumentas for the case A = PSL(2,p) (the last paragraph in the proof of Lemma 4.2), we have that r satisfies the condition in Table 6. Note that since 16 divides | PGL(2, p) x Z21 and |A : Aa| = 4n, we have (A, Aa) = (PGL(2,p) x Z2, D10). Suppose that A = PSL(2,25). Since rN is A/N-arc-transitive, we have that 5 • 390 I |A/N|. By checking the maximal subgroup of PSL(2, 25) (see Atlas [3] for example), we have that A/N = A = PSL(2,25). It follows that A = SL(2, 25) or PSL(2, 25) x Z2. If A = PSL(2,25) x Z2, then by Example 3.2, r = C780 in Table 1, where 1 < i < 3. If A = SL(2,25), then by Magma [1], no graph r exists. Suppose that A = J1. Similarly, since rN is A/N-arc-transitive, we have that 5 • 2926 | | A/N|. By checking the maximal subgroup of J1 (see Atlas [3] for example), we have that A/N = A = J1. Since the Schur multiplier of J1 is Z1, A = N.J1 = J1 x Z2. By Example 3.2, r = C5852 in Table 1. □ Finally, suppose that r > 2. We first prove the following lemma. Lemma 4.4. Let E be a graph. Assume that E is isomorphic to one of the graphs appearing in Lemma 2.7, in Lemma 4.2 or in Lemma 4.3. If M is an arc-transitive subgroup of Aut E, then M contains the derived subgroup of Aut E. Proof. Let E be a graph and isomorphic to one of the graphs appearing in Lemma 2.7, in Lemma 4.2 or in Lemma 4.3. Let M be an arc-transitive subgroup of B = Aut E. Then B = MBaß, where (a, ß) G AE. In particular, m := |B : M| divides |Baß|. Assume first that E is isomorphic to one of the graphs appearing in Lemma 2.7. Then, in the first three rows of Table 3, we have that M has index at most two, and for the fourth row M has index at most four, so in particular, M contains B'. For the last two rows, we have that m | 12. Since there is no faithful representation of B in degree m for 2 < m < 12, we have I < m < 2 and so M also contains B'. Now assume that E is isomorphic to one of the graphs appearing in Lemma 4.2 or in Lemma 4.3. Then B is isomorphic to one of the groups PSL(2,p), PGL(2,p), PSL(2,p) x Z2, PGL(2,p) x Z2, J1, J1 x Z2 or PSL(2, 25) x Z2 withp > 29. If B = J1, then M has index at most two. If B = J1 x Z2, then M has index at most 12. If B = PSL(2, 25) x Z2, then M has index at most four. For these three cases, by a similar argument as above, we also have M contains B'. If B = PSL(2,p), then sincep | n and 20n | |M|, by Lemma 2.2, M < Zp : Zp-i or M = B = PSL(2,p). If M < Zp : Zp-i, then M = Zp : Z; for some II ■. Thus, M has a normal subgroup, say S = Zp, which has more than three orbits on VE. It then follows from Theorem 2.6 that the normal quotient graph ES is M/S-arc-transitive, a contradiction occurs as M/S = Z; is cyclic. Hence, M < Zp : Zp-i and so M = B' = PSL(2,p). If B = PGL(2,p), then since 20n | |M|, by Lemma 2.3, M < Zp : Zp-1, M < PSL(2,p) or M = B = PGL(2,p). With a similar argument, we can conclude that M > B' = PSL(2,p). Similarly, we can further show that M > B' = PSL(2,p) for the case B = PSL(2,p) x Z2 or PGL(2,p) x Z2. □ Now assume that A has a soluble minimal normal subgroup N = Zr for r > 2. Lemma 4.5. Assume that A has a soluble minimal normal subgroup N = Zr for r > 2. Then the normal quotient rN is not isomorphic to any graph appearing in Lemma 2.7, Lemma 4.2 or Lemma 4.3. B. Ling et al.: Pentavalent symmetric graphs of order four times an odd square-free integer 93 Proof. Suppose to the contrary that rN is isomorphic to one of the graphs appearing in Lemma 2.7, Lemma 4.2 or Lemma 4.3. Let M/N = (Aut rN )',andlet Q := {PSL(2,p), Ji, PSL(2, 25), A5 }. By checking the graphs appearing in Lemma 2.7, in Lemma 4.2 or in Lemma 4.3, we have that Aut rN is isomorphic to one of the groups PSL(2, p), PGL(2, p), PSL(2,p) x Z2, PGL(2,p) x Z2, J1, J1 x Z2, PSL(2, 25) x Z2 or A5 x D10. Thus, m/n is isomorphic to one of the groups in Q. Since the order of the Schur multiplier of a group in Q is less than or equal to 2 (see [10, Theorem 7.1.1] for PSL(2,p) and Atlas [3] for the others) and r > 2, we have that M' G Q. By Theorem 2.6, A/N < Aut rN is transitive on ArN. It follows from Lemma 4.4 that A/N contains the derived subgroup of Aut Tn , that is, M/N < A/N. Since M/N < Aut rN, we have M/N < A/N. Therefore, M' char M < A, it implies that M' 2. Proof. Suppose that, on the contrary, A has a soluble minimal normal subgroup N = Zr with r > 2. We prove the lemma by induction on the order of r. Assume first that n = pqt has three prime factors. (Note that, by Table 3, the conclusion of Lemma 4.6 does not hold for n = pq.) Without loss of generality, we may assume that r = t. Then rN is a pentavalent symmetric graph of order 4pq. By Lemma 2.7, rN is isomorphic to one of the graphs in Table 3, which contradicts to Lemma 4.5. Assume next that n has at least four prime factors. Note that Aut rN is insoluble. If Aut rN has no nontrivial soluble normal subgroup, then rN is isomorphic to one of the 94 Ars Math. Contemp. 16 (2019) 97-109 graphs in Lemma 4.2, which contradicts to Lemma 4.5. If Aut rN has a soluble minimal normal subgroup NV, then we can also conclude that NV = Zf with f a prime. If f > 2, then by induction, no such rN exists, a contradiction. If f = 2, then rN is isomorphic to one of the graphs appearing in Lemma 4.3, which also contradicts to Lemma 4.5. This completes the proof of the Lemma. □ References [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235-265, doi:10.1006/jsco.1996.0125. [2] P. J. Cameron, G. R. Omidi and B. Tayfeh-Rezaie, 3-designs from PGL(2,q), Electron. J. Combin. 13 (2006), #R50, http://www.combinatorics.org/ojs/index.php/ eljc/article/view/v13i1r50. [3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985. [4] L. E. Dickson, Linear Groups: With an Exposition of the Galois Field Theory, Dover Publications, New York, 1958. [5] Y.-Q. Feng, J.-X. Zhou and Y.-T. Li, Pentavalent symmetric graphs of order twice a prime power, Discrete Math. 339 (2016), 2640-2651, doi:10.1016/j.disc.2016.05.008. [6] D. Gorenstein, Finite Simple Groups, University Series in Mathematics, Plenum Publishing, New York, 1982. [7] S.-T. Guo and Y.-Q. Feng, A note on pentavalent s-transitive graphs, Discrete Math. 312 (2012), 2214-2216, doi:10.1016/j.disc.2012.04.015. [8] S.-T. Guo, J.-X. Zhou and Y.-Q. Feng, Pentavalent symmetric graphs of order 12p, Electron. J. Combin. 18 (2011), #P233, http://www.combinatorics.org/ojs/index.php/ eljc/article/view/v18i1p2 3 3. [9] X.-H. Hua, Y.-Q. Feng and J. Lee, Pentavalent symmetric graphs of order 2pq, Discrete Math. 311 (2011), 2259-2267, doi:10.1016/j.disc.2011.07.007. [10] G. Karpilovsky, The Schur Multiplier, volume 2 of London Mathematical Society Monographs New Series, The Clarendon Press, Oxford, 1987. [11] C. H. Li, Z. P. Lu and G. Wang, Arc-transitive graphs of square-free order and small valency, Discrete Math 339 (2016), 2907-2918, doi:10.1016/j.disc.2016.06.002. [12] C. H. Li and J. Pan, Finite 2-arc-transitive abelian Cayley graphs, European J. Combin. 29 (2008), 148-158, doi:10.1016/j.ejc.2006.12.001. [13] Y. Li and Y.-Q. Feng, Pentavalent one-regular graphs of square-free order, Algebra Colloq. 17 (2010), 515-524, doi:10.1142/s1005386710000490. [14] J. Pan, B. Ling and S. Ding, On prime-valent symmetric graphs of square-free order, Ars Math. Contemp. 15 (2018), 53-65, doi:10.26493/1855-3974.1161.3b9. [15] J. Pan and Y. Liu, There exist no arc-regular prime-valent graphs of order four times an odd square-free integer, Discrete Math. 313 (2013), 2575-2581, doi:10.1016/j.disc.2013.08.009. [16] J. Pan, B. Lou and C. Liu, Arc-transitive pentavalent graphs of order 4pq, Electron. J. Combin. 20 (2013), #P36, http://www.combinatorics.org/ojs/index.php/eljc/ article/view/v2 0i1p3 6. [17] C. E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. LondonMath. Soc. 47 (1993), 227-239, doi:10.1112/ jlms/s2-47.2.227. B. Ling et al.: Pentavalent symmetric graphs of order four times an odd square-free integer 95 [18] M. Suzuki, Group Theory II, volume 248 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1986, translated from the Japanese. [19] D.-W. Yang, Y.-Q. Feng and J.-L. Du, Pentavalent symmetric graphs of order 2pqr, Discrete Math. 339 (2016), 522-532, doi:10.1016/j.disc.2015.09.009. [20] J.-X. Zhou and Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010), 1725-1732, doi:10.1016/j.disc.2009.11.019. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 97-109 https://doi.org/10.26493/1855-3974.1350.990 (Also available at http://amc-journal.eu) The pairing strategies of the 9-in-a-row game Lajos Gyorffy *, Geza Makay, Andras Pluhar University of Szeged, 6720 Dugonics ter 13., Szeged, Hungary Received 16 March 2017, accepted 15 January 2018, published online 18 September 2018 One of the most useful strategies for proving Breaker's win in Maker-Breaker Positional Games is to find a pairing strategy. In some cases there are no pairing strategies at all, in some cases there are unique or almost unique strategies. For the k-in-a-row game, the case k = 9 is the smallest (sharp) for which there exists a Breaker winning pairing (paving) strategy. One pairing strategy for this game was given by Hales and Jewett. In this paper we show that there are other winning pairings for the 9-in-a-row game, all have a very symmetric torus structure. While describing these symmetries we prove that there are only a finite number of non-isomorphic pairings for the game (around 200 thousand), which can be also listed up by a computer program. In addition, we prove that there are no "irregular", non-symmetric pairings. At the end of the paper we also show a pairing strategy for a variant of the 3-dimensional k-in-a-row game. Keywords: Positional games, k-in-a-row game, pairing strategies, symmetries. Math. Subj. Class.: 05C65, 05C15 1 Introduction 5-in-a-row is one of the most well known positional games, and its study inspired several deep results in this field. For a very thorough introduction of these, see Beck [2, 3]. In the classical version two players take the squares of a gridpaper (integer lattice), alternately, and the first who achieves five in a row, i.e. five consecutive squares in a vertical, horizontal or diagonal direction, wins the game. John Nash [4] invented the "strategy stealing" argument showing that in these type of games the first player either wins the game or it is a draw. It explains the notion of the so-called Maker-Breaker (M-B) version of a game; in these, Maker wins by achieving the original goal, while Breaker wins by preventing Maker * Corresponding author. E-mail addresses: lgyorffy@math.u-szeged.hu (Lajos Gyorffy), makayg@math.u-szeged.hu (Geza Makay), pluhar@inf.u-szeged.hu (Andras Pluhar) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 98 ArsMath. Contemp. 16(2019)97-109 to doing so. There is a connection between the normal (Maker-Maker) and the M-B versions of a game: If the first player wins the normal game, she wins the M-B one, as well. If Breaker wins the M-B game, then the second player can draw the normal game. However, the reverse statements are not true, see the Tic-Tac-Toe game. In summary, the M-B version is easier for Maker because she does not need to act Breaker's moves. Allis et al. [1] solved the 5-in-a-row game for the 19 x 19 and 15 x 15 boards: the first player wins. However, the case of infinite board is still open (in the normal version). It is natural to ask then what happens in the k-in-a-row game, where the winning condition is to get k consecutive squares in a row. The first result in that direction is due to C. Shannon and H. Pollak [4] who showed that Breaker wins the 9-in-a-row. Later A. Hales and R. Jew-ett even gave a winning pairing strategy for Breaker. A. Brouwer, under the pseudonym T. G. L. Zetters published in [11] (as a solution to a problem by Guy and Selfridge [6]) that Breaker wins the 8-in-a-row on the infinite board. The cases k = 6,7 are still open, although it is widely believed that those are both draws. (Of course for k < 4 Maker wins easily.) On the other hand, it was shown that there are no pairing strategies for k < 8, see [5, 10]. The concept of pairing strategies were useful for other games, the best known are the Harary games. Here Maker's goal is to get a given polyomino on the infinite board; most cases solved by A. Blass, see [4]. The notorious case of "Snaky" is still unsolved, although there are partial results for it. Csernenszky et al. [5] proved a relative existence theorem of pairing strategies for the Snaky: if a pairing is good for the Snaky, it is good for the polyomino P5 that consists of five consecutive squares vertically or horizontally (but not diagonally). They also managed to give all possible pairings for P5, there are two of those and those are not appropriate for Snaky. As we mentioned, there is a Hales-Jewett winning pairing strategy for Breaker in the 9-in-a-row, see [3, 4] and also in Figure 1. The easiest way to see that the Hales-Jewett pairing blocks all lines of the board is that this pairing is an extension of a domino pairing from the 8 x 8 torus to the whole board. The torus lines consist of not only the rows and columns, but all diagonals, continuing on the opposite side when reaching the border of the 8 x 8 board. Since no one has published different pairings for the 9-in-a-row,1 the highly symmetric structure of the Hales-Jewett pairing, and the other examples of uniqueness or quasi uniqueness of pairings in similar problems; it is natural to think this is the only possible solution. It turned out that this belief is very far from reality, as we found lots of new ones and will exhibit a few in the following sections. Another belief was that all pairings must be torical extensions of their 8 x 8 section. Somehow surprisingly, this belief is not true either; there are lot of solutions which are connected to the 16 x 16 torus, but are not extensions of the pairings of a 8 x 8 torus. In the next sections, we define pairings precisely and give some conditions for their existence and structure. We will show that all solutions are either the extension of the pairings of a 8 x 8 torus (there are 194 543 non-isomorphic ones) or some combinations of those resulting in 16 x 16 toric solutions. At the end of the paper, we prove a special case of the conjecture of Kruczek and Sundberg [8] about the existence of pairings in higher dimensions. 1 According to [4], Selfridge also produced a Hales-Jewett pairing, but that pairing is either not different from the known Hales-Jewett pairing or left unpublished. L. GyOrffy et al.: The pairing strategies of the 9-in-a-row game 99 Since in our paper k-in-a-row type games play an important role, we define Hk, the hypergraph of the k-in-a-row games. Definition 1.1. The vertices of the k-in-a-row hypergraph Hk are the squares of the infinite (chess)board, i.e., the infinite square grid. The edges of the hypergraph Hk are the k-element sets of consecutive squares in a row horizontally, vertically or diagonally. We refer to the whole infinite rows as lines. 2 Pairing strategies Given a hypergraph H = (V, E), where V = V(H) and E = E(H) C P(H) = {S : S C V} are the set of vertices and edges, respectively. A bijection p: X ^ Y, where X,Y C V(H) and X n Y = 0 is a pairing on the hypergraph H. An (x, p(x)) pair blocks an A e E(H) edge, if A contains both elements of the pair. If the pairs of p block all edges, we say that p is a good pairing of H. Pairings are one way to show that Breaker has a winning strategy in hypergraph games. A good pairing p for a hypergraph H can be turned to a winning strategy for Breaker in the M-B game on H: following p on H in a M-B game, for every x e X U Y element chosen by Maker, Breaker chooses p(x) or in case of x e Y vice versa (if x e X U Y, then Breaker can choose an arbitrary vertex). Hence Breaker can block all edges and wins the game. Since our main topic is the 9-in-a-row game, this will be the first illustration to pairings and pairing strategies. The following result is due to Hales and Jewett [4, 7]: Theorem 2.1. Breaker wins the 9-in-a-row M-B game by a pairing strategy, i.e., there exists a good pairing for the 9-in-a-row. Proof. Figure 1 is an extension of a pairing of an 8 x 8 torus (framed), where the pairs have a periodicity 8 in every line. Hence, the pairing blocks all 9-in-a-row edges. □ / \ / \ / 1 \ \ / / 1 \ f / \ \ \ / \ / / / / \ \ / / \ \ \/ / \ \ / / \ \ / / \ \ / \ \ / 1 \ \ / / / / \ \ / / \ \ s/ / 1 \ \ / / 1 V \ f / \ / \ \ / \ / / / / \ \ / / \ \ / 1 \ / 4- 1 \ Figure 1: Hales-Jewett pairing blocks the 9-in-a-row. 100 Ars Math. Contemp. 16 (2019) 97-109 A pairing is a domino pairing or rather a match(-stick) pairing on the square grid, if all pairs consist of only neighboring cells (horizontally, vertically or diagonally), called dominoes. Note that the pairing in Figure 1 is a domino pairing. A counting type proposition [5] showed that there is no good pairing strategy for the k-in-a-row hypergraph, if k < 9. We will use this proposition, so we formulate the exact statement. For a hypergraph H let d2 (H) (briefly d2) be the greatest number of edges that can be blocked by two vertices of H, i.e., d2 is the maximal co-degree. Proposition 2.2 ([5]). If there is a good pairing p for the hypergraph H = (V, E), then d2|X |/2 > |G| must hold for all X c V, where G = {A : A e E, A c X}. Proof. We will refer to X as a sub-board. The edges of G can be blocked only by pairs coming from X. There are at most |X|/2 such pairs of p on the sub-board of size |X|. Since a pair blocks maximum d2 edges, |X |/2 pairs block maximum d2|X |/2. So, if there are more edges on the sub-board, there cannot be a good pairing. □ With the help of Proposition 2.2, we can conclude that there is no pairing strategy for Hk if k < 9. In the hypergraph Hk, d2 = k - 1 because a pair blocks at most k - 1 edges and this happens if and only if the pair is a domino. If X is an n x n sub-board for sufficiently large n, then |G| = 4n2 + O(n) because four edges start from every square (a vertical, a horizontal and two diagonal, except at the border). By Proposition 2.2 we get (k — 1)n2/2 > 4n2 + O(n); that is, k > 9 + O(1/n). One can even compute O(n) exactly: O(n) = —48n +128. Hales and Jewett gave a pairing for k = 9, see [4] or Figure 1. However, there are neither different solutions nor claims of the uniqueness of the Hales-Jewett pairing in the literature. Our main goal is to decide about these questions. 3 Conditions for good pairings of the 9-in-a-row Consider an n x n square sub-board of the infinite board. Proposition 2.2 gives (k — 1)n2/2 > 4n2 + O(n) which implies k > 9 + O(1/n). It suggests that one must use the pairs "optimally" to block H9 that is a pair should block the maximum possible edges of H9. We make the notion of optimality more precise as follows. Definition 3.1. A pairing is optimal if: 1. Every pair blocks exactly k — 1 edges. 2. There are no overblockings, i.e., every edge is blocked by exactly one pair. 3. There is no empty square, i.e., every square is contained in a pair. Corollary 3.2. Let us consider an optimal good pairing for H9. This pairing is then a domino pairing in which the dominoes are following each other by 8-periodicity in each line and all squares are covered by a pair. Proof. The first point of Definition 3.1 implies that the pairing is a domino pairing, while the second gives the 8-periodicity since otherwise it would cause either overblocking or resulting in an unblocked edge. The lack of empty squares is just the repetition of the third condition. □ L. GyOrffy et al.: The pairing strategies of the 9-in-a-row game 101 Definition 3.3. We call a square of a pairing anomaly where the 8-periodicity is violated, a non-domino type pair or an empty square appears in the pairing. Of course the Hales-Jewett pairing is anomaly-free. Remark 3.4. Because of the O(n), there might be anomalies even in a good pairing of H9.2 However, in Section 6 we will show that a good pairing of H9 is always anomaly-free. The first step towards this is the following lemma: Lemma 3.5. For every good pairing of H9 there is an arbitrarily big, anomaly-free square sub-board. Proof. Let us take any n x n sub-board X and cut it up to smaller /n/100 x /n/100 sub-boards. According to Proposition 2.2, there are at most 48n - 128 anomalies in X. Hence, most of the 10000n sub-squares of X must be anomaly-free. □ From now on we describe the structure of anomaly-free pairings of H9. Let us divide a good pairing of H9 into 8 x 8 sub-boards and designate one that we call Central square, shortly C. We keep only the (domino) pairs touching C and examine where pairs should be on the neighboring 8 x 8 sub-boards of C. In order to talk about these sub-boards we call the 8 x 8 sub-boards Eastern (E), North-Eastern (NE) etc., while for the individual squares of the sub-boards the usual algebraic chess notations (A1 to H8) are used, see Figure 2. Lemma 3.6. Suppose we have an anomaly-free good pairing of H9 and we have nine 8 x 8 squares, (C, E, NE,...) as above. The horizontal and vertical dominoes touching the Central square C appear on the same places in all eight neighboring sub-boards of C. The diagonal dominoes also must appear on the sub-boards NE, NW, SW, SE in the same places. However, while the diagonal pattern of C may extend to the other four subboards, namely the E, S, W, N, it cannot be guaranteed. That is: the whole plane is the periodic copies either of the 8 x 8 sub-board C or the 16 x 16 square consisting of the sub-boards C, S, SE, E. Proof. It suffices to check the following five steps. We designate a general square in a 8 x 8 sub-board by Xi e {A1,..., H8} according to the chess notation. If a domino d covers the same pair of squares e.g., in the C and E square, we say that d extends to E from C. 1. Because of the 8-periodicity of the domino pairs on horizontal (vertical) lines, the pairs of C extend uniquely to the same places of W and E (N and S respectively). The slope +1 diagonal dominoes extend similarly to SW and NE, while the slope -1 diagonal dominoes to SE and NW. 2. To see the horizontal (vertical) extension of dominoes to N and S (W and E respectively) we need a little case study. We have already seen that the vertical dominoes of C extend to north and south. Suppose for example that there is a vertical domino v at the Xi square of C. If the Xi square of W is covered by a slope +1 (or -1) 2 A pairing with anomalies might be called "quasi-crystal" referring to the highly symmetric, crystal like appearance of known anomaly-free examples such as the Hales-Jewett or the ones shown in [5]. 102 Ars Math. Contemp. 16 (2019) 97-109 \ NW / 1 N \ NE / " W ■ l E _ _ \ ( - 1 + y SW / 1 s \ SE / 1 Figure 2: The extension of a pairing of C. diagonal domino, then the 8-periodicity implies that the Xi square of N (or S) is also covered by a diagonal domino. This is a contradiction because we know from the previous point, that the Xi square of N is covered by a copy of the vertical domino v. The same is true for the sub-board E. If the X« square of W (or E) is covered by a horizontal domino, then C should contain the copy of that horizontal domino at X« by 8-periodicity, which is also a contradiction. We get that the vertical domino v in C extends to W and E, moreover, by 8-periodicity v extends to SW, NW, NE, SE, too. So, we have seen that the vertical dominoes of C extend to all its eight neighboring sub-boards. The same is true for the horizontal dominoes of C. 3. Let us check the diagonal dominoes. At the first and second step all slope +1 diagonal dominoes of C extend to SW and NE. Since there are no empty squares or overblockings, the remaining squares in SW and NE can be covered only by -1 slope diagonal dominoes. The same is true for +1 slope diagonal dominoes in SE and NW. That is so far, all dominoes of C extend to the SW, SE, NE, NW, furthermore, the vertical and horizontal dominoes of C extend to S, E, N and W. 4. We can see that the diagonal dominoes of C do not necessarily extend to the subboards S, E, N, W (colored by black in Figure 2). However, by 8-periodicity the diagonal pairs of E extend to S, N and W, that is the black sub-boards S, E, N, W have the exactly same structure of pairs. □ Remark 3.7. The diagonal dominoes of C may extend to the sub-boards S, E, N, W, and then all 8 x 8 sub-boards of the infinite board are the exact copy of C. However, it is possible that there are two different diagonal structures on the whole board, one in the L. GyOrffy et al.: The pairing strategies of the 9-in-a-row game 103 C, NW, NE, SE and SW types 8 x 8 sub-boards (colored by white in Figure 2) and a different diagonal structure in the sub-boards S, E, N and W (black ones). We will see a few examples in the next section. Definition 3.8. A pairing of the infinite board (or of an anomaly-free sub-board) is k-toric if it is an extension of a k x k torus, but not for a smaller value. Now we can summarize the previous lemmas and remarks in one central theorem: Theorem 3.9. Suppose we have an anomaly-free good pairing of H9. Then that pairing is either 8-toric or 16-toric. Proof. Lemma 3.6 and Remark 3.7 gives the proof of the Theorem. □ Observation 3.10. There are 8-toric good pairings of H9 that are not isomorphic to the Hales-Jewett pairing. Figure 3: Some other pairings for 9-in-a-row. Proof. The extensions of the 8 x 8 pairings in Figure 3 to the infinite board result in three different 8-toric pairings. Note that the pairings on the left have reflectional symmetry, while the pairing on the right has rotational symmetry. □ It is somehow surprising that there exist also some 16-toric pairings of H9. To understand their structure we refine the argument of the proof of Lemma 3.9 in the next section. 4 Diagonal alternating cycles The 8-toric and 16-toric good pairings of H9 can be considered as special perfect matchings of graphs based on H9. The vertex sets are the basic tori, and each vertex is connected to the eight neighbors of the square it represents. A domino of a pairing is an edge, and the whole pairing is not only a perfect matching but has the additional property that it contains exactly one edge (domino) from each torus line. It is well known that the union of two perfect matchings on the same vertex set consist of parallel edges and alternating cycles. So if we take the (graph theoretic) union of two good pairings (e.g. of C and W) which have the same horizontal and vertical edges, then the non-trivial alternating cycles contain only diagonal edges. Identifying the vertices in the case of non-isomorphic GC and GW the system of diagonal alternating cycles gives the possible ways to get the 16-toric good pairings. We arrive to the following simple corollary. 104 Ars Math. Contemp. 16 (2019) 97-109 Corollary 4.1. If there exists a 16-toric good pairing for H9, then we can derive two 8-toric good pairings from it (in case ofnon-isomorphic GC and GW) differing only in some diagonal cycles. ■ Figure 4: Diagonal alternating cycles give 16-toric pairing (left) and some -1 slope diagonal torus lines (right). Theorem 4.2. An 8-toric solution C gives a 16-toric solution if and only if another 8-toric solution W exists, differing in some diagonal dominoes, such that their union gives a system of diagonal alternating cycles. There are only two possible systems of diagonal alternating cycles which are shown in Figure 5; the left and middle ones. Figure 5: The diagonal alternating cycles. Proof. Since there is exactly one domino in each torus line of an arbitrarily chosen 8 x 8 square sub-board of a 8-toric solution, then the alternating cycles coming from the diagonal dominoes of the union of C and W must meet the torus lines either in zero or two dominoes. (If they meet only in one, then there will be an unblocked torus line in C or W. Meeting more than two times would mean overblocking.) An easy case study gives that only the systems of diagonal alternating cycles of Figure 5 may come into consideration. However, the third one would make a horizontal line (namely the 1-9) impossible to be blocked by a domino. □ Remark 4.3. There are only two different systems of alternating cycles, but it is possible that there is more than one such system in one 16-toric pairing. In that case we can deduce L. GyOrffy et al.: The pairing strategies of the 9-in-a-row game 105 more than two (four or eight) 8-toric pairings from that 16-toric one. An example is shown in the right of Figure 6. Observation 4.4. There exist good pairings for H9 containing the first or the second type of (the systems of) diagonal alternating cycles. Proof. In Figure 6, one can see examples of the statement. Taking bold (thin) pairs of the alternating cycle for C (W) we get a 16-toric pairing. Naturally this 16-toric pairing is not 8-toric. □ Figure 6: Examples of the alternating circles. 5 Pairings of the 8 torus We have seen that pairings on the anomaly-free sub-boards are either 8-toric or 16-toric. Since the 16-toric solutions can be reduced to 8-toric ones or conversely, they can be constructed from 8-toric solutions we examine only the later ones in detail. Definition 5.1. The 8 x 8 Maker-Breaker torus game is played on the 64 squares of the discrete torus, where there are 32 winning sets; the eight rows and columns and the diagonal torus lines of slope ±1 (see the right side of Figure 4). We will call 78 the hypergraph of that game. Observation 5.2. An arbitrary good 8-toric pairing for H9 provides a good pairing for 78. Remark 5.3. The reverse is not true, since 78 has good pairings which are not domino types. However, considering only domino pairings we can always extend a good pairing of 78 into a good 8-toric pairing of H9. To find all good domino pairings for the 8 x 8 torus is a finite task, which is not hard using a computer. However, one has to check the torus symmetries to list the non-isomorphic pairings, which gives the difficulty of the problem. The number of non-isomorphic domino type good pairings is 194 543, which turns out to be a prime. The pairings themselves can be downloaded at the page [9]. 6 There is no quasi crystal pairing for the infinite board We have an open problem left: are there any pairings for H9 with anomalies? Note that on a n x n sub-board there can be O(n) anomalies which might result in infinitely many (and possibly untraceable) solutions. Fortunately, this is not the case as we will see. 106 Ars Math. Contemp. 16 (2019) 97-109 Lemma 6.1. A given anomaly-free pairing of a large enough square sub-board can be extended uniquely to the whole plane. Proof. We have seen that all anomaly-free pairings of a square sub-board is an extension of a domino pairing of either a 8 x 8 or a 16 x 16 torus. Continuing the extension to the whole plane gives a good pairing. □ Lemma 6.2. Let us assume that a pairing of the whole plane is an extension of an anomaly-free half-plane R. Then the whole pairing is anomaly-free. Proof. To prove by contradiction, assume that we have an extension AL containing anomalies. Let AF be the anomaly-free extension of the half-plane pairing that exists by Lemma 6.1. Obviously AL is not equal to AF. Figure 7: There are no quasicrystals. Let us take a square with an anomaly which is one of the closest to R, and denote it by q. As it is pictured in Figure 7 we may assume that the border line of the half-plane R is vertical and the pairing AL is anomaly-free left to the square q. Let AF(q) be the domino covering the square q in AF. If AF(q) is placed horizontally and q is the right half of it, then AL does not contain the domino AF(q) at the square q which leaves a 9-in-a-row edge unblocked by AL. A similar argument to diagonally placed dominoes shows that AF(q) can be nothing but a vertical domino. Let us take the six squares above and below AF(q). Because of 8-periodicity, there are no other squares containing a vertical pair in AF covering these 12 squares, but there must be a half of a vertical pair on those places (e.g. in s) in AL, because of the blocking condition. The domino AF(s) is either horizontal or diagonal, and since AF(s) is not in AL, it results in an unblocked horizontal or diagonal edge in AL. □ To answer the main question at the beginning of this section, we will need the ideas of the previous lemma. Theorem 6.3. An anomaly-free pairing of a big enough square sub-board extends uniquely and anomaly-free to the whole board. L. GyOrffy et al.: The pairing strategies of the 9-in-a-row game 107 Figure 8: The extension of an anomaly-free pairing. Proof. Fix a good pairing for H9 and take an m x m sub-board B that is anomaly-free; this exists by Lemma 3.5. The pairing on B extends anomaly-free to a large part of the right side of B, like in Lemma 6.2. The extension surely contains the right-angled triangle whose hypotenuse length is m - 16, and touches the right side of B, see Figure 8. The argument of Lemma 6.2 does not work next to the top and the bottom of B, since there are no diagonal dominoes there in B which were used before. Doing the same trick to extend the pairing on the other sides of B, that results in a bigger (the size is about (A/2m - 16) x (V2m - 16)) rotated square. Repeating this procedure, we can see that the anomaly-free pairing of B is forced to extend to the whole plane. □ 7 A pairing strategy in 3D Kruczek and Sundberg [8] conjectured upper bounds matching with the lower bound of Proposition 2.2 for k-in-a-row type games in d dimension. Conjecture 7.1 ([8]). In the Maker-Breaker game on Zd where there is a finite set S c Zd of winning line direction-vectors, Breaker has a pairing strategy that allows him to win if the length of each winning line is at least 2|S| + 1, i.e., Breaker has a winning pairing-strategy for the game k-in-a-row if k > 2|S| + 1. The special case of the plane gives back that Breaker has winning pairing strategy in the k-in-a-row if and only if k > 9. The higher dimensional versions are mainly open. One possible form of the M-B game in 3-dimension is when the winning directions are given by 13 vectors: {(0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,0,1), (1,1,0), (0,1,-1), (1,0,-1), (1,-1,0), (1,1,1), (1,1, -1), (1,-1,1), (-1,1,1)}. Here Proposition 2.2 implies that k should be at least 27 to have a good pairing. According to Conjecture 7.1, we may expect good pairings for k = 27. We have examined a related problem in 3-dimension. If the directions of winning lines are given by three vectors: {(0,0,1), (0,1,0), (1,0,0)}, then one expects pairing strategies if k > 7. (In other words, this is a Harary-type game [4] in 3-dimension, where the winning polyomino is the P7, i.e. the seven connected consecutive cubes in a row.) In fact, a computer search confirms this expectation, see Figure 9. This is a domino pairing of 3-dimensional torus type, we give the pairing on the 6 x 6 x 6 torus in layers. The horizontal and vertical pairs of the same layer are obvious, while the pairs between the layers are denoted by points and circles. 108 Ars Math. Contemp. 16 (2019) 97-109 Figure 9: A good pairing of the 3D 7-in-a-row. 8 Conclusion We have shown some new pairings for H9. We have proved that a good pairing for H9 is either 8-toric or 16-toric. There are 194 543 pairings which are 8-toric, and it is possible to construct 16-toric good pairings of H9 from some of those. There are no good pairings on the infinite board containing anomalies. References [1] L. V. Allis, H. J. van den Herik and M. P. H. Huntjens, Go-Moku solved by new search techniques, Comput. Intell. 12 (1996), 7-23, doi:10.1111/j.1467-8640.1996.tb00250.x. [2] J. Beck, Van der Waerden and Ramsey type games, Combinatorica 1 (1981), 103-116, doi: 10.1007/bf02579267. [3] J. Beck, Combinatorial Games: Tic-Tac-Toe Theory, volume 114 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2008, http://www. cambridge.org/97 8 05214 610 0 9. [4] E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways for Your Mathematical Plays, Volume 3, A K Peters/CRC Press, Natick, Massachusetts, 2nd edition, 2003. [5] A. Csernenszky, R. R. Martin and A. Pluhar, On the complexity of chooser-picker positional games, Integers 12 (2012), 427-444, doi:10.1515/integ.2011.113. [6] R. K. Guy and J. L. Selfridge, Problems and solutions: Problems dedicated to Emory P. Starke: S10, Amer. Math. Monthly 86 (1979), 306-306, doi:10.2307/2320758. [7] A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222-229, doi:10.2307/1993764. [8] K. Kruczek and E. Sundberg, A pairing strategy for tic-tac-toe on the integer lattice with numerous directions, Electron. J. Combin. 15 (2008), #N42, http://www.combinatorics. org/ojs/index.php/eljc/article/view/v15i1n42. L. GyOrffy et al.: The pairing strategies of the 9-in-a-row game 109 [9] G. Makay, 9-in-a-row game, http://www.math.u-szeged.hu/ -makay/amoba/, accessed on 6 December 2016. [10] P. Mukkamala and D. Paivolgyi, Asymptotically optimal pairing strategy for tic-tac-toe with numerous directions, Electron. J. Combin. 17 (2010), #N33, http://www. combinatorics.org/ojs/index.php/eljc/article/view/v17i1n33. [11] T. G. L. Zetters, Problems and solutions: Solutions of problems dedicated to Emory P. Starke: S10, Amer. Math. Monthly 87 (1980), 575-576, doi:10.2307/2321433. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 111-117 https://doi.org/10.26493/1855-3974.1317.745 (Also available at http://amc-journal.eu) Weight choosability of oriented hypergraphs Marcin Anholcer * Faculty of Informatics and Electronic Economy, Poznan University of Economics and Business, al. Niepodleglosci 10, 61-875 Poznan, Poland Bartlomiej Bosek Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Prof. S. Lojasiewicza 6, 30-348 Krakow, Poland Jaroslaw Grytczuk t Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland Received 4 February 2017, accepted 22 May 2018, published online 19 September 2018 The 1-2-3 conjecture states that every simple graph (with no isolated edges) has an edge weigthing by numbers 1,2, 3 such that the resulting weighted vertex degrees form a proper coloring of the graph. We study a similar problem for oriented hypergraphs. We prove that every oriented hypergraph has an edge weighting satisfying a similar condition, even if the weights are to be chosen from arbitrary lists of size two. The proof is based on the Combinatorial Nullstellensatz and a theorem of Schur for permanents of positive semi-definite matrices. We derive several consequences of the main result for uniform hypergraphs. We also point on possible applications of our results to problems of 1-2-3 type for non-oriented hypergraphs. Keywords: Oriented hypergraphs, 1-2-3 conjecture, combinatorial nullstellensatz, list weighting. Math. Subj. Class.: 05C15, 05C65, 05C78 * Corresponding author. t Supported by the Polish National Science Center, decision nr DEC-2012/05/B/ST1/00652. E-mail addresses: m.anholcer@ue.poznan.pl (Marcin Anholcer), bosek@tcs.uj.edu.pl (Bartlomiej Bosek), j.grytczuk@mini.pw.edu.pl (Jaroslaw Grytczuk) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 112 Ars Math. Contemp. 16 (2019) 97-109 1 Introduction The famous 1-2-3 conjecture, posed by Karonski, Luczak, and Thomason [8], states that every simple graph (with no isolated edges) has an edge weighting with integers 1,2,3 such that no two adjacent vertices get the same weighted degrees. This innocently looking problem remains open for more than ten years, despite serious attacks based on various techniques, including the celebrated Combinatorial Nullstellensatz of Alon [1] (see [2, 10, 12]). The best result up to now [6] confirms the conjecture when the set of allowable weights includes also 4 and 5. There were also many variants considered involving lists, orientations, and most recently hypergraphs (see [2, 3, 5, 7]). It is known, for instance, that any oriented graph has a weighting with numbers 1 and 2 only, such that the resulting weighted degrees are different for every pair of adjacent vertices [2, 9]. In this paper we consider the list version of the 1-2-3 conjecture for oriented hypergraphs. A hypergraph H on the set of vertices V is a family E of non-empty subsets of V, called the edges of H. A hypergraph H is k-uniform if each of its edges has size exactly k. So, a 2-uniform hypergraph is just a simple graph. Let IH denote the incidence graph of a hypergraph H, that is, a bipartite graph with color classes V and E, whose edges are of the form ve, with v G V, e G E, whenever v G e. Now, by an orientation of a hypergraph H we mean any function ^: E(IH) ^ C assigning non-zero complex "signs" to the edges of the graph IH. The cumulated degree of a vertex v G V is defined by Dv ^(ev). If the range of the mapping ^ is confined to the set {-1, +1}, or more generally, to the set of complex roots of unity, then we get a natural generalization of traditional orientation of a simple graph. The orientation ^ is conveniently represented by the oriented incidence matrix X = [^ev] of dimension |E| x |V|, where ^ev = ^(ev) if ev G E(IH), and ^ev = 0, otherwise. By an oriented hypergraph we mean a hypergraph H together with some fixed orientation Suppose now that each edge e G E of an oriented hypergraph (H, is assigned a complex weight we. Then the resulting weighted degree of a vertex v G V is computed as Wv ^evWe. e3v For each e g E we now define We = £ ^v Wv , vGe where x* denotes the complex conjugate of x. Observe that in case of a usual oriented graph, We is exactly the difference between weighted degrees of both ends of e. We say that the weighting w is virtuous if for each edge e G E we have We = 0. Suppose that a list of complex numbers Le is assigned to each e G E. We say that an oriented hypergraph H is t-weight choosable if for any lists satisfying |Le| > t we are able to choose weights we G Le so that w is a virtuous weighting of H. Our main theorem extends the results of [2] and [9] in the following way. M. Anholcer et al.: Weight choosability of oriented hypergraphs 113 Theorem 1.1. Every oriented hypergraph is 2-weight choosable. In the next section we will give an algebraic proof of this theorem based on the Combinatorial Nullstellensatz. It is inspired by the approach applied in [9]. In the last section we will speculate on possible consequences of this result for non-oriented hypergraphs, in particular, to the recent intriguing conjecture of Karotiski, Kalkowski, and Pfender [7]. 2 Proof of the main result First recall the celebrated Combinatorial Nullstellensatz of Alon [1]. Theorem 2.1 (Alon). Let K be an arbitrary field, and let F = F (x1,x2,..., xn) be a polynomial in K[x1,x2,..., xn]. Suppose that the total degree of f is n=1 ti, where each ti is a nonnegative integer, and suppose that the coefficient of\{rn=1 x-i is nonzero. If S1,S2,... ,Sn are subsets of K, with |Si| > ti, thenthereare s1 G S1, s2 G S2, ..., sn G Sn so that F (81,82,...,sn)=0. Let A = (aij )nxn be a square matrix with complex entries. Recall that the permanent of A is defined by: n per(A) = ^ IK«*), aeSni=1 where Sn denotes the group of permutations of the set {1, 2, . . . , n}. This is seemingly almost the same as the determinant of A, only the signs of permutations a are ignored. The following classic result of Schur [11] gives a relation between permanent and determinant of a Hermitian matrix. Theorem 2.2 (Schur). If A is a positive semi-definite Hermitian matrix, then per(A) > det(A), with equality if and only if A is diagonal or A has a zero row. Proof of Theorem 1.1. Let H be a hypergraph with n vertices, m edges, and fixed orientation Assume that a list Le with two complex numbers has been assigned to each edge e g E of H, as well as a complex variable xe. Let us define for each vertex v g V of H, and ^ ^ Mev xe e3v Pe 'y ] Mev Pv for each e G E. Finally, let us define the complex multivariate polynomial ph=n pe. e • eEE We see that H is 2-weight choosable if Ph (Wei ,We2 , . . .,Wem ) = 0 pv 114 Ars Math. Contemp. 16 (2019) 97-109 for some choice of weights we G Le, with e G E. Each monomial in PH has total degree m. We are going to show that the coefficient of n Xe eEE is nonzero. This will finish the proof by Theorem 2.1. Let us expand PH. We have: Pe "y ] Mev Pv "y ] Mev ^ ] Mfv xf ^ ] Mev ^ ] Mf v xf . vEe vEe f3v v£e f£E The sums in the last expression are independent, thus we can write Pe = E Mev Mfv xf = ^M*ev Mfv xf . vEe f EE f EE vGe Let mef = ^ MevMfv . vEe Let M be the m x m matrix consisting of the mef. From the above definition it follows that M = XX*, where X is the oriented incidence matrix of H, and X* is its conjugate transpose. We have Pe = E mef Xf > fEE and Ph = H Pe = HE mef Xf. eEE eEE fEE It follows that the coefficient of n xe eE E is equal to per(M). So, it is enough to prove that per(M) = 0 (see the permanent lemma in [1]). To get this we will apply Schur's theorem. First notice that for any complex vector z we have zMz* = zXX*z* = zX(zX)* = |zX|2 . As the last number is real and non-negative, M is positive semi-definite. Notice also that by assumption H has no empty edges and Mev =0 for all v G e. Therefore all entries on the main diagonal of M are strictly positive real numbers. In particular, M has no zero row. Hence, if M is not diagonal, then by Schur's theorem we get per(M) > det(M) > 0. If M is diagonal, then per(M) = det(M) = H mee > 0. eEE This completes the proof. □ M. Anholcer et al.: Weight choosability of oriented hypergraphs 115 3 Some applications for uniform hypergraphs We give two applications of Theorem 1.1 extending some results from [2, 5] and [9]. For simplicity we confine ourselves to uniform hypergraphs with specific orientations defined as follows. Let k > 2 be a fixed positive integer and let Uk denote the multiplicative group of k-th complex roots of unity. If e is a primitive root in Uk, then we may write Uk = {1, e, e2,..., ek-1}. Let H be a k-uniform hypergraph. Consider a canonical orientation of H given by the mapping ^: E(IH) ^ Uk such that ^(eu) = ^(ev) for every edge e G E and any two vertices u, v G e. Notice that for k = 2 we get the traditional orientation of a simple graph. Recall that a coloring of the vertices of a hypergraph is proper if no edge is monochromatic. Corollary 3.1. Every canonically oriented k-uniform hypergraph H has an edge weighting with numbers 1, 2 such that weighted vertex degrees give a proper coloring of H. Proof. Let H be a given k-uniform hypergraph and let ^ be any canonical orientation of H. Assign the lists Le = {1,2} to all edges of H. Then by Theorem 1.1 there exists a virtuous edge weighting w such that we G {1, 2} for every e G E. We claim that this weighting satisfies the assertion of the corollary. Suppose on the contrary that this is not the case, and that there is some edge e = {v1, v2,..., vk} such that all weighted degrees Wv* are equal: * Wv! = • • • = Wvfc = W. This implies that kk We = ^ M(evi)*Wv* = W ^ M(evi)* = 0, i=i i=i since k ^ M(evi)* = 1 + e + ••• + ek-1 = 0. i= 1 This contradicts the virtue of the weighting w. □ Corollary 3.2. Every k-uniform hypergraph H has a canonical orientation such that cumulated vertex degrees give a proper coloring of H. Proof. Let H be a given k-uniform hypergraph and let ^ be any of its canonical orientations. Assign the list Le = {1,e} to every edge e G E. Then by Theorem 1.1 there exists a virtuous edge weighting w such that we G {1, e} for every e G E. Consider now a new orientation ^ defined by ^'(ev) = ^(ev)we for every edge ev of the incidence graph IH. We claim that this orientation satisfies the assertion of the corollary. Suppose on the contrary that this is not the case, and that there is some edge e = {v1, v2,..., vk} with all cumulated degrees Dv* equal in orientation Dvt = • • • = Dvfc = D. This implies that also weighted degrees Wv* are equal to D, since Wv* = M(evi)we = ^'(evi) = Dv* = D. e3v* e3v* 116 Ars Math. Contemp. 16 (2019) 97-109 In consequence, we get k We = )WVi = D^(evi)*=0, =1 which contradicts the fact that w is virtuous. □ 4 Discussion We conclude the paper with pointing on some possible applications of our results to non-oriented problems of "1-2-3" type. Actually, the original 1-2-3 conjecture can be stated in a setting similar to virtuous weightings of oriented hypergraphs. To see this consider a bipartite graph B on bipartition classes X and Y with some signing ^: E(B) ^ {-1, +1}. Let w : X ^ Z be a weighting of one part of B. Then for every vertex y G Y we may define the induced weight of y by Wy = w(x)^(xy), xeN (y) where N(y) denote the set of neighbors of y. A weighting w is called half-virtuous if Wy = 0 for every y G Y .A natural problem is to find for a given signed graph B, the least integer k such that B has a half-virtuous weighting with w(x) G {1, 2,..., k} for each x g X .It is not hard to see that 1-2-3-conjecture is equivalent to the statement that certain signed bipartite graphs arising from simple graphs have half-virtuous weightings with weights in the range {1,2,3}. Consider a simple graph G and the related bipartite graph BG on bipartition classes X = Y = E(G), with xy G E(BG) whenever x and y are incident edges of G. An appropriate signing ^ is defined so that for a fixed y G Y, the edges xy, x'y G E(BG) have the same sign if and only if x and x' are incident in G to the same end of y. A half-virtuous weighting of BG is then equivalent to a weighting of the edges of G with different sums over edges incident to opposite ends of any given edge. It is also possible that our results could be useful in a recently introduced version of the 1-2-3 conjecture for (non-oriented) hypergraphs [7]. Let H be a k-uniform hypergraph and let w denote a weighting of its edges. For every vertex v define its weighted degree by Wv =53 we. Recall that a proper coloring of H is a coloring of its vertices such that no edge is monochromatic. A hypergraph H is r-weight colorable if there is a weighting w : E(H ) ^ {1,2,..., r} such that the weighted degrees Wv form a proper coloring of H. The following conjecture is stated in [4] (see also [7]). Conjecture 4.1. Every k-uniform hypergraph (k > 2) with no isolated edges is 3-weight colorable. The conjecture holds for random uniform hypergraphs in a stronger sense that even non-weighted degrees give a proper coloring, as proved recently in [4]. Another strong support for validity of the conjecture is given in [7], where it is proved that every nontrivial hypergraph (not only uniform) is (2,3)-weight colorable. This means that a proper M. Anholcer et al.: Weight choosability of oriented hypergraphs 117 coloring of a hypergraph is obtained by using weights {1,2, 3} for the edges, and weights {1, 2} for the vertices, with weighted vertex degrees computed by a formula: This statement in restriction to simple graphs was formerly proved by Kalkowski in his PhD thesis. Then the result was extended to the list version by Wong and Zhu [12] who applied the Combinatorial Nullstellensatz. This encourages us to conclude the paper with the following general conjecture. Conjecture 4.2. Every k-uniform hypergraph (k > 2) with no isolated edges is 3-weight choosable. References [1] N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999), 7-29, doi:10. 1017/s0963548398003411. [2] T. Bartnicki, J. Grytczuk and S. Niwczyk, Weight choosability of graphs, J. Graph Theory 60 (2009), 242-256, doi:10.1002/jgt.20354. [3] O. Baudon, J. Bensmail and É. Sopena, An oriented version of the 1-2-3 conjecture, Discuss. Math. Graph Theory 35 (2015), 141-156, doi:10.7151/dmgt.1791. [4] P. Bennett, A. Dudek, A. Frieze and L. Helenius, Weak and strong versions of the 1-2-3 conjecture for uniform hypergraphs, Electron. J. Combin. 23 (2016), #P2.46, http://www. combinatorics.org/ojs/index.php/eljc/article/view/v2 3i2p4 6. [5] M. Borowiecki, J. Grytczuk and M. Pilâniak, Coloring chip configurations on graphs and digraphs, Inform. Process. Lett. 112 (2012), 1-4, doi:10.1016/j.ipl.2011.09.011. [6] M. Kalkowski, M. Karonski and F. Pfender, Vertex-coloring edge-weightings: towards the 1-2-3-conjecture, J. Comb. Theory Ser. B 100 (2010), 347-349, doi:10.1016/j.jctb.2009.06.002. [7] M. Kalkowski, M. Karonski and F. Pfender, The 1-2-3-conjecture for hypergraphs, J. Graph Theory 85 (2017), 706-715, doi:10.1002/jgt.22100. [8] M. Karonski, T. Luczak and A. Thomason, Édge weights and vertex colours, J. Comb. Theory Ser.B 91 (2004), 151-157, doi:10.1016/j.jctb.2003.12.001. [9] M. Khatirinejad, R. Naserasr, M. Newman, B. Seamone and B. Stevens, Digraphs are 2-weight choosable, Electron. J. Combin. 18 (2011), #P21, http://www.combinatorics.org/ ojs/index.php/eljc/article/view/v18i1p21. [10] J. Przybylo and M. Wozniak, Total weight choosability of graphs, Electron. J. Combin. 18 (2011), #P112, http://www.combinatorics.org/ojs/index.php/eljc/ article/view/v18i1p112. [11] I. Schur, Uber endliche Gruppen und Hermitesche Formen, Math. Z. 1 (1918), 184-207, doi: 10.1007/bf01203611. [12] T.-L. Wong and X. Zhu, Évery graph is (2, 3)-choosable, Combinatorica 36 (2016), 121-127, doi:10.1007/s00493-014-3057-8. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 119-139 https://doi.org/10.26493/1855-3974.1490.eea (Also available at http://amc-journal.eu) The Doyen-Wilson theorem for 3-sun systems* Giovanni Lo Faro , Antoinette Tripodi Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universita di Messina, Messina, Italia Received 25 September 2017, accepted 18 April 2018, published online 20 September 2018 A solution to the existence problem of G-designs with given subdesigns is known when G is a triangle with p = 0,1, or 2 disjoint pendent edges: for p = 0, it is due to Doyen and Wilson, the first to pose such a problem for Steiner triple systems; for p = 1 and p = 2, the corresponding designs are kite systems and bull designs, respectively. Here, a complete solution to the problem is given in the remaining case where G is a 3-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor. Keywords: 3-sun systems, embedding, difference set. Math. Subj. Class.: 05B05, 05B30 1 Introduction If G is a graph, then let V(G) and E(G) be the vertex-set and edge-set of G, respectively. The graph Kn denotes the complete graph on n vertices. The graph Km \ Kn has vertex-set V(Km) containing a distinguished subset H of size n; the edge-set of Km \ Kn is E(Km) but with the edges between the n distinguished vertices of H removed. This graph is sometimes referred to as a complete graph of order m with a hole of size n. Let G and r be finite graphs. A G-design of r is a pair (X, B) where X = V(r) and B is a collection of isomorphic copies of G (blocks), whose edges partition E(r). If r = Kn, then we refer to such a design as a G-design of order n. A G-design (Xi, Bi) of order n is said to be embedded in a G-design (X2, B2) of order m provided X1 C X2 and B1 C B2 (we also say that (X1, B1) is a subdesign (or subsystem) of (X2, B2) or (X2, B2) contains (X1, B1) as subdesign). Let N(G) denote the set of integers n such that there exists a G-design of order n. A natural question to ask is: given n, m G N(G), with m > n, and a G-design (X, B) of order n, does exists a G-design of order m containing (X, B) as subdesign? Doyen and Wilson were the first to * Supported by I.N.D.A.M. (G.N.S.A.G.A.). E-mail addresses: lofaro@unime.it (Giovanni Lo Faro), atripodi@unime.it (Antoinette Tripodi) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 120 Ars Math. Contemp. 16 (2019) 141-155 pose this problem for G = K3 (Steiner triple systems) and in 1973 they showed that given n, m G N (K3) = {v : v = 1,3 (mod 6)}, then any Steiner triple system of order n can be embedded in a Steiner triple system of order m if and only if m > 2n +1 or m = n (see [3]). Over the years, any such problem has come to be called a "Doyen-Wilson problem" and any solution a "Doyen-Wilson type theorem". The work along these lines is extensive ([1, 4, 5, 6, 7, 8, 9, 10, 13]) and the interested reader is referred to [2] for a history of this problem. In particular, taking into consideration the case where G is a triangle with p = 0,1, 2, or 3 mutually disjoint pendent edges, a solution to the Doyen-Wilson problem is known when p = 0 (Steiner triple systems, [3]), p =1 (kite systems, [9, 10]) and p = 2 (bull designs, [4]). Here, we deal with the remaining case (p = 3) where G is a 3-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor, by giving a complete solution to the Doyen-Wilson problem for G-designs where G is a 3-sun (3-sun systems). 2 Notation and basic lemmas The 3-sun consisting of the triangle (a, b, c) and the three mutually disjoint pendent edges {a, d}, {b, e}, {c, f} is denoted by (a, b, c; d, e, f). A 3-sun system of order n (briefly, 3SS(n)) exists if and only if n = 0,1,4, 9 (mod 12) and if (X, S) is a 3SS(n), then |S| = (see [14]). Let n, m = 0,1,4,9 (mod 12), with m = u + n, u > 0. The Doyen-Wilson problem for 3-sun systems is equivalent to the existence problem of decompositions of K„+n \ Kn into 3-suns. Let r and s be integers with r < s, define [r, s] = {r, r + 1,..., s} and [s, r] = 0. Let = [0, u — 1] and H = {ro^ ro2, ..., rot}, H fl = 0. If S = (a, b, c; d, e, f) is a 3-sun whose vertices belong to U H and i G Zu, let S + i = (a + i, b + i, c + i; d + i, e + i, f + i), where the sums are modulo u and ro + i = ro, for every ro G H. The set (S) = {S + i : i G Zu} is called the orbit of S under and S is a base block of (S). To solve the Doyen-Wilson problem for 3-sun systems we use the difference method (see [11, 12]). For every pair of distinct elements i, j G Zu, define |i — j|u = min{|i — j|,u — |i — j|} and set D„ = {|i — j|„ : i, j G Z„} = {1, 2,..., |_fj}. The elements of are called differences of . For any d G Du, d = u, we can form a single 2-factor {{i, d + i} : i G Zu}, while if u is even and d = u, then we can form a 1-factor {{i, i + u} : 0 < i < u — 1}. It is also worth remarking that 2-factors obtained from distinct differences are disjoint from each other and from the 1-factor. If D C D„, denote by (Z„ U H, D) the graph with vertex-set V = Z„ U H and the edge-set E = {{i, j} : |i — j|u = d, d G D} U {{ro, i} : ro G H, i G Zu}. The graph (Zu U H, Du) is the complete graph K„+i \ Kt based on U H and having H as a hole. The elements of H are called infinity points. Let X be a set of size n = 0,1,4, 9 (mod 12). The aim of the paper is to decompose the graph (Zu U X, Du) into 3-suns. To obtain our main result the (Zu U X, Du) will be regarded as a union of suitable edge-disjoint subgraphs of type (Zu U H, D) (where H C X may be empty, while D C Du is always non empty) and then each subgraph will be decomposed into 3-suns by using the lemmas given in this section. From here on suppose u = 0,1, 3, 4, 5, 7, 8, 9,11 (mod 12). Lemmas 2.1-2.4 give decompositions of subgraphs of type (Zu U H, D) where D G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 121 contains particular differences, more precisely, D = {2}, D = {2,4} or D = {1, u}. Lemma 2.1. Let u = 0 (mod 4), u > 8. Then the graph (Z„ U {toi, to2}, {2}} can be decomposed into 3-suns. Proof. Consider the 3-suns (toi, 2 + 4«, 4«; 3 + 4«, 4 + 4«, to2), (to2, 3 + 4«, 1 + 4«; 2 + 4«, 5 + 4«, toi), for « = 0,1,..., u - 1. □ Lemma 2.2. Let u = 0 (mod 12). Then the graph (Z„ U {toi, to2, to3, to4}, {2}} can be decomposed into 3-suns. Proof. Consider the 3-suns (toi, 12«, 2 + 12«; 7 + 12«,, to3, to4), (toi, 4+ 12«, 6 + 12«; 9 + 12«, to3, to4), (toi, 8 + 12«, 10 + 12«; 11 + 12«, to3, to4), (to2, 2 + 12«, 4 + 12«; 1 + 12«, to3, to4), (to2, 6 + 12«, 8 + 12«; 7 + 12«, to3, to4), (to2, 10 + 12«, 12 + 12«; 11 + 12«, to3, to4), (to3, 1 + 12«, 3 + 12«; 9 + 12«, toi, to2), (to3, 5 + 12«, 7 + 12«; 11 + 12«, toi, 9 + 12«), (to4, 3 + 12«, 5 + 12«; 1 + 12«, toi, to2), (to4, 9 + 12«, 11 + 12«; 7 + 12«, to2, 13 + 12«), for « = 0,1,..., 12 - 1. □ Lemma 2.3. The graph (Z„ U {toi, to2, to3, to4}, {2,4}}, u > 7, u = 8, can be decomposed into 3-suns. Proof. Let u = 4k + r, with r = 0,1,3, and consider the 3-suns (toi, 4 + 4«, 6 + 4«; 5 + 4«, 8 + 4«, to4), (to2, 5 + 4«, 7 + 4«; 6 + 4«, 9 + 4«, toi), (to3, 6 + 4«, 8 + 4«; 7 + 4«, 10 + 4«, to2), (to4, 7 + 4«, 9 + 4«; 8 + 4«, 11 + 4«, to3), for « = 0,1,..., k - 3, k > 3, plus the following blocks as the case may be. If r = 0, (toi, 0, 2; 1, 4, to4), (to2, 1, 3; 2, 5, toi), (to3, 2,4; 3, 6, TO2), (to4, 3, 5; 4, 7, to3), (toi, 4k - 4, 4k - 2; 4k - 3,0, to4), 122 Ars Math. Contemp. 16 (2019) 141-155 If r = 1, If r = 3, (to2, 4k - 3,4k - 1; 4k - 2,1, toi), (to3, 4k - 2,0; 4k - 1, 2, to2), (to4, 4k - 1,1; 0, 3, to3). (toi, 0, 2; 1, 4, to2), (to2, 1, 3; 0, 5, toi), (to3, 2,4; 3, 6, to2), (to4, 3, 5; 4, 7, to3), (toi, 4k - 4, 4k - 2; 4k - 3, 4k, to2), (to2, 4k - 3, 4k - 1; 4k, 0, tox), (to3, 4k - 2, 4k; 4k - 1,1, toi), (to4, 4k - 1, 0; 4k - 2, 2, to3), (to4, 4k, 1; 2, 3, to3). (toi, 0, 2; 1, 4, to4), (to2, 1, 3; 2, 5, toi), (TO3, 2,4; 3, 6, TO2), (to4, 3, 5; 4, 7, to3), (toi, 4k - 4, 4k - 2; 4k - 3,4k, to4), (to2, 4k - 3, 4k - 1; 4k - 2,4k + 1, toi), (to3, 4k - 2, 4k; 4k - 1, 4k + 2, to2), (to4, 4k - 1, 4k + 1; 4k, 0, to3), (toi, 4k, 4k + 2; 4k + 1,1, to4), (to2, 4k + 1, 0; 4k + 2, 2, to4), (to3, 4k + 2,1; 0, 3, to4). With regard to the difference 4 in Z7, note that |4|7 obtained for k = 1 and r into 3-suns. 3 and the seven distinct blocks 3 gives a decomposition of (Z7 U {toi, to2, to3, to4}, {2,3}} □ Lemma 2.4. Let u = 0 (mod 3), u > 12. Then the graph (Zu U {toi, to2, ..., to8}, {1, U }} can be decomposed into 3-suns. Proof. If u = 0 (mod 6) consider the 3-suns: (toi, 2i, u + 2i; 2u + 2i, to5, to6), i = 0,1, u - 1 6 1, u _ 1 6 1, (toi, 1 + 2i, u + 1 + 2i; 2u + 1 + 2i, to6, to5), i = 0,1,.. (to2, 2u + 2i, u + 2i; 2 + 2i, 2i, to5), i = 0,1,..., u - 2, (to2, 2u + 1 + 2i, u + 1 + 2i; 3 + 2i, 1 + 2i, to6), i = 0,1,..., u - 2, (to2, u - 2, 2u - 2; 0, u - 2, to5), G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 123 - - 1 6 1 to2,U - 1, 2- - 1; 1, - - 1, to6), to3, 2i, 1 + 2i; 2- + 2i, to7, to8), i = 0,1,..., - - 1, TO3, - + 2i, - + 1 + 2i; 2- + 1 + 2i, TO7, tos), i = 0,1, to4, 1 + 2i, 2 + 2i; 2- + 2 + 2i, to7, to8), i = 0,1,..., - - 1, to4, - + 1 + 2i, - + 2 + 2i; 2- + 1 + 2i, to7, to8), i = 0,1,..., - - 1, to5, 2- + 2i, 2- + 1 + 2i; 1 + 2i, to7, to8), i = 0,1,..., - - 1, to6, 2- + 3 + 2i, 2- + 4 + 2i; 2 + 2i, to7, to8), i = 0,1,..., - - 2, (to6, 2- + 1, 2- +2; 2-, to7, to8). If u = 3 (mod 6) consider the 3-suns: (toi, 2i, - + 2i; 2- + 2i, to5, to6), i = 0,1,..., , TOi TO2 TO2 TO2 TO2 TO3 TO3 TO3 TO3 to4 to4 TO5 TO6 TO6 to7 m-9 6 , 1 + 2i, - + 1 + 2i; 2- + 1 + 2i, to6, to5), i = 0,1, 2- + 2i, - + 2i; 2 + 2i, 2i, to5), i = 0,1,..., , u - 1, 2- - 1; 0, - - 1, to5), 2- + 1 + 2i, - + 1 + 2i; 3 + 2i, 1 + 2i, to6), i = 0,1,..., , u - 2, 2- - 2; 1, - - 2, to6), 2i, 1 + 2i; 2- + 2i, to7, to8), i = 2, 3,..., , 0,1; 2-, to6, TO8), 2, 3; 2- + 2, TO6, tos), - + 1 + 2i, - + 2 + 2i; 2- + 1 + 2i, to7, to8), i = 0,1,... 1 + 2i, 2 + 2i; 2- + 2 + 2i, to7, to8), i = 0,1,..., , - + 2i, - + 1 + 2i; 2- + 1 + 2i, to7, to8), i = 0,1,..., , 2- + 2i, 2- + 1 + 2i; 1 + 2i, to7, to8), i = 0,1,..., , 2- + 1 + 2i, 2- + 2 + 2i; 4 + 2i, to7, to8), i = 0,1,..., ^g15, u - 2,u - 1; 2-, to7, to8), u - 1,0; 2, to5, to8). m-9 6 , □ Lemmas 2.5 - 2.9 allow to decompose (Zm U H, D) where u is even and D contains the difference -. Lemma 2.5. Let u be even, u > 8. Then the graph (Zm U {to1, to2, to3}, {1, -}) can be decomposed into 3-suns. Proof. Consider the 3-suns (to1, 2i, 1 + 2i; - + 2 + 2i, - + 2i, to3), i = 0,1, (toi, - - 2, - - 1; -, u - 2, TO3), (to2, 1 + 2i, - + 1 + 2i; 2i, 2 + 2i, to1), i = 0,1, (TO3, - + 1 + 2i, - + 2i; 2i, - + 2 + 2i, TO2), i = - 4 - 2, - - 1 4 1, 0,1,...,- - 1. □ Lemma 2.6. Let u = 0 (mod 12). Then the graph (Zm U {to1, to2, to3, to4}, {1, -}) can be decomposed into 3-suns. 124 Ars Math. Contemp. 16 (2019) 141-155 Proof. Consider the 3-suns (toi, 6«, f + 6«; 4 + 6«, to3, to2), (toi, 1 + 6«, f + 1 + 6«; 5 + 6«, to4, to2), (toi, 2 + 6«, f +2 + 6«; f + 3 + 6«, to4, to3), (to2, 1 + 6«, 6«; f + 3 + 6«, to3, to4), (to2, 2 + 6«, 3 + 6«; f +4 + 6«, 1 + 6«, to4), (to2, 5 + 6«, 4 + 6«; f +5 + 6«, 6 + 6«, 3 + 6«), (to3, 3 + 6«, f + 3 + 6«; 2 + 6«, toi, f + 2 + 6«), (to3, 4 + 6«, f +4 + 6«; f + 6«, to4, toi), (to3, 5 + 6«, f +5 + 6«; f + 1 + 6«, to4, toi), (TO4, f + 1 + 6«, f + 2 + 6«; f +3 + 6«, f +6«, TO2), (TO4, f +4 + 6«, f + 5 + 6«; f +6«, f + 3 + 6«, f + 6 + 6«), for « = 0,1,..., if - 1. □ Lemma 2.7. Let u be even, u > 8. Then the graph (Zf U {toi, to2, ..., to6}, {1, f }} can be decomposed into 3-suns. Proof. Consider the 3-suns (toi, 2«, 1 + 2«; f +2 + 2«, f + 2«, TO3), « = 0,1,..., f - 2, (toi, f - 2, f - 1; f, u - 2, TO3), (to2, 1 + 2«, f + 1 + 2«; 2«, to6, toi), « = 0,1,..., f - 1, (TO3, f + 1 + 2«, f +2«; 2«, TO6, TOf), « = 0,1,..., f - 1, (to4, 1 + 2«, 2 + 2«; f + 2 + 2«, to5, to6), « = 0,1,..., f - 1, (TO5, f + 1 + 2«, f + 2 + 2«; 2 + 2«, TO4, TO6), « = 0,1,..., f - 1. □ Lemma 2.8. Let u = 0 (mod 12). Then the graph (Zf U {toi, to2, ..., to7}, {1, f}} can be decomposed into 3-suns. Proof. Consider the 3-suns (toi, 6«, f + 6«; 4 + 6«, to7, to2), (toi, 1 + 6«, f + 1 + 6«; f + 3 + 6«, to7, to4), (toi, 2 + 6«, f + 2 + 6«; f + 5 + 6«, to5, to2), (to2, 3 + 6«, f + 3 + 6«; 6«, toi, to4), (to2, 4 + 6«, f + 4 + 6«; 2 + 6«, to7, toi), (to2, 5 + 6«, f + 5 + 6«; f + 1 + 6«, toi, to7), (to3, 6«, 1 + 6«; f + 6«, to5, to6), (to3, 2 + 6«, 3 + 6«; f + 2 + 6«, to7, to6), (to3, 4 + 6«, 5 + 6«; f + 5 + 6«, to5, to6), (to4, 1 + 6«, 2 + 6«; f + 6 + 6«, to2, to6), (to4, 3 + 6«, 4 + 6«; f + 4 + 6«, to7, to6), G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 125 (to4, 5 + 6«, 6 + 6«; f + 5 + 6«, to7, to6), (tos, f +6«, f + 1 + 6«; 1 + 6«, TO7, tos), (tos, f + 2 + 6«, u + 3 + 6«; 3 + 6«, to7, tos), (tos, f + 4 + 6«, u + 5 + 6«; 5 + 6«, TO7, f + 6 + 6«), (to6, f + 1 + 6«, f + 2 + 6«; f + 5 + 6«, TO7, to4), (to6, f +3 + 6«, f +4 + 6«; f + 6 + 6«, TO7, tos), for « = 0,1,..., 12 - 1. □ Lemma 2.9. Let u = 0 (mod 12). Then the graph (Zf U {toi, to2, tos}, {1,2, f}} can be decomposed into 3-suns. Proof. Consider the 3-suns (toi, 6«, 1 + 6«; f + 1 + 6«, f + 6«, 3 + 6«), (toi, 2 + 6«, 3 + 6«; f + 5 + 6«, f +2 + 6«, 5 + 6«), (toi, 4 + 6«, 5 + 6«; f + 2 + 6«, f +4 + 6«, 7 + 6«), (toi, f + 3 + 6«, f + 4 + 6«; f + 6«, f +2 + 6«, to2), (to2, 1 + 6«, f + 1 + 6«; f +3 + 6«, 2 + 6«, f + 2 + 6«), (to2, 3 + 6«, 4 + 6«; 2 + 6«, f + 3 + 6«, 6 + 6«), (to2, 5 + 6«, f + 5 + 6«; f +2 + 6«, 6 + 6«, f + 6 + 6«), (tos, 2 + 6«, 6«; 1 + 6«, 4 + 6«, to2), (tos, f + 2 + 6«, f + 6«; 4 + 6«, f + 4 + 6«, TO2), (tos, f + 1 + 6«, f + 3 + 6«; 3 + 6«, f +6«, f + 5 + 6«), (tos, f + 5 + 6«, f + 4 + 6«; 5 + 6«, f +7 + 6«, f + 6 + 6«), for « = 0,1,..., 12 - 1. □ The following lemma "combines" one infinity point with one difference d = f, f such that gcdft d) = 0 (mod 3) (therefore, u = 0 (mod 3)). Lemma 2.10. Let u = 0 (mod 3) and d G Du \ { f, f} such that p = gcdfM d) = 0 (mod 3). Then the graph (Zf U {to}, {d}} can be decomposed into 3-suns. Proof. The subgraph (Zf, {d}} can be decomposed into f cycles of length p = 3q, q > 2. If q > 2, let (xi, x2,..., xSq) be a such cycle and consider the 3-suns (to, X2+3i, X3+3i; X7+3i, Xi+Si, X4+Si), for « = 0, 1, . . . , q - 1 (where the sum is modulo 3q). If q = 2, let (xij),x2j),xSj),x4j),x5j),x6j)), j =0,1,..., f - 1, be the 6-cycles decomposing (Zf, {d}} and consider the 3-suns (nn x(j) x(j)-x(j + i) x(j) x(j)) , x2 , xs ; xi , xi , x4 ), x(j) x(j)-x(j+i) x(j) x(j)) x^ , x6 ; x4 , x4 , xi ), for j =0,1,..., f - 1 (where the sums are modulo f). □ 126 Ars Math. Contemp. 16 (2019) 141-155 Subsequent Lemmas 2.11-2.14 allow to decompose (Zu U H, D), where |H| = 1, 2, 3,5, |D| = 6 - |H| and f G D; here, u and D are any with the unique condition that if D contains at least three differences d^ d2, d3, then d3 = d2 — d1 or di + d2 + d3 = u. Lemma 2.11. Let di, d2, d3, d4, d5 G Du\{ U} such that d3 = d2 — di or di +d2 +d3 = u. Then the graph (Zu U {to}, {di, d2, d3, d4, d5}) can be decomposed into 3-suns. Proof. If d3 = d2 — di, consider the orbit of (di, d2,0; to, d2 + d5, d4) (or (di, d2, 0; to, d2 + d5, —d4), if d2 + d5 = d4) under Zu. If di + d2 + d3 = u, consider the orbit of (—di, d2, 0; to, d2 + d5, d4) (or (—di, d2, 0; to, d2 + d5, —d4), if d2 + d5 = d4) under Zu. □ Lemma 2.12. Let di, d2, d3, d4 G Du \ {u} such that d3 = d2 — di or di + d2 + d3 = u. Then the graph (Zu U {to1, to2}, {di, d2, d3, d4}) can be decomposed into 3-suns. Proof. Consider the orbit of (d1, d2,0; to1, to2, d4) or (—d1, d2,0; to1, to2, d4) under Zu when, respectively, d3 = d2 — d1 or d1 + d2 + d3 = u. □ Lemma 2.13. Let d1, d2, d3 G Du \ {|} such that d3 = d2 — d1 or d1 + d2 + d3 = u. Then the graph (Zu U {to1, to2, to3}, {d1, d2, d3}) can be decomposed into 3-suns. Proof. Consider the orbit of (d1, d2,0; to1, to2, to3) or (—d1, d2,0; to1, to2, to3) under Zu when, respectively, d3 = d2 — d1 or d1 + d2 + d3 = u. □ Lemma 2.14. Let d G Du \ {}, the graph (Zu U {to1, to2, to3, to4, to5}, {d}) can be decomposed into 3-suns. Proof. The subgraph (Zu, {d}) is regular of degree 2 and so can be decomposed into l-cycles, l > 3. Let (x1, x2,..., x;) be a such cycle. Put l = 3q + r, with r = 0,1, 2, and consider the 3-suns with the sums modulo l (toi, xi+3i, X2+3i; X3+3i, TO4, TO5), (TO2, X2+3i, X3+3j; X4+3j, TO4, TO5), (TO3, X3+3j, X4+3j; X5+3j, TO4, TO5), ., q — 2, q > 2, plus the following blocks as the case may be. for i = 0,1,. If r = 0, If r = 1, (TO1, X3q-2, X3q-i; X3q, TO4, TO5), (TO2, X3q_i, X3q; Xi, TO4, TO5), (TO3,X3q,Xi; X2, TO4, TO5) . (TO1, X3q_2, X3q_i; X3q+1, TO4, TO5), (TO2,X3q_1, X3q; Xi, TO4, TOi), (TO3, X3q, X3q+i; X2 , TO4, TO2), (TO5, X3q+i, Xi; X3q, TO4, TO3). G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 127 If r = 2, («1, X3q-2, X3q-i; X3q+2, »4, »5), (»2, X3q-1, X3q; Xl, «4, «5), («3,X3q,X3q+i; X2, »1, »2), (»4, X3q+1, X3q+2; X3q, »1, »3), (»5,X3q+2,Xi; X3q+1, »2, »3). □ Finally, after settling the infinity points by using the above lemmas, if u is large we need to decompose the subgraph (Zu, L), where L is the set of the differences unused (difference leave). Since by applying Lemmas 2.1 - 2.13 it could be necessary to use the differences 1, 2 or 4, while Lemma 2.14 does not impose any restriction, it is possible to combine infinity points and differences in such a way that the difference leave L is the set of the "small" differences, where 1, 2 or 4 could possibly be avoided. Lemma 2.15. Let a e {0,4,8} and u, s be positive integers such that u > 12s + a. Then there exists a decomposition of (Zu, L) into 3-suns, where: i) a = 0 and L = [1, 6s]; ii) a = 4 and L = [3, 6s + 2]; iii) a = 8 and L = [3, 6s + 4] \ {4, 6s + 3}. Proof. i) Consider the orbits (Sj) under Zu, where Sj = (5s + 1 + j, 5s - j, 0; 3s, s, u -2 - 2j), j =0,1,...,s - 1. ii) Consider the orbits in i), where (S0) is replaced with the orbit of (6s+1, 4s, 0; s, 9s, 6s + 2). iii) Consider the orbits in i), where the orbits (S0) and (S1) are replaced with the orbits of (6s + 1, 4s, 0; s, 9s, 6s + 4) and (5s + 2, 5s - 1, 0; 3s, s, 6s + 2). □ 3 The main result Let (X, S) be a 3SS(n) and m = 0,1,4, 9 (mod 12). Lemma 3.1. If (X, S) is embedded in a 3-sun system of order m > n, then m > | n +1. Proof. Suppose (X, S) is embedded in (X', S'), with |X'| = m = n + u (u positive integer). Let c be the number of 3-suns of S' each of which contains exactly i edges in X' \ X. Then J2i=1 i x ci = (2) and J25=1(6 - i)cj = u x n, from which it follows 6c2 + 12c3 + 18c4 + 24c5 +30c6 = "(5"-2"-5) andsou > |n +1 andm > |n +1. □ By previous Lemma: 1. if n = 60k + 5r, r = 0, 5, 8, 9, then m > 84k + 7r +1; 2. if n = 60k + 5r + 1, r = 0, 3,4, 7, then m > 84k + 7r + 3; 3. if n = 60k + 5r + 2, r = 2, 7,10,11, then m > 84k + 7r + 4; 4. if n = 60k + 5r + 3, r = 2, 5, 6, 9, then m > 84k + 7r + 6; 128 Ars Math. Contemp. 16 (2019) 141-155 5. if n = 60k + 5r + 4, r = 0,1,4, 9, then m > 84k + 7r + 7. In order to prove that the necessary conditions for embedding a 3-sun system (X, S) of order n in a 3-sun system of order m = n + u, u > 0 are also sufficient, the graph (Zu U X, Du) will be expressed as a union of edge-disjoint subgraphs (Zu U X, Du) = (Z„ U X, D) U (Zu, L), where L = D„ \ D is the difference leave, and (Z„ U X, d) (if necessary, expressed itself as a union of subgraphs) will be decomposed by using Lemmas 2.1-2.14, while if L = 0, (Zu, L) will be decomposed by Lemma 2.15. To obtain our main result we will distinguish the five cases 1.-5. listed before by giving a general proof for any k > 0 with the exception of a few cases for k = 0, which will be indicated by a star * and solved in Appendix. Finally, note that: a) u = 0,1,4, or 9 (mod 12), if n = 0 (mod 12); b) u = 0, 3, 8, or 11 (mod 12), if n = 1 (mod 12); c) u = 0, 5, 8, or 9 (mod 12), n = 4 (mod 12); d) u = 0, 3,4, or 7 (mod 12), if n = 9 (mod 12). Proposition 3.2. For any n = 60k + 5r, r = 0, 5, 8,9, there exists a decomposition of Kn+u \ Kn into 3-sunsfor every admissible u > 24k + 2r + 1. Proof. Let X = {«^ to2,..., «60fc+5r}, r = 0, 5, 8, 9, and u = 24k + 2r + 1 + h, with h > 0. Set h = 12s + l, 0 < l < 11 (l depends on r), and distinguish the following cases. Case 1: r = 0, 5, 8,9 and l = 0 (odd u). Write (Zu U X, Du) = (Zu U X, D) U (Zu, L), where D = [6s + 1,12k + r + 6s], |D| = 12k + r, and L = [1,6s], and apply Lemmas 2.14 and 2.15. Case 2: r = 0, 9 and l = 8 (odd u). Write (Zu U X, Du) = (Zu U j^i, <^2, «3}, {2, 6s + 3, 6s + 5}) U (Zu U {<4}, {1}) U (Zu U {<5}, {6s + 4}) U (Zu U (X \ {«1, «2, «3, «4, «5}), D') U (Zu, L), where D' = [6s + 6,12k + r + 6s + 4], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.13, 2.10, 2.14 and 2.15. Case 3: r = 5, 8 and l = 4 (odd u). Write (Zu U X, Du) = (Zu U {«1, «2, «3, «4}, {2,4}) U (Zu U {«5}, {1}) U (Zu U (X \ {«1, «2, «3, «4, «5}), D') U (Zu, L), where D' = [6s + 3,12k + r + 6s + 2] \ {6s + 4}, |D'| = 12k + r - 1, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.3, 2.10, 2.14 and 2.15. Case 4: r = 0, 8 and l = 3 (even u). Write (Zu U X, Du) = (Zu U {«1, «2, «3}, {1, u}) U (Zu U {«4, «5}, {2}) U (Zu U (X \ {«1, «2, «3, «4, «5}), D') U (Zu, L), G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 129 where D' = [6s + 3,12k + r + 6s + 1], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.1, 2.14 and 2.15. Case 5: r = 0 and l =11 (even u). Write (Zu u X, Du> = (Zu u {»1, »2, »3}, {1, 2, u}) U (Zu U {»4, »5}, {4, 6s + 3, 6s + 5, 6s + 7}> U (Zu U (X \ {»1, »2, »3, »4, »5}), D'> U (Zu, L), where D' = [6s + 6,12k + 6s + 5] \ {6s + 7}, |D'| = 12k - 1, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.9, 2.12, 2.14 and 2.15. Case 6: r = 5 and l = 1 (even u). Write (Zu U X, Du> = (Zu U {»1, »2, . . . , »6}, {1, u}> U (Zu U {»7, »8, »9, »10}, {2}> U (Zu U (X \ {»1, »2,..., »10}), D'> U (Zu, L>, where D' = [6s + 3,12k + 6s + 5], |D'| = 12k + 3, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.2, 2.14 and 2.15. Case 7: r = 5,9 and l = 9 (even u). Write (Zu U X, Du> = (Zu U {»1, »2, »3}, {1, u}> U (Zu U {»4, »5}, {2, 6s + 3, 6s + 4, 6s + 5}> U (Zu U (X \ {»1, »2, »3, »4, »5}), D'> U (Zu, L>, where D' = [6s + 6,12k + r + 6s + 4], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.12, 2.14 and 2.15. Case 8: r = 8 and l = 7 (even u). Write (Zu U X, Du> = (Zu U {»1, »2, »3}, {1, 2, u}> U (Zu U {»4}, {4}> U (Zu U {»5}, {6s + 5}> U (Zu u (X \ {»1, »2, »3, »4, »5}), D' >U(Zu,L>, where D' = [6s + 3,12k + 6s + 11] \ {6s + 4, 6s + 5}, |D'| = 12k + 7, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.9, 2.10, 2.14 and 2.15. Case 9: r = 9 and l = 5 (even u). Write (Zu U X, Du> = (Zu U {»1, »2, »3}, {1, u}> U (Zu U {»4}, {2}> U (Zu U {»5}, {4}> U (Zu U (X \ {»1, »2, »3, »4, »5}), D'> U (Zu, L>, where D' = [6s + 3,12k + 6s + 11] \ {6s + 4}, |D'| = 12k + 8, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.5, 2.10, 2.14 and 2.15. □ 130 Ars Math. Contemp. 16 (2019) 141-155 Proposition 3.3. For any n = 60k + 5r + 1, r = 0,3,4, 7, there exists a decomposition of \ K„ into 3-sunsfor every admissible u > 24k + 2r + 2. Proof. Let X = (œj, œ2, ..., œ60k+5r+i}, r = 0, 3, 4, 7, and u = 24k + 2r + 2 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases. Case 1: r = 0, 3 and l = 1 (odd u). Write (Zu u X, Du) = (Zu U (œ}, (6s + 2}) U (Zu U (X \ (œ}), D') U (Zu, L), where D' = [6s + 1,12k + r + 6s +1] \ (6s + 2}, |D'| = 12k + r, and L = [1,6s], and apply Lemmas 2.10, 2.14 and 2.15. Case 2: r = 0, 3,4,7 and l = 9 (odd u). Write (Zu U X, Du) = (Zu U (œi, œ2, «a}, (1, 6s + 3, 6s + 4}) U (Zu U («4, œ5, œ6}, (2, 6s + 5, 6s + 7}) U (Zu U (X \ (œi, «2,..., «6}), D') U (Zu, L), whereD' = [6s + 6,12k + r + 6s + 5]\(6s + 7}, |D'| = 12k + r- 1, andL = [3, 6s + 2], and apply Lemmas 2.13, 2.14 and 2.15. Case 3: r = 4*, 7 and l = 5 (odd u). Write (Zu U X, Du) = (Zu U (œi, œ2, œa, »4}, (2,4}) U (Zu U (»5}, (1}) U (Zu U (œ6}, (6s + 8}) U (Zu U (X \ (œi, œ2,..., œ6}), D') U (Zu, L), where D' = [6s + 3,12k + r + 6s + 3] \ (6s + 4,6s + 8}, |D'| = 12k + r - 1, and L = [3,6s + 4] \ (4, 6s + 3}, and apply Lemmas 2.3, 2.10, 2.14 and 2.15. Case 4: r = 0,4 and l = 6 (even u). Write (Zu U X, Du) = (Zu U (œi, œ2, œa}, (1, f}) U ( Zu U (œ 4 , œ 5 , œ 6 } , ( 2 , 6 s + 3 , 6 s + 5 }) U (Zu U (X \ (œi, œ2,..., œ6|), D') U (Zu, L), where D' = [6s + 4,12k + r + 6s + 3]\(6s + 5}, |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.13, 2.14 and 2.15. Case 5: r = 0 and l = 10 (even u). Write (Zu U X, Du) = (Zu U (œi, œ2,..., œ6}, (1, u}) U (Zu U (œr, œg, œg, œio}, (2}) U (Zu U (œii}, (4, 6s + 3, 6s + 5, 6s + 6, 6s + 7}) U (Zu U (X \ (œi, œ2,..., œii}), D') U (Zu, L), where D' = [6s + 8,12k + 6s + 5], |D'| = 12k - 2, and L = [3,6s + 4] \ (4,6s + 3}, and apply Lemmas 2.7, 2.2, 2.11, 2.14 and 2.15. G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 131 Case 6: r = 3,7 and l = 0 (even u). Write (Zu u X, Du> = (Zu u {toi, TO2,..., TOe}, {1, u}> U (Z„ U (X \ {toi, ^2,..., TOe}), D'> U (Z„, L>, where D' = {2} U [6s + 3,12k + r + 6s], |D'| = 12k + r - 1, and L = [3,6s + 2], and apply Lemmas 2.7, 2.14 and 2.15. Case 7: r = 3 and l = 4 (even u). Write (Zu U X, Du > = (Zu U {TOi, TO2, TO3, TO4}, {1, u }> U (Zu U {TO5}, {2}> U (Zu U {TOe}, {6s + 5}> U (Zu U (X \ {TOi, TO2,..., TOe}), D'> U (Z„, L>, where D' = [6s + 3,12k + 6s + 5] \ {6s + 5}, |D'| = 12k + 2, and L = [3, 6s + 2], and apply Lemmas 2.6, 2.10, 2.14 and 2.15. Case 8: r = 4 and l = 2 (even u). Write (Zu U X, Du > = (Zu U {TOi, TO2, TO3, TO4}, {1, f }> U (Zu U {TO5, TOe}, {2}> U (Zu U (X \ {TOi, TO2, . . . , TOe}), D'> U (Zu, L>, where D' = [6s + 3,12k + 6s + 5], |D'| = 12k + 3, and L = [3, 6s + 2], and apply Lemmas 2.6, 2.1, 2.14 and 2.15. Case 9: r = 7 and l = 8 (even u). Write (Zu U X, Du> = (Zu U {TOi, TO2, TO3}, {1, 2, u}> U (Zu U {TO4, TO5, TOe}, {4, 6s + 3, 6s + 7}> U (Zu U (X \ {TOi, TO2,..., TOe}), D'> U (Zu, L>, where D' = [6s + 5,12k + 6s + 11] \ {6s + 7}, |D'| = 12k + 6, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.9, 2.13, 2.14 and 2.15. □ Proposition 3.4. For any n = 60k + 5r + 2, r = 2, 7,10,11, there exists a decomposition of Kn+u \ Kn into 3-sunsfor every admissible u > 24k + 2r + 2. Proof. Let X = {to1, to2,..., TOe0k+5r+2}, r = 2, 7,10,11, and u = 24k + 2r + 2 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases. Case 1: r = 2,11 and l = 3 (odd u). Write (Zu U X, Du> = (Zu U {TOi}, {6s + 2}> U (Zu U {TO2}, {6s + 4}> U (Zu U (X \{TOi, TO2}),D'>U(Zu,L>, 132 Ars Math. Contemp. 16 (2019) 141-155 where D' = [6s + 1,12k + r + 6s + 2] \{6s + 2,6s + 4}, |D'| = 12k + r, and L = [1, 6s], and apply Lemmas 2.10, 2.14 and 2.15. Case 2: r = 2, 7,10,11 and l = 7 (odd u). Write (Z„ U X, D„} = (Z„ U {TO1, TO2}, {1, 2, 6s + 3, 6s + 4}} U (Z„ U (X \ {toi, TO2}), D'} U (Z„, L}, where D' = [6s + 5,12k + r + 6s + 4], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.12, 2.14 and 2.15. Case 3: r = 7,10 and l =11 (odd u). Write (Z„ U X, D„} = (Z„ U {toi, TO2, TO3}, {1, 6s + 3, 6s + 4}} U (Z„ U {TO4, TO5, TOe}, {2, 6s + 5, 6s + 7}} U (Z„ U {to7}, {6s + 8}} U (Z„ U (X \ {TOi, TO2, ..., tot}), D'} U (Z„, L}, where D' = [6s + 6,12k + r + 6s + 6] \ {6s + 7,6s + 8}, |D'| = 12k + r - 1, and L = [3,6s + 2], and apply Lemmas 2.13, 2.10, 2.14 and 2.15. Case 4: r = 2 and l = 6 (even u). Write (Z„ U X, D„} = (Z„ U {TOi, TO2, TO3, TO4}, {1, u}} U (Z„ U {TO5, TOe, Tot}, {2, 6s + 3, 6s + 5}} U (Z„ U (X \ {TOi, TO2, ..., tot}), D'} U (Z„, L}, where D' = [6s + 4,12k + 6s + 5] \ {6s + 5}, |D'| = 12k + 1, and L = [3, 6s + 2], and apply Lemmas 2.6, 2.13, 2.14 and 2.15. Case 5: r = 2,10 and l = 10 (even u). Write (Z„ U X, D„} = (Z„ U {TOi, TO2,..., TOe}, {1, u}} U (Z„ U {tot}, {2, 6s + 3, 6s + 4, 6s + 5, 6s + 6}} U (Z„ U (X \ {TOi, TO2,..., tot}), D'} U (Z„, L}, where D' = [6s + 7,12k + r + 6s + 5], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.11, 2.14 and 2.15. Case 6: r = 7,11 and l = 4 (even u). Write (Z„ U X, D„} = (Z„ U {TOi, TO2, TO3}, {1, u}} U (Z„ U {TO4, TO5, TOe, Tot}, {2,4}} U (Z„ U (X \ {TOi, TO2,..., Tot}), D'} U (Z„, L}, G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 133 where D' = [6s + 3,12k + r + 6s + 2] \ {6s + 4}, |D'| = 12k + r - 1, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.5, 2.3, 2.14 and 2.15. Case 7: r = 7 and l = 8 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2, tos}, {1, u}> u (Zu U {TO4, TO5, TOe}, {2, 6s + 3, 6s + 5}) U (Zu U {tot}, {6s + 7}) U (Zu U (X \ {toi, ^2,..., tot}), D') U (Zu, L), where D' = [6s + 4,12k + 6s + 11] \{6s + 5,6s + 7}, |D'| = 12k + 6, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.13, 2.10, 2.14 and 2.15. Case 8: r =10 and l = 2 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2,..., TOe}, {1, u}) U (Zu U {tot}, {2}) U (Zu U (X \ {toi, TO2,..., tot}), D') U (Zu, L), where D' = [6s + 3,12k + 6s + 11], |D'| = 12k + 9, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.10, 2.14 and 2.15. Case 9: r =11 and l = 0 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2, ..., tot}, {1, u }) U (Zu U (X \ {toi, TO2,..., tot}), D') U (Zu, L), where D' = {2} U [6s + 3,12k + 6s + 11], |D'| = 12k + 10, and L = [3,6s + 2], and apply Lemmas 2.8, 2.14 and 2.15. □ Proposition 3.5. For any n = 60k + 5r + 3, r = 2,5,6, 9, there exists a decomposition of Kn+u \ Kn into 3-sunsfor every admissible u > 24k + 2r + 3. Proof. Let X = {to1, to2,..., TOeok+5r+3}, r = 2, 5, 6, 9, and u = 24k + 2r + 3 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases. Case 1: r = 2,5, 6, 9 and l = 4 (odd u). Write (Zu U X, Du) = (Zu U {TO1, TO2, TO3}, {1, 6s + 3, 6s + 4}) U (Zu U (X \ {TO1, TO2, TO3}) where D' = {2} U [6s + 5,12k + r + 6s + 3], |D'| = 12k + r, and L apply Lemmas 2.13, 2.14 and 2.15. Case 2: r = 2,5 and l = 8 (odd u). Write (Zu U X, Du) = (Zu U {to1, to2}, {1, 6s + 3, 6s + 4, 6s + 5}) U (Zu U {TO3}, {2}) U (Zu U (X \ {TO1, TO2, tos}), D') U (Zu, L), , D' )U(Zu,L), = [3,6s + 2], and 134 Ars Math. Contemp. 16 (2019) 141-155 where D' = [6s + 6,12k + r + 6s + 5], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.12, 2.10, 2.14 and 2.15. Case 3: r = 6, 9 and l = 0 (odd u). If s = 0, then write (Zu u X, D„) = (Zu u {to1, TO2,..., TOg}, {1, s}) u (Z„U (X \{TO1, TO2,..., TOg}),D'), where D' = [2,12k + r +1] \ {u }, |D'| = 12k + r - 1, and apply Lemmas 2.4 and 2.14. If s > 0, then write (Zu U X, Du) = (Zu U {TO1, TO2, tos}, {1, 5s, 5s + 1}) U (Zu U {TO4, TO5, Tog}, {2, 6s + 1, 6s + 3}) U (Zu u {to7}, {6s + 2}) U (Zu U {TOg}, {6s + 4}) U (Zu U (X \ {toi, ..., TOg}), D') U (Zu, L), where D' = {2s + 1,4s} U [6s + 5,12k + r + 6s + 1], |D'| = 12k + r - 1, and L = [3, 6s] \ {2s + 1,4s, 5s, 5s + 1}, and apply Lemmas 2.13, 2.10 and 2.14 to decompose the first five subgraphs, while to decompose the last one apply Lemma 2.15 i) and delete the orbit (So). Case 4: r = 2, 6 and l = 1 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2, TO3}, {1, f}) U (Zu U (X \ {TO1, TO2, TOs}), D') U (Zu, L), where D' = {2} U [6s + 3,12k + r + 6s + 1], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.5, 2.14 and 2.15. Case 5: r = 2* and l = 5 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2,..., tog}, {1, u}) U (Zu U {TO7, TOg, TO9, TO10}, {2}) U (Zu U {to 11, to 12, to 13}, {4, 6s + 3, 6s + 7}) U (Zu U (X \ {TO1, TO2, . . . , TO 13}), D') U (Zu, L), whereD' = [6s + 5,12k + 6s + 5]\{6s + 7}, |D'| = 12k, andL = [3, 6s + 4]\{4, 6s + 3}, and apply Lemmas 2.7, 2.2, 2.13, 2.14 and 2.15. Case 6: r = 5, 9 and l = 7 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2,..., tog}, {1, u}) U (Zu U {to7, to8}, {2, 6s + 3, 6s + 4, 6s + 5}) U (Zu U (X \ {TO1, TO2, . . . , TOg}), D') U (Zu, L), G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 135 where D' = [6s + 6,12k + r + 6s + 4], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.12, 2.14 and 2.15. Case 7: r = 5 and l =11 (even u). Write (Zu U X, Du} = (Zu U {toi, TO2, TO3, TO4}, {1, U}} U (Zu U {TO5, TO6}, {2, 6s + 3, 6s + 5, 6s + 6}} U (Z„ U {TO7}, {4}} U (Zu U {TOg}, {6s + 7}} U (Zu U (X \ {to 1, TO2,..., TO8}), D'} U (Zu, L}, where D' = [6s + 8,12k + 6s + 11], |D'| = 12k + 4, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.6, 2.12, 2.10, 2.14 and 2.15. Case 8: r = 6 and l = 9 (even u). Write (Zu U X, Du} = (Zu U {TO1, TO2, TO3}, {1, u}} U (Zu U {TO4, TO5, TO6}, {2, 6s + 3, 6s + 5}} U (Z„ U {TO7}, {4}} U (Z„ U {TOg}, {6s + 7}} U (Z„ U (X \ {toi, to2, ..., TOg}), D'} U (Z„, L}, where D' = [6s + 6,12k + 6s + 11] \ {6s + 7}, |D'| = 12k + 5, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.5, 2.13, 2.10, 2.14 and 2.15. Case 9: r = 9 and l = 3 (even u). Write (Zu U X, Du} = (Zu U {toi, TO2, TO3}, {1, 2, u}} U (Zu U (X \ {toi, TO2, TO3}), D'} U (Zu, L}, where D' = [6s + 3,12k + 6s + 11], |D'| = 12k + 9, and L = [3, 6s + 2], and apply Lemmas 2.9, 2.14 and 2.15. □ Proposition 3.6. For any n = 60k + 5r + 4, r = 0,1,4, 9, there exists a decomposition of Kn+„ \ Kn into 3-sunsfor every admissible u > 24k + 2r + 3. Proof. Let X = (œj, œ2,..., œ60k+5r+4}, r = 0,1,4, 9, and u = 24k + 2r + 3 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases. Case 1: r = 0,1*, 4, 9 and l = 2 (odd u). Write (Zu U X, Du} = (Zu U{TO1, TO2, TO3, TO4}, {2,4}}U (Zu U (X \ {TO1, TO2, TO3, TO4}), D'} U (Zu, L}, where D' = {1, 6s + 3} U [6s + 5,12k + r + 6s + 2], |D'| = 12k + r, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.3, 2.14 and 2.15. 136 Ars Math. Contemp. 16 (2019) 141-155 Case 2: r = 0, 9 and l = 6 (odd u). Write (Zu U X, D„) = (Zu U {to1, TO2, TO3}, {1, 6s + 3, 6s + 4}) U (Z„ U {TO4}, {2}) U (Zu U (X \ {toi, to2, , TO4}), D') U (Zu, L), where D' = [6s + 5,12k + r + 6s + 4], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.13, 2.10, 2.14 and 2.15. Case 3: r = 1,4 and l = 10 (odd u). Write (Zu U X, Du) = (Zu U {to1, to2}, {1, 6s + 3, 6s + 5, 6s + 6}) U (Zu U {TO3}, {2}) U (Zu U {TO4}, {6s + 4}) U (Zu U (X \ {TO1, TO2, TO3, TO4}), D') U (Z„, L), where D' = [6s + 7,12k + r + 6s + 6], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.12, 2.10, 2.14 and 2.15. Case 4: r = 0,4 and l = 5 (even u). Write (Zu U X, Du) = (Zu U {^1, TO2,..., TOe}, {1, u}) U (Zu U {to7, to8, to9}, {2, 6s + 3, 6s + 5}) U (Z„ U (X \ {^1, TO2,..., to9}), D') U (Z„, L), whereD' = [6s + 4,12k + r + 6s + 3]\{6s + 5}, |D'| = 12k + r- 1, andL = [3, 6s + 2], and apply Lemmas 2.7, 2.13, 2.14 and 2.15. Case 5: r = 0 and l = 9 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2, TO3, TO4}, {1, u}) U (Zu U {TO5, TOe, to7}, {2, 6s + 3, 6s + 5}) U (Zu U {TOg}, {4}) U (Zu U {TO9}, {6s + 7}) U (Zu U (X \ {TO1, to2, ..., TO9}), D') U (Zu, L), whereD' = [6s+6,12k+6s+5]\{6s+7}, |D'| = 12k-1,andL = [3,6s+4]\{4,6s+3}, and apply Lemmas 2.6 , 2.13, 2.10, 2.14 and 2.15. Case 6: r = 1 and l = 7 (even u). Write (Zu U X, Du) = (Zu U {TO1, TO2, . . . , TO7}, {1, u}) U (Zu U {TOg, TO9}, {2,4, 6s + 3, 6s + 5}) U (Zu U (X \ {TO1, TO2, ..., TO9}), D') U (Zu, L), where D' = [6s + 6,12k + 6s + 5], |D'| = 12k, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.8, 2.12, 2.14 and 2.15. G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 137 Case 7: r =1,9 and l = 11 (even u). Write (Z„ u X, D„> = (Z„ u {»1, »2, »3}, {1, u}) u (Z„ U {»4}, {2, 4, 6s + 3, 6s + 5, 6s + 6}> U (Z„ U (X \ {»1, »2, »3, »4}), D'> U (Z„, L>, where D' = [6s + 7,12k + r + 6s + 6], |D'| = 12k + r, and L = [3, 6s + 4] \{4,6s + 3}, and apply Lemmas 2.5, 2.11, 2.14 and 2.15. Case 8: r = 4 and l = 1 (even u). Write (Z„ U X, D„> = (Z„ U {»1, »2, »3, »4}, {1, u}> u (Z„ U (X \ {»1, »2, »3, »4}), D'> U (Z„, L>, where D' = {2} U [6s + 3,12k + 6s + 5], |D'| = 12k + 4, and L = [3,6s + 2], and apply Lemmas 2.6, 2.14 and 2.15. Case 9: r = 9 and l = 3 (even u). Write (Z„ U X, D„> = (Z„ U {»1, »2, »3}, {1, u}> U (Z„ U {»4}, {2}> U (Z„ U (X \ {»1, »2, »3, »4}), D'> U (Z„, L>, where D' = [6s + 3,12k + 6s + 11], |D'| = 12k + 9, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.10, 2.14 and 2.15. □ Combining Lemma 3.1 and Propositions 3.2 - 3.6 gives our main theorem. Theorem 3.7. Any 3SS(n) can be embedded in a 3SS(m) if and only if m > 7n +1 or m = n. References [1] D. E. Bryant and C. A. Rodger, The Doyen-Wilson theorem extended to 5-cycles, J. Comb. Theory Ser. A 68 (1994), 218-225, doi:10.1016/0097-3165(94)90101-5. [2] D. E. Bryant and C. A. Rodger, On the Doyen-Wilson theorem for m-cycle systems, J. Combin. Des. 2 (1994), 253-271, doi:10.1002/jcd.3180020405. [3] J. Doyen and R. M. Wilson, Embeddings of Steiner triple systems, Discrete Math. 5 (1973), 229-239, doi:10.1016/0012-365x(73)90139-8. [4] C.-M. Fu, Y.-L. Lin, S.-W. Lo, Y.-F. Hsu and W.-C. Huang, The Doyen-Wilson theorem for bull designs, Discrete Math. 313 (2013), 498-507, doi:10.1016/j.disc.2012.11.023. [5] H.-L. Fu and C. C. Lindner, The Doyen-Wilson theorem for maximum packings of Kn with 4-cycles, Discrete Math 183 (1998), 103-117, doi:10.1016/s0012-365x(97)00080-0. [6] H.-L. Fu, C. C. Lindner and C. A. Rodger, Two Doyen-Wilson theorems for maximum packings with triples, Discrete Math. 178 (1998), 63-71, doi:10.1016/s0012-365x(97)81817-1. 138 Ars Math. Contemp. 16 (2019) 141-155 [7] H. Gao and J. Wang, Doyen-Wilson theorem for perfect hexagon triple systems, Discrete Math. 311 (2011), 1006-1014, doi:10.1016/j.disc.2011.03.004. [8] W.-C. Huang and W.-C. Yang, The Doyen-Wilson theorem for extended directed triple systems, Ars Combin. 84 (2007), 77-83. [9] G. Lo Faro and A. Tripodi, The Doyen-Wilson theorem for kite systems, Discrete Math. 306 (2006), 2695-2701, doi:10.1016/j.disc.2006.03.074. [10] G. Lo Faro and A. Tripodi, Embeddings of A-fold kite systems, A > 2, Australas. J. Combin. 36 (2006), 143-150, https://ajc.maths.uq.edu.au/pdf/3 6/ajc_v36_p143. pdf. [11] R. Peltesohn, Eine Losung der beiden Heffterschen Differenzenprobleme, Compositio Math. 6 (1939), 251-257, http://www.numdam.org/item?id=CM_1939_6_251_0. [12] G. Stern and H. Lenz, Steiner triple systems with given subspaces; another proof of the DoyenWilson-theorem, Boll. Un. Mat. Ital. A Serie 5 17 (1980), 109-114. [13] J. Wang, Perfect dexagon triple systems with given subsystems, Discrete Math. 309 (2009), 2930-2933, doi:10.1016/j.disc.2008.07.004. [14] J.-X. Yin and B.-S. Gong, Existence of G-designs with | V(G)| = 6, in: W. D. Wallis, H. Shen, W. Wei and L. Zhu (eds.), Combinatorial Designs and Applications, Marcel Dekker, New York, volume 126 of Lecture Notes in Pure and Applied Mathematics, 1990 pp. 201-218. G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems 139 Appendix • n = 21, u = 12s +15 Write (Zu u X, D„) = (Zu u {TO1, TO2, TOS, TO4}, {2,4}) U (Zu U {to5}, {1}) U (Zu U {tob}, {6s + 7}) U (Zu U (X \ {toi, TO2,..., tob}), {6s + 3, 6s + 5, 6s + 6}) U (Zu, L), where L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.3, 2.10, 2.14 and 2.15. • n =13, u = 12s +12 Write (Zu U X, Du) = (Zu U {toi, TO2,..., Tob}, {1,6s + 6}) U (Zu U {TO7, TOg, TOq, TO10}, {2}) U (Zu U {TO11, TO12, to 13}, {4, 6s + 3, 6s + 5}) U (Zu, L), where L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.7, 2.2, 2.13 and 2.15. • n = 9, u = 12s + 7 Write (Zu U X, Du) = (Zu U {TO1, TO2, Tos, TO4}, {2,4}) U (Zu U {TO5, TO6, TO7, TOg, TOq}, {1}) U (Zu, L), where L = [3,6s + 3] \ {4}, and apply Lemmas 2.3, 2.14 and decompose (Zu, L) as in Lemma 2.15 iii), taking in account that |6s + 4|12s+7 = 6s + 3. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 141-155 https://doi.org/10.26493/1855-3974.1460.fd6 (Also available at http://amc-journal.eu) The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace Irene Sciriha * Department of Mathematics, Faculty of Science, University of Malta, Msida, Malta Didar A. Ali Department of Mathematics, Faculty of Science, University ofZakho, Duhok, Iraq John Baptist Gauci Department of Mathematics, Faculty of Science, University of Malta, Msida, Malta Khidir R. Sharaf Department of Mathematics, Faculty of Science, University ofZakho, Duhok, Iraq Received 9 August 2017, accepted 8 September 2018, published online 5 November 2018 Two connected labelled graphs Hi and H2 of nullity one, with identical one-vertex deleted subgraphs Hi - zi and H2 - z2 and having a common eigenvector in the nullspace of their 0-1 adjacency matrix, can be overlaid to produce the superimposition Z. The graph Z is Hi + z2 and also H2 + zi whereas Z + e is obtained from Z by adding the edge {zi, z2}. We show that the nullity of Z cannot take all the values allowed by interlacing. We propose to classify graphs with two chosen vertices according to the type of the vertices occurring by using a 3-type-code. Out of the 27 values it can take, only 9 are hypothetically possible for Z, 8 of which are known to exist. Moreover, the SSP molecular model predicts conduction or insulation at the Fermi level of energy for 11 possible types of devices consisting of a molecule and two prescribed connecting atoms over a small bias voltage. All 11 molecular device types are realizable for general molecules, but the structure of Z and of Z + e restricts the number to just 5. Keywords: Nullity, core vertices, key-graphs, superimposition, circuit. Math. Subj. Class.: 05C50, 15A18, 47N70 * Corresponding author. Homepage: http://staff.um.edu.mt/isci1/ E-mail addresses: irene.sciriha-aquilina@um.edu.mt (Irene Sciriha), didar.ali@uoz.edu.krd (Didar A. Ali), john-baptist.gauci@um.edu.mt (John Baptist Gauci), khidir.sharaf@uoz.edu.krd (Khidir R. Sharaf) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 142 Ars Math. Contemp. 16 (2019) 141-155 1 Introduction The graphs we consider are simple, that is they are undirected with no multiple edges or loops. The 0-1 adjacency matrix G = (a j) of a labelled graph G on n vertices is a n x n matrix such that a j = 1 if there is an edge between the vertices i and j, and aij = 0 otherwise. The degree of a vertex v is the number of non-zero entries in the vth row (or column) of G. A graph is singular if G has zero as an eigenvalue, and nonsingular otherwise. The multiplicity of zero in the spectrum of G is the nullity n = n(G) of the graph G. A kernel eigenvector x of G is a nonzero vector that satisfies Gx = 0. The nullspace ker(G) of G is generated by a basis of n linearly independent kernel eigenvectors. Thus, a graph G is singular if and only if dim(ker(G)) > 1. A core vertex (CV) of G corresponds to a nonzero entry in some kernel eigenvector. The set of CVs is an invariant of G, that is, it is independent of the basis chosen for ker(G) [14, 15, 16]. A vertex which is not a CV is a core-forbidden vertex (CFV), recently referred to as a Fiedler vertex [1, 10]. Proposition 1.1 characterizes a CV in a singular graph, and Corollary 1.2 is its direct consequence for graphs of nullity one. Proposition 1.1 ([18]). Let G + u be a graph obtained from G by adding a vertex u. Then n(G + u) = n(G) + 1 if and only if u is a CV of G + u. Corollary 1.2 ([17]). Let v be a CV of a graph G of nullity one. Then the graph G — v is non-singular. We make use of a result on the nullity of graphs derived from Cauchy's Interlacing Theorem for real symmetric matrices. Theorem 1.3 ([11, p. 119]). Let v be any vertex of a graph G on n > 2 vertices. Then n(G) — 1 < n(G — v) < n(G) + 1. Theorem 1.3 permits the nullity of a graph to change by at most one on the deletion or addition of a vertex. Thus, a vertex u in a graph G can be one of three types, depending on the difference of the nullity of G — u from the nullity of G. Following the terminology used in [4], a vertex u is a CV, a middle core-forbidden vertex (CFVm¡d) or an upper core-forbidden vertex (CFVupp) if the nullity of the graph G — u obtained from G upon deleting the vertex u is n(G) — 1, n(G), or n(G) + 1, respectively. It follows from Proposition 1.1 that CFVs are vertices corresponding to a zero entry in each kernel eigenvector in the nullspace of G. For the eigenvalue zero, CFVs were renamed F-vertices. For the specific case of CFVupp, they were renamed and P-vertices [2]. Whether electricity flows through a molecule or not is mainly determined by the types of the two vertices (atoms of the molecule) chosen as terminals with a bias voltage across them [5]. From the definitions of the possible types of vertices in a graph, the following result is immediate. Lemma 1.4. Let Hi and H2 be two graphs such that n(Hi) = n(H2). Then n(Hi — zi) = n(H2 — z2) if and only if zi and z2 are of the same type in Hi and in H2, respectively. 1.1 Superimpositions To explore the structure of singular graphs, basic subgraphs of nullity one that are found in singular graphs are constructed in [14, 15, 16]. In Proposition 4.3 of [16], it is proved that a I. Sciriha et al.: The conductivity of superimposed key-graphs with a common ... 143 singular graph of nullity n has n induced subgraphs of nullity one having the least possible number of vertices (called singular configurations). The kernel eigenvectors are key to determining the substructures that make a singular graph. Focus is placed on singular graphs of nullity one; otherwise distinct singular configurations which are induced subgraphs in a singular graph of nullity more than one may be masked by others belonging to linearly independent kernel eigenvectors. The vertices of a singular configuration corresponding to the nonzero entries in a kernel eigenvector x are the CVs of G and the remaining vertices, if any, are CFVupp [16]. By Proposition 1.1, deleting a CV reduces the nullity, whereas deleting an CFVupp increases the nullity. The question then arises: what are the conditions that need to be satisfied by a graph H of nullity one so that, for some vertex v, the graph H + v retains nullity one and has the same nonzero entries of a kernel eigenvector as H ? The investigations in this paper stem from the quest to answer this question. First we fix some notation. Two labelled graphs Gi and G2 are identical if they are isomorphic to a labelled graph G and have the same labelling as G; we write G1 = G2 = G. We consider pairs of graphs of nullity one which have a common kernel eigenvector for some labelling of their vertices, and such that each of the two graphs have a vertex which, when deleted, yields two identical graphs. One such example is illustrated in Figure 1, where x = (1, -1,1, -1,0,0,0)* is a kernel eigenvector of both H1 and H2 with the labelling shown, such that when the vertex labelled 7 is deleted from each, the resulting graphs are identical. 5 ¿> à 6 Figure 1: A pair of non-isomorphic graphs G + 7 of nullity one having a common kernel eigenvector x = (1, -1,1, -1,0,0, 0)4. 1 2 3 4 1 3 6 7 5 2 If two graphs H and H2 are not isomorphic, but the deletion of a vertex zi of H yields a graph identical to that obtained by deleting a vertex z2 from H2, then the difference in the dimensions of the nullspaces of Hi and H2 is bounded as given in the following theorem. Theorem 1.5. Let Hi and H2 be two graphs having vertices zi and z2, respectively, such that G = Hi — zi = H2 — z2. Then |n(Hi) — n(H2)| = 2 when one of the vertices is a CV and the other is an CFVupp, and |n(Hi) — n(H2)| < 1, otherwise. Proof. By Theorem 1.3, n(G)-1 < n(Hi) < n(G) + 1 andn(G)-1 < ^(#2) < n(G) + 1. Thus, |n(Hi)-n(H2)| < 2. Equality holds only when, without loss of generality, n(Hi) = n(G) - 1 and n(H2) = n(G) + 1, in which case zi is an CFVupp in Hi and z2 is a CV in H2. □ In the sequel, let Hi and H2 be two connected labelled graphs of order n > 3 whose 0-1 adjacency matrix has nullity one with a common kernel eigenvector x such that, for 144 Ars Math. Contemp. 16 (2019) 141-155 some vertex zi in Hi and some vertex z2 in H2, Hi - zi = H2 - z2 = G. The graphs Hi and H2 are termed key-graphs. It follows immediately that the label of z1 in H1 is the same as that of z2 in H2. We choose the labels of the vertices zi and z2 to be the last in the two graphs. The superimposition of the key-graphs Hi and H2 is the graph Z obtained from G by adding both vertices zi and z2 adjacent to the same neighbours as those of zi in Hi and z2 in H2. The graph Z + e is obtained from Z by adding the edge e = ziz2. In the next section, we look at some examples so that the concept of superimpositions and its possible effects on the type of vertices of a graph becomes clearer. 1.2 Motivation A conjugated hydrocarbon molecule has a n-system where each carbon atom contributes a delocalized electron in the neutral molecule. The Huckel/Tight-Binding model simplifies Schrodinger's equation to Ax = Ex where A is the adjacency matrix of the carbon skeleton of the molecule, x represents a molecular orbital and E is the orbital energy. Since carbon has a valency of four, chemical graphs for n-systems have at most three sigma bonds per atom (edges meeting at any vertex). In this article we extend our study to any graph where the vertex degree (or valency) can be larger than three. In chemistry, the role of the electrons in the molecule is crucial in determining the physical and chemical properties of the molecule. The discrete energy levels that an electron may occupy within a molecule are the solutions to Schrodinger's time-independent equation in quantum mechanics. The wave function as a solution of Schrodinger's equation predicts the electron probability density, which in Huckel theory is a sum of orbital densities. The Hamiltonian for the n-atomic molecular system turns out to be a linear function of the 0-1 n x n adjacency matrix G of the labelled molecular graph G, whose eigenvalues give the energy E of the electron orbitals. The non-zero entries of G correspond to the sigma bonds between pairs of atoms. In this article we investigate the change in nullity when forming Z and Z + e. The conduction of electricity through a molecular graph with a bias electrical potential across two vertices L and R depends on the nullities of G and of three of its induced subgraphs obtained by deleting L and R separately and jointly [4, 13]. Note that the deletion of a CFV typically preserves the chemical nature of the graph (unless it is a cut vertex), but addition typically does not. In Section 5, conductivity of molecular devices is discussed for examples of molecular graphs of the form Z and Z + e. In Figure 2, the four vertices 1, 2,3 and 4 are CVs in both H and Z, but the vertex 5 is a CFVupp in H, whereas each of the vertices 5 and 6 is a CFVmid in Z. Each vertex of Z + e is a CV. Both H and Z have nullity one and, since H has a kernel eigenvector (1,1, -1, —1,0)4 and Z has a kernel eigenvector (1,1, -1, —1,0,0)4, there is a kernel eigenvector of H - 5 with the same nonzero entries as for H and Z. It is interesting to note that Z is obtained by superimposing two isomorphic copies Hi and H2 of the singular configuration H, but Z itself is not a singular configuration. It is thus natural to ask whether Hi and H2 need to be isomorphic (as in the example discussed above) to retain nullity one in Z obtained from H1 by adding the vertex z2. Also, is this a condition that H1 and H2 must satisfy so that the nonzero entries of a kernel eigenvector are preserved in Z? The graph Z shown in Figure 3 is obtained by superimposing the two graphs of Fig- I. Sciriha et al.: The conductivity of superimposed key-graphs with a common ... 145 1 2 1 2 1 2 4 3 4 3 4 3 H Z Z + e Figure 2: Two copies of H, of nullity one, are induced subgraphs of the superimposition Z, also of nullity one. The graph Z + e is of nullity two. For all three graphs, there is a vector in the respective nullspace with the same nonzero part 1,1, -1, -1 associated with the first four labelled vertices. ure 1. Adding the edge e between z1 = 7 and z2 = 8 produces Z+e. The nullity of Z is two whereas that of Z + e is one. This example shows that H1 and H2 need not be isomorphic for the nullity to be one in Z + e. A kernel eigenvector of Z + e is (1, -1,1, —1,0,0,0,0)4, and thus the nonzero entries of a kernel eigenvector of H1 and H2 are also preserved, even though H1 and H2 are not isomorphic. Figure 3: The graph Z + e with nullity one, having a kernel eigenvector (1, -1,1, -1,0,0,0,0)4, is obtained from the superimposition Z (which has nullity two) of the graphs in Figure 1. Observe that the nullities of Z and of Z + e are different in the two examples discussed above. The vertices z1 in H1 and z2 in H2 in both examples are CFVupp. They become CFVmid in Z in the example of Figure 2 and also in Z + e in the example of Figure 3. However, z1 and z2 become CV in Z + e in the example of Figure 2 and also in Z in the example of Figure 3. The following results follow immediately from the definitions of CFVmid and CV. Proposition 1.6. The vertices z1 and z2 are CFVmid in the superimposition Z (respectively in Z + e) if and only if n(Z ) = 1 (respectively n(Z + e) = 1). Proposition 1.7. The vertices z1 and z2 are CV in the superimposition Z (respectively in Z + e) if and only if n(Z ) = 2 (respectively n(Z + e) = 2). As we shall show, graphs satisfying Propositions 1.6 and 1.7 exist. However, is it possible that both z1 and z2 be CFVupp in Z or in Z + e? Do the types of the vertices z1 146 Ars Math. Contemp. 16 (2019) 141-155 and z2 determine the type in Z or Z + e? We shall investigate all possible combinations of the vertex type of z1 in H and z2 in H2. By Lemma 1.4, the type of the vertex z1 in H1 and of the vertex z2 in H2 is the same. Moreover, as we shall see in Lemmas 2.2 and 3.1, vertices z1 and z2 are of the same type in Z and of the same type in Z + e (the type in the latter graph Z + e possibly different from that in the former Z). We thus propose a 3-type-code1 where a type is denoted by: 1. C if it corresponds to a core vertex; 2. M if it corresponds to a middle core-forbidden vertex; and 3. U if it corresponds to an upper core-forbidden vertex. The code consists of an ordered string of three types and, thus, it has three available positions, namely y1, y2 and y3. Each of the positions y1, y2 and y3 is filled with the symbol C, M or U, depending on the type of the vertices z1 and z2 in the key-graphs, in Z and in Z+e, in that order. The 3-type-code presents 27 classes of graphs. Algebraic considerations show that only 9 may exist. The case when the two vertices z1 and z2 are both CFVs in the respective key-graphs is discussed in Section 2, yielding 8 possible classes of graphs. In Section 3, the case when they are both CVs produces just one class of graphs. For the graphs {Z} and {Z + e}, what factors determine that the nullity of a graph remains unchanged on deleting a vertex? When does the type of a pair of adjacent vertices remain unchanged after deleting the edge between them? These questions are answered in Section 4. Chemical implications for the conductivity of a molecule which has a graph that is a superimposition are discussed in Section 5. 2 Core-forbidden vertices in the key-graphs In this section, the vertices zi and z2 are CFVs in the key-graphs Hi and H2, respectively. Thus, the last entry of the common kernel eigenvector x of H1 and of H2 (which corresponds to z1 and z2) is zero. We write x = where v = 0. Letting z1 and z2 denote the characteristic vectors representing the adjacencies of z1 and z2 to the vertices of G, we obtain and Hix = Hi — H2X = H2 H- = f G Zi 0 J = V zi4 0 = f G Z2 0 J V Z24 0 v "0 v "F Gv Zi4v Gv Z2*v 0 0, (2.1) (2.2) for some v = 0. The following lemma explores the nullspaces of Z and of Z + e. On adding a vertex to the key graph Hi, of nullity one, the graph Z or Z + e produced is never non-singular. Lemma 2.1. If z1 and z2 are CFVs in H1 and in H2, respectively, then (i) (x, 0)4 = (v, 0, 0)4 is a kernel eigenvector of both Z and Z + e; 1 Different three letter acronyms are proposed in [6, 7] to classify classes of molecular graphs as conductors or insulators with respect to the graph-theoretical distance between two connecting vertices of the graph across which there is a small bias voltage. I. Sciriha et al.: The conductivity of superimposed key-graphs with a common ... 147 (ii) 1 < ) < 2 and 1 < n(Z + e) < 2. Proof. Let Z be the adjacency matrix of the graph Z and let W be the adjacency matrix of Z + e, where z1 and z2 are respectively the nth and (n + 1)th labelled vertices of Z and of Z + e. (i) Since and Z W G Zi z2 0 I = zi4 0 0 0 ^ Z24 0 0 G zi z2 0 I = zi4 0 1 0 ^ z24 1 0 Gv ziiv Z2tV Gv z^v Z2tV then by (2.1) and (2.2), (v, 0,0)* is a kernel eigenvector of Z and of W. (ii) By Theorem 1.3, 0 < n(Z) < 2 and 0 < n(Z + e) < 2. From (i) above, n(Z) > 1 and n(Z + e) > 1. Thus, 1 < n(Z) < 2 and 1 < n(Z + e) < 2. □ Next we show that the vertices z1 and z2 must be of the same type in each of the graphs Z and in Z + e, and that they cannot be CFVupp. Lemma 2.2. Let z1 and z2 be CFVs in H1 and in H2, respectively. Then in each of the graphs Z and Z + e, the two vertices z1 and z2 are either both CFVmid or both CV. Proof. Suppose first that z1 and z2 are not of the same type in Z. Then, deleting z1 from Z yields the graph H2 which has a different nullity from the graph H1 obtained on deleting z2 from Z, a contradiction since n(H1) = n(H2) = 1. A similar argument yields that the type of vertices z1 and z2 in Z + e must be the same. From Lemma 2.1,1 < n(Z) < 2 and 1 < n(Z + e) < 2, and thus by Propositions 1.6 and 1.7, z1 and z2 are either both CFVmid or both CV. □ Remark 2.3. From Lemma 2.2 it follows that when z1 and z2 are CFVs in the key-graphs, each of the two positions y2 and in the 3-type-code can be filled in two ways, namely C and M. Therefore, there are only eight possible different classes of the 3-type-code graphs having the first position y1 filled with either M or U. In the case when both z1 and z2 are CFVs in the key-graphs, we have the following necessary and sufficient condition. Theorem 2.4. Let z1 and z2 be CFVs in H1 and in H2, respectively. In the graph Z or Z + e, z1 and z2 are CV if and only if they correspond to nonzero entries in exactly one kernel eigenvector of the basis of the nullspace of the graph Z or Z + e. Proof. By Proposition 1.7, the dimension of the nullspace of Z and of Z + e is two. From Lemma 2.1, (x, 0)* = (v, 0,0)* is a kernel eigenvector of Z and of Z + e. Since z1 and z2 are CV in Z, they correspond to nonzero entries in a kernel eigenvector (yx, a1, ^1)t of Z, for a1 =0 and = 0. A similar argument holds for Z + e. □ We note that although there are 18 possible 3-type-code classes of graphs when the first entry of the code is not C, Lemma 2.2 restricts the number of possible classes to just 8. 148 Ars Math. Contemp. 16 (2019) 141-155 3 Core vertices in key-graphs In this section we show that for the case when z1 and z2 are both CV in the key-graphs H1 and H2, respectively, only one 3-type-code class may occur. This case is completely different from the case discussed in Section 2 in that, as we prove in Proposition 3.2 and Theorem 3.5, the nullity of each of the graphs Z and Z + e can take only one value and it is not the same value in the two graphs. Recall that H1 and H2 have a common kernel eigenvector generating their nullspace. Since z1 and z2 are CVs, the last entry of a common kernel eigenvector x of H1 and of H2 is nonzero, that is x = ( —— ) for v = 0 and a = 0. Thus, letting z1 and z2 denote the characteristic vectors representing the adjacencies of z1 and z2 to the vertices of G, we obtain and "ar H G Zi ^ (v] ( Gv + az1 zi4 0 ) [a) V zi4v _ f G z2 a - V z24 0 Gv + az2 tl Z2 v 0 0, (3.1) (3.2) for some v = 0 and a = 0. An argument similar to that used in the proof of Lemma 2.2 yields the following result. Lemma 3.1. Let z1 and z2 be CVin H1 and in H2, respectively. Then in each of the graphs Z and Z + e, the two vertices z1 and z2 are of the same type. The unique value that the dimension of the nullspace of Z can take is given next. Proposition 3.2. If z1 and z2 are CVin H1 and in H2, respectively, then n(Z) = 2. Proof. Let Z be the adjacency matrix of the graph Z, where zi and z2 are the nth and (n + 1)th columns corresponding to the characteristic vectors of z1 and z2, respectively. Since Z M G zi z2 a ] = zit 0 0 0 z2t 0 0 and Z G zi z2 0 ] = zit 0 0 a z2t 0 0 then by (3.1) and (3.2), (v, a, 0)4 and (v, 0, a)1 are two linearly independent kernel eigenvectors of Z and hence n(Z) > 2. By Theorem 1.3, 0 < n(Z) < 2. Thus n(Z) = 2, and (v, a, 0)4 and (v, 0,a.)* span ker(Z). □ A consequence which has important implications on the construction of Z and, eventually, of Z + e, is the following. Corollary 3.3. If z1 and z2 are CV in H1 and in H2, respectively, then z1 and z2 are duplicates in Z and H1 = H2. I. Sciriha et al.: The conductivity of superimposed key-graphs with a common ... 149 Proof. Since (v, a, 0)4 and (v, 0, a)4 are kernel eigenvectors of Z, then (0, a, —a)4 is also a kernel eigenvector of Z. Thus z1 = z2. Hence, z1 and z2 are duplicate vertices in Z, implying that H1 and H2 are equivalent graphs. □ The dimension of the nullspace of Z + e turns out to be different from that of Z. The result is stated in Theorem 3.5 and the proof follows from Corollary 3.3 and the following lemma. We remark that the graph H in the following lemma plays the role of each of the key-graphs H1 and H2, and hence equations (3.1) and (3.2) still hold for H. Lemma 3.4. Let z1 be a CV in a graph H of nullity one and let Z be obtained from H by duplicating the vertex z1 to obtain a new vertex z2. Then n(Z + e) =0, where e is the edge z1 z2 . Proof. Let W be the adjacency matrix of Z + e, where z1 and z2 are the nth and (n + 1)th columns corresponding to the characteristic vectors of z1 and z2, respectively. Let x = , where v = 0 and a = 0, be a kernel eigenvector of H and let G = H — z1. From (3.1) and (3.2), it follows that W(v, a, 0)4 = (0,0,a)4 and W(v, 0,a)4 = (0,a, 0)4, and thus neither (v, a, 0)4 nor (v, 0, a)4 are kernel eigenvectors of Z + e. We claim that Z + e does not have any kernel eigenvectors. For, suppose (u, p, J)4 is a kernel eigenvector of Z + e. Since z1 and z2 are duplicates in Z, and hence co-duplicates in Z + e, then W G z1 z1 p I =I z^ 0 1 5 1 0 Gu + (p + ¿)zi ziiu + 5 zi fu + p implying that p = Thus, a kernel eigenvector of Z + e must be of the form (u, p, p)4. If p = 0, then Gu = 0 and hence (u, 0)4 is another kernel eigenvector of H which is linearly independent of x = (v, a)4, a contradiction since n(H) = 1. Thus p = 0 and we can choose p = a such that an eigenvector of Z + e is (w, a, a)4. Thus, W w G z1 z1 a =I z^ 0 1 a 1 0 Gw + 2az1 a I = I z1tw + a a / \ z1tw + a But from (3.1), Gv + az1 = 0 and thus G(w - 2v) = 0. Hence • either w — 2v = 0, in which case v = i w. From (3.1), z1tv = 0, implying that z1tw = 0 and hence a = 0, a contradiction; • or w — 2v is a kernel eigenvector of G, in which case n(G) > 1, a contradiction since zi is a CV in H and n(G) = n(H — zi) = 0. Hence, n(Z + e) = 0. □ Lemma 3.4 is now applied to the particular case when Z + e is obtained from the superimposition of Hi and H2 with core vertices zi and z2, respectively. Theorem 3.5. If zi and z2 are CV in Hi and in H2, respectively, then n(Z + e) = 0 and zi and z2 are both CFVupp in Z + e. 0 0 150 Ars Math. Contemp. 16 (2019) 141-155 Proof. By Corollary 3.3, z1 and z2 are duplicate vertices. The first part of the result follows by applying Lemma 3.4. Also, since n(Z + e) = 0, then z1 and z2 cannot be CV in Z + e. Noting that n(Z + e - z1) = n(H2) = 1 and n(Z + e - z2) = n(H1) = 1, we get that z1 Remark 3.6. Proposition 1.7, Proposition 3.2 and Theorem 3.5 imply that when z1 and z2 are CV in the key-graphs, each of the two positions y2 and in the 3-type-code can be filled in only one way, namely C in the position y2 and U in the position y3. Therefore, there is only one possible class of the 3-type-code graphs having C in its first position y1, namely CCU. Moreover the two key graphs H1 and H2 are induced subgraphs in both Z and Z + e. 4 Three-type-code Were it not for the restrictions of Lemmas 1.4, 2.2 and 3.1, the type of vertices would allow 81 classes of graphs for Z and another 81 for Z + e. These Lemmas allow only 27 potential classes and by eigenvector techniques, even these are further restricted to just nine with a specific 3-type-code. In Figure 4, three molecular graphs {Z + e} that are not chemical are presented for the each type of vertex z1 in H1. Figure 4: Graphs Z + e, having type M, U and C respectively, for the vertex z1 in H1. Except for the case UCC (that is, when {z1, z2} are CFVupp in H1 and in H2 and CV in Z and in Z + e), examples for all the remaining eight possible 3-type-codes graphs are known to exist (see Table 1). It is worth noting that among the eight graphs {Z + e} and the associated graphs {Z} drawn in Table 1, six are chemical graphs. The occurrence, or otherwise, of the UCC class remains open. Table 1 illustrates the different types of z1 and z2 in {H1, H2}, in Z and in Z + e, the associated code, the corresponding nullities of Z and of Z + e, and an example of a possible graph Z + e (when existence is known). Observe that although the interlacing theorem allows three values for the nullity of Z, this value can never be zero. At this stage, we can provide answers to the questions we posed at the end of Section 1.2 for the subclasses of graphs {Z} and {Z + e}. (i) On deleting the vertex z1 or z2 from Z, the nullity remains unchanged only for MMM, MMC, UMM and UMC out of the nine possibilities for the 3-type-code with z1 and z2 CFVmid in Z. Similarly, on deleting the vertex z1 or z2 from Z + e, the nullity remains unchanged for MMM, MCM, UMM and UCM. and z2 are both CFVupp in Z + e. □ (ii) On deleting the edge e = z1z2 in Z + e, the type of the vertices z1 and z2 remains unchanged when the 3-type-code is one of MMM, MCC, UMM and, possibly, UCC. I. Sciriha et al.: The conductivity of superimposed key-graphs with a common ... 151 Table 1: All possible cases of superimpositions {Z} and the derived class {Z+e} of graphs. Type of z1 and of z2 in Code n(Z) n(Z + e) Example of Z + e Hi & H2 Z Z + e CFVmid CFVmid CFVmid MMM 1 1 1 CFVmid CV MMC 1 2 CV CFVmid MCM 2 1 1 CV CV MCC 2 2 Sk / ^2 Jz\ CFVUpp CFVmid CFVmid UMM 1 1 Zi 1 CFVmid CV UMC 1 2 zi 1 CV CFVmid UCM 2 1 0- CV CV UCC 2 2 (not known) CV CV CFVUpp CCU 2 0 r 152 Ars Math. Contemp. 16 (2019) 141-155 We observe that no 3-type-code has U in both positions y2 and y3. This can be explained since if z1 and z2 are CFVupp in Z and n(H1) = n(H2) = 1, then n(Z) = 0, which never occurs by Lemma 2.1 and Proposition 3.2. Another point worth noting is that, in the case where zi and z2 are CFV in the key-graphs, the results do not depend on whether they are upper or middle. Thus, the type of the core-forbidden vertices z1 and z2 in H1 and H2 is not a factor that determines their type in Z and in Z + e. Could it be that distinguishing between the types U and M for z1 and z2 in H1 and H2 would determine the existence or otherwise of UCC? 5 Electrical conductivity A model device consists of the molecule with a pair of semi-infinite wires attached to it, so that a voltage can be applied across the molecule. The molecule, the wires and the contact atoms are represented by an augmented molecular graph with vertices for atoms and with edges for the sigma bonds. Left and right wires are represented by two special source and sink vertices L and R outside the molecule, which are then in contact with the molecule through single (usually distinct) vertices (contact atoms) labelled L and R. Coulomb and resonance integrals are assigned to the wires and molecule-wire contacts. This model gives a Huckel/Tight-Binding model for ballistic currents which is the simplest version of the SSP (Source-and-Sink Potential) model for ballistic conduction through simple molecular electronic devices [8, 9, 12]. The approximations lead to a non-Hermitian set of linear equations of order n + 2, with an implicit dependence of the SSP matrix entries on the eigenvalue. Linear algebraic techniques are used to describe the solutions of the larger problem in terms of characteristic polynomials derived from the original n x n adjacency matrix. Conduction or insulation of the unsaturated molecular device can then be predicted. The criteria for conduction at zero energy (the Fermi or non-bonding level) depend on the changes in nullity when the contact vertices L and R are deleted from the molecular graph on n vertices, separately and together [3, 5, 19]. 5.1 Transmission A molecular device G can be considered as a graph on n vertices with two prescribed vertices L and R connected by wires to two vertices L and R outside the molecule. The transmission at energy E, from the sink R, of ballistic electrons entering at the source L, can be expressed in terms of the characteristic polynomials s(E), u(E), t(E) and v(E) of G, G — L, G — R and G — L — R, respectively, as functions of E. To determine whether a device conducts or bars conduction at the Fermi level of energy (E = 0), it suffices to consider the possible nullity signatures as an ordered quadruple (gs = n(G),gu = n(G — L),gt = n(G — R), gv = n(G — L — R)). Cauchy's inequalities for the eigenvalues of real symmetric matrices and of their principal submatrices lead to the interlacing theorem for graphs. As a consequence, the change in the nullity on deleting a vertex can be at most one. Hence relative to gs, each of gu and gt can take 3 values whereas gv can take 5. Thus the quadruple signature can in principle take 45 values with respect to gs. However interlacing, device symmetry, and the Jacobi-Sylvester theorem (that is u(E)t(E) — s(E)v(E) is a perfect square j^R, where Jir is the LR^ entry of the adjugate of EI — A) restrict the number from 45 to just 11 [5, 19]. Table 2 gives the signatures of all possible classes of n-conjugated devices and their conducting/insulating properties at the I. Sciriha et al.: The conductivity of superimposed key-graphs with a common ... 153 Fermi level of energy. Table 2: The conductivity of all devices (G, L, R), their variety [19] and case [5]. Signature(gs ,9t,9u,9v ) Nullity of G Variety Case Transmission Two CVs 1 (9s, 9s - 1,gs - 1,gs - 2) no > 2 1(i) 11 Insulator (.9s, 9s - 1,gs - 1,gs) no > 1 1(ii) 9 Conductor (9s, 9s - 1,9s - 1,9s - 1) no > 1 1(iii) 10 Conductor CV and CFV 2 (9s, 9s + 1,9s - 1,9s) no > 1 2a 5 Insulator (9s, 9s, 9s - 1,9s - 1) no > 1 2b 8 Insulator Two CFVs 3 (9s, 9s + 1,9s + 1, 9v) 3a (9s, 9s + 1,9s + 1,9s) 3a(i) 2 Conductor (9s, 9s + 1,9s + 1,9s +2) 3a(ii) 1 Insulator (9s, 9s + 1, 9s, 9v) 3b (9s, 9s + 1,9s, 9s + 1) 3b(i) 3 Insulator (9s, 9s + 1,9s, 9s) 3b(ii) 4 Conductor (9s, 9s, 9s, 9v) 3c (9s, 9s, 9s, 9s + 1) 3c(i) 6 Conductor (9s, 9s, 9s, 9s) 3c(ii) 7 (9s, 9s, 9s, 9s) & ja (0) = 0 3c(iiA) 7i Conductor (9s, 9s, 9s, 9s) & ja (0) = 0 3c(iiB) 7ii Insulator 5.2 A superimposition device The superimposition Z and the derived graph Z + e have a structure that restricts the number of device classes to which they can belong. Their signature can be determined from Table 1. All the 11 device classes are realizable by molecular graphs. Table 3 shows that, of these, the superimpositions Z may be of only 5 cases and the derived graphs Z + e may also be of 5 cases. Both Z and Z + e can be of case 7 which assumes conductivity or insulating properties according to the vanishing or otherwise of jZlZ2(0). From Table 2, the derived graph Z + e may be an insulator only for case 7 when both z1 and z2 are middle core-forbidden vertices in their respective key-graphs H1 and H2. Apart from case 7, a superimposition Z is an insulator only when both z1 and z2 are core vertices in their respective key-graphs. Table 3: All possible cases of superimpositions {Z} and the derived class {Z} of graphs. Type of zi and of in Code Signature(Z) Signature(Z + e) Til & z Z + e Variety Case Variety Case CFVmid CFVmid MMM (9s, 9s, 9s, 9s) 3c/7 (9s, 9s, 9s, 9s) 3c/7 CFVmid CFVmid CV MMC (9s, 9s, 9s, 9s) 3c/7 (9s,9s-i,9s-i,9s-i) lm/10 CV CFVmid MCM (9s, 9s~ 1,9s -1,9s-I) lm/10 (9s, 9s, 9s, 9s) 3c/7 CV CV MCC (9s,9s-i,9s-i,9s-i) lm/10 (9s, 9s~ 1,9s -1,9s-I) lm/10 CFVmid CFVmid UMM (9s, 9s, 9s, 9s + 1) 3«/6 (9s, 9s, 9s, 9s + 1) 3«/6 CFVupp CFVmid CV UMC (9s, 9s, 9s, 9s + 1) 3«/6 (9s, 9s - 1,9s ~ 1,9s) lu/9 CV CFVmid UCM (9s, 9s - 1,9s ~ 1,9s) lu/9 (9s, 9s, 9s, 9s + 1) 3«/6 CV CV UCC (9s, 9s - 1,9s ~ 1,9s) lu/9 (9s, 9s - 1,9s ~ 1,9s) lu/9 cv CV CFVupp CCU (9s, 9s~ 1,9s -1,9s li/11 (9s,9s + 1,9s + 1,9s) Zai/2 I. Sciriha et al.: The conductivity of superimposed key-graphs with a common ... 155 References [1] D. A. Ali, J. B. Gauci, I. Sciriha and K. R. Sharaf, Coalescing Fiedler and core vertices, Czechoslovak Math J. 66 (2016), 971-985, doi:10.1007/s10587-016-0304-8. [2] M. Andelic, C. M. da Fonseca and R. Mamede, On the number of P-vertices of some graphs, Linear Algebra Appl. 434 (2011), 514-525, doi:10.1016/j.laa.2010.09.017. [3] P. W. Fowler, B. T. Pickup, T. Z. Todorova, M. Borg and I. Sciriha, Omni-conducting and omni-insulating molecules, J. Chem. Phys. 140 (2014), 054115, doi:10.1063/1.4863559. [4] P. W. Fowler, B. T. Pickup, T. Z. Todorova, R. De Los Reyes and I. Sciriha, Omni-conducting fullerenes, Chem. Phys. Lett. 568-569 (2013), 33-35, doi:10.1016/j.cplett.2013.03.022. [5] P. W. Fowler, B. T. Pickup, T. Z. Todorova and W. Myrvold, A selection rule for molecular conduction, J. Chem. Phys. 131 (2009), 044104, doi:10.1063/1.3182849. [6] P. W. Fowler, I. Sciriha, M. Borg and B. T. Pickup, Molecular graphs and molecular conduction: the d-omniconductors, to appear. [7] P. W. Fowler, I. Sciriha, M. Borg, V. E. Seville and B. T. Pickup, Near omni-conductors and insulators: Alternant hydrocarbons in the SSP model of ballistic conduction, J. Chem. Phys. 147 (2017), 164115, doi:10.1063/1.4995544. [8] A. Goker, F. Goyer and M. Ernzerhof, Bond dissociation and correlation effects in molecular electronic devices, J. Chem. Phys. 129 (2008), 194901, doi:10.1063/1.3013815. [9] F. Goyer and M. Ernzerhof, Correlation effects in molecular conductors, J. Chem. Phys. 134 (2011), 174101, doi:10.1063/1.3581096. [10] I.-J. Kim and B. L. Shader, On Fiedler- and Parter-vertices of acyclic matrices, Linear Algebra Appl. 428 (2008), 2601-2613, doi:10.1016/j.laa.2007.12.022. [11] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964. [12] B. T. Pickup and P. W. Fowler, An analytical model for steady-state currents in conjugated systems, Chem. Phys. Lett. 459 (2008), 198-202, doi:10.1016/j.cplett.2008.05.062. [13] B. T. Pickup, P. W. Fowler, M. Borg and I. Sciriha, A new approach to the method of source-sink potentials for molecular conduction, J. Chem. Phys. 143 (2015), 194105, doi:10.1063/1. 4935716. [14] I. Sciriha, On the construction of graphs of nullity one, Discrete Math. 181 (1998), 193-211, doi:10.1016/s0012-365x(97)00036-8. [15] I. Sciriha, On the rank of graphs, in: Y. Alavi, D. R. Lick and A. Schwenk (eds.), Combinatorics, Graph Theory, and Algorithms, Vol. II, New Issues Press, Kalamazoo, Michigan, 1999 pp. 769-778, proceedings of the 8th Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms, and Applications, dedicated to the memory of Paul Erdos, held at Western Michigan University, Kalamazoo, Michigan, June 3-7, 1996. [16] I. Sciriha, A characterization of singular graphs, Electron. J. Linear Algebra 16 (2007), 451462, doi:10.13001/1081-3810.1215. [17] I. Sciriha, Coalesced and embedded nut graphs in singular graphs, Ars Math. Contemp. 1 (2008), 20-31, doi:10.26493/1855-3974.20.7cc. [18] I. Sciriha, Maximal core size in singular graphs, Ars Math. Contemp. 2 (2009), 217-229, doi: 10.26493/1855-3974.115.891. [19] I. Sciriha, M. Debono, M. Borg, P. W. Fowler and B. T. Pickup, Interlacing-extremal graphs, Ars Math. Contemp. 6 (2013), 261-278, doi:10.26493/1855-3974.275.574. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 157-171 https://doi.org/10.26493/1855-3974.997.7ef (Also available at http://amc-journal.eu) Regular polygonal systems Jurij Kovic * Institute for Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana and FAMNIT, University of Primorska, Glagoljaska 8, 6000 Koper, Slovenia Received 19 December 2015, accepted 4 March 2018, published online 18 November 2018 Let M = M(Q) be any triangle-free tiling of a planar polygonal region Q with regular polygons. We prove that its face vector f (M) = (f3, f4, f5,...), its symmetry group S(M) and the tiling M itself are uniquely determined by its boundary angles code ca(M) = ca(Q) = (t1,... ,tr), a cyclical sequence of numbers ti describing the shape of Q. Keywords: Regular polygonal system, boundary code, face vector, symmetry group, reconstructibility from the boundary. Math. Subj. Class.: 05B40, 05B45 1 Introduction Systems of regular (planar or spherical) polygons joined edge to edge arise in various contexts (in tilings, in polyhedral maps, in nature, in chemistry, in art). A rich theory of such systems may be developed. Researchers usually focus on some particular class of such systems (defined by some conditions), try to determine all its elements and explore various questions related to their combinatorial description, parameters, enumeration, characterization, classification, coding, etc. To unify the investigations of such objects and to emphasize their common characteristics we propose a general concept of a regular polygonal system and make some first few steps towards a general theory of such systems. Here we give definitions, examples and remarks; the general reconstruction problem is presented in Section 2; results are gathered in Section 3. A polygonal system M is a (finite or infinite) incidence structure M = (V, E, F) whose elements are called vertices, edges and faces: faces are abstract polygons - cyclical sequences of vertices (v1,..., vn), and edges are pairs of vertices [vi, vi+1}. Two faces are *This work is supported in part by the Slovenian Research Agency (research program P1-0294 and research project N1-0032). E-mail address: jurij.kovic@siol.net (Jurij Kovic) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 158 Ars Math. Contemp. 16 (2019) 141-155 incident if they share an edge, two edges are incident if they share a vertex. If M consists only of n-gons it is called a n-system; if the number of these n-gons is m, it is a nm-system. If all these faces are congruent polygons X, it is a monohedral system denoted Xm. If M consists only of ni-gons, n2-gons, ..., nk-gons, it is called a (ni,..., nk)-system. The face vector (or just the f-vector) of the polygonal system M is the sequence f (M) = (f3, f4, f5,...) where f (i) = fi(M) are the numbers of its faces with i edges. We use also the notation f (M) = (3f (3), 4f(4), 5f (5),...). A planar or spherical polygon P is called regular if there is a cyclic group G = (R | Rn = I) of rotations acting transitively on the vertices and edges of P, and if its boundary dP is simply connected (thus we exclude star polygons, as in Kepler solids). A regular polygonal system M = M(Q) consists of regular polygons joined edge to edge, covering a polygonal planar or spherical region Q. The symmetry group of M = M(Q) is the group of the rotations and reflections of Euclidean space E2 or E3 preserving the incidences in M. A code c(X) of a given mathematical object X is any (not necessarily discrete) structure from which X can be completely reconstructed (up to isomorphism). A boundary code of a given m-dimensional object X is any code c(dX) of the shape of its (m - 1)-dimensional boundary dX. The boundary angles code of the planar or spherical regular polygonal system M = M(Q) covering (tiling) a polygonal region Q is the cyclical sequence Ca(M) = Ca(Q) = (ti,t2,.. .,tr) of the angles ti = 180° - a G (-180°, 180°), where ai are the interior angles between adjacent edges Ai-1Ai and AiAi+1 of Q. The boundary faces-edges code Cf,e(M) is the cyclical sequence of symbols f (i)e(i), where f (i) is the number of edges of i-th boundary face and e(i) is the number of the boundary edges of this face (see examples in Appendix A). A few examples and remarks will help us better understand these definitions and the motivation for them. In chemistry, the mathematical model of benzenoid molecules are called benzenoid systems or polyhexes (composed of regular hexagons). Similar systems are polydiamonds (composed of equilateral triangles) andpolyominoes (composed of squares) (see Figure 1). The motivation for the boundary angles code ca comes from the "turtle geometry" [1]. The definition of boundary faces-edges code cf e is motivated by the boundary codes of various n-systems presented below. Several boundary codes exist for the planar regular hexagonal systems B. One of them is the boundary edges code [6]. This code ce(B) = (k1,k2,... ,kr) is a cyclical sequence counting the numbers ki of boundary edges in boundary hexagons; we travel around the boundary in the clockwise direction, starting at any hexagon, and having the interior of B always at our right (see [2, 7]). The code for the regular planar triangular systems (or 3-systems) T may be defined exactly in the same way - as the boundary edges code ce(T) counting the number of boundary edges in boundary triangles of T. Thus, for the 3-system T in Figure 1 its code is ¿aEbnra Figure 1: Planar regular n3-systems for n = 3,4,5,6. ce(T) = (2,1, 2). J. Kovic: Regular polygonal systems 159 The case of the planar square systems (or 4-systems) S is trickier. To get the right numbers in ce(S) we must count also the "zeros" of boundary edges in boundary vertices adjacent to no boundary edge. We can use also a simple boundary vertices-faces code cVff (S), a cyclical sequence counting for each boundary vertex how many squares (1, 2 or 3) are incident with that vertex. For the first 4-system S in Figure 1 we have ce(S) = (2,3,0, 3) and cVff (S) = (1,2,1,1, 3,1,1,2). There is also a code for the regular pentagonal systems P (described in [3]) that actually counts the numbers of vertices incident only with boundary edges. For the second 5-system from Figure 1 this code is (3,3, 2). One can easily imagine polygonal systems composed of regular polygons with different numbers of edges, and also non-planar generalizations of these concepts. Example 1.1. The face vector of the planar regular (3,4, 5)-system M in Figure 2 is f (M) = (33,4i, 52) = T3S1P2, since there are three triangles T (f3 = 3), one square S (f4 = 1) and two pentagons P (f5 = 2). Its symmetry group S(M) is generated by one reflection (over the vertical axis). Figure 2: A polygonal system M with the symmetry group S(M) = Z2. Example 1.2. The boundary faces-edges code of the planar system M in Figure 3 is Cf,e(M) = (54,40, 32,42). The boundary angles codes ca(M) of the spherical systems with the same Cf,e(M) depend on the size of the spherical triangle T. Increasing the size of T the interior angles a increase as well, hence the angles tj = 180° - a decrease. If T is big enough, the angle between the pentagon and the triangle vanishes and we get a region with Cf,e(M) = (53, 3i, 42). Figure 3: A planar regular polygonal (31, 41, 51 )-system M with the boundary angles code c0(M) = (120°, 30°, 90°, -18°, 72°, 72°, 72°, -78°). The maps of Platonic solids are examples of regular 3-systems (tetrahedron 34, octahedron 38, icosahedron 320), 4-systems (cube 46) and 5-systems (dodecahedron 512). The 160 Ars Math. Contemp. 16 (2019) 141-155 polygonal system of the square pyramid is (B, C, D, E), (A, B, C), (A, C, D), (A, D, E), (A, E, B), thus it is a (3,4)-system. The system (A, B, C, D), (A, B, D, C) represents the Mobius band tiled with two quadrilaterals. In a polygonal system more than two faces may share the same edge, as in the "3-page book" (A, B, C, D), (A, B, D, E), (A, B, F, G), which is a 3-system, too. 2 The reconstruction problem The first motivation for this research was the realization that it is possible to generalize the method that worked so well for benzenoid systems B: "Encode the boundary of a system as a cyclical sequence of numbers, and then obtain various information about B from its boundary code b(B)" to other planar polycyclic molecules. Thus b(B) was used to find various relations between the parameters of B, to determine its symmetry scheme, to calculate its face count, etc. (as in [5, 7, 8]). With this motivation in mind we have in the Introduction already defined two boundary codes cej(M) and ca(M). Now we can ask: Is it possible to reconstruct the face vector f (M), the symmetry group S(M) or even the whole planar regular polygonal system M = M(Q) from its boundary code Cf,e(M) or ca(M)? Obviously, some regular polygonal systems M are reconstructible from the chosen boundary code of the region Q covered by M. How to characterize such systems? This question may be stated in a more general form: Given some class C of objects X whose boundary is determined by some boundary code c(dX) characterize those X from C that can be reconstructed from its boundary code, in other words, find X for which c(dX) is also the code c(X) of X. The code is not necessarily a cyclical sequence of numbers. The boundary of any simply connected polycube P (a solid composed of cubes of unit length joined face to face) may be coded with a graph G(P) whose vertices are boundary square faces of P and whose edges are labeled with angles (0, n/2, -n/2) between adjacent boundary faces. The number of cubes in P, the symmetry group S(P) and the polycube P itself are obviously determined by its boundary. However, their actual reconstruction from the graph G(P) is complicated. The code is not always a discrete structure. In analysis, a differentiable function f is reconstructible from its derivative f' by the Newton-Leibniz formula up to an additive constant C. Likewise, a harmonic function f: Q ^ R is determined by its values on the boundary of Q. Not all the codes of the same object contain the same amount of information. For example, two similar planar triangles have different codes (lengths a, b, c and a',b', c'), but the same boundary code (consisting of angles a, p, y). In some cases the chosen boundary code c(dX) contains some additional information about the structure of X that cannot be deduced from dX alone. Thus it may happen that it is possible to reconstruct X from the code of dX, although it is not possible to reconstruct X only from dX. Example 2.1. Let Q be a planar polygonal region composed of a regular 12-gon and a square sharing one side. There are two tilings M1 and M2 of Q with 12 squares S and 20 equilateral triangles T, having the same boundary angles code (90°, 90°, 60°, 30°, 30°, 0°, 30°, 30°, 30°, 0°, 30°, 30°, 30°, 0°, 30°, 30°, 30°, -90°), but different boundary faces-ed- J. Kovic: Regular polygonal systems 161 ges codes (see Figure 4): ce,f (Mi) = (43, 4o, 3i, 4i, 3i, 3o, 3i, 4i, 3i, 4i, 4i, 3i, 4i, 3i, 3o, 3i, 4i, 3i, 4i, 4o), Ce,f (M2) = (43, 3o, 4i, 3i, 4i, 4i, 3i, 4i, 3i, 3o, 3i, 4i, 3i, 4i, 4i, 3i, 4i, 3i, 3o, 3o). Hence, it is not possible to reconstruct M from the shape of the boundary dQ, encoded by ca(Q). However, M is reconstructible from Cf,e(M), since there are only two possible tilings Mi and M2 of Q. In this sense, Cf,e(M) is a stronger code (containing more information), and ca(M) = ca(Q) is weaker (containing less information). Figure 4: Two different planar (3,4)-systems M1 and M2 with the same boundary and with the same face vector (320,412). Remark 2.2. Two closed planar polygonal regions Q and Q' may have the same boundary angles code ca = (t^ ... , tr), but if the ratio of the lengths of the corresponding edges AjAi+1 and AjAj+1 are not all the same, then the shapes of these regions are different. Note that the lengths of the edges of a region Q tiled by regular polygons joined edge to edge are the same. This is the reason why the boundary angles code ca(Q) suffices to describe the shape of the boundary of Q, tiled by regular polygons.1 3 Results In this Section we show: if M is a regular triangle-free tiling of a planar polygonal region Q, then we can use ca(M) to find the face vector f (M) (Theorems 3.1 and 3.8), to determine the symmetry group S(M) (Theorem 3.12), or even to completely reconstruct M = M(Q) (Theorem 3.5). Theorem 3.1. Let M = M(Q) be a planar regular (3,4)-system covering the polygonal region Q. Then the boundary angles code ca(M) = ca(Q) determines the face vector f (M) of M. Proof. The area of Q can be calculated from the boundary angles code ca(Q) = (t 1,..., tn) as follows: fix the coordinates of two adjacent vertices A1 = (0,0) and A2 = (1,0) of Q, use vectors to find the coordinates of other vertices Aj, triangulate Q and sum the areas 1 However, as we see in Figure 4, the tiling M (Q) may not be uniquely determined by the shape of its boundary, although it is tiled by regular polygons. 162 Ars Math. Contemp. 16 (2019) 141-155 of all these triangles. The area of the regular n-gon Pn with side 1 is Area(Pn) — n ■ cot(360°/n). These numbers are incommensurable at least for n — 3,4 since Area(P3) — a/3/4, Area(P4) — 1. Now it is easy to see that the integer solutions of the equation xi(%/3)/4 + y1 — x2(\/3)/4 + y2 are possible only if x1 — x2, y1 — y2. Thus f (M) is determined by ca(M). Solving the equation x1^v/3)/4 + y1 — Area(M) is easy, since the calculated expression for Area(M) must appear in this form, from which we just read x1 and y1. □ Theorem 3.2. The sum of the angles in the boundary angles code ca(Q) — (t1, t2, ... ,tr) of a planar polygonal region Q is t1 +t2 + ■ ■ ■ + tr — 360°. Proof. This follows from the formula J2™=i a = (n — 2) • 180° for the sum of the interior angles of a n-gon: J2 "=1 ti = 2 "=1 (180° — a). Another proof in the context of the "turtle geometry" is given in [1, p. 175]. □ Regular (planar or spherical) n-gons are usually denoted by the symbol {n} or just n. The vertex type of the interior vertex of the regular polygonal system M is defined as the cyclical sequence (a.b.c.... ) of the faces a, b, c,... surrounding it. The planar vertex types in this notation are listed in [4]. It is easy to check the following very useful observation ([4, p. 60]). Theorem 3.3. If the planar regular n1-gon, ..., nr -gon surround a vertex without gaps and overlaps, then 3 < r < 6 and (n1 — 2)/n1 + • • • + (nr — 2)/nr = 2, hence there are 21 types of vertices surrounded by regular polygons in a plane without gaps or overlaps: 3.3.3.3.3.3, 3.3.3.6, 3.3.3.4.4, 3.3.4.3.4, 3.3.4.12, 3.4.3.12, 3.3.6.6, 3.6.3.6, 3.4.4.6, 3.4.6.4, 3.7.42, 3.9.18, 3.8.24, 3.10.15, 3.12.12, 4.4.4.4, 4.5.20, 4.6.12, 4.8.8, 5.5.10, 6.6.6. Theorem 3.4. The possible (interior or boundary) vertex types in spherical regular polygonal systems are (if we exclude spherical 2-gons): • 5 triangles: 3.3.3.3.3; • 4 triangles: 3.3.3.3.4, 3.3.3.3.5; • 3 triangles: 3.3.3, 3.3.3.4, 3.3.3.5; • 2 triangles: 3.3, 3.3.m, 3.m.3 (m > 4), 3.3.4.5, 3.4.3.5, 3.3.5.5; • 1 triangle: 3, 3.m (m > 4), 3.4.n, 3.n.4 (n > 4), 3.5.n, 3.n.5 (n > 5); • 0 triangles: m, 4.m, 4.4.m, 4.m.4 (m > 4), 4.5.n, 4.n.5 (19 > n > 4), 5.5.5. Proof. We just use the fact that for the interior angle an of the spherical n-gon it holds that 180° > an > 180° — 360°/n, and check all the possible cases. A similar treatment of faces is given in Appendix B. □ Theorem 3.5. Let M = M(Q) be a planar regular polygonal system covering the polygonal region Q. If M contains no triangles then it is reconstructible from its boundary angles code ca(M). Likewise, M is reconstructible from ca(M) also in the case M is without squares and hexagons. J. Kovic: Regular polygonal systems 163 Proof. The theorem is certainly true if M contains only one face. Now suppose it is true for the systems with m or less faces and let M = M(Q) be a system with m + 1 faces. Since the sum of the r inner angles a of Q is J2¿=1 a = (r - 2) • 180°, at least one a must be smaller than 180°. The vertex A of such a < 180° is incident with at most three regular polygons: with two or three triangles, with one triangle and one square or pentagon or with two squares, or with a single n-gon with the inner angle (n - 2) • 180°/n. Hence, if M is triangle-free, then the angle a < 180° is incident with only one n-gon Pn whose interior angle is (n - 2) • 180°/n = Oj. Removing this Pn from M we get a smaller system, whose tiling is unique, by the induction hypothesis. Hence the position of each polygon in M is uniquely determined. Similarly, if there are no squares and hexagons in M, then every interior angle a < 180° is incident either with a single polygon Pn (and then we proceed as before) or with a triangle and a pentagon. But there is no planar interior vertex type containing faces 3 and 5 (see Theorem 3.3). Therefore both the vertices of the edge AC shared by a boundary triangle ABC and a pentagon are boundary vertices (see Figure 5). If we interchange the Figure 5: Left: A (3, 5)-system with two boundary components and with the cyclical symmetry group C15. Right: An illustration of the fact that such system cannot have two different tilings. positions of two adjacent boundary faces 3 and 5 along the adjacent boundary edges then the interchanged face 5 covers a neighbourhood of C (as in Figure 5 right). So there is only one possible tiling of M in the neighbourhood of all boundary points incident with 3.5 or 5.3. Removing this triangle and pentagon we get a smaller region for which the theorem is true by the induction hypothesis. □ Remark 3.6. It may happen that the boundary of M - Pn is no longer a simple closed curve (see Figure 6); it may have crossings and it may have more than one component. However, by repeating the process of removing the polygons corresponding to interior angles a < 180° from the system and calculating the boundary angles code of the smaller system we may find the exact locations of each face in the system algorithmically. For example, removing a boundary square P4 from M changes the boundary code from ca (M) = (tl,t2, . . . , ti-i,tj = «4 = 90°, tj+1, . . . ,tr) into Ca(M - P4) = (¿1, ¿2,^-1 + «4, «4 -180°, a4 + ti+1,..., tr), as in Figure 6. For n > 5 we get a more complicated formula for ca(M - Pn), dependent on how many successive angles tj, tj+1,... ti+k in ca(M) are equal to (n - 2) • 180°/n. Just as in the planar case, there are polygonal regions on the sphere admitting more 164 Ars Math. Contemp. 16 (2019) 141-155 Figure 6: Removing of a boundary face may produce a system that is no more face-connected. than one regular tiling, too. A spherical pentagon, covered by five regular triangles sharing a vertex of the spherical icosahedron is an example of such a region. Theorem 3.7. Let M be a spherical regular polygonal system M without triangles and squares. If the boundary of M has only one component, then M is reconstructible from its boundary faces-edges code Cf,e(M). Proof. By Theorem 3.4 M has no interior points. Hence there can be no cycle composed of adjacent faces in M, and Cf,e(M) completely describes the system M. □ Theorem 3.8. The face vector f (M) = (f3, f4, f5,...) and consequently also the number of faces f = f3 + f4 + f5 + ■ ■ ■ of any regular planar polygonal system M = M (Q) without hexagons or triangles and with all sides of length 1 is uniquely determined by the area of Q, and this area is uniquely determined by its boundary angles code ca(Q) = ca(M). Proof. If M = M(Q) has at most two faces, then its face vector is obviously determined by its boundary code, except in the case of a hexagon, which may be decomposed into six triangles. Suppose the theorem is true for any system with n faces. Take a system M with n+2 faces. In every boundary vertex with a positive angle ti we can repeat the procedure of taking out the polygons of M as in the proof of Theorem 3.5. In the boundary vertices with the interior angles filled with 3.4, 4.3, 3.5, 5.3, we take away from M both combinations and get two smaller systems M* and M** composed of n faces which must have the same area, hence they have the same face vector by the induction hypothesis. Therefore M has the unique face vector, too. □ Theorem 3.9. The symmetry group S(M) of a planar or spherical regular polygonal system M = M(Q) is a subgroup of S(Q) = S(dQ). Proof. Any rotation or reflection preserving M preserves the region Q, tiled by the polygons of M. The symmetry group of the boundary of Q is obviously isomorphic to the symmetry group of Q. □ A symmetry of the boundary of Q does not necessarily induce a symmetry of M = M(Q). For example, if we glue together the two 12-gons from Figure 4 (without the added top square) along the vertical edge, we get a region with two reflection symmetries (over the vertical axis and over the horizontal axis), yet its tiling has only one reflection symmetry (over the horizontal axis). J. Kovic: Regular polygonal systems 165 Theorem 3.10. If there is only one regular polygonal system M = M(Q) covering the region Q then the groups S(M) and S(Q) are isomorphic. Proof. If the tiling M of Q with regular polygons is unique, then every symmetry of Q automatically induces an automorphism of M. □ Lemma 3.11. Let Q be a planar polygonal system with given lengths of its edges li = AiAi+1 and with the boundary angles code c(Q) = (t\,... ,tr). (i) If ti+k = tk and if li+k = lfc for some k > 2, then Q has a rotational symmetry for the angle 360o/(r/k). (ii) If ca(Q) = (t1,... ,tr) = (tr,... ,t1) and if the cyclical sequence ci = (l1, l2,..., lr) of the lengths li of the boundary edges AiAi+1 has a reflection symmetry, then Q has a reflection symmetry. Proof. (i) This is obviously true for k = 1, for in that case we have ca(Q) = (t,t,... ,t), hence all ti are the same, and all li are the same, hence Q is a regular polygon Pr invariant for the rotation for the angle 360o/r. In the case k = 2 we have ca(Q) = (t, s,t,s,...,t, s). Removing from Q the congruent triangles AA1A2A3, AA3A4A5,..., AA2n-1A2nA1 we get a region with r/k = r/2 boundary edges where all ti are the same and all li the same (case k = 1) and for which we know (i) is true; hence Q has the rotation for the angle 360o/(r/2), too. Similarly, for the k > 3 we remove r/k = n congruent triangles from Q to get a smaller region with r — r = nk — n = n(k — 1) boundary angles and with a period k — 1, hence by the induction hypothesis having a rotation Rn where n = k = "^l^. Hence Q has the rotation for the angle 360o/(r/k), too. (ii) Reflection symmetry of the sequence cl (Q) may have three forms2: (a) ci = (y,z,...,b,a,a,b,..., y, z), (b) ci = (x,y,z, .. . ,b,a,a,b,.. ., y, z), (c) ci = (x,y,z, .. . ,b,a,a,b,.. ., y, z, w), and the same holds for the cyclical sequences of the lengths li. Obviously (ii) is true if Q has 3,4, 5 or 6 sides, since the only possible cases of cl (Q) are: (x, y, y), (y, z, z, y), (x, y, y, z), (x, y, z, z, y), (x, y, z, z, y, w) and (x, y, z, z, y, x). Now we can use a simple induction argument to see that if Q has more sides than 6, then we can cut it in two pieces with the boundary angles code of the types (c) or (b), and each of these two pieces has the same reflection symmetry (whose axis is the symmetral of the same line XY), hence Q has the same reflection symmetry (see Figure 7 middle). This is clear for regions of the type (a); the regions of the types (b) and (c) are obtained from a region of the type (a) by glueing one or two triangles to it. □ Theorem 3.12. Let M = M(Q) be a regular planar polygonal system without triangles covering the region Q. Then the symmetry group S(M) of M and the center of the rotation R" are determined by its boundary angles code ca(M) as follows: (i) if ca(M) has a reflection symmetry ca(M) = (t1,... ,tr) = (tr,... ,t1) then M has a reflection symmetry; 2The letters x,y, z,a,b,... are used here to denote lengths li of boundary edges of Q. 166 Ars Math. Contemp. 16 (2019) 141-155 Figure 7: Left: a region with rotational symmetry; middle: a region with reflection symmetry; right: a region whose all boundary angles are identical (equal to 120°, as in a regular hexagon), yet it has no symmetry. (ii) if ca(M) = (tl,...,tr) is a periodical cyclical sequence: ti+k = ti for some k > 2 then M has a rotational symmetry for the angle 360° /(r/k); (iii) if f = 1 (mod n) then the center of the rotation Rn is in a face; (iv) if f = 0 (mod n) and n > 2 then the center of the rotation is in the vertex (if n > 2); (v) if f = 0 (mod n) and n = 2 then the center of the rotation is in an edge. Proof. By Theorem 3.5, M = M(Q) is uniquely determined by its boundary angles code ca(M) = ca(Q). By Theorem 3.9, the group of symmetries of M is isomorphic to the group of symmetries of the boundary of Q. Now (i) and (ii) follow directly from Lemma 3.11. Indeed, if the boundary angles code ca(M) = ca(Q) of length r remains the same after the shift k (mod m), then r = kn is divisible by k and Q has a rotational symmetry Rn (see Lemma 3.11). Likewise, the reflection symmetry of ca (Q) implies the reflection symmetry of Q, since all the edges of Q are of the same size (since they are tiled by the regular polygons of M joined edge to edge) and we can apply Lemma 3.11. Now we use again S(Q) = S(M), implied by Theorem 3.5. The center of the rotational symmetry Rn of any planar regular polygonal system can be in a vertex (this is possible only in the cases when n = 3,4,6), in an edge center (this is possible only if n = 2) or in a center of a m-gon, where n divides m. If there are no triangles in the system, and if n is odd, then the center of the rotation Rn is in a face center. If the rotational symmetry Rn has the center in a m-gon, then fm = 1 (mod n) while the number of all other i-gons fi is divisible by n, hence f = 1 (mod n). If the center of the rotation is in a vertex or in an edge center, then fi = 0 (mod n) for every i, hence f = 0 (mod n). By Theorem 3.8 the face vector of M without triangles may be obtained from the boundary angles code ca(M) so the numbers fi are known, hence we can easily distinguish between the two possible cases with rotational symmetry: f = 1 (mod n) and f = 0 (mod n). If f = 0 (mod n) and n > 2 then the center of the rotation must be in a vertex. If f = 0 (mod n) and n = 2 then the center of the rotation must be in an edge. □ Remark 3.13. So the information hidden in the boundary angles code suffices to find out whether the center of rotation is located in a face, in a vertex or in the middle of an edge.3 3Another possible approach to this question is to calculate the center of rotation directly via orbit barycenters, and compare its coordinates with the coordinates of all the vertices, face centers and edge centers of the tiling M. J. Kovic: Regular polygonal systems 167 4 Summary and open questions We proposed a concept of a regular polygonal system, a mathematical model of chemical (planar or non-planar) molecules composed of chains of atoms forming regular n-gonal cycles of equal or various length (n = 3,4, 5, 6,... ). We proved that the structure (and hence the symmetry group) of any regular planar polygonal system M without triangles or without squares and hexagons is reconstructible from its boundary angles code ca(M ). The proof of Theorem 3.1 implicitly uses a simple incommensurability argument. This type of argument can be generalized as in the following definition and lemma: Definition 4.1. The quantities q1,q2,... ,qn are called incommensurable, if they satisfy the following condition: if two linear combinations of these quantities with integer coefficients are the same, i.e. J2n=i aiqi = n=i Mi, then their corresponding coefficients must be the same, i.e. ai = bi. From this definition immediately follows: Lemma 4.2. If any quantity q can be expressed as an integer linear combination of incommensurable quantities, then the coefficients ai are uniquely determined by q. Conjecture 4.3. The areas of all planar regular polygons with the same side length (except the triangle or hexagon) are incommensurable quantities. Conjecture 4.4. The volumes of all Platonic and Archimedean solids with the same side are incommensurable quantities. If one could prove Conjecture 4.3, this would automatically prove also Theorem 3.8. However, the converse is not true. If one could prove Conjecture 4.4 this would prove another conjecture: Conjecture 4.5. If we glue together copies of Platonic and Archimedean solids (with unit sides) face to face then we can find the number of each of them just by knowing the volume ofsuch composed solid. References [1] H. Abelson and A. diSessa, Turtle Geometry: The Computer as a Medium for Exploring Mathematics, MIT Press Series in Artificial Intelligence, MIT Press, Cambridge, Massachusetts, 1981. [2] A. T. Balaban, Chemical graphs-VII: Proposed nomenclature of branched cata-condensed benzenoid polycyclic hydrocarbons, Tetrahedron 25 (1969), 2949-2956, doi:10.1016/ s0040-4020(01)82827-1. [3] M. Deza, P. W. Fowler and V. P. Grishukhin, Allowed boundary sequences for fused polycyclic patches and related algorithmic problems, J. Chem. Inf. Comput. Sci. 41 (2001), 300-308, doi: 10.1021/ci000060o. [4] B. Grunbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Company, New York, 1987. n i= 1 168 Ars Math. Contemp. 16 (2019) 141-155 [5] I. Gutman and S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, SpringerVerlag, Berlin, 1989, doi:10.1007/978-3-642-87143-6. [6] P. Hansen, C. Lebatteux and M. Zheng, The boundary-edges code for polyhexes, J. Mol. Struct. THEOCHEM 363 (1996), 237-247, doi:10.1016/0166-1280(95)04139-7. [7] J. Kovic, How to obtain the number of hexagons in a benzenoid system from its boundary edges code, MATCH Commun. Math. Comput. Chem. 72 (2014), 27-38, http://match.pmf.kg. ac.rs/electronic_versions/Match72/n1/match72n1_2 7-38.pdf. [8] J. Kovic, T. Pisanski, A. T. Balaban and P. W. Fowler, On symmetries of benzenoid systems, MATCH Commun. Math. Comput. Chem. 72 (2014), 3-26, http://match.pmf.kg.ac. rs/electronic_versions/Match72/n1/match7 2n1_3-2 6.pdf. [9] N. J. A. Sloane (ed.), The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org. J. Kovic: Regular polygonal systems 169 Appendix A Planar regular (3,4)-systems These systems may be classified by two parameters: the numbers /3 and /4 of triangular and square faces, as in Figure 8. As we see, the number of these systems grows very quickly with the total number of faces / = /3 + /4. There are 16 such systems with 1, 2 or 3 faces. In Figure 9 they are arranged by the increasing total number of faces. We give also their boundary faces-edges codes cf,e and boundary angles codes ca. Observe that in all these cases all the faces are boundary faces. Since for the planar regular polygonal ( 3,4)-systems all their boundary angles are multiples of 30°, we give the boundary angles codes also in the form of multiples of 30°. The corresponding boundary faces-edges code and boundary angles codes of these ( 3,4)-systems, ordered as in Figure 9, are: cf,e (M ) c0(M ) (33 ) (120,120,120)° = 30° 4,4,4) (44 ) (90, 90, 90, 9o)° = 30° (3, 3, 3, 3) (32 32) (60,120, 60,120)° = 30° (2, 4, 2, 4) (43 32) (120, 30, 90, 90, 30)° = 30° 4,1, 3, 3,1) (43 43) (90, 0, 90, 90, 0, 90)° = 30° (3, 0, 3, 3, 0, 3) (32 3i, 32 3o) (60, 60,120,0,120)° = 30° (2, 2, 4, 0, 4) (32 3o, 43 3i) (-30, 90, 90, 30, 60,120)° = 30° -1, 3, 3,1, 2,4) 32 , 3i, 43, 3o) (90, 90, -30,120, 60, 30)° = 30° 3, 3,-1,4, 2,1) (32, 4i, 32, 4i) (120, 30, 30,120, 30, 30)° = 30° 4,1,1, 4,1,1) (32, 4o, 32, 42) (120, -30,120, 30, 90, 30)° = 30° 4,-1, 4,1, 3,1) (43, 4o, 32, 42) (90, -60,120, 30, 90, 0, 90)° = 30° 3, -2, 4,1, 3,0, 3) (42, 32, 4o, 43) (30,120, -60, 90, 90, 0, 90)° = 30° 1,4, -2, 3, 3,0, 3) (43, 4i, 32, 4i) (90, 0, 30,120, 30, 0, 90)° = 30° 3,0,1, 4,1,0, 3) (43, 3o, 43, 3i) (90, 90, -60, 90, 90, 30, 30)° = 30° 3, 3, -2, 3, 3,1,1) (43, 4i, 43, 4i) (90, 90, 0, 0, 90, 90,0,0)° = 30° 3, 3, 0, 0, 3, 3, 0, 0) (43, 4o, 43, 42) (90, 90, -90, 90, 90, 0, 90,0)° = 30° 3, 3, -3, 3, 3,0, 3, 0) /4 =0 /4 = 1 /4=2 /4=3 /3 = 0 □ m irr: r /3 = 1 A Ù àitû m> etc. Figure 8: Regular planar ( 3,4)-systems classified by two parameters. 170 Ars Math. Contemp. 16 (2019) 141-155 au & fatn & & Q rf) fVi m> -OO* rm:!; <6 Figure 9: Planar regular polygonal (3,4)-systems with f = f3 + f4 < 3 faces. Appendix B Types of faces The idea to classify the types of hexagons in a benzenoid system with respect to their contacts to adjacent faces in a system [2, 7] may be generalized to any n-gonal faces in any polygonal system as follows: Definition B.1. The type of the n-gonal face f in a polygonal system M is the cyclical binary sequence (bi,..., bn) where b = 0 if the ¿-th edge of f is incident to at least one other face of M and b = 1 if it is a boundary edge of f. The number of types of n-gons having k entries 1 in the binary code is denoted T(n, k). The number of possible types of a n-gon is denoted T(n). Theorem B.2. The following relations hold: (i) T (n) equals the number of binary cyclical sequences of length n. (ii) T(3) = 4, T(4) = 6, T(5) = 8, T(6) = 13. Proof. (i) This is obvious, and trivially implies also relations T (n, k) = T (n, n - k) and T (n) = T (n, 0) + T (n, 1) + • • • + T (n, n). To each type t of a n-gon exists the opposite type t* with the entries 6* = 1 - 6j, hence t** = t. For (ii) see Figure 10. Summing the adjacent entries 1 and ignoring the entries 0 we can, at least for the triangles T, squares S and pentagons P, denote their possible boundary J. Kovic: Regular polygonal systems 171 types like this: To = (0,0,0), Ti = (1,0,0), T2 = (1,1,0) = T1Î, T3 = (1,1,1)= T0, So = (0,0, 0, 0), S1= (1, 0,0,0), S2 = (1,1,0,0) = S2, Si+i = (1,0,1, 0) = SÎ+1, S3 = (1,1,1,0) = SÎ, S4 = (1,1,1,1), Po = (0,0, 0, 0,0), P1= (1, 0,0,0,0), P2 = (1,1,0,0, 0), P1+1 = (1,0,1,0,0), P1+2 = (1, 0,1,1,0) = P1 P3 = (1,1,1,0, 0) = P2*, P4 = (1,1,1,1,0) = Pj\ P5 = (1,1,1,1,1). The inner faces are T0 = (0,0,0), S0 = (0,0, 0, 0), P0 = (0,0,0,0, 0). The 13 types of boundary hexagons H with 1,2,3,4 or 5 boundary edges, e.g. H1+1+1 = (1,0,1,0,1,0) or H2 = (1,1,0,0,0,0) are given in [2] and more precisely classified in [7]. □ Remark B.3. The sequence T(n) (although defined only for n > 3) corresponds to the sequence A000029 in OEIS [9]. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 173-182 https://doi.org/10.26493/1855-3974.1017.0ac (Also available at http://amc-journal.eu) F-WORM colorings of some 2-trees: partition vectors Julian D. Allagan Department of Mathematics and Computer Science, Elizabeth City State University, Elizabeth City, North Carolina, USA Vitaly Voloshin Department ofMathematics and Geomatics, Troy University, Troy, Alabama, USA Received 24 January 2016, accepted 28 May 2018, published online 21 November 2018 Suppose F = {F1,... ,Ft} is a collection of distinct subgraphs of a graph G = (V,E). An F-WORM coloring of G is the coloring of its vertices such that no copy of each subgraph Fj G F is monochrome or rainbow. This generalizes the notion of F-WORM coloring that was introduced recently by W. Goddard, K. Wash, and H. Xu. A (restricted) partition vector ((a,..., Q) is a sequence whose terms Zr are the number of F-WORM colorings using exactly r colors, with a < r < p. The partition vectors of complete graphs and those of some 2-trees are discussed. We show that, although 2-trees admit the same partition vector in classic proper vertex colorings which forbid monochrome K2, their partition vectors in K3-WORM colorings are different. Keywords: 2-tree, maximal outerplanar, partition, Stirling numbers. Math. Subj. Class.: 05C15, 05C10 1 Preliminaries A partition a of a set S is a set of nonempty subsets or blocks of S such that each element of S is in exactly one of the subsets of S. The number of blocks of a is its rank and a partition of rank r is simply called an r-partition. For instance, the Stirling number of the second kind, {"} counts the number r-partitions of the set [n] = {1,2,... ,n}. Consider the mapping c: S ^ [x] being an x-coloring of the elements of S. A subset A C S is said to be monochrome if all of its elements share the same color and A is rainbow if all of its elements have different colors. As such, a coloring c(S) is a partition of the set E-mail addresses: aallagan@gmail.com (Julian D. Allagan), vvoloshin@troy.edu (Vitaly Voloshin) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 174 Ars Math. Contemp. 16 (2019) 141-155 S since all of the elements of S are assigned a color; elements that share the same color belong to the same block (monochrome subsets), and different blocks are used for those with distinct colors (rainbow subsets). Let G = (V, E) denote a simple graph and suppose F = {Fi,..., Ft} is a collection of some distinct subgraphs Fj C G, 1 < i < t. An F-WORM coloring of G is the coloring of the vertices of G such that no copy of each subgraph Fj is monochrome or rainbow. When F has only one member, say F, we write F-WORM coloring; this special case was first introduced by W. Goddard, K. Wash and H. Xu, and independently studied by Cs. Bujtas and Zs. Tuza [5, 7, 8, 12, 13]. We note that this coloring requirement makes sense only if each Fj G F is of order three or more. However, for a generalization purpose if some Fj is of order 2, we allow only rainbow copies of Fj in order to meet the classic proper (vertex) coloring requirement. Suppose H = (V, E) is a hypergraph. If |e| = s for each hyperedge e G E, then H is said to be s-uniform. Given any vertex coloring of H, if no e G E is monochrome, H is called a D-hypergraph or simply a hypergraph. When no e G E is rainbow, H is called a cohypergraph. In the event no e G E is monochrome or rainbow, then H is called a bihypergraph. Moreover, if G is a hypergraph and each subgraph Fj = Er, the null graph on r-vertices, then an F-WORM coloring of G is a proper (vertex) coloring of an r-uniform bihypergraph; Cs. Bujtas and Zs. Tuza [7, 8] also noted this strong relation between F-WORM coloring and mixed hypergraph colorings, a theory that was first introduced by the second author [25, 26]. Thus, the notion of F-WORM colorings generalizes several well known coloring constraints. Given an F-WORM coloring, the sequence ((a,..., (p) whose terms, (r, are the number of r-partitions is called a (restricted) partition vector, with a < r < p. In general, partition vectors have some added benefits in the study of log-concave and unimodal sequences which often arise in algebra, combinatorics, computer science, even in probability and statistics (see for e.g., [2,4, 11]). A sequence of non-negative terms (a0,... ,an) is called log-concave if a2 > aj-1aj+1 for i = 1,..., n - 1. Such sequence is also said to be unimodal if it has no gap (i.e., there is no i with aj-1 = 0, aj =0 and aj+1 = 0) and there is an index 0 < j < n such that a0 < ... < aj > ... > an. Further, partition vectors are closely related to colorings; each zr gives the number of F-WORM colorings using exactly r colors, in which case a and P are the lower and upper chromatic numbers, respectively. In [7], it is shown that it is NP-hard to determine a and it is NP-complete to decide whether or not a graph G admits a K3-WORM coloring using k > 2 colors. Moreover, the integer set S = {x : a < x < P} commonly known as feasible set, has been the subject of numerous research publications (see e.g., [6, 15, 18, 27]). We note that the term chromatic spectrum has also been used for feasible set in some of the aforementioned literatures. Further, we call the rank-generating function £ a(G|F; x) = ^ Cfcxfc k=a the restricted partition polynomial of G subject to an F-WORM coloring. Note that, when xj is replaced by the falling factorial power x- = x(x - 1)(x - 2) • • • (x - i + 1), the polynomial £ a(G|F; x) = Y, Zkxk k=a counts all F-WORM colorings using at most x colors. Some variants of restricted partition polynomials have been well studied. For instance when G = En, a(G|g; x) is the J. D. Allagan and V. Voloshin: F -WORM colorings of some 2-trees: partition vectors 175 Bell polynomial which is a widely studied tool in combinatorial analysis [9, 24]. Also, a(G|K2 ; x) has been recently called Stirling polynomial [11] although it was first introduced by Korfhage as a-polynomial [17]. In particular, when written in the falling factorial power of x, a(G|K2; x) = x(G; x) is the well known chromatic polynomial [3, 22]. Thus, a restricted partition polynomial extends both the chromatic polynomial and the Stirling polynomial of graphs. In this paper, in Section 2, we determine the partition vectors of some mixed hypergraphs. Later, in Section 3, we investigate K3-WORM colorings of some 2-trees. We find that, while 2-trees admit the same partition vector given any (classic) proper vertex coloring, it is not true for their K3-WORM colorings. To support this argument, we present two non-isomorphic members of 2-trees which have different partition vectors. In Section 4, we conclude this paper with F-WORM colorings when F includes a family of 2 or more graphs such as Path, Star or Cycle. 2 Coloring Kn with forbidden monochrome or rainbow subgraphs We begin by establishing a connection between Ks-WORM coloring of a complete graph Kn and mixed hypergraph colorings. Theorem 2.1. The partition vector in a Ks-WORM coloring of Kn is (([^J,..., Cs-i), where Zr = {n} for all 3 < s 3 of its vertex set is called an s-uniform complete hypergraph. Corollary 2.2. The partition vector of an s-uniform complete bihypergraph is (([^ J,.. Zs-1), where Zr = {n} for all s > 3. Removing either restriction on r gives each of the next result. Corollary 2.3. The partition vector of an s-uniform complete hypergraph is (C[-^- j ,..., Zs), where Zr = {n} for all s > 2. Corollary 2.4. Thi partition vector of an s-uniform complete cohypergraph is (Z1,..., Zs-1), where Zr = {n} for all s > 3. 3 Partition vectors of some 2-trees As a generalization of a tree, a k-tree on n vertices (with 1 < k < n) is a graph which arises from a Kk by adding n — k > 1 new vertices, each joined to a Kk in the old graph; this process generates several non-isomorphic k-trees, k > 1. Figure 1 depicts four non-isomorphic 2-trees on 6 vertices. K-trees are chordal graphs which are known to admit at least one simplicial elimination ordering ([10]). Recall, a graph is chordal if it does not contain an induced cycle of length 4 or more. The characterization of families of graphs by forbidden subgraphs is an old tradition in graph theory and k-trees, despite 176 Ars Math. Contemp. 16 (2019) 141-155 (a) 3-sun (b) fan (c) snake (d) 0(1, 2, 2, 2, 2) Figure 1: Some non-isomorphic 2-trees. being ubiquitous, have yet to be fully classified even in the case when k = 1. Adding some additional restrictions on the coloring of certain subgraphs besides K2 and En may help in the analysis of the structure of the graphs that contain them. To help support this claim, we begin with the partition vectors in the coloring of 2-trees when monochrome K2 are forbidden. These vectors do not characterize any member of k-trees, since non-isomorphic k-trees do share the same partition vector as shown, later, in Corollary 3.2. Proposition 3.1. The equality n — k holds for all 1 < k < n. xk (x - k)n-k = y, \ '¡-l t=k+1 Proof Since xn = £"=1{"}x-, this implies that (x - k)n-k = {"T7(x - k)- and xk(x - k) n-k n-k = E t=1 n-k E t=1 E t=k + 1 n - k t n - k t n - k tk x(x - 1) • • • (x - k - 1)(x - k)- „t+k giving the result. □ Corollary 3.2. The partition vector of any k-tree on n — k simplicial vertices such that no K2 is monochrome is (Cfe+i,..., Cn), where Çr = {n-^} • Proof. It is easy to see that the left side of the equality of the formula in Proposition 3.1 is that of the chromatic polynomial of any k-tree, k > 1. The result follows from the right side of that equality. □ x J. D. Allagan and V. Voloshin: F -WORM colorings of some 2-trees: partition vectors 177 We note the quantity } = } ^ is known for counting the number of k-nonconse-cutive r-partitions of n elements (see e.g., [16]); a partition of the set {1,... ,n} is said to be k-nonconsecutive whenever x and y are in the same block, |x - y| > k. Recall that a graph is called outerplanar if it can be embedded in the plane in such a way that every vertex lies on the outer cycle. A maximal outerplanar (MOP) graph is an outerplanar graph with a maximum number of edges [21]. Graphs such as 3-sun, fan and snake are some well known MOPs; these graphs are depicted in Figures 1(a), 1(b) and 1(c), respectively. Laskar and Mulder [19,20] characterized MOPs as the intersection of any two of the following graphs: chordal, path-neighborhood, and triangle graphs T(G) which are trees. Recall, a path-neighborhood graph is a graph in which every vertex neighborhood induces a path and the triangle graph T(G) of G has the triangles of G as its vertices, and two vertices of T(G) are adjacent whenever their corresponding triangles in G share an edge [1]. Simply put, MOPs are members of 2-trees. Here, by considering K3-WORM colorings of 2-trees, we have found that their partition vectors are uniquely determined and the process reveals that MOPs are 2-trees with the characteristic that every edge is shared by at most two triangles. Theorem 3.3. Suppose G is a 2-tree such that its triangle graph is a path. Then the number of colorings of its vertices such that no triangle is monochrome or rainbow is P(G)= ^ 4>(n - 1j)x(x - 1)j, i(n — 1,j) -1,j + an-1,l n- j +j ifj < f O-n-1, \ otherwise and the values ai,j 's satisfying, (i) a11 = 1,ai1 = 2 for each 2 < i < n — 1 and, for each k (ii) 1,..., L2J, a2k, j = aik-1 ,j + a^k-1 ,j+k-1 if 2 < j < k a2k-1,j-k if k +1 < j < i (iii) a2k+1,j a2k,j + a2k,j+k-1 if 2 < j < k 1 a2k,j-k-1 if j = k + 1 if k + 2 < j < i. Proof. Suppose G = (V, E) is a 2-tree on n > 3 whose triangle graph is a path. Then, there exists a simplicial elimination ordering n = {u1,... ,un}, such that v,i is adjacent to the edge with endpoints (ui-1, ui-2). Let G1 := u1, G2 := u1u2, and Gi := Gi-1 U {ui} where ui is adjacent to the pair (ui-1,ui-2) in Gi for all i > 3. Suppose c is any coloring of G and denote by P(G) the restricted number of colorings of G. For n = 3, we count the colorings when u1 u2 is rainbow and when u1 u2 is monochrome, separately. If we denote a f 178 Ars Math. Contemp. 16 (2019) 141-155 A1 = x(x-1) andB1 = x thenP(G3) = A1 + (x- 1)Bi + Ai. Set A2 := Ai + (x- 1)Bi and B2 := Ai and clearly A2 and B2 count the number of colorings where c(u3) = c(u2) and c(u3) = c(u2), respectively. For all n > 3, at each iteration, we separate the terms that count c(uj) = c(ui_i) from those that count c(uj) = c(ui_i), giving the recursion P(G„) = A„_i + B„_i, where A„_i := A„_2 + (x - 1)B„_2 and A„_i := A„_2. Now use Ai and Bi as basis for the previous recursion and record at each iterative step the coefficients ai ,j's of each expression (x - 1)k, for 1 < i, j < n - 1. By letting aiji = 1, it is easy to verify that the coefficients ai}j's satisfy the conditions (i) - (iii). For instance, when n = 3, a2ji = 2, a2,2 = ai,i = 1. Now, define an (n - 1) x (n - 1) matrix A whose entries are the coefficients aiij-'s of P(Gi+i) with P(G2) = x(x - 1). It follows that P := xA • Q, where P and with Thus, P(G2) P (G3) , A = P (Gn)_ Q= q1 = (x - 1)1 Q2 = = (x - 1)1 = ai,i a2,i a2,2 an-i,i an-1,2 . . . an-in-1 Q = [Q1 I Q2]T, (x - 1)T^f11 (x - 1)L^f1 J and where E an_i,k(x - 1)k + £ an_i,k(x - 1)fc_r^i1 ^ = i 2, the sequence {bn} satisfies the shifted Fibonacci recurrence given by bi = 3, b2 = 5 and bn = bn-1 + bn-2, for n > 3. 180 Ars Math. Contemp. 16 (2019) 141-155 3. If each triangle of G is replaced by a hyperedge (of size 3), the previous result also gives the partition vector of several nonlinear 3-uniform acyclic bihypergraphs, which include the complete 3-uniform interval bihypergraphs [26]; 3-uniform bihypergraphs often appear in communication models for cyber security [14]. Obviously, there are other members of 2-trees who have 3 or more triangles sharing the same edge as subgraphs. Here, we present the other extremal case of 2-trees when all triangles share a single edge, say u1u2. This 2-tree, often denoted by 6(1,2,..., 2), is a member of the well known n-bridge graphs. See Figure 1(d) for an example of a 5-bridge. Note that 6(1, 2,..., 2) is a maximal planar graph but not a MOP, for all n > 5. Corollary 3.6. Suppose G = 6(1, 2,..., 2), an (n — 1)-bridge graph on n > 3 vertices. The partition vector of a K3-WORM coloring of G is (Z2,..., Cn-i) where 2n-2 + 1 if r = 2 Zr n 2\ otherwise. r1 Proof. Count the number of colorings when the shared edge wiw2 is monochrome and when it is rainbow, giving x(x - 1)n-2 + 2n-2x(x - 1) colorings. Now apply Proposition 3.1 (when k = 1) to the terms of the expression to obtain the result. □ We leave it to the reader to verify that the previous values in the partition vector when G = 6(1,2,..., 2) are different from those of MOPs, for all n > 5. 4 Conclusion We've shown that while 2-trees admit the same partition vector given any proper vertex coloring, it is not the case with their K3-WORM colorings. We hope these results indicate the importance of WORM colorings in general in the analysis of the structures of some well-known graphs which could not be classified with the usual proper vertex colorings. For a potential future research, we introduce some generalizations of F-WORM colorings when F includes multiple graphs such as Path, Star or Cycle. In the next results, Cn, K1n-1, and Pn denote an n-cycle, an n-star, and an n-path that includes a fixed vertex (apex) u1, respectively. Corollary 4.1. Suppose G is a fan on n > 4 vertices. If G has a K3-WORM coloring then G admits an F-WORM coloring with F = |Ps*,K1ji,Cr,6(1,n1,n2)} where s > 4, Ln—1J < t < n — 1, r > 3, and 2 < n1 < n2 such that n1 + n2 < n. Proof. Suppose G is a fan on n > 4 vertices which we can construct as follow: start with a triangle, say (u1,u2,u3), and iteratively add n — 3 new vertices such that each additional vertex uj is adjacent to the pair (u1, ui-1), for i = 4,..., n. Assume G admits a K3-WORM coloring. (i) Observe that for s > 4, every path P* C G contains the subgraph u1uiui+1 for some i (2 < i < n — 2). If some 3-path (that includes u1 ) is monochrome/rainbow then the triangle (u1,ui,ui+1) is monochrome/rainbow, violating the K3-WORM coloring assumption. Hence G admits a PS*-WORM coloring for all s > 4. J. D. Allagan and V. Voloshin: F -WORM colorings of some 2-trees: partition vectors 181 (ii) By letting the vertices of K1jt C G be all the vertices of G, it follows that t < n - 1. Now, consider the coloring such c(ui) = c(u2k) and c(ui) = c(u2k+1) for k = 1,..., [n-1 ]. Clearly, such coloring does not violate our assumption of K3-WORM coloring of G. Hence, the lower bound of t is satisfied by letting the vertices of K1jt be {u1, u2} U {u2k+1 : k = 1,..., [n-11}, which guarantees a K1jt-WORM coloring for all t > [ 1. (iii) For r > 4, since every cycle Cr C G includes the apex u1, there exists an s < r such that P* C Cr, with 4 < s < r < n. From (i), G admits a Cr-WORM coloring. The case when r = 3 is trivial. (iv) Likewise, since 6(1, n1, n2) contains C1+q C G with q e {n1, n2}, the result follows from (iii) that, for all 2 < n1 < n2 such that n1 + n2 < n, G admits a 6(1, n1, n2)- WORM coloring. □ Note that the converse of the statement in Corollary 4.1 is not true. Using a similar argument as in the previous proof establishes the next result; recall, a snake (see Figure 1(c)) is a 3-sun-free maximal outerplanar graph with at least four vertices. Corollary 4.2. Suppose G is a snake on n > 4 vertices. If G has a K3-WORM coloring then G admits an F-WORM coloring, where F = {Cr, 6(1, 2,2)} with 3 < r < n. References [1] R. Balakrishnan, Triangle graphs, in: R. Balakrishnan, H. M. Mulder and A. Vijayakumar (eds.), Graph Connections, Allied Publishers Limited, New Delhi, 1999 p. 44, proceedings of the conference held at Cochin University of Science and Technology, Cochin, January 28-31, 1998. [2] E. Bertin and R. Theodorescu, On the unimodality of discrete probability measures, Math Z. 201 (1989), 131-137, doi:10.1007/bf01162000. [3] G. D. Birkhoff and D. C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60 (1946), 355-451, doi:10.1090/s0002-9947-1946-0018401-4. [4] F. Brenti, Expansions of chromatic polynomials and log-concavity, Trans. Amer. Math. Soc. 332 (1992), 729-756, doi:10.1090/S0002-9947-1992-1069745-7. [5] Cs. Bujtas, E. Sampathkumar, Zs. Tuza, M. S. Subramanya and C. Dominic, 3-consecutive C-colorings of graphs, Discuss. Math. Graph Theory 30 (2010), 393-405, doi:10.7151/dmgt. 1502. [6] Cs. Bujtas and Zs. Tuza, Uniform mixed hypergraphs: the possible numbers of colors, Graphs Combin. 24 (2008), 1-12, doi:10.1007/s00373-007-0765-5. [7] Cs. Bujtas and Zs. Tuza, K3-WORM colorings of graphs: lower chromatic number and gaps in the chromatic spectrum, Discuss. Math. Graph Theory 36 (2016), 759-772, doi:10.7151/dmgt. 1891. [8] Cs. Bujtas, Zs. Tuza and V. Voloshin, Hypergraph colouring, in: L. W. Beineke and R. J. Wilson (eds.), Topics in Chromatic Graph Theory, Cambridge University Press, Cambridge, volume 156 of Encyclopedia of Mathematics and its Applications, pp. 230-254, 2015, doi: 10.1017/cbo9781139519793.014. [9] C. B. Collins, The role of Bell polynomials in integration, J. Comput. Appl. Math. 131 (2001), 195-222, doi:10.1016/s0377-0427(00)00274-0. [10] G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71-76, doi: 10.1007/bf02992776. 182 Ars Math. Contemp. 16 (2019) 141-155 [11] D. Galvin and D. T. Thanh, Stirling numbers of forests and cycles, Electron. J. Com-bin. 20 (2013), #P73, http://www.combinatorics.org/ojs/index.php/eljc/ article/view/v2 0i1p7 3. [12] W. Goddard, K. Wash and H. Xu, WORM colorings forbidding cycles or cliques, Congr. Numer. 219 (2014), 161-173. [13] W. Goddard, K. Wash and H. Xu, WORM colorings, Discuss. Math. Graph Theory 35 (2015), 571-584, doi:10.7151/dmgt.1814. [14] A. Jaffe, T. Moscibroda and S. Sen, On the price of equivocation in byzantine agreement, in: D. Kowalski and A. Panconesi (eds.), PODC'12, ACM Symposium on Principles of Distributed Computing, ACM, 2012pp. 309-318, doi:10.1145/2332432.2332491, proceedings of the 2012 ACM Symposium on Principles of Distributed Computing held in Funchal, Madeira, Portugal, July 16-18, 2012. [15] T. Jiang, D. Mubayi, Zs. Tuza, V. Voloshin and D. B. West, The chromatic spectrum of mixed hypergraphs, Graphs Combin. 18 (2002), 309-318, doi:10.1007/s003730200023. [16] Zs. Kereskenyi-Balogh and G. Nyul, Stirling numbers of the second kind and Bell numbers for graphs, Australas. J. Combin. 58 (2014), 264-274, https://ajc.maths.uq.edu.au/ pdf/58/ajc_v58_p264.pdf. [17] R. R. Korfhage, 2, and the paw graph P. Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 185 For a subset S C V(G) we define £(S,G) = E dG(v). ves Let vi, v2,..., vn be an ordering of the vertices of G such that d(vi) < d(v2) < • • • < d(vn). The annihilation number a(G) is the largest integer k such that d(vj) < m(G). Equivalently, a = a(G) is the only integer satisfying both a a+1 ^ d(vi) < m(G) and ^ d(v,) > m(G) + 1. i=i i=i It is clear by definition that every independent set1 A satisfies J2veA d(v) < m(G) and consequently, the annihilation number is an upper bound on the independence number [11]. The annihilation number was first introduced by Pepper in [12]. The 'annihilation process', which is referred to in this original definition, is very similar to the 'Havel-Hakimi process' (see [7] and [11] for exact descriptions). In general, a set S of vertices is called an annihilation set if J2veS d(v) < m(G); and S is an optimal annihilation set, if |S| = a(G) and max{d(v) | v G S} < min{d(u) | u G V(G) \ S}. In particular, if G is a connected graph on at least 3 vertices, any optimal annihilation set of G contains all leaves. Assuming that S is an optimal annihilation set, we introduce the following notations. First, denote by d* (G) (or simply by d*) the minimum vertex degree over the set V(G) \ S. Note that d*(G) = d(va(G)+1), and consequently, the value of d*(G) is independent from the choice of the optimal annihilation set S. The following conjecture can be found in a slightly different form in Graffiti.pc [4], and was later reformulated in [5]. Conjecture 1.1 ([4, 5]). If G is a connected nontrivial graph, then Yt(G) < a(G) + 1. (1.1) By definition, every graph satisfies a(G) > |_^^J. Hence, the formulas given for Yi(C„) above show that each cycle Cn satisfies the conjecture. Further, if ¿(G) > 3, it was observed that the total domination number is at most |^J [1, 3, 13, 14]. Hence, if ¿(G) > 3, then Yt(G) < a(G) clearly holds, even if G is disconnected. Therefore, it is interesting to study this conjecture for graphs with small minimum degree, i.e. ¿(G) G {1, 2}. So far, Conjecture 1.1 has been proved for only one further important graph class. The following result was established by Desormeaux, Haynes, and Henning in 2013. Theorem 1.2 ([5]). If T is a nontrivial tree, then Yt(T) < a(T) + 1, and the bound is sharp. 1A set A c V(g) is called an independent set if it induces an edgeless subgraph in G. The largest cardinality of such a vertex set is the independence number of g and denoted by a(G). 186 Ars Math. Contemp. 16 (2019) 141-155 A similar result was proved by Desormeaux, Henning, Rail, and Yeo [6] for the 2-domination number of trees. Very recently, a different proof was given for the same statement by Lyle and Patterson [10]. Namely, their result can be obtained if we replace the total domination number with the 2-domination number in Theorem 1.2. In this paper we prove Conjecture 1.1 over two further graph classes, namely for cactus graphs and block graphs. These are two natural generalizations of trees and also, for a cactus graph G we have ¿(G) < 2 and there exist block graphs with small minimum degree. Remark that cactus and block graphs are well-studied classes with several applications, see for instance [2]. Our main results are the following ones. Theorem 1.3. If G is a nontrivial cactus graph, then Yt(G) < a(G) + 1. Theorem 1.4. If G is a nontrivial block graph, then Yt(G) < a(G) + 1. To formulate and to prove our results we will use the following function f defined for every finite graph G as f (G) = n(G) + 3m(G) + ni(G), where n1 (G) denotes the number of leaves in G. Remark that f is strictly monotone in the sense that if G' is a proper subgraph of G, then f (G') < f (G). Indeed, n(G') + m(G') < n(G) + m(G) clearly holds, and 2m(G') + n1(G') < 2m(G) + n1(G) is true because the deletion of an edge may result in at most two new leaves. Also note that we have f (G) > 7 for any nontrivial, finite and connected graph G. The paper is organized as follows. In Section 2, we establish several lemmas which will be referred to in later proofs. In Section 3 and 4 we prove Theorem 1.3 and 1.4, respectively. In the last section we discuss the sharpness of our main theorems and arise some related problems. 2 Preliminary results Here we present some preliminary results on how we can obtain a smaller graph G' from G (mainly, by deleting some edges and/or vertices from G) such that Yt(G') < a(G') + 1 implies Yt(G) < a(G) + 1. First we consider changes related to vertices of small degree. Lemma 2.1. Assume that G is a connected graph on at least three vertices and it fulfills at least one of the following properties: (i) d*(G) < 2; (ii) G has a strong support vertex; (iii) G contains an induced path vu^^w such that d(u1) = d(u2) = d(u3) = 2; (iv) G contains a path u1 u2u3v such that u1 is a leaf and d(u2) = d(u3) = 2; (v) G contains two adjacent support vertices. Then, there exists a nontrivial connected graph G' with f (G') < f (G) such that jt (G') < a(G') + 1 implies Yt(G) < a(G) + 1. Moreover, if G is a cactus graph, then G' can be chosen to be a cactus graph as well; and if G is a block graph, in cases (ii) -(v), G' can be chosen to be a block graph. Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 187 Proof. Since trees and cycles satisfy Conjecture 1.1, we may suppose throughout that G is neither a tree nor a cycle. (i) First assume that d*(G) < 2. Since G is neither a tree nor a cycle, there exists a vertex v G V(G) with d(v) > 3 which is incident to a cycle. Let e = vu be an edge from that cycle. Clearly, G' = G - e is connected, f (G') < f (G) and m(G') = m(G) - 1. The deletion of an edge does not decrease the total domination number. This establishes 7t(G) < 7t(G'). Consider now an optimal annihilation set S' of G'. By definition, it satisfies £(S',G') < m(G') = m(G) - 1. If u,v G S' then £(S', G) = £(S',G') < m(G) -1; if S' contains exactly one of u and v,thenj](S',G) = £(S',G') + 1 < m(G). In either case a(G) > |S'| = a(G') follows. In the third case u, v G S' and J2(S', G) = XXS', G') + 2 < m(G) + 1. Let V12 denote the set of vertices which have degree 1 or 2 in G. Our assumption d*(G) < 2 implies X(V1j2, G) > m(G) + 1. Since d(v) > 3, we have X(Vi,2 U {v}, G) > m(G) + 4. Therefore, (Vi,2 U {v}) g S' implies that we have a vertex v* g V1j2 which is not contained in S'. If v is replaced with v* in S', we obtain a set S with X(S, G) < ^(S', G) - 1 < m(G). This proves a(G) > |S| = a(G'). If G' satisfies (1.1), we may conclude that the same is true for G: In the sequel of the proof we will assume that d* (G) > 3. (ii) Assume that a vertex v G V(G) has two neighbors u1 and u2 which are leaves in G. Since v remains a support vertex in G' = G - {u1}, it is contained in every total dominating set of G'. This implies 7t(G') = 7t(G). On the other hand, every optimal annihilation set of G contains u1 and hence a(G') < a(G). Then, f (G') < f (G), and Yt(G') < a(G') + 1 implies 7t(G) < a(G) + 1. (iii) If vu1u2u3w is an induced path in G and d(u1) = d(u2) = d(u3) = 2, consider the graph G' = G- {u1,u2,u3} + vw. Observe that n(G') = n(G) - 3, m(G') = m(G) - 3, n1(G') = n1(G) and hence, f (G') = f (G) - 12. Let D' be an optimal total dominating set of G' and define D as follows: In either case, D is a total dominating set in G. Hence, 7t(G) < 7t(G') + 2. Consider next an optimal annihilation set S" of G'. Since dG(v) = dG> (v) and dG(w) = dG> (w), E(S',G) = Z(S',G') < m(G') = m(G) - 3. Our assumption d*(G) > 3 implies that every vertex x with degree d(x) < 2 is contained in every optimal annihilation set of G. Hence, either S' U {«i, u2, u3} is a subset of an optimal annihilation set of G and a(G) > a(G') + 3, or there is a vertex v* G S' with d(v*) > 3. In the latter case, consider S = (S'\{v*}) U{ui, «2, «3}, and observe that £(S,G) < ^(S',G) - 3 + 3 • 2 < m(G). Therefore, a(G) > |S| = |S'| + 2 = a(G') + 2. If G' satisfies inequality (1.1), we have and that proves the statement for property (iii). (iv)Letm1m2m3vbeapathinGsuchthatd(u1) = 1 andd(u2) = d(u3) = 2. SinceGis connected and not a path, G' = G-{m1,m2,m3} is nontrivial, and we have f (G') < f (G). Yt(G) < Yt(G') < a(G') + 1 < a(G) + 1. Yt(G) < Yt(G') + 2 < a(G') +3 < a(G) + 1 188 Ars Math. Contemp. 16 (2019) 141-155 If D' is an optimal total dominating set of G', then D = D' U {u2, u3} totally dominates all vertices in G. Thus, Yt(G) < |D| < Yt(G') + 2. Next, we choose an optimal annihilation set S' in G' and consider three cases concerning v and S'. • If d(v) = 2, then G contains three consecutive degree-2 vertices and, as we have already proved it in (iii), there exists a graph G' with the required property. • If v G S', then E(S',G) = £(S',G'), and£(S',G) < m(G') = m(G) - 3. Hence, S = S'U{ui, «2} satisfies £(S, G) = X(S', g) + 3 < m(G), and a(G) > a(G') + 2. This, together with the assumption Yt(G') < a(G') + 1, establishes inequality (1.1) for G. • In the last case we assume that both v G S' and d(v) > 3 hold. Then, J2(S', G) = £(S',G') + 1 < m(G') +1 = m(G) - 2. We define S = (S' \{v}) U{u,u2,u3} and observe that X(S, G) = X(S',G) - d(v) + 5 < m(G). Hence, S is an annihilation set in G and we may conclude a(G) > |S| > a(G') + 2. The statement of the lemma is proved by the following chain of inequalities: 7t(G) < Yt(G') + 2 < a(G') + 3 < a(G) + 1. (v) Let u and v be two leaves in G with support vertices u' and v' respectively such that uu', vv', u'v' g E(G). Since G is not a path, at least one of these two support vertices, say u', is of degree of at least 3. Then, we define G' = G - uu' + uv and observe that f (G') = f (G) - 1. Let D' be an optimal total dominating set of G'. Since v is a support vertex in G', v g D' must hold. Moreover, since NG/ (u) C NG/ (v'), we can choose D' such that u does not belong to it. Then, D = (D' \ {v}) U {u'} is a total dominating set in G. Hence, 7t(G) < |D| = |D'| = 7t(G'). By construction, every vertex has the same degree in G as in G' with the two exceptions v and u', for which dG (u') = dG> (u') + 1 and dG(v) = dG> (v) - 1. Hence, any optimal annihilation set S' of G' satisfies one of the following cases. • If u',v G S' or u',v G S', then (S', G) = £(S',G') < m(G') = m(G). Therefore, a(G) > |S'| = a(G'). • If v G S' and u' G S', the^(S',G) = £(S',G') - 1 < m(G') - 1 = m(G) - 1. Therefore, a(G) > |S'| = a(G'). • If u' G S' and v G S', then £(S', G) = ^(S', G') + 1 < m(G') + 1 = m(G) + 1. We define S = (S' \ {u'}) U {v}. By our assumption, dG(u') > 3 and so, we have XXS, G) = X(S', G) - dG(u') + 1 < m(G) + 1 - dG(u') + 1 < m(G) - 1. This implies a(G) > a(G'). We have seen that for all possible cases a(G') < a(G) and 7t(G) < 7t(G'). Together with the condition that G' satisfies (1.1), these imply 74(G) < Yt (G') < a(G') + 1 < a(G) + 1. At the end of the proof we remark that all the above transformations result in a cactus graph G', if G was of the same type. Further, with the only exception of (i), the obtained graphs stay block graphs if G is a block graph. □ Lemma 2.2. (i) For an integer £ > 3, let Q = Ke be a complete subgraph of the connected graph G such that Q contains exactly one vertex, say x, of degree larger than £ - 1. Assume further that G' = G - (V(Q) \ {x}) satisfies Yt(G') < a(G') + 1. Then, Yt(G) < a(G) + 1 follows. Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 189 (ii) Let C be a cycle in a connected graph G such that C contains exactly one vertex which is of degree larger than 2. Then, there exists a nontrivial connected graph G' with f (G') < f (G) such that 7t(G') < a(G') + 1 implies 7i(G) < a(G) + 1. Moreover, if G is a cactus graph, then G' can be chosen to be a cactus graph as well. Proof. (i) We suppose d(x) > £ > 3 and V(Q) = jvi, v2,..., x}. By definition, m(G') = m(G) - Q. For any total dominating set D' of G', D' U{x} is a total dominating set of G. Hence, Yt(G) < 7t(G') + 1. Now, let S' be an optimal annihilation set in G'. • If x G S', we define S = (S'\{x}) U {v1,..., vy t j+1}. Since dG>(x) > 1, we E(S, G) < £(S', G') - 1 + (L3J +1) (£ - 1). 3 • If x G S', let S = S' U {v1,..., vytj}. Then, since £ > 3, we have E(S, G) < £(S', G') + |jj (£ - 1) < £(S', G') - 1 + ( |jj + (£ - 1). Observe that in either case and for every £ > 3, the relation |S| > |S'| + 1 holds. Moreover, as J2(S', G') < m(G'), we may estimate J2(S, G) as follows: £(S,G) < m(G') - 1+( 3 +^(£ - 1) = m(G) - ( 2 ) - 1 + + 1 (£ -1) = m(G) - 1H2 - - 1 + 1 < m(G). Here, the last inequality can be directly checked for £ =3, 4 and 5. If £ > 6, this clearly follows from f - 3 - 1 > 0. We conclude that S is an annihilation set in G and therefore, a(G) > |S| > |S'| + 1 = a(G') + 1. Together with the condition given in (i) for G', Yt(G) < Yt(G') + 1 < a(G') +2 < a(G) + 1 follows. This finishes the proof of (i). (ii) Since C3 = K3, it suffices to prove (ii) for cycles C = Ce of length £ > 4. If d* (G) < 2 or £ > 6, Lemma 2.1(i) and 2.1(iii) establish the statement. Henceforth, we will suppose that d*(G) > 3 and £ = 4 or 5. Let xv1... v^_1x be the cycle C such that d(x) > 3. First, assume that £ + dG(x) > 8; i.e., at least one of £ = 5 and dG (x) > 4 holds. Let G' = G - (V(C) \ {x}) and let D' be an optimal total dominating set of G'. Observe that D = D' U {v2, v3} is a total dominating set in G and consequently, Yt(G) < Yt(G') + 2. Now, fix an optimal annihilation set S' in G' and consider the following two subcases. • If x G S', we have E(S', G) = £(S', G') < m(G') = m(G) - £. Then, we define S = S' U{v1,v2} and observe that £(S,G) = ¿(S',G) + 2 • 2 < m(G) - £ + 4 < m(G). This proves a(G) > |S| = a(G') + 2. • If x G S', we have £(S', G) = £(S', G') + 2 < m(G') + 2 = m(G) - £ + 2. In this case, consider S = (S' \ {x}) U {v1, v2, v3}. For this set, X)(S, G) < ^(S', G) - dG(x) + 3 • 2 < m(G) - £ - dG(x) + 8 < m(G) 190 Ars Math. Contemp. 16 (2019) 141-155 holds under the present assumption £ + dG(x) > 8. Therefore, we have a(G) > |S| = a(G') +2. In either subcase, if G' satisfies (1.1), we may conclude that Yt(G) < Yt(G') +2 < a(G') + 3 < a(G) + 1. In the other case, C = C4 and dG(x) = 3. Here, we define G' = G - V(C). Since dG (x) = 3, G' is connected. If G' consists of only one vertex, Yt(G) = 2 < a(G) + 1 can be proved directly. Hence, we may assume that G' is nontrivial. Let D' be an optimal total dominating set in G' and observe that, also in this case, D = D' U {v2,v3} is a total dominating set in G. Hence, Yt(G) < Yt(G') + 2. On the other hand, let S' be an optimal annihilation set in G'. Since there is at most one edge between S' and V(C), ¿(S', G) < ¿(S', G') + 1 < m(G') + 1 = m(G) - 4. Moreover, for S = S' U {v^ v2}, we obtain ¿(S, G) = ¿(S', G) + 4 < m(G), from which a(G) > a(G') + 2 follows. Thus, if G' satisfies (1.1), the desired inequality Yt(G) < a(G) + 1 holds again. □ The analogue of the following proof was given by Desormeaux et al. [5] inside the proof of Theorem 1.2. There, both H and T were restricted to be a tree. Here, we restate and prove the lemma in a more general form, where H can be an arbitrary connected graph. Lemma 2.3. Let H be a nontrivial connected graph and T be a tree such that V(H) n V (T) = 0. Suppose that w G V (H), u G V (T), and v is a leaf in T such that d(u, v) > 3. If G is obtained from H and T by identifying w and u, there exists a connected graph G' with f (G') < f (G) such that Yt(G') < a(G') + 1 implies Yt(G) < a(G) + 1. Proof. First note that the statement follows from Lemma 2.1 (i) if d*(G) < 2. Hence, we may suppose that d*(G) > 3. Assume that T is rooted in u and choose a leaf v1 G V (T) which is of maximum distance from u. Let v2 be the parent of v1, and v3 be the parent of v2. By assumption, d(u, v1) > 3 and hence, v4 = u (i = 1,2,3). We will consider graphs G' obtained from G by removing a set of vertices from V(T) in such a way that G' will stay connected. Throughout, S' will denote an optimal annihilation set in G'. If v2 is a strong support vertex, Lemma 2.1 (ii) implies the statement. So, we may suppose that v1 is the only leaf of the support vertex v2. Since v1 is of maximum distance from u, d(v2) = 2 also follows. Remark that the same is true for any other leaf and its support vertex, if the leaf is of maximum distance from u. Suppose that d(v3) > 3 and let G' = G - {v1,v2}. So m(G') = m(G) - 2. If v3 is a support vertex in G', then v3 belongs to a minimum total dominating set D' of G'. If v3 is not a support vertex, then every child of v3 is a support vertex of degree 2. If a leaf-neighbor of a child of v3 belongs to D', then we can simply replace it in D' with the vertex v3. In either case, we may assume that v3 g D'. Thus the set D = D' U {v2} is a total dominating set of G, and so Yt(G) < |D| = |D'| + 1 = Yt(G') + 1. Independently of whether vertex v3 lies in S' or not we have ¿(S', G) < ¿(S', G') + 1 < m(G') + 1 = m(G) - 1. Consider S = S' U {vj. Then ¿(S, G) = ¿(S', G) + d(v1) < m(G), implying that a(G) > |S| = |S'| + 1 = a(G') + 1. By assumption, we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') + 1 < a(G') + 2 < a(G) + 1. Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 191 So, we may suppose that d(v3) = 2. Now we have three consecutive vertices v1, v2,v3 with degrees d(v1) = 1 and d(v2) = d(v3) = 2. Thus, by Lemma 2.1(iv), there exists a graph G' with f (G') < f (G) which satisfies the statement. □ The following lemmas will be needed to cover two specific cases in the proofs of Theorems 1.3 and 1.4. Therefore, we give the proof for both cases here. Lemma 2.4. Let H and F = S(K1,/) be two vertex-disjoint graphs with n(H) > 3 and t > 2. Assume that w is a vertex of H such that H — {w} is connected and u is the central vertex of the subdivided star F .If G is the graph obtained from H and F by identifying w and u, and Yt(G — V(F)) < a(G — V(F)) + 1, then Yt(G) < a(G) + 1. Proof. Suppose the subgraph F of G is rooted in u. We denote with vi,... ,v£ the children of u, and with w1,...,w/ the leaves. By our assumption, G' = G — V (F) = G — {u,v1,... ,v/,w1,..., wf} is a nontrivial connected graph, and m(G') = m(G)—dG(u) — t. If D' is a minimum total dominating set of G', then D = D' U {u,v1,..., vg} is a total dominating set of G,and hence Yt(G) < |D| = ID'l+t+1 = jt(G')+t+1. Now, consider an optimal annihilation set S' in G'. Independently of whether the vertices in NG> (u) are inside S' or not, we have £(S', G) 2. Then, we have a(G) > |S| = |S'| +1 +1 = a(G')+1 +1. By assumption, we have that jt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') + t +1 < a(G') + t + '2 < a(G) + 1. □ Lemma 2.5. Let H and P be two vertex-disjoint graphs, where P is the paw graph and H is a nontrivial connected graph. Moreover, let z be a vertex of H and let x be a vertex of P with dP (x) = 2. Assume that G is the graph obtained from H and P by identifying z and x. Then, there exists a connected graph G' with f (G') < f (G) such that jt(G') < a(G') + 1 implies Yt(G) < a(G) + 1. Proof. If H = K2, then G is a graph of order 5 satisfying Yt(G) = 2 and a(G) = 3. Thus, (1.1) holds for G. From now on, we assume that n(H) > 3. We denote the neighbors of x in P with u and w, and let v be the leaf neighbor of u. Two subcases will be considered depending on the degree d(x) of x in G. First suppose that d(x) = 3. Denote the third neighbor of x outside P with y. Since H had at least three vertices, y is not a leaf, and hence G' = G — V(P) = G — {x, u, v, w} is not a trivial graph. Also, m(G') = m(G) — 5. If D' is a minimum total dominating set of G', then D = D' U {u, w} is a total dominating set of G, and hence Yt(G) < |D| = |D'| + 2 = Yt(G') + 2. If S' is an optimal annihilation set of G', we have (S', G) < J2(S', G') + 1 < m(G') + 1 = m(G) — 4. Let S = S' U {u,v}. Then Y,(.S,G) = J2(S', G) + d(u) + d(v) < m(G) — 4 + 3+1 = m(G), and we have a(G) > |S| = |S'| + 2 = a(G') + 2. Then, jt(G') < a(G') + 1 implies Yt(G) < Yt(G') +2 < a(G') + 3 < a(G) + 1. (2.1) Now, suppose d(x) > 4. In this case let G' = G—{u, v, w}, and so m(G') = m(G)—4. If D' is a minimum total dominating set of G', then D = D' U {u, w} is a total dominating set of graph G, and hence Yt(G) < D = ID'I + 2 = Yt(G') + 2. Now, let S' be an optimal annihilation set in G'. If x <£ S', then J2(S', G) = J2(S', G'). In this case, let 192 Ars Math. Contemp. 16 (2019) 141-155 S = S'U{u,v}. Then ^(S,G) = £(S',G) + d(u) + d(v) < m(G) -4 + 3 + 1 = m(G), and we have a(G) > |S| = |S'| + 2 = a(G') + 2. If 7i(G') < a(G') + 1, the chain (2.1) of inequalities verifies the statement. But, if x e S', then £(S',G) = £(S',G') +2 < m(G') + 2 = m(G) - 4 + 2 = m(G) — 2. In this case, let S = (S'\{x}) U {u, v, w}. Since d(x) > 4 we have £(S, G) = £(S', G) — d(x) + 3+1 + 2 < m(G) — 2 — 4 + 6 = m(G) implying that a(G) > |S| = |S'| + 2 = a(G') + 2. By assumption we have that 7i(G') < a(G') + 1. Therefore, we get again (2.1) which proves the lemma. □ 3 Cactus graphs Recall that a cactus graph is a connected graph such that any two of its cycles are pairwise edge-disjoint. If the cactus graph does not contain any cycles, then it is a tree. Let C1 and C2 be two cycles in the cactus graph. We define d (C\C2) = min{d(u,v) | u e V (C1) ,v e V (C2)}, where d(u, v) denotes the distance between vertices u and v. Let x1 e V (C^ and x2 e V (C^ be two vertices such that d(x1, x2) = d (C\ C2). Then we call x1 and x2 exit-vertices of cycles C1 and C2, respectively. A cycle is said to be an outer cycle if it has at most one exit-vertex. If a cactus graph is not a tree, then by the definition of a cactus graph it must contain at least one outer cycle. Note that a cactus graph, which is neither a tree nor a cycle, does not contain exit-vertices if and only if it is unicyclic. In this case, we will take an arbitrary vertex of the unique cycle whose degree is at least 3 for the role of the exit-vertex x. In the right-hand side graph of Figure 2, we have three possibilities for the choice of that vertex x (either x1 or x2 or x3). In both cases, whether a cactus graph has one or more cycles, vertex x will always have degree d(x) > 3. Figure 2: Two examples of cactus graphs. The first one has three outer cycles (C1, C2, C4), its exit-vertices are filled with black. The second cactus graph is unicyclic with one outer cycle, and has no exit-vertices. In this section we prove Conjecture 1.1 for cactus graphs. Recall the corresponding statement. Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 193 Theorem 1.3. If G is a nontrivial cactus graph, then Yt(G) < a(G) + 1. Proof. We proceed by induction on the value of function f (G) > 7. For f (G) = 7 we have G = K2, and Yt(K2) = 2 = a(K2) + 1. For the inductive hypothesis, let f (G) > 8 and assume that for every nontrivial cactus graph G' with f (G') < f (G) we have Yi(G') < a(G') + 1. If G is a tree, then by Theorem 1.2 the result follows. Also, if G is a cycle, the statement is true. Thus, we may suppose that G contains at least one cycle as a proper subgraph. We denote with Ck, k > 3, an outer cycle of G. Through most part of the proof, we will consider cactus graphs G' formed from G by removing a set of vertices in such a way that graph G' will still be a connected cactus graph and consequently f (G') < f (G) will hold. Throughout, S' will denote an optimal annihilation set in G'. We consider two cases. Case 1: All vertices from V(Ck)\{x} have degree 2. Lemma 22(ii) and our inductive hypothesis together imply that Yt(G) < a(G) + 1. Case 2: There exists a vertex from V(Ck )\{x} that has degree at least 3. Since V(Ck)\{x} contains some vertices of degree at least 3, and Ck is an outer cycle, there are trees attached to those vertices. Suppose, we root all trees in the vertices V(Ck)\{x} to which these trees are attached. Amongst those trees we consider the tree T with the largest height h(T) = max{d(u, v) | u = V(Ck) n V(T), v e V(T)}. Denote this maximum height with h > 1 and let u be the vertex of V(Ck)\{x} to which tree T is attached. We consider three subcases. Case 2.1: h > 3. Since h > 3, there exists a leaf v e V(T) such that d(u, v) = h > 3. By Lemma 2.3 and our inductive hypothesis, graph G satisfies (1.1). Case 2.2: h = 2. We only need to consider the four cases shown in Figure 3. All other cases for h = 2 can be proved with the help of Lemma 2.1(ii) and 2.1(v). (a) (b) (c) (d) Figure 3: Cases for h = 2. We first start with the case in Figure 3(a). In this case, we have a subdivided star Ki^, I > 2, attached to the outer cycle, and hence, by Lemma 2.4 and our inductive hypothesis for G' = G - V(S(Km)), graph G satisfies (1.1). Next, we consider the case in Figure 3(b). Vertex u has only one path of length 2 attached to it, i.e. d(u) = 3. We suppose that u has a neighbor u1 in V(Ck)\{x} with 194 Ars Math. Contemp. 16 (2019) 141-155 degree d(ui) = 2. We denote with v the only child of u, and with w the only child of v. Let G' = G — {u, u1, v, w}, and so m(G') = m(G) — 5. If D' is a minimum total dominating set of G', then D = D' U{u,v} is a total dominating set of graph G, and hence 7t(G) < |D| = |D'| +2 = Yt(G' ) + 2. Independently of whether the neighbors of u and u1 in G' are inside S' or not, wehavej](S',G) < £(S',G') + 2 < m(G') + 2 = m(G) — 3. Let S = S' U {v, w}. Then £(S, G) = £(S', G) + d(v) + d(w) < m(G) — 3 + 2 + 1 = m(G), and we have a(G) > |S| = |S' |+2 = a(G')+2. Applying our inductive hypothesis to G', we have that Yt (G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +2 < a(G') + 3 < a(G) + 1. We proceed with the case in Figure 3(c). Vertex u has again only one path of length 2 attached to it, i.e. d(u) = 3. We suppose that u has a neighbor u1 in V(Ck)\{x} with degree d(u1) = 3, and a path of length 1 attached to it. Denote its child with v1. We also denote with v the only child of u, and with w the only child of v. Let G' = G — {u, v, w, u1, v1}, and so m(G') = m(G) — 6. If D' is a minimum total dominating set of G', then D = D' U {u, v, u1} is a total dominating set of G, and hence Yt(G) < |D| = |D'| + 3 = Yt(G') + 3. Independently of whether the neighbors of u and u1 in G' are inside S' or not, we have £(S', G) < E(S', G') + 2 < m(G') + 2 = m(G) — 4. Let S = S' U {v, w, v1}. Then £(S, G) = £(S', G) + d(v) + d(w) + d(v1) < m(G) — 4 + 2 + 1 + 1 = m(G), and we have a(g) > |S| = |S'| + 3 = a(G') + 3. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +3 < a(G') + 4 < a(G) + 1. The last case to consider is the one shown in Figure 3(d). Denote with u1,..., uk-1 all vertices of V(C)\{x}. Each of those vertices must have one path of length 2 attached to it, i.e. d(uj) = 3 for every i G {1,..., k — 1}, since otherwise this case would be covered by one of the previous three cases. Clearly, vertices u1 and uk-1 are neighbors of x. Denote for every i G {1,..., k — 1} with v4 the only child of u4, and with w4 the only child of v4. We consider two subcases. First, suppose that d(x) = 3. Denote the third neighbor of x outside Ck with y. If vertex y was a leaf, then we could exchange vertex x with one of u/s, and use the proof for the case in Figure 3(c). Hence, we may assume that y is not a leaf and graph G' = G — {x,u1,... ,uk-1,v1,... ,vk-1, w1,... ,wk-1} is not a trivial cactus graph. Also, m(G') = m(G) — 3k +1. If D' is a minimum total dominating set of G', then D = D' U {u1,..., uk-1, v1,..., vk-1} is a total dominating set of G, and hence Yt(G) < |D| = |D'| + 2k — 2 = Yt(G') + 2k — 2. Independently of whether y is inside S' or not we have (S',G) < ^(s', G') + 1 < m(G') + 1 = m(G) — 3k + 2. Let 5 = S'U{v1,..., vfc_1, w1,..., wfc_1}. Then £(S, G) = E(S', G)+2(k—1) + (k—1) < m(G) — 3k + 2 + (3k — 3) = m(G) — 1, and we have a(G) > |S| = |S'| + 2k — 2 = a(G') + 2k — 2. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') + 2k — 2 < a(G') + 2k — 1 < a(G) + 1. Now, suppose that d(x) > 4. Let G' = G — {u1,..., uk-1, v1,..., vk-1, w1,..., wk-1}, and so m(G') = m(G) — 3k + 2. If D' is a minimum total dominating set of G', then D = D' U {u1,..., uk-1, v1,..., vk-1} is a total dominating set of G, and hence Yt(G) < |D| = |D'| + 2k — 2 = Yt(G') + 2k — 2. If x £ S', the^(S',G) = ^(S',G'). Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 195 In this case, let S = S' U {vi,..., vfc_i, wi,..., wfc_i}. Then £(S,G) = £(S',G) + 2(k - 1) + (k -1) < m(G) -1, and we have a(G) > |S| = |S'| + 2k - 2 = a(G') + 2k - 2. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, 7t(G) < 7i(G') + 2k - 2 < a(G') + 2k - 1 < a(G) + 1. If x € S', then £(S', G) = £(S', G') + 2 < m(G') + 2 = m(G) - 3k + 2 + 2 = m(G) - 3k + 4. In this case, let S = (S'\{x}) U {u1, v1,..., vk_1, w1,..., wk_1}. Since d(x) > 4 we have £(S,G) = £(S', G)-d(x)+d(u1)+2(k-1) + (k-1) < m(G)-3k+ 4 -4 + 3 + 3(k - 1) = m(G), implying that a(G) > |S| = |S'| + 2k - 2 = a(G') + 2k - 2. By our inductive hypothesis, we have that jt (G') < a(G') + 1. Therefore, 7i(G) < Yt(G') + 2k - 2 < a(G') + 2k - 1 < a(G) + 1. Case 2.3: h = 1. It suffices to consider only those cases shown in Figure 4. Note that all other cactus graphs with h =1 would involve two leaves at distance of at most 3, and hence these cases can be reduced to the direct application of Lemma 2.1(ii) and 2.1(v). (a) (b) (c) Figure 4: Cases for h = 1. First, consider Figure 4(a). Here, we assume that vertex u has degree d(u) = 3, and its neighbors in V(Ck)\{x}, namely u1 and u2, are of degree 2. Denote the child of u with v. In this case we want u1 and u2 to be different from the exit-vertex x. Let G' = G - {u, v, u1, u2}, and so m(G') = m(G) - 5. If D' is a minimum total dominating set of G', then D = D' U{u,u4} with i = 1 or i = 2 is a total dominating set of G, and hence Yt(G) < |D| = |D'|+2 = Yt(G')+2. Independently of whether the neighbors of u1 and u2 in G' are inside S' or not, we have E(S',G) < E(S',G')+2 < m(G')+2 = m(G) - 3. Let S = S' U{u1,v}. Then £(S,G) = ^(s', G) + d(u1) + d(v) < m(G) - 3 + 2 + 1 = m(G), and we have a(G) > |S| = |S'|+2 = a(G' )+2. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +2 < a(G') + 3 < a(G) + 1. We proceed with the case in Figure 4(b). Denote with u the vertex of V(Ck)\{x} with one path of length 1 attached to it, i.e. d(u) = 3, and let v be its only child. One of the neighbors of u must clearly be vertex x because otherwise we would have the case in Figure 4(a). Suppose that all other vertices in V(Ck )\{x}, denote them with w1,..., wk_2, have degree 2. 196 Ars Math. Contemp. 16 (2019) 141-155 First suppose that k = 3. In this case x, u, v and wi induce the paw graph. Then, by Lemma 2.5 and our inductive hypothesis, graph G satisfies (1.1). Suppose that k > 4. Let G' = G — {u, v, w1,w2}, and so m(G') = m(G) — 5. Remark that G' remains a cactus graph. If D' is a minimum total dominating set of G', then D = D' U{u,w1} is a total dominating set of G,and hence Yt(G) < |D| = |D'| +2 = 7t(G') + 2. Independently of whether x and w3 is inside S' or not we have J2 (S', G) < ¿(S', G') + 2 < m(G') + 2 = m(G) — 3. Let S = S' U {v, wj. Then (S, g) = XXS', G) + d(v) + d(w1) < m(G) — 3+1 + 2 = m(G), and we have a(G) > |S| = |S'| + 2 = a(G') + 2. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') +1. Therefore, Yt(G) < Yt(G') +2 < a(G') + 3 < a(G) + 1. We finish with the case in Figure 4(c). Denote with u1 and u2 two vertices in V(Ck)\{x} each with one path of length 1 attached to it, i.e. d(u1) = d(u2) = 3, and let v1 and v2 be the only child of u1 and u2, respectively. The exit-vertex x must be the neighbor of both u1 and u2 because otherwise we would have the case in Figure 4(a). We denote all vertices in V(Ck)\{x} between vertex u1 and u2 with w1,..., wk-3. Those vertices have all degree 2. If k = 3, the statement follows immediately from the hypothesis and Lemma 2.1 (v), since in this case the support vertices of v1 and v2 are adjacent. Thus, we first suppose that k = 4. Let G' = G — {u1, v1, u2, v2, w1}, and so m(G') = m(G) — 6. If D' is aminimum total dominating set of G', then D = D' U {x, u1, u2} is a total dominating set of G, and hence Yt(G) < |D| = |D' | + 3 = Yt(G') + 3. Independently of whether x G S' or x G S', we have X(S',G) < jr(S',G')+2 < m(G') + 2 = m(G)—4. LetS = S'U{v1,v2,w1}. Then £(S, G) = £(S', G) + d(v1) + d(v2) + d(w1) < m(G) — 4 + 1 + 1 + 2 = m(G), and we have a(G) > |S| = |S'| + 3 = a(G') + 3. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +3 < a(G') + 4 < a(G) + 1. Now, suppose that k = 5. We make a similar cut than the one for k = 4. Let G' = G — {u1, v1, u2, v2, w1, w2}, and so m(G') = m(G) — 7. If D' is aminimum total dominating set of G', then D = D' U {x, u1, u2} is a total dominating set of G, and hence Yt(G) < |D| = |D'| + 3 = Yt(G') + 3. For any optimal annihilation set S' of G', we hav^(S',g) < ^(S', G') + 2 < m(G') + 2 = m(G) — 5. Let S = S'U{v1,v2,w1}. Then ¿(S,G) = ^(S',G) + d(v1) + d(v2) + d(w1) < m(G) — 1, and a(G) > |S| = |S'| + 3 = a(G') + 3 follows. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +3 < a(G') + 4 < a(G) + 1. For the last case, let k > 6. We have three consecutive vertices w1, w2, w3 with degree d(w1) = d(w2) = d(w3) = 2. Furthermore, vertices u1 and w4 (or u1 and u2, if k = 6) are not adjacent. Thus, by Lemma 2.1 (iii) and our inductive hypothesis, graph G satisfies (1.1). These cover all possible cases which can occur in a cactus graph which is neither a tree nor a cycle. Hence, Conjecture 1.1 is true for the family of cactus graphs. □ Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 197 4 Block graphs Recall that a block graph is a connected graph in which every 2-connected component (block) is a clique. Block graphs have minimum degree at least 3 if its building blocks are complete graphs Kk, k > 4. Thus, Conjecture 1.1 obviously holds for them. On the other hand, block graphs also contain blocks K2 and K3, and therefore, it clearly makes sense to study Conjecture 1.1 on block graphs. We proceed with a similar definition than the one for cactus graphs. If all cliques in a block graph are K2, then it is a tree. For every k > 3 we will call complete graph Kk a complex clique. Let K1 and K2 be two complex cliques in the block graph. We define d (K 1,K2) = min{d(u, v) | u G V (K1) ,v G V (K2)}, where t)(u, v) denotes the distance between vertices u and v. Let x1 G V (K^ and x2 g V (K2) be two vertices such that d(x1, x2) = d(K1, K2). Then we call x1 and x2 exit-vertices of complex cliques K1 and K2, respectively. Notice that a complex clique might not have any exit-vertices if it is the only complex clique in the block graph. A complex clique will be called an outer complex clique if it has at most one exit-vertex. If a block graph is not a tree, then by the definition of a block graph it must contain at least one outer complex clique. Now, we are ready to present a proof of Theorem 1.4. Recall its statement. Theorem 1.4. If G is a nontrivial block graph, then Yt(G) < a(G) + 1. Proof. We proceed by induction on the value of function f (G). For f (G) = 7 we have G = K2, and Yt(K2) = 2 = a(K2) + 1. For the inductive hypothesis, let f (G) > 8 and assume that for every nontrivial block graph G' with f (G') < f (G) we have Yi(G') < a(G') + 1. If G does not contain complex cliques, then it is a tree, and by Theorem 1.2 the result follows. Also, if G is a complete graph, i.e. G = Ke, I > 2, we have 7t(Ke) = 2 < a(Ke) + 1. Thus, we may suppose that G is neither a tree nor a complete graph, but contains at least one complex clique as a proper subgraph. We denote with Kk an outer complex clique of G. Similarly as in the proof for cactus graphs, all outer complex cliques in the figures will be drawn with an exit-vertex x even though a unique complex clique in a block graph does not have one. In the latter case, we denote with x an arbitrary vertex of clique Kk whose degree is at least k. In both cases, whether a block graph has one or more complex cliques, vertex x will have degree d(x) > k. Through most part of the proof, we will consider block graphs G' formed from G by removing a set of vertices in such a way that graph G' will still be a connected block graph and consequently f (G') < f (G) will hold. Throughout, S' will denote an optimal annihilation set in G'. We consider two cases. Case 1: All vertices from V(Kk )\{x} have degree k - 1. Let u1,..., uk-1 be vertices from V(Kk)\{x} with degree k - 1. By Lemma 2.2(7), and inductive hypothesis for G' = G - {u1,..., uk-1}, graph G satisfies (1.1). Case 2: There exists a vertex from V(Kk)\{x} that has degree at least k. Since V(Kk)\{x} contains vertices of degree at least k, and Kk is an outer complex clique, there are trees attached to those vertices. Suppose, we root all trees in the vertices V(Kk)\{x} to which these trees are attached. Amongst those trees we consider the tree 198 Ars Math. Contemp. 16 (2019) 141-155 T with the largest height h. Let u be the vertex of V(Kk)\{x} to which this tree T is attached. We split the problem into three subcases. Case 2.1: h > 3. Since h > 3, there exists a leaf v G V(T) such that d(u, v) = d > 3. By Lemma 2.3 and our inductive hypothesis, graph G satisfies (1.1). Case 2.2: h = 2. We only need to consider cases shown in Figure 5. All other cases for h = 2 can be proved directly with the help of Lemma 2.1(h) and 2.1(v). (a) (b) Figure 5: Cases for h = 2. We start with the case in Figure 5(a) and suppose that there exists a subdivided star S(Ki^), I > 2, attached to the outer complex clique. By Lemma 2.4 and our inductive hypothesis for G' = G - V(S(Kii^)), graph G satisfies (1.1). In the case shown in Figure 5(b), there are vertices in V(Kk)\{x} such that a path of length 2 is attached to them. We denote such vertices with ui,..., ua. Since h = 2, we must have at least one such vertex. Thus, a G {1,..., k - 1}. For each i G {1,..., a} we denote with ui the child of u, and with ui' the child of ui. Also, we may suppose that at most one vertex in V(Kk)\{x} has a path of length 1 attached to it. If we had more such vertices, then we would have two adjacent support vertices and we could prove the statement by referring to Lemma 2.1 (v). Hence, denote this vertex with v and let b denote the Boolean value whether it exists in V(Kk)\{x} or not, i.e. b G {0,1}. We denote the child of v with v'. There may also be some vertices in V(Kk)\{x} without a path attached to them. Denote them with w,..., wc, c G {0,..., k - 2}. Clearly, we have a + b + c = k — 1. Let G' = G — {ui,... ua, ui,... ua, u'',..., ua', v,v',wi,..., wc}, and so m(G') = m (G) - (ZU—1 + 2a + b). If D' is a minimum total dominating set of G', then D = D' U {u,..., ua, u', ..., ua, v} is a total dominating set of G, and hence Yt(G) < |D| = |D'| + 2a + b = Yt(G') + 2a + b. Independently of whether x is inside S' or not we have E(S',G) < £(S',G') + (k - 1) = m(G') + (k - 1) = m(G) - f k(k - 1) + 2a + ^ + k - 1. Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 199 Let S = S' U {u1,..., «a, u'i,... ua, v'} and observe that ^(S, G) = £(S', G) + d(u 1) + • • • + d(<) + d(u 1') + • • • + d(<) + d(v') < m(G) - ^ k(k- 1) + 2a + b^ + k - 1 + 3a + b. First, suppose that 1 < a < k - 2 holds. Then, m(G) - ( k(k - 1) +2a + b ) + k - 1 + 3a + b = m(G) - 1 k2 + 3k + a - 1 2 J 2 2 1 5 1 < m(G) - 1 k2 + -k - 3 = m(G) - ^k - 2)(k - 3) < m(G). Similarly, under the conditions a = k - 1 > 3, the following relations hold: k(k - 1) , O,. , A , L ! , O. , l.^^/TA 1 i_2 , 51 m(G) - —+ 2a + bj + k - 1 + 3a + b < m(G) - -k2 + -k - 2 < m(G). In both cases we get J](S, G) < m(G), which implies a(G) > |S| = |S'| + 2a + b = a(G') + 2a + b. By our inductive hypothesis, G' satisfies Conjecture 1.1. Consequently, 7t(G) < 7i(G') + 2a + b < a(G') + 2a + b +1 < a(G) + 1. What remains is to establish the statement for k = 3 and a = k - 1=2. We consider two subcases. First, suppose that d(x) = 3. Denote the third neighbor of x outside K3 with y. If vertex y was a leaf, then we could exchange vertex x either with u1 or u2, and apply the proof for the case a = 1 = k - 2. Hence, we may assume that y is not a leaf, and therefore, graph G' = G - {x, u 1, u 1, u '/, u2, u2, u2'} is not a trivial block graph. Also, m(G') = m(G) - 8. If D' is a minimum total dominating set of G', then D = D' U {u i, u 1, u2, u2} is a total dominating set of graph G, and hence Yt (G) < | D | = |D' | + 4 = Yt(G') + 4. Independently of whether y is inside S' ornotwehav^J] (S',G) < X)(S',G') + 1 < m(G') +1 = m(G) -8 + 1 = m(G) -7. Let S = S'U{u1, u'/, u2, u2'}. Then £(S,G) = ^(S',G) + d(u') + d(u^') + d(u2) + d(u2') < m(G) -7+2+1 + 2+1 = m(G) - 1, and we have a(G) > |S| = |S'| + 4 = a(G') + 4. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +4 < a(G') + 5 < a(G) + 1. Now, suppose that d(x) > 4. Let G' = G - {u 1, u 1, u", u2, u2, u2'}, and so m(G') = m(G) - 7. If D' is a minimum total dominating set of G', then D = D' U {u 1, u 1, u2, u2} is a total dominating set of G, and hence Yt(G) < |D| = |D'| + 4 = Yt(G) + 4. If x S', then ^(S',G) = ^(S',G'). In this case, let S = S'U{u1 ,u 1',u2,u2'}. ThenJ](S, G) = ^(S', G) + d(u 1) + d(u 1') + d(u2) + d(u2') < m(G) - 7 + 2 + 1+2 2+1 = m(G) - 1, and we have a(G) > |S| = |S'| + 4 = a(G') + 4. Applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +4 < a(G') + 5 < a(G) + 1. If x € S', then ^(S',G) = E(S',G') + 2 < m(G') + 2 = m(G) - 7 + 2 = m(G) - 5. In this case, let S = (S'\{x}) U {u 1, u1, u 1', u2, u2'}. Since d(x) > 4 we hav^(S, G) = 200 Ars Math. Contemp. 16 (2019) 141-155 X)(S',G)-d(x)+d(u1 )+d(u1)+d(u1')+d(u2)+d(u2') < m(G)-5-4+3+2+1+2+1 = m(G), implying that a(G) > |S| = |S'| + 4 = a(G') + 4. Applying again our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +4 < a(G') + 5 < a(G) + 1. Case 2.3: h = 1. We need to consider only one case which is shown in Figure 6. As we have already seen in Case 2.2, all other cases for h =1 can be proved with the help of Lemma 2.1 (ii) and 2.1(v). P Figure 6: The case for h = 1. We may also suppose that there is at most one vertex in V(Kk)\{x} which has a path of length 1 attached to it. If we had more such vertices, then we would have adjacent support vertices and we could prove this case with Lemma 2.1(v). Denote this vertex with u and its child with v. There are also vertices in V(Kk)\{x} without a path attached to them. Denote them with w1,..., wk-2. Let G' = G — {u, v, w1,..., wk-2}, and so m(G') = m(G) — (fc(fc-1) + 1). If D' is a minimum total dominating set of G', then D = D' U{x,u} is a total dominating set of G,and hence Yt(G) < |D| = |D'| + 2 = Yt(G') + 2. Independently of whether x is inside S' or not we have J2 (S', G) < J2 (S', G') + k — 1 < m(G') + k — 1 = m(G) — (+ 1) + k — 1. Let S = S' U {v,w}. Then, £(S, G) = ^(S', G) + d(v) + d(W1) < m(G) ^+ ^ + k — 1 + 1 + k — 1. For k > 4, this gives 15 G) < m(G) — ^k2 + 5k — 2 < m(G). Hence, a(G) > |S| = |S'| + 2 = a(G') + 2 and applying our inductive hypothesis to G', we have that Yt(G') < a(G') + 1. Therefore, Yt(G) < Yt(G') +2 < a(G') + 3 < a(G) + 1. We end the proof with k = 3. In this case, x, u, v and w1 induce the paw graph and, by Lemma 2.5 and our inductive hypothesis, graph G satisfies (1.1). We have considered all possible cases which can occur in a block graph which is neither a tree nor a complete graph. Hence, Conjecture 1.1 is true over the family of block graphs. □ 2 Cs. Bujtas andM. Jakovac: Relating the total domination number and the annihilation ... 201 5 Concluding remarks To show that our main results, namely Theorem 1.3 and 1.4, are sharp, we remark that trees are included in both classes. Therefore, we may refer to the family of trees characterized in [5] which satisfy Conjecture 1.1 with equality. We may also observe that even cycles Cn, where n = 2 (mod 4), have Yt(Cn) = n + 1 and a(Cn) = n. Also, there are other cactus graphs which are neither trees nor cycles, but satisfy 7t(G) = a(G) + 1. Take two vertex-disjoint cycles C6, and connect any vertex from the first cycle and any vertex from the second cycle with a path of length 3. We get a cactus graph G on n = 14 vertices and m =15 edges. It is easy to see that jt (G) = 8 and a(G) = 7. Thus, Theorem 1.3 holds with equality for the graph G constructed this way. One can also use other cycles Cn with n = 2 (mod 4) and connect them with different paths to obtain other extremal examples. Hence, the following characterization problem remains open. Problem 5.1. Characterize cactus graphs G which satisfy Yt(G) = a(G) + 1. For block graphs, already the following question might be interesting. Problem 5.2. Does there exist a block graph G which is neither a tree nor a K3 but satisfies Yt(G) = a(G) + 1? References [1] D. Archdeacon, J. Ellis-Monaghan, D. Fisher, D. Froncek, P. C. B. Lam, S. Seager, B. Wei and R. Yuster, Some remarks on domination, J. Graph Theory 46 (2004), 207-210, doi:10.1002/ jgt.20000. [2] B. Brimkov and I. V. Hicks, Memory efficient algorithms for cactus graphs and block graphs, Discrete Appl. Math. 216 (2017), 393-407, doi:10.1016/j.dam.2015.10.032. [3] V. Chvatal and C. McDiarmid, Small transversals in hypergraphs, Combinatorica 12 (1992), 19-26, doi:10.1007/bf01191201. [4] E. DeLaVina, Written on the Wall II, http://cms.dt.uh.edu/faculty/ delavinae/research/wowII/. [5] W. J. Desormeaux, T. W. Haynes and M. A. Henning, Relating the annihilation number and the total domination number of a tree, Discrete Appl. Math. 161 (2013), 349-354, doi:10.1016/j. dam.2012.09.006. [6] W. J. Desormeaux, M. A. Henning, D. F. Rall and A. Yeo, Relating the annihilation number and the 2-domination number of a tree, Discrete Math. 319 (2014), 15-23, doi:10.1016/j.disc. 2013.11.020. [7] J. R. Griggs and D. J. Kleitman, Independence and the Havel-Hakimi residue, Discrete Math. 127 (1994), 209-212, doi:10.1016/0012-365x(92)00479-b. [8] M. A. Henning, A survey of selected recent results on total domination in graphs, Discrete Math. 309 (2009), 32-63, doi:10.1016/j.disc.2007.12.044. [9] M. A. Henning and A. Yeo, Total Domination in Graphs, Springer Monographs in Mathematics, Springer, New York, 2013, doi:10.1007/978-1-4614-6525-6. [10] J. Lyle and S. Patterson, A note on the annihilation number and 2-domination number of a tree, J. Comb. Optim. 33 (2017), 968-976, doi:10.1007/s10878-016-0019-7. 202 Ars Math. Contemp. 16 (2019) 141-155 [11] R. Pepper, On the annihilation number of a graph, in: V. Zafiris, M. Benavides, K. Gao, S. Hashemi, K. Jegdic, G. A. Kouzaev, P. Simeonov, L. Vladareanu and C. Vobach (eds.), AMATH'09 Proceedings of the 15th American Conference on Applied Mathematics, World Scientific and Engineering Academy and Society (WSEAS), Stevens Point, Wisconsin, USA, 2009 pp. 217-220, held in Houston, USA, April 30 - May 02, 2009. [12] R. D. Pepper, Binding Independence, Ph.D. thesis, University of Houston, Houston, Texas, 2004, https://search.proquest.com/docview/305196562. [13] S. Thomasse and A. Yeo, Total domination of graphs and small transversals of hypergraphs, Combinatorica 27 (2007), 473-487, doi:10.1007/s00493-007-2020-3. [14] Zs. Tuza, Covering all cliques of a graph, Discrete Math. 86 (1990), 117-126, doi:10.1016/ 0012-365x(90)90354-k. /^creative ^commor ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 203-213 https://doi.org/10.26493/1855-3974.1406.cc1 (Also available at http://amc-journal.eu) The validity of Tutte's 3-flow conjecture for some Cayley graphs* Milad Ahanjideh, Ali Iranmanesh Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran Received 19 May 2017, accepted 12 June 2018, published online 24 November 2018 Abstract Tutte's 3-flow conjecture claims that every bridgeless graph with no 3-edge-cut admits a nowhere-zero 3-flow. In this paper we verify the validity of Tutte's 3-flow conjecture on Cayley graphs of certain classes of finite groups. In particular, we show that every Cayley graph of valency at least 4 on a generalized dicyclic group has a nowhere-zero 3-flow. We also show that if G is a solvable group with a cyclic Sylow 2-subgroup and the connection sequence S with |S | > 4 contains a central generator element, then the corresponding Cayley graph Cay(G, S) admits a nowhere-zero 3-flow. Keywords: Nowhere-zero flow, Cayley graph, Tutte's 3-flow conjecture, connection sequence, solvable group, nilpotent group. Math. Subj. Class.: 05C25, 05C21, 20D10 1 Introduction Let D be an orientation of a graph r and let k be a positive integer. A k-flow on a graph r is a pair (D, f) where f is an integer valued function f: E(r) ^ Z such that |f (e)| < k for every e e E(r), and for every v e V(r), E f (e)= E f (e). e£E(v) + eeE(v)- *The authors would like to thank Professor Martin Skoviera and Professor Saieed Akbari for their useful comments which improved the paper. Also we thank the anonymous referees for reading the paper carefully and making the useful suggestions. E-mail addresses: ahanjidm@gmail.com (Milad Ahanjideh), iranmanesh@modares.ac.ir (Ali Iranmanesh) ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 204 Ars Math. Contemp. 16(2019)203-213 where E(v)+ and E(v) are the all edges with tails at v and heads at v, respectively. A nowhere-zero k-flow (abbreviated a k-NZF) is a pair (D, f) such that for every e G E(r), f (e) = 0. The following conjecture is due to Tutte and is known as Tutte's 3-flow conjecture: Conjecture 1.1 (Tutte's 3-flow conjecture [8, 9]). Every bridgeless graph with no 3-edge-cut has a 3-NZF. Although Tutte's 3-flow conjecture has been studied by many authors, it is still widely open. Let G be a finite group with identity 1 and S = (s1; s2,..., sn) be a sequence of elements of G \ {1} such that the mapping sj ^ s-1 permutes the entries of S. We call S a connection sequence (note that all entries of S are distinct unless stated otherwise). A Cayley graph, denoted by Cay(G, S), is a graph whose vertex set is G with adjacency defined by g ~ h if and only if g-1h G S, for every g, h G G. We see at once that if S generates G, then Cay(G, S) is connected. Alspach et al. [1] conjectured that every Cayley graph of valency at least 3 has a nowhere-zero 4-flow. They also showed their conjecture to be true for solvable groups. Their result was significantly strengthened and extended by Nedela and Skoviera to a much wider class of groups [5]. By combining the fact that a k-valent Cayley graph is k-edge-connected graph with the fact that every 4-edge-connected graph has a 4-NZF [2], we deduce that every Cayley graph of valency at least 4 has a 4-NZF. Thus the question about the existence of a nowhere-zero 4-flow is interesting only for cubic Cayley graphs. Since 4-regular graphs admit a nowhere-zero 2-flow, the important question about flows on Cayley graphs of valency greater than 3 is whether every Cayley graph of valency at least 5 has a nowhere-zero 3-flow. In other words, it is interesting to verify whether Tutte's 3-flow conjecture holds on such Cayley graphs. In [6], it has been proved that every abelian Cayley graph of valency k, where k > 4, admits a 3-NZF. Nanasiova and Skoviera [4] improved the above result to Cayley graphs on a group G whose Sylow 2-subgroup is the direct factor of G, and as a consequence, they showed that every Cayley graph of valency at least 4 on a nilpotent group has a 3-NZF. Recently, Yang and Li [11] showed the same fact for a Cayley graph on a dihedral group, and L. Li and X. Li [3] verified Tutte's 3-flow conjecture for Cayley graphs on generalized dihedral groups and generalized quaternion groups. In this paper, we investigate Tutte's 3-flow conjecture for Cayley graphs on a solvable group with a suitable normal subgroup (Theorems 3.1 and 3.2 and Remark 3.5) and as a consequence of these theorems, we show that every Cayley graph of valency at least 4 on a generalized dicyclic group satisfies Tutte's 3-flow conjecture. By using Theorem 3.6 we can obtain the results of [3] and [11] by a different method. In [4], the authors showed that a Cayley graph of valency at least 4 with the connection sequence containing a central involution admits a 3-NZF. In Theorem 3.6, we extend this result to the case when Sylow 2-subgroups of G are cyclic and the connection sequence of G contains a central generator element. As a consequence of this theorem, we show that if a Cayley graph of valency at least 4 on a solvable group G, with a cyclic Sylow 2-subgroup, admits a 3-NZF, then every Cayley graph of valency at least 4 on the direct product of G and a nilpotent group admits a 3-NZF. M. Ahanjideh and A. Iranmanesh: The validity ofTutte's 3-flow conjecture for some ... 205 2 Notation and preliminaries The terminology and notation used in this paper are standard both in group theory and graph theory, see for instance [7, 10]. An element g of G is called an involution if g has order 2. Let Z(G) be the center of a group G. We say that an element x of G is central if x G Z(G). The group generated by a sequence S is denoted by (S) and the element x G G is named a generator element of G in S if (S \ {x}) = (S). For integers m, n > 2, a cycle of length n and a path of length m - 1 are denoted by Cn and Pm, respectively. For an integer m > 3 and for n G Zm, the Cayley graph Cay(Zm, {-1, 1, -n, n}) will be denoted by C(m, n). Let N be a subgroup of G and x belongs to a left transversal set of N in G. The image of Cay(N, S) under left translation by x is denoted by x Cay(N, S). The Cartesian product Hi □ H2 of graphs Hi and H2 is a graph such that V(Hi) x V(H2) is its vertex set and any two vertices (u, u') and (v, v') are adjacent in Hi □ H2 if and only if either u = v and u'v' G E(H2) or u' = v' and uv G E (Hi). Set L = P„ □ K2, where V (P„) = {1,2, ...,n} and V (K2) = {1,2}. The Mobius ladder MLn is a graph obtained by adding the edges (12)(n1) and (11)(n2) to L. Also, by adding the edges (11)(n1) and (12)(n2) to L, we obtain a graph is called the circular ladder CLn. In fact CLn = Cn □ K2. Any graph isomorphic to either CLn or MLn for some n will be referred to as a closed ladder. It is easy to check that the circular ladder is bipartite if and only if n is even while the Mobius ladder is bipartite if and only if n is odd. Lemma 2.1 ([4, Theorems 3.3 and 4.3]). Let Cay(G, S) be a Cayley graph of valency k, where k > 4. If S contains a central involution, then Cay(G, S) has a 3-NZF. In particular, if G is nilpotent, then Cay(G, S) has a 3-NZF. Lemma 2.2 ([4, Proposition 4.1]). Let G be a group, H be a normal subgroup of G and let S be a connection sequence with no intersection with H. If Cay(G/H, S/H) has a 3-NZF, then so does Cay(G, S). Note that in Lemma 2.2, according to the paragraph before Proposition 4.1 in [4], for distinct elements s,t G S, we regard sH and tH as distinct elements of S/H. So, Cay(G/H, S/H) may have parallel edges even when Cay(G, S) is simple and |S/H| = |S|. Lemma 2.3 ([6, Theorem 1.1]). Every abelian Cayley graph of valency k, where k > 4, admits a 3-NZF. Lemma 2.4 ([6, Proposition 2.5]). Let m, n > 3 be integers. Then the graph Cn □ Cm □ K2 admits a 3-NZF. Lemma 2.5 ([6, Proposition 2.6]). Let m, n > 3 be two integers such that m > n > 1 and m > 3. Then the graph C(m, n) □ K2 admits a 3-NZF. Lemma 2.6 ([6, Corollary 2.2]). A regular bipartite graph of valency at least 2 admits a 3-NZF. Lemma 2.7 ([10, page 308]). A cubic graph has a 3-NZF if and only if it is bipartite. Lemma 2.8. Let G be a group and N be a subgroup of G of index 2. Then Cay(G, S \ (S n N)) is bipartite. 206 Ars Math. Contemp. 16 (2019) 141-155 Proof. Since the index of N in G is 2, there exists d e G \ N such that G = N U dN. So, we can consider the vertices of Cay(G, S) as two partitions N and dN. Since for every m, n e N, m and n are adjacent, and dm and dn are adjacent if and only if m-1n e S n N, we obtain that Cay(G, S \ S n N) is a bipartite graph with partite sets N and dN. □ Lemma 2.9 ([10, page 308]). A graph has a 2-NZF if and only if it is an even graph. Remark 2.10. According to the above lemma, for discussion about a nowhere-zero 3-flow in a Cayley graph with a connection sequence S, it is enough to investigate the case when |S| is odd. Remark 2.11. Let G be a group and N be a subgroup of G. Let T = {x1;..., xt}, where t e N, be a left transversal set of N in G. If S is a connection sequence of N such that Cay(N, S) is connected, then {x, Cay(N, S) : 1 < i < t} is the set of connected components of Cay(G, S). For every x, where i e {1,..., t}, Cay(N, S) and x, Cay(N, S) are isomorphic, because for every m, n e N, x,m ~ x,n (in x, Cay(N, S n N)) if and only if (xjm)-1(xjn) e S n N if and only if m-1n e S n N if and only if m ~ n (in Cay(N, S n N)). Thus if Cay(N, S) has a 3-NZF, then Cay(G, S) has a 3-NZF. Hence for finding a 3-NZF in Cay(G, S), we reduce to find a 3-NZF in Cay(N, S). 3 Main results In this section we show the validity of Tutte's 3-flow conjecture for a solvable group with a suitable normal subgroup. As examples, we show the same result for Cayley graphs on generalized dicyclic groups, generalized dihedral groups and quaternion groups. We also prove that every Cayley graph Cay(G, S) on a solvable group G with a cyclic Sylow 2-subgroup such that the connection sequence S contains a central generator element, admits a 3-NZF. Theorem 3.1. Let G be a solvable group, N be a subgroup of G of index 2 and let S be a connection sequence of G such that |S| > 5 is odd and S n Z(N) = 0. If (1) Cay(N, S n N) admits a 3-NZF and (2) for every d e S \ N, d-1(S n N)d = S n N, then Cay(G, S) has a 3-NZF. Proof. Without loss of generality, we can assume that there exists an element d e S \ N, because otherwise S c N and by Condition (1), we could conclude that Cay(G, S) has a 3-NZF. Thus, there is d e S \ N. Note that |S| is odd. We continue the proof in the following two cases: Case 1. If |S n N| is odd, then since |S \ (S n N)| = |S| \ |S n N| is even, Lemma 2.9 shows that Cay(G, S \ (S n N)) admits a 3-NZF. Also by Condition (1), Cay(N, S n N) admits a 3-NZF, and so does Cay(G, S) = Cay(G, S \ (S n N)) U Cay(G, S n N). M. Ahanjideh and A. Iranmanesh: The validity ofTutte's 3-flow conjecture for some ... 207 Case 2. If |S n N| is even, then the proof will be divided into two subcases: Subcase 1. Assume that |S\(S nN )| > 2. By Lemma 2.8, Cay(G, S\(S nN)) is bipartite. So Lemma 2.6 shows that Cay(G, S \ (S n N)) admits a 3-NZF. Since Cay(G, S n N) admits a 3-NZF, we deduce that Cay(G, S) has a 3-NZF. Subcase 2. Assume that |S \ (S n N)| = 1. Thus {S \ (S n N)} = {d}, so O(d) = 2 and it is not hard to check that G is the semidirect product of N and (d). We want to show that Cay(N, S n N) □ Cay((d), {d}) = Cay(G, S). For this purpose, we define ^: Cay(N, S n N) □ Cay((d), {d}) ^ Cay(G, S) such that ^(m, x) = mx for every m G N and x G (d). Since G is the semidirect product of N and (d), it is obvious that ^ is a bijective function. Now we will show that ^ is homomorphism. For every m, n G N and x,y G (d), we have: (m,x) - (n,y) (in Cay(N, S n N) □ Cay((d), {d})) if and only if m = n, x — y or n — m, x = y. We should check the following cases: (1) If m = n, x = 1 and y = d, then (^(m, x))-1 ^>(n, y) = m-1nd = d G S. Thus ^>(m, x) — ^>(n, y) in Cay(G, S). (2) If m = n, x = d and y =1, then (^(m, x))-1^(n, y) = d-1m-1n = d G S. Thus ^>(m, x) — ^>(n, y) in Cay(G, S). (3) If m — n and x = y =1, then m-1n G S n N. Thus (^(m, x))-1^(n, y) = (mx)-1(ny) = m-1n G N n S .So ^(m, x) — ^(n, y) in Cay(G, S). (4) If m — n and x = y = d, then m-1n G S n N. Thus (^(m, x))-1^(n, y) = d-1(m-1n)d G d-1 (S n N)d = N n S C S. So ^(m, x) — ^(n,y) in Cay(G,S). Now, let t1 — t2 in Cay(G, S). Since G is the semidirect product of N and (d), there exist m, n G N and x, y G (d) such that t1 = mx and t2 = ny. We continue the proof in the following cases: (i) If x =1 and y = d, then m-1nd = t-1t2 G S\(SnN) = {d}. Therefore, m-1n = 1 and so m = n. From this, we have ^-1(t1) = (m, x) — (n, y) = ^-1(t2). (ii) If x = d and y = 1, the above reason shows that ^-1(t1) = (m, x) — (n, y) = ^-1(t2). (iii) If x = y =1, then m-1n = t-1t2 G S n N. Therefore m — n in Cay(N, S n N) and hence ^-1(t1) = (m, x) — (n, y) = ^-1(t2). (iv) If x = y = d, then d-1m-1nd = t-1t2 G d(S n N)d-1 = S n N. Therefore m-1n G d(S n N)d-1 = S n N and hence, ^-1(t1) = (m,x) — (n, y) = ^-1(t2). These show that Cay(N, SnN) □ Cay((d), {d}) = Cay(G, S). Now, suppose that the theorem is false, and let G be the smallest group satisfying the hypothesis and Cay(G, S) does not admit a 3-NZF. Note that |S| > 5. We examine the following possibilities: Subcase 2.1. If there is y G S n Z(N) of order n > 2 such that d-1yd G {y,y-1}, then since Z(N) is normal in G, the assumption guarantees the existence of an element z G S n Z(N) such that d-1yd = z. Since O(d) = 2, we see that d-1zd = y. 208 Ars Math. Contemp. 16(2019)203-213 Thus (y, y-1, z, z-1) < (y, y-1,z, z-1, d). If G = (y, y-1,z, z-1,d), then by our assumption on G, Cay((y, y-1, z, z-1 ,d), {y, y-1,z, z-1,d}) admits a 3-NZF. Thus since IS \ {y, y-1,z, z-1, d}| is even, we get that Cay(G, S) admits a 3-NZF. This is a contradiction. Therefore, we can assume that G = (y,y-1,z,z-1,d), N = (y,y-1,z,z-1), S = {y,y-1,z,z-1,d} and S n N = {y,y-1, z, z-1}. Let K be a minimal normal subgroup of G such that K < Z(N). If K n S = 0, then N/K < G/K with [G/K : N/K] = 2 and Z(N/K) n S/K = 0. Note that IS/K| = 5 and |(S n N)/K| = 4. So Cay(N/K, (S n N)/K) admits a 3-NZF. Also IG/K| < |G|. Thus our assumption on G leads us to see that Cay(G/K, S/K) admits a 3-NZF, and so does Cay(G, S) by Lemma 2.2. This is a contradiction. Thus K n S = 0. Without loss of generality, we can suppose that y G K, so d-1yd = z G K. Therefore, K = N. This forces N to be cyclic or elementary abelian. Thus either N = (y) or N = (S n N) = (y) x (z) and hence, either z = yl and Cay(N, N n S) = Cay((y), {y, y-1,y\ y-i}) = C(n, i) or Cay(N, N n S) = Cay((y), {y, y-1}) □ Cay((z), {z, z-1}) = C„ □ Cn. Note that Cay(G, S) = Cay(N, S n N) □ K2. Thus Cay(G, S) is isomorphic to either C(n, i) □ K2 or (Cn □ Cn) □ K2. So Lemmas 2.5 and 2.4 guarantee that Cay(G, S) admits a 3-NZF. This is a contradiction. Subcase 2.2. If S n Z(N) contains an involution y such that d-1 yd = y, then there exists an element z G S n Z (N ) such that d-1yd = z. Therefore, (y, z) is an elementary abelian 2-group of order 4. So Cay((y, z, d), {y, z, d}) is the circular ladder CL4 (see Figure 1) which is bipartite and hence, it admits a 3-NZF. Also, Cay(G, S\{y, z, d}) admits a 3-NZF, and so does Cay(G, S). This is a contradiction. Figure 1: The circular ladder CL4. Subcase 2.3. Suppose that for every y G Z(N) n S, d-lyd G {y/y-1}. Applying the above argument shows that there exists an element y G Z(N) n S such that (y) is a minimal normal subgroup of G. If the order of y is 2, then y is a central involution and hence, Cay(G, S) admits a 3-NZF. This is a contradiction. Thus the order of y is an odd prime number. Now if N n S contains an element z such that O(z) > 3 and d-1zd G {z, z-1}, then applying the same argument as that of used in Subcase 2.1 leads us to get a contradiction. Now suppose that there exists an element z g (SnN)\{y,y-1} such that O(z) > 3 and d-1zd G {z, z-1}. So our assumption on G allows us to assume that S = {y, y-1,z, z-1 ,d-1zd, d-1z-1d, d}. Let K be anormal subgroup of G containing y such that K < N and it is maximal with the property K n (S \ {y, y-1}) = 0. If M/K is a minimal normal subgroup of G/K such that M/K < N/K, then our assumption on K shows that M n (S \ {y, y-1}) = 0. Without loss of generality, we can assume that z G M. Since M is normal in G, we deduce that d-1zd G M and hence, S — {d} C M. Thus M = N. Set S1 = {z,z-1,d,-1zd,dr1z-1d,d,}. Moreover M/K = N/K is M. Ahanjideh and A. Iranmanesh: The validity ofTutte's 3-flow conjecture for some ... 209 abelian and normal in G/K of index 2 such that S1/K \ (S1/K n M/K) = {dK} and dK(S1/K n M/K)dK = (S1/K n M/K). By our assumption on G, Cay(G/K, S1/K) admits a 3-NZF. But S1 n K = 0, so Lemma 2.2 shows that Cay(G, S1) admits a 3-NZF. In addition, since |S \ S11 = 2, Cay(G, S \ S1) admits a 3-NZF and hence, Cay(G, S) admits a 3-NZF. This is a contradiction. Finally, let N n S contain an element z of order 2. Since |S n N| is even, our assumption on G allows us to assume that there exists an involution w G (S n N) \ {z} such that G = (y, y-1, z, w, d). Since z, w are distinct involutions, we have that either (z, w) is an elementary abelian 2-group of order 4 or a dihedral group. We can see at once that Cay((w, z, d), {w, z, d}) is a circular ladder CLk, for some even number k, which is bipartite. Therefore, Cay((w, z, d), {w, z, d}) admits a 3-NZF, and so does Cay(G, S). This is a contradiction. This shows that Cay(G, S) admits a 3-NZF, as desired. □ Theorem 3.2. Let G be a group, N be an abelian subgroup of G of index 2 and let S be a connection sequence of G such that |S| > 4. If there exists d G S \ (S n N) such that d-1(S n N)d = S n N, then Cay(G, S) admits a 3-NZF. Proof. First, assume that |S n N| > 4. By Lemma 2.3, Cay(N, S n N) has a 3-NZF. Since |G/N| = 2, we can assume that G/N = (dN), and hence for every y G S \ (S n N), yN G (dN). Thus there exists t G N such that y = td and for every s G S n N and y G S \ (S n N), y-1sy G S n N. (3.1) So the Conditions (1) and (2) of Theorem 3.1 are fulfilled and hence Cay(G, S) admits a 3-NZF. Now, we assume that |S n N| < 3. The proof falls naturally into several parts. If |S n N| =0, then by Lemma 2.8, Cay(G, S) is bipartite, and hence Lemma 2.6 shows that Cay(G, S) admits a 3-NZF. Moreover, if |S n N| = 2, then Lemma 2.9 forces Cay(N, S n N) to admit a 3-NZF. Also by (3.1), for every s G S n N, y-1sy = d-1sd G S n N. So Theorem 3.1 completes the proof. Therefore, |S n N| G {1, 3}. We consider these possibilities in the following cases: Case 1. Assume that |S n N| = 1. So S n N = {x}. Clearly, O(x) = 2 and d-1xd = x-1 = x. Also, for every y G S \ (S n N), we have yN G (dN) and hence, y = md for some m G N. Therefore, we can see y-1xy = x. Thus x G Z((S)) is of order 2. Hence by Lemma 2.1, we have Cay((S), S) admits a 3-NZF, and so does Cay(G, S). Case 2. Assume that |S n N| = 3. We continue the proof in two subcases: Subcase 1. Let S n N = {x, y, y-1}, where O(x) = 2 and O(y) > 3. Since d-1xd G S n N and O(d-1xd) = 2, the same argument as that of used in Case 1 completes the proof. Subcase 2. Let S n N = {x, y, z}, where O(x) = O(y) = O(z) = 2. First, assume that none of the elements in S n N generates by the other ones. Since x, y, z are of order 2 and N is abelian, we have (N n S) = {xy z1 | 1 < i, j, k < 2} = (x) x (y) x (z) < N. It is easy to check that Cay( (N n S), S n N) is bipartite (similar to Figure 1) and hence by Lemma 2.6, Cay(N, S n N) admits a 3-NZF. The rest of the proof runs as the case when |S n N| > 4. 210 Ars Math. Contemp. 16(2019)203-213 Otherwise, without loss of generality, assume that S n N — {x, y,xy}. Set S1 — {d, d-1}. Note that |S| is odd. Thus |S \ ((S n N) U Si)| — 0 or 2k where k G N. Set S2 — S \ ((S n N) U S1) and H — ((S n N) U S1). In fact, Cay(G, S2) U Cay(G, (S n N) U S1) — Cay(G, S) and Cay(G, S2) admits a 3-NZF. So it is sufficient to find a 3-NZF in Cay(G, (SnN) US1). We know that d-1xd G S n N. If d-1xd — x, then since N is abelian, we have x G Z(H) and its order is 2, so the proof is complete by Lemma 2.1. Now, assume that d-1xd — y. Since N — dN G G/N and |G/N| — 2, we have O(dN) — 2, and hence d2 G N. It follows that x — d2xd-2 — dyd-1. Therefore, Thus xy G Z(H) and O(xy) — 2. Lemma 2.1 shows that Cay(H, (S n N) U S1) admits a 3-NZF, and so does Cay(G, (S n N) U S1), as desired. The same reasoning can be applied In the following we show that Theorem 3.2 guarantees the existence of a 3-NZF in a Cayley graph on a generalized dicyclic group. Example 3.3. Let H be an abelian group, having a specific element y G H of order 2. A group G is called a generalized dicyclic group, Dic(H, y), if it is generated by H and an additional element x. Moreover, we have [G : H] — 2, x2 — y and x-1ax — a-1 for every a G H .It is easy to see that every Cayley graph of valency at least 4 on Dic(H, y) has a 3-NZF by applying Theorem 3.2. Note that in [3, 11], as the main theorems, it is showed that the graphs mentioned in Example 3.4 admit nowhere-zero 3-flows. Example 3.4. (1) Let H be an abelian group. The generalized dihedral group DH is a group of order 2|H| generated by H and an element p where p G H, p2 — 1 and p-1hp — h-1 for all h G H. We see at once that every Cayley graph of valency at least 4 on DH satisfies the conditions of Theorem 3.2, and hence it admits a 3-NZF. In particular, G — (x, a | an — x2 — 1, x-1ax — a-1) is a special case of DH, where H — (a), p — x and it is called a dihedral group and denoted by D2n. (2) Let G — (z, a | an — z2,an — 1,z-1az — a-1) which is called a generalized quaternion group, denoted by Q4n. Note that G is a special case of a generalized dicyclic group where (a) and z play the roles of H and x, respectively. Thus every Cayley graph of valency at least 4 on Q4n admits a 3-NZF. Remark 3.5. Let G be a group, N be a normal subgroup of G of an odd index at least 3 and S be a connection sequence of G such that |S| > 4. Assume that T — {x1;..., x2k+1} is a left transversal set of N in G and Cay(N, S n N)) has a 3-NZF. Note that by Remark 2.11, d 1xyd — d 1xdd 1yd — yx — xy. to the case d-1 xd — xy. □ Cay(G, S) — x, Cay(N, S n N) I U Cay(G, S \ (S n N)). i=1 M. Ahanjideh and A. Iranmanesh: The validity ofTutte's 3-flow conjecture for some ... 211 By the assumption, for every i e {1,..., 2k + 1}, x, Cay(N, S n N) admits a 3-NZF. For finding a 3-NZF in Cay(G, S), it is enough to find a 3-NZF in Cay(G, S \ (S n N)). If |S \ (S n N)| is odd, then there exists y e S \ (S n N) such that O(y) =2 and hence yN e G/N and O(yN) = 2. So we have 2 | |G/N|. This is impossible. Thus |S \ (S n N) | is even and hence Cay(G, S \ (S n N)) admits a 3-NZF by Lemma 2.9. Therefore if Cay(N, S n N) has a 3-NZF, then so does Cay(G, S). Theorem 3.6. Let G be a solvable group with a cyclic Sylow 2-subgroup and let S be a connection sequence of G with |S| > 4. If there exists an element x e Z (G) n S such that x is a generator element of G in S, then Cay(G, S) admits a 3-NZF. Proof. Suppose that G is the smallest counterexample satisfies the above conditions, but Cay(G, S) does not admit a 3-NZF. Without loss of generality, we can assume that |S| = 5 and x e Z(G) n S. Thus O(x) > 3 by Lemma 2.1. If there exists u e Z(G) such that (u)nS = 0, then |S/(u)| = |S|, x(u) e Z(G/(u))nS/(u) and |G/(u)| < |G|. Ifx(u) isa generator element of G/(u) in S/(u), then by our assumption, Cay(G/(u), S/(u)) admits a 3-NZF. Lemma 2.2 forces Cay(G, S) to admit a 3-NZF, a contradiction. Thus x(u) is not a generator element. Therefore, there exist an element t e (S \ {x, x-1}) and i e N such that xu! = t and hence t e Z(G). If there exists t1 e (t) n S, then as stated above, we can see that O(t1) > 3. Thus Z(G) n S = {x, x-1,t1,t-1}. Therefore |G/Z(G)| e {1,2} and hence, G/Z(G) is cyclic. So G is an abelian group. This forces Cay(G, S) to admit a 3-NZF, a contradiction. Thus (t) n S = 0. Moreover, we can see at once that x(t) is a generator element of G/(t) in S/(t), |S/(t)| = |S| and |G/(t)| < |G|. Therefore, our assumption forces Cay(G/ (t), S/ (t)) to admit a 3-NZF, and so does Cay(G, S) by Lemma 2.2. This is a contradiction. So for every u e Z(G), we have (u) n S = 0. We continue the proof in two cases: Case 1. Suppose that |Z(G)| is even. So there exists w e Z(G) of order 2. By our assumption, (w) n S = 0, and hence S contains a central involution. Lemma 2.1 shows that Cay((S), S) admits a 3-NZF, and so does Cay(G, S). This is a contradiction. Case 2. Let |Z(G)| be odd. Since |S| = 5, S contains an involution y. We continue the proof in three subcases: Subcase 1. Suppose that |S n Z(G)| is odd, so Z(G) contains an involution. This is a contradiction, because |Z(G)| is odd. Subcase 2. Suppose that |Z(G)nS| = 2. So we have Z(G)nS = {x,x-1}, where O(x) is an odd prime number p. Therefore, (x) is a cyclic subgroup of order p. By the assumption, x e (S \ {x, x-1}) and hence, we deduce that G = (x) x M, where M = (S \ {x, x-1}) is a maximal subgroup of G. Let N be a minimal normal subgroup of G such that N < M. So N is an elementary abelian q-group, where q is a prime number. If N n S = 0, then x(N) e Z(G/N) n S/N is a generator element of G/N in S/N, |G/N| < |G| and |S/N| = 5. Thus by our assumption on G, Cay(G/N, S/N) admits a 3-NZF, and so does Cay(G, S). This contradicts our assumption. If N n S = 0, then the proof falls naturally into several parts: (a) If y e N n S such that O(y) = 2, then 2 | |N|. Since N is elementary abelian, we get that N is an elementary abelian 2-group. Thus | N| = 2 by the assumption. Therefore y e N < Z(G), and hence |Z(G)| is even. This is a contradiction. 212 Ars Math. Contemp. 16(2019)203-213 (b) If N n S = {z,z-1}, where O(z) > 3, then S \{x,x-1} = {z,z-1,y}. Since N is an elementary abelian q-group where q is a prime number, we get O(z) = q = 2. So y G N .If yz = zy, then G is an abelian group and hence, Lemma 2.3 forces Cay(G, S) to admit a 3-NZF, a contradiction. If yz = zy and O(yz) = 2, then we have yzy = z-1. Thus L = (x, x-1, z, z-1) < G = (x, x-1, z, z-1, y). Therefore, [G : L] = 2 and L < G. We thus get that Cay(G, S) admits a 3-NZF by Theorem 3.1. This is a contradiction. Now, suppose that yz = zy and O(yz) > 3. Since O(z) = q, z G N and |M/N| = |(yN)| = 2, we have |M| = 2q4, where t G N. If O(yz) = qn, where n < t, then yz G N. So y G N, a contradiction. Suppose that O(yz) = 2qn where n < t. Since gcd(2, qn) = 1, there exist k, s G Z such that 2s + kqn = 1. So, O((yz)2s) = qn and O((yz)fcq") = 2. Thus we have (yz)2s G N. Since z G N and N is abelian, we can see that (yz)2sy = y(yz)2s. Therefore (yz)2s G Z(M) < Z(G). Thus ((yz)2s) is a normal subgroup of G and ((yz)2s) < N .So z G N = ((yz)2s) < Z (M) < Z (G) and hence yz = zy. This is a contradiction with the above statements. Subcase 3. Suppose that |S n Z(G)| = 4. Since |S| = 5, we can see |S|\|S n Z(G)| = 1. It follows that [(S) : (S n Z(G))] = 2. So (S)/((S n Z(G))) is a cyclic group. On the other hand, (S n Z(G)) < Z((S)). Therefore (S) is abelian, and hence Lemma 2.3 yields that Cay((S), S) admits a 3-NZF, and so does Cay(G, S), a contradiction. □ Corollary 3.7. Let G be a solvable group such that the Sylow 2-subgroups of G are cyclic and every Cayley graph of valency at least 4 on G admits a 3-NZF. If H is a nilpotent group, then every Cayley graph of valency at least 4 on G x H admits a 3-NZF. Proof. Suppose that H is the smallest nilpotent group such that Cay(G x H, S) does not admit a 3-NZF. Note that by the assumption on G, we have H = 1. If there exists 1 = t G Z(H) such that (t) n S = 0, then since (t) < G x H, our assumption on H shows that Cay((G x H)/ (t), S/ (t)) admits a 3-NZF. So Lemma 2.2 forces Cay(G x H, S) to admit a 3-NZF. This is a contradiction. Thus for every t G Z(H), (t) n S = 0. If |H| is even, then S contains a central involution and hence, Lemma 2.1 shows that Cay(G x H, S) admits a 3-NZF, a contradiction. Thus |H| is odd. Let the order of t G Z(H) n S be odd. If |H n S| is odd, then 2 | |H|. This is a contradiction. If |H n S| = 2, then H n S = {x, x-1} and hence, Z(H) n S = {x, x-1} and O(x) is a prime number. Since G is solvable, we can assume that K is a normal subgroup of G x H such that K < G and K is maximal with the property that S n K = 0. If G = K, then (G x H)/G is nilpotent and |S/G| = |S|, and hence, Cay((G x H)/G, S/G) admits a 3-NZF, and so does Cay(G x H, S). This is a contradiction. Thus G = K and for a minimal normal subgroup M/K of (G x H)/K such that M/K < G/K, we have M n S = 0. So one of the following possibilities occurs: (I) Suppose that M n S contains an involution z. Then 2 | |M/K|. Since M/K is elementary abelian and the Sylow 2-subgroups of G are cyclic, we have M/K = (zK) and hence (zK) < Z((G x H)/K). Therefore, Lemma 2.1 shows that Cay((G x H)/K, S/K) admits a 3-NZF, and so does Cay(G x H, S) by Lemma 2.2, a contradiction. (II) If M n S does not contain any involution, then |M n S| is an even number. Since |S| is odd, we get that S \ (M n S) contains an involution z. But |H| is odd, so z G G. Let S1 = (M n S) U {z,x,x-1}. We have (S1) = (M n S,z) x (x) and |S1| > 5 is an odd number. Thus Theorem 3.6 shows that Cay((S1), S1) admits a 3-NZF, so M. Ahanjideh and A. Iranmanesh: The validity ofTutte's 3-flow conjecture for some ... 213 does Cay(G x H, Si). Since |S \ Si| is even, Cay(G x H, S) admits a 3-NZF, a contradiction. If |H n S| > 4, then there exists an element x G S such that O(x) = 2. Since |H| is odd, we have x G H n S and the Sylow 2-subgroups of G x H are the Sylow 2-subgroups of G and hence, x G G. Therefore x G CGxH(H n S), the centralizer of H n S in G x H, and hence x G Z((H n S) x (x)). So Lemma 2.1 forces Cay((H n S) x (x), (H n S) U {x}) to admit a 3-NZF, so does Cay(G x H, (H n S) U {x}). But |S \ ((H n S) U {x}) | is even, So Cay(G x H, S) admits a 3-NZF, a contradiction. □ Corollary 3.8. If L is a nilpotent group, then for every generalized dihedral group DH, the Cayley graph of valency at least 4 on DH x L admits a 3-NZF. Proof. Let DH be the smallest generalized dihedral group such that the Cayley graph of valency at least 4 on DH x L does not admit a 3-NZF. If |H | is odd, then the Sylow 2-subgroups of Dh are cyclic, and hence Corollary 3.7 shows that Cay(DH x L, S) admits a 3-NZF, a contradiction. If |H| is even, then H contains a central involution t. If t G S, then Lemma 2.1 shows that the Cayley graph of valency at least 4 on DH x L admits a 3-NZF, a contradiction. If t G S, then by our assumption, Cay((DH x L)/(t), S/(t)) admits a 3-NZF. It follows that Cay(DH x L, S) admits a 3-NZF by Lemma 2.2. This is impossible. These contradictions show that every Cayley graph of valency at least 4 on DH x L admits a 3-NZF. □ References [1] B. Alspach, Y. Liu and C. Zhang, Nowhere-zero 4-flows and Cayley graphs on solvable groups, SIAMJ. Discrete Math. 9 (1996), 151-154, doi:10.1137/s0895480193258017. [2] F. Jaeger, Flows and generalized coloring theorems in graphs, J. Comb. Theory Ser. B 26 (1979), 205-216, doi:10.1016/0095-8956(79)90057-1. [3] L. Li and X. Li, Nowhere-zero 3-flows in Cayley graphs on generalized dihedral group and generalized quaternion group, Front. Math. China 10 (2015), 293-302, doi:0.1007/ s11464-014-0378-2. [4] M. Nanasiova and M. Skoviera, Nowhere-zero 3-flows in Cayley graphs and Sylow 2-subgroups, J. Algebraic Combin. 30 (2009), 103-111, doi:10.1007/s10801-008-0153-0. [5] R. Nedela and M. Skoviera, Cayley snarks and almost simple groups, Combinatorica 21 (2001), 583-590, doi:10.1007/s004930100014. [6] M. Potocnik, P. Skoviera and R. Skrekovski, Nowhere-zero 3-flows in abelian Cayley graphs, Discrete Math. 297 (2005), 119-127, doi:10.1016/j.disc.2005.04.013. [7] J. S. Rose, A Course on Group Theory, Cambridge University Press, Cambridge, 1978. [8] W. T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954), 80-91, doi:10.4153/cjm-1954-010-9. [9] W. T. Tutte, On the algebraic theory of graph colorings, J. Comb. Theory 1 (1966), 15-50, doi:10.1016/s0021-9800(66)80004-2. [10] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 1996. [11] F. Yang and X. Li, Nowhere-zero 3-flows in dihedral Cayley graphs, Inform. Process. Lett. 111 (2011), 416-419, doi:10.1016/j.ipl.2011.01.017. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ars mathematica contemporanea 16 (2019) 215-235 https://doi.org/10.26493/1855-3974.1437.6e8 (Also available at http://amc-journal.eu) Edge-transitive bi-p-metacirculants of valency p Yan-Li Qin, Jin-Xin Zhou * Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China Received 7 July 2017, accepted 23 February 2018, published online 20 December 2018 Let p be an odd prime. A graph is called a bi-p-metacirculant on a metacyclic p-group H if admits a metacyclic p-group H of automorphisms acting semiregularly on its vertices with two orbits. A bi-p-metacirculant on a group H is said to be abelian or non-abelian according to whether or not H is abelian. By the results of Malnic et al. in 2004 and Feng et al. in 2006, we see that up to isomorphism, the Gray graph is the only cubic edge-transitive non-abelian bi-p-metacirculant on a group of order p3. This motivates us to consider the classification of cubic edge-transitive bi-p-metacirculants. Previously, we have proved that a cubic edge-transitive non-abelian bi-p-metacirculant exists if and only if p = 3. In this paper, we give a classification of connected edge-transitive non-abelian bi-p-metacirculants of valency p, and consequently, we complete the classification of connected cubic edge-transitive non-abelian bi-p-metacirculants. Keywords: Bi-p-metacirculant, edge-transitive, inner-abelian p-group. Math. Subj. Class.: 05C25, 20B25 1 Introduction Given a group H, let R, L and S be three subsets of H such that R-1 = R, L-1 = L and RUL does not contain the identity element of H. The bi-Cayley graph over H with respect to the triple (R, L, S), denoted by BiCay(H, R, L, S), is the graph having vertex set the union H0 U H1 of two copies of H, and edges of the form {h0, (xh)0}, {h1, (yh)1} and {h0, (zh)1} with x G R, y G L, z G S and h0 G H0, h1 G H1 representing a given h G H .It is easy to see that a graph is a bi-Cayley graph over a group H if and only if it admits H as a semiregular automorphism group with two orbits. * Supported by the National Natural Science Foundation of China (11671030) and the Fundamental Research Funds for the Central Universities (2015JBM110). E-mail addresses: yanliqin@bjtu.edu.cn (Yan-Li Qin), jxzhou@bjtu.edu.cn (Jin-Xin Zhou) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 216 Ars Math. Contemp. 16 (2019) 141-155 Let r = BiCay(H, R, L, S). For g G H, define a permutation R(g) on the vertices of r by the rule hf(g) = (hg)i, Vi G Z2, h G H. Then R(H) = {R(g) | g G H} is a semiregular subgroup of Aut(r) which is isomorphic to H and has H0 and Hi as its two orbits. When R(H) is normal in Aut(r), the bi-Cayley graph r = BiCay(H, R, L, S) is said to be normal (see [24]). When NAut(r) (R(H)) is transitive on the edge set of r, we say that r is normal edge-transitive (see [7]). Bi-Cayley graphs are useful in constructing edge-transitive graphs (see [7, 24]). However, it is difficult in general to decide whether a bi-Cayley graph is edge-transitive. So it is natural to investigate the edge-transitive bi-Cayley graphs over some given groups. Note that metacylic groups are widely used in constructing graphs with some kinds of symmetry, see, for example, [1, 11, 12, 13, 14, 18]. (A group G is called metacyclic if it contains a cyclic normal subgroup N such that G/N is cyclic.) In this paper, we shall focus on the bi-Cayley graphs over a metacyclic p-group with p an odd prime. For convenience, a bi-Cayley graph over a (resp. non-abelian or abelian) metacyclic p-group is simply called a (resp. non-abelian or abelian) bi-p-metacirculant. Note that the Gray graph [6], the smallest cubic semisymmetric graph, is a non-abeian bi-3-metacirculant of order 2 • 33. Malnic et al. in [8,17] gave a classification of cubic edge-transitive graphs of order 2p3 for each prime p. Actually, it is easy to prove that every cubic edge-transitive graphs of order 2p3 is a bi-Cayley graph over a group of order p3. Rather than describe the classification in detail, we would simply like to point out one striking feature: except the Gray graph, there do not exist other cubic edge-transitive non-abelian bi-p-metacirculants of order 2 • p3 for every odd prime p. This seems to suggest that cubic edge-transitive non-abelian bi-p-metacirculants are rare. Motivated by this, we are going to consider the following problem: Problem 1.1. Classify cubic edge-transitive non-abelian bi-p-metacirculants for every odd prime p. In [19], we gave a partial answer to this problem. We first proved that a cubic edge-transitive non-abelian bi-p-metacirculant exists if and only if p = 3, and then we gave a classification of cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic p-group for each odd prime p. (A non-abelian group is called an inner-abelian group if all of its proper subgroups are abelian.) In view of this, to solve Problem 1.1, it suffices to classify cubic edge-transitive non-abelian bi-3-metacirculants. Naturally, the following problem arises. Problem 1.2. Classify edge-transitive non-abelian bi-p-metacirculants of valency p for every odd prime p. The following is the main result of this paper which gives a solution of Problem 1.2. Theorem 1.3. Let p be an odd prime, and let r be a connected edge-transitive non-abelian bi-p-metacirculants of valency p. Then p = 3 and r is isomorphic to one of the following graphs: (i) rr = BiCay(Gr, 0, 0, {1, a, a-ib}), Gr = (a, b | a3r+1 = b3r = 1, b-iab = ai+^ ) , Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 217 (ii) £r = BiCay(Hr, 0, 0, {1, b, b-1a}), Hr = ^a,b | a3r+1 = b3^1 = 1, b-1ab = a1+3^ , where r is a positive integer. Remark 1.4. The graphs rr and £r are actually those graphs what we have found in [19]. By [19], rr is semisymmetric while £r is symmetric. To the best of our knowledge, the graphs rr form the first known infinite family of cubic semisymmetric graphs of order twice a power of 3. From the above theorem and [19, Theorem 1], we may immediately obtain the following result which gives a solution of Problem 1.1. Corollary 1.5. Let p be an odd prime. A connected cubic non-abelian bi-p-metacirculant is edge-transitive if and only if it is isomorphic to one the graphs given in Theorem 1.3. Remark 1.6. The classification of cubic edge-transitive bi-Cayley graphs on abelian groups has been given in [10, 23]. So our result actually completes the classification of all cubic edge-transitive bi-p-metacirculants for each odd prime p. 2 Preliminaries 2.1 Definitions and notation Throughout this paper, groups are assumed to be finite, and graphs are assumed to be finite, connected, simple and undirected. For the group-theoretic and the graph-theoretic terminology not defined here we refer the reader to [4, 21]. Let G be a permutation group on a set Q and take a e Q. The stabilizer Ga of a in G is the subgroup of G fixing the point a. The group G is said to be semiregular on Q if Ga = 1 for every a e Q and regular if G is transitive and semiregular. For a positive integer n, denote by Zn the cyclic group of order n and by Z*n the multiplicative group of Zn consisting of numbers coprime to n. For a finite group G, the full automorphism group and the derived subgroup of G will be denoted by Aut(G) and G', respectively. Denote by exp(G) the exponent of G. For any x e G, denote by o(x) the order of x. For two groups M and N, N x M denotes a semidirect product of N by M .A non-abelian group is called an inner-abelian group if all of its proper subgroups are abelian. For a graph r, we denote by V(r) the set of all vertices of r, by E(r) the set of all edges of r, by A(r) the set of all arcs of r, and by Aut(r) the full automorphism group of r. For u,v e V(r), denote by {u, v} the edge incident to u and v in r. If a subgroup G of Aut(r) acts transitively on V(r), E(r) or A(r), we say that r is G-vertex-transitive, G-edge-transitive or G-arc-transitive, respectively. In the special case when G = Aut(r) we say that r is vertex-transitive, edge-transitive or arc-transitive, respectively. An arc-transitive graph is also called a symmetric graph. A graph r is said to be semisymmetric if r is regular and is edge- but not vertex-transitive. 218 Ars Math. Contemp. 16(2019)203-213 2.2 Quotient graph Let r be a connected graph with an edge-transitive group G of automorphisms and let N be a normal subgroup of G. The quotient graph rN of r relative to N is defined as the graph with vertices the orbits of N on V(T) and with two orbits adjacent if there exists an edge in r between the vertices lying in those two orbits. Below we introduce two propositions of which the first is a result of [15, Theorem 9]. Proposition 2.1. Let p be an odd prime and r be a graph of valency p, andlet G < Aut(T) be arc-transitive on r. Then G is an s-arc-regular subgroup of Aut(T) for some integer s. If N < G has more than two orbits in V(T), then N is semiregular on V(T), rN is a symmetric graph of valency p with G/N as an s-arc-regular subgroup of automorphisms. In view of [16, Lemma 3.2], we have the following proposition. Proposition 2.2. Let p be an odd prime and r be a graph of valency p, andlet G < Aut(T) be transitive on E(T) but intransitive on V(T). Then r is a bipartite graph with two partition sets, say V0 and Vl. If N < G is intransitive on each of V0 and Vi, then N is semiregular on V(T), rN is a graph of valency p with G/N as an edge- but not vertex-transitive group ofautomorphisms. 2.3 Bi-Cayley graphs Proposition 2.3 ([23, Lemma 3.1]). Let r = BiCay(H, R, L, S) be a connected bi-Cayley graph over a group H. Then the following hold: (1) H is generated by Ru£u S. (2) Up to graph isomorphism, S can be chosen to contain the identity of H. (3) For any automorphism a of H, BiCay(H, R, L, S) = BiCay(H, Ra, La, Sa). (4) BiCay(H, R, L, S) = BiCay(H, L, R, S-i). Let r = BiCay(H, R, L, S) be a bi-Cayley graph over a group H. Recall that for each g e H, R(g) is a permutation on V(r) defined by the rule hf{9) = (hg)i, Vi e Z2,h,g e H, and R(H) = {R(g) | g e H} < Aut(r). For an automorphism a of H and x,y,g e H, define two permutations on V(r) = H0 U Hl as following: Sa,x,y: ho ^ (xha)i, hi ^ (yha)o, Vh e H, Va,9: h0 ^ (ha)0, hL ^ (gha)i, Vh e H. Set I = {6aiXy I a e Aut(H) s.t. Ra = x-iLx, La = y-iRy, Sa = y-1S-ix}, F = { n, we have „m pm pm (xy)p = xp yp , Vx, y G G. (2) For any positive integer k and for any x,y G G, xpk = ypk ^ (x-1y)pk = 1 ^ (xy-1)pk = 1. Proof. By [22, Theorem 2.1], it suffices to prove the items (1) and (2). Since G' is cyclic, (1) follows from [9, Chapter 3, §10, Theorem 10.2 (c) and Theorem 10.8 (g)]. Item (2) follows from [9, Chapter 3, § 10, Theorem 10.2 (c) and Theorem 10.6 (a)]. □ Lemma 3.2. Let p be an odd prime, and let H be a metacyclic p-group generated by a, b with the following defining relations: aPm = bPn = 1, b-1ab = a1+Pr, where m, n, r are positive integers such that r < m < n + r. Then the following hold: (1) For any i G Zpm, j g Zpn, we have aibj = bai(1+pr)j. (2) For any positive integer k and for any i G Zpm, j g Zpn, we have (bjai)k = bkjai ^ks=0(1+pr)sj. 2t (3) For any positive integers t, k and any element x of H, if xp = 1, then x(1+pt)k = x1+k-pt. (4) The subgroup of H of order p is one of the following groups: (ar-) , (bpn-1 ai'pm-1) (i' G Zp). 220 Ars Math. Contemp. 16(2019)203-213 Proof. From [19, Lemma 14 (1) -(2)], we have the items (1) - (2). For (3), the result is clearly true if k = 1. In what follows, assume k > 2. Since xp = 1, we have xp = 1. Then = xc0'(pt)° • xc1(Pt)1 • xc2-(p4)2 • • • xck•(pt)k = x • (xpt)C1 • (xP2t)C2 ••• (xPkt)Ck — x • xkpt = x1+kpt, and so (3) holds. (Here for any integers N > l > 0, we denote by C'N the binomial coefficient, that is, C'N = n^N-iy..) For (4), let Q1(H) = (x G H | o(x) = p}. Since H is ametacyclic p-group, by [2, Exercise 85], we have Q1(H) = Zp x Zp. It implies that H has p +1 subgroups of order p. Furthermore, the subgroup of H of order p is one of the following groups: (a,Pm-1) , (bP"-1 ai Pm-1) (i' G Zp), as required. □ 4 Inner-abelian bi-p-metacirculants of valency p In this section, we focus on edge-transitive bi-Cayley graphs over inner-abelian metacyclic p-groups of valency p. For convenience, a bi-Cayley graph over an inner-abelian metacyclic p-group is simply called an inner-abelian bi-p-metacirculant. In [19, Theorem 2], we gave a classification of cubic edge-transitive inner-abelian bi-p-metacirculants. Proposition 4.1 ([19, Theorem 2]). Let r be a connected cubic edge-transitive bi-Cayley graph over an inner-abelian metacyclic 3-group H. Then H = Gr or Hr, and r = rr or £r, where the groups Gr, Hr, and the graphs rr, £r are defined as in Theorem 1.3. In particular, H/H' = Z3r X Z3r or Z3r x Z3r + 1 . In this section, we shall prove the following theorem. Theorem 4.2. Let H be an inner-abelian metacyclic p-group with p an odd prime, and let r be a connected edge-transitive bi-Cayley graph over H of valency p. Then p = 3, and r is isomorphic to one of the graphs given in Theorem 1.3. 4.1 Two technical lemmas Lemma 4.3. Let p be an odd prime and let r be a connected edge-transitive graph of valency p. If G < Aut(r) is transitive on the edges of r, then for each v G V(r), |Gv | = pm with (m,p) = 1. Proof. Since G is transitive on the edges of r, for each v G V(r), the order of a vertex stabilizer Gv must be divisible by p. Suppose, by way of contradiction, that | Gv | is divisible by p2. Let G*v be the subgroup of Gv fixing the neighborhood r(v) of v in r pointwise. (1+Pt) - ,Jc°-ifc-(pt)°+ci-ifc-1-(pt)1+c2-ifc-2-(pt)2H—hck•i°^(pt)fc] x =x Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 221 Then Gv/G*v < Sp, forcing that p | \G*v |. Then G* contains an element a of order p. Note that each orbit of (a) has length either 1 or p. Since (a) fixes v and each vertex in r(v), the connectedness of r implies that each orbit of (a) has length 1, and so a = 1, a contradiction. □ Lemma 4.4. Let H be a p-group with p an odd prime, and let r = BiCay(H, R, L, S) be a connected edge-transitive bi-Cayley graph of valency p. Then (1) r is normal edge-transitive, R = L = 0, and S = {1, h, hha,..., hha ■ ■ ■ h°P 2} for some 1 = h G H and a G Aut(H) satisfying hhaha ■ ■ ■ haP = 1 and o(a) \ p; (2) if H has a characteristic subgroup K such that H/K is isomorphic to Zpm x Zpn, then \m — n| < 1. Proof. Let A = Aut(r), and let P be a sylow p-subgroup of A such that R(H) < P. Since r is edge-transitive, Lemma 4.3 gives that |A| = |R(H)|^ p ■ m, where (p, m) = 1. It follows that |P| = p|R(H)|, and hence P < Na(R(H)). Furthermore, for any e G E(r), we have |A : Ae| = |E(r)| = p|R(H )|,andso |Ae| = m. It follows that Pe = P nAe = 1, and hence |P : Pe| = |P| = p|R(H)| = |E(r)|. Thus, P is transitive on the edges of r. Thus, r is normal edge-transitive. Let N = Na(R(H)). Then N is transitive on the edges of r. Since R(H) < N, the two orbits H0, Hi of R(H) do not contain any edge of r, and so R = L = 0. By Proposition 2.3, we may assume that 1 G S. Since N is transitive on the edges of r and r has valency p, Nlo has an element of order p for some a G Aut(H) and 1 = h G H. Furthermore, cyclically permutes the elements in r(10). So we have r(10) = {1i,hi, (hha)i,..., (hha ■■■ haP-2 )i} and hhah"2 ■■■ haP-1 = 1. This implies that P2 S = {1,h,hha,...,hha ■■■ ha }, and haP = h. Since r is connected, one has H = (S) = (ha% | 0 < i < p — 1). As h°P = h, ap is a trivial automorphism of H. Consequently, we have o(a) = 1 or p and (1) is proved. For (2), without loss of generality, assume that H/K = Zpm x Zpn with m > n, where K is a characteristic subgroup of H. Let T = (R(x) G R(H) | xpn G K). Then T is characteristic in R(H) and R(H)/T = Zpm-n. Propositions 2.1 and 2.2 implies that the quotient graph rT of r relative to T is a graph of valency p with N/T as an edge-transitive group of automorphisms. Clearly, R(H)/T is semiregular on V(rT) with two orbits and R(H)/T < N/T, so rT is a normal edge-transitive bi-Cayley graph over R(H)/T = Zpm-n of valency p. So to complete the proof, it suffices to show that if H = Zpm then m < 1. Suppose to the contrary that H = Zpm with m > 2. Since H = (ha | 0 < i < p — 1), we have H = (h). Let ha = hx for some A G Z*pm. Then 1 = hhaha2 ■ ■ ■ h°P-1 = h1+x+x2+ -+XP-1, and then 1 + A + A2 + ■■■ + Ap-1 = 0 (mod pm). 222 Ars Math. Contemp. 16(2019)203-213 It follows that Ap = 1 (mod pm), and hence A = 1 (mod p). Let A = kp +1 for some integer k. Since m > 2, we have 1 + (kp + 1) + (kp + 1)2 + • • • + (kp + 1)p-1 = 0 (mod p2). It follows that 1 + (kp + 1) + (2kp + 1) +-----+ ((p - 1)kp + 1) = 0 (mod p2), and hence p + ^p(p — 1)kp = 0 (mod p2). A contradiction occurs. □ 4.2 Proof of Thorem 4.2 Throughout this subsection, we shall always let H be an inner-abelian metacyclic p-group with p an odd prime, and r be a connected edge-transitive bi-Cayley graph over H of valency p. In view of Lemma 4.4(1) and since H is inner abelian, we may make the following assumption throughout this subsection. Assumption 4.5. r = BiCay(H, 0, 0, S), where S = {1, h, hha,..., hha • • • haP 2} for some 1 = h G H and a G Aut(H) satisfying hhaha • • • haP = 1 and o(a) = p. Proof of Theorem 4.2. Suppose to the contrary that p > 3. Since H is an inner-abelian metacyclic p-group, by elementary group theory (see also [20] or [3, Lemma 65.2]), we may assume that H = ^a, b | apt+1 = bP' = 1,b-1ab = , where t > 1, s > 1. Note that H/H' = (aH') x (bH') = Zpt x Zps. By Lemma 4.4, we have H/H' = (aH') x (bH') = Zpt x Zpt, Zpt x Zpt+i or Zpt x Zpt-i. If H/H' = (aH') x (bH') = Zpt x Zpt-i, then s = t - 1 and H = ^a, b | apt+1 = bpt- = 1,b-1ab = . Let T = (R(x) | x € H,xpt-1 = 1). Then T is characteristic in R(H) and R(H)/T is isomorphic to Zp2. However, by the proof of Lemma 4.4, this is impossible. If H/H' = (aH') x (bH') = Zpt x Zpt, then s = t and H = ( a, b I ap ^a, b | aPt+1 = bp = 1,b-1ab = apt+1) where t > 1. We shall show that this is impossible in Lemma 4.6. If H/H' = (aH') x (bH') = Zpt x Zpt+1, then s = t +1 and H = (a, b | aPt+1 = bPt+1 = 1, b-1ab = a^1) , where t > 1. We shall show that this is impossible in Lemma 4.7. □ Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 223 Lemma 4.6. If H = (a, b | apt+l = bpt = 1, b-1ab = apt+1^ (t > 0), then p = 3. Proof. Suppose to the contrary that p > 3. We first define the following four maps. Let Y : a ^ a1+p, b ^ b, S: a ^ a,b ^ b1+p, a: a ^ a,b ^ bap, t : a ^ ba, b ^ b. Let x1 = a1+p, x2 = x3 = a, x4 = ba, y1 = y4 = b, y2 = b1+p and = bap. Since H is an inner-abelian metacyclic-p group, by Proposition 3.1 and a direct computation, we have o(xi1) = o(a) = pt+1, o(yil) = o(b) = pt and it is direct to check that xil and yil have the same relations as do a and b, where i1 G {1,2, 3,4}. Moreover, for any i1 G {1, 2, 3,4}, we have (xil, yil} = H. It follows that each of the above four maps induces an automorphism of H. Set P = (a, y, S, t}. By a direct computation, we have o(y) = pt, o(S) = pt-1 and o(a) = o(t) = pt. Furthermore, yS = Sy, Y-1aY = ap+1 and S-1aS = ae with £(p +1) = 1 (mod pt). As both y and S fixes the subgroup (b} while a does not, one has (a, y, S} = (a} x ((y} x (S}) = Zpt x (Zpt x Zpt-l). Observing that (a, y, S} fixes the subgroup (a} setwise but t does not, it follows that (a, y, S} n (t} = 1, and hence |P| > p4t-1. In view of [13, Theorem 2.8], Aut(H) has a normal Sylow p-subgroup of order p4t-1. It follows that P = (a, y, S, t} is the unique Sylow p-subgroup of Aut(H). In particular, we have P = (y}(S}(a}(T}. Recall that S = {1, h, hha, ...,hha ■■■ haP-2}. Assume that h = buav for some u G Zpt and v G Zpt+l. Since H = (S}, we have o(h) = exp(H). It follows that (v,p) = 1. Then the map : a ^ av,b ^ b induces an automorphism of H. Let ip = (Tu(p1 )-1. Then ip G Aut(H) and hv = a. By Proposition 2.4(3), we have that r = F = BiCay(H, 0, 0,SLet P = p-1ap. Then G Aut(r') cyclically permutates the elements in r'(10). It follows that S^ = {1, a, aaP, aaPaP", .. ., aaPaP" ■ ■ ■ a^-2}, and aapaP ■ ■ ■ apP l = 1. Clearly, o(P) = o(a) = p, so P G P. We assume that P = yiSjaktl for some i, k, I G Zpt and j G Zpt-l. By Lemma 3.2(2)-(3) and Proposition 3.l(l), we have P : | b ^ (b ■ (bla)pk )(1+p)j = b(1+p)j (1+pkl)a(1+p)jpk (4.1) Let U1(H) = {xp | x G H}. Then ^(H) < Z(H) and a bla ■ w P: U b / (4.2) \bh^b -w' for some w, w' G öi(H). Since r' is connected, by Proposition 2.3, we have H = By Proposition 3.1(1), it follows that (l,p) = 1. We shall finish the proof by the following steps. 224 Ars Math. Contemp. 16(2019)203-213 Step 1: t > 1. Suppose to the contrary that t = 1. Then H = (a, b | ap = bp = 1, b-1ab = a1+p). We shall first show that for any r > 1, a8" = brl a1+1 r(r-1)klP+irP (4.3) By Equation (4.1) we have ja ^ bla1+ip ' |b ^ bakp So Equation (4.3) holds when r =1. Now assume that r > 1 and ar-1 = b(r-1)'a1+1 (r-1)(r-2)fcip+j(r-1)p. By a direct computation, we have = (b(r-1)la1+ 2 (r-1)(r-2)fcip+i(r-1)p)8 a = (bafcP)(r-1)'(b'a1+ip)1+ 2 (r-1)(r-2)fcip+i(r-1)p = b(r-1)l a(r-1)lkp b'a1+1 i(r-1)2-(r-1)]fcip+irp = b(r-1)l+la1+[ 2 (r-1)2- 1 (r-1) + (r-1)]fcip+irp _ bri„1+1 r(r-1)klp+irp By induction, we have Equation (4.3). Now we show that for any r > 1, a • a8 • • • = b2r(r+1)la(r+1) + [6r(r+1)(2r+1)i+ 2r(r+1)i+ 6(r-1)r(r + 1)fci]p. (4.4) By Equation (4.3) and Lemma 3.2(1)&(3), we have a • a8 = a • bla1+iP = bla(1+P)!a1+iP = bla1+lPa1+iP = bla2+(l+i)P So Equation 4.4 holds when r = 1. Now assume that r > 1 and a • a8 • • • a8r-1 = b2 (r-1)War+[6 (r-1)r(2r-1)l+ 2 (r-1)ri+1 (r-2)(r-1)rfci]p. By a direct computation, we have 8 82 8r aa8a8 •• • a8 = b 1 (r-1)rlar+[ 1 (r-1)r(2r-1)i +1 (r-1)ri+1 (r-2)(r-1)rfci]p _ bWa1 +1 r(r-1)fcip+irp = b 1 r(r+1)l a{r+[ 6 (r-1)r(2r-1)l+1 (r-1)ri+ 6 (r-2)(r-1)rfci]p}-(1+rip) + 1 + 2 r(r-1)fcip+irp = b 1 r (r+1) l ar (1+rlp) + [ 6 (r-1)r(2r-1)i+ 2 (r-1)ri+1 (r-2)(r-1)rfci]p+1+ 2 r(r-1)fcip+irp = b2r(r+1)la(r+1)+[ 1 (r-1)r(2r-1)+r2]lp+[2 (r-1)r+r]ip+[6 (r-2)(r-1)+ 2r(r-1)]rfcip = b2r(r+1)la(r+1)+[6r(r+1)(2r+1)l+ 2r(r + 1)i+ 6 (r-1)r(r+1)fci]p By induction, we have Equation (4.4). Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 225 Since p is a prime and p > 3, by Equation (4.4), we have «aßaß2 • • • aßP-1 = b 1 (p-1)p'ap+l 1 (p-i)p(2p-i)i+1 (p-i)pi+ 6(p-2)(p-1)pfci]p = «p = ^ a contradiction. Step 2: A final contradiction Let 02(H) = {xp2 | x € H}. Then Ö2(H) < Z(H). By Equation (4.1), we have a1+ip • to, bß = b1+jp+pklapk • to', for some to, to' € Ö2(H). Let m = i/ (mod p), n = i (mod p), f = j + k/ (mod p) for some m, n, f € Zp. Then fa ^ bmp+'anp+1 • to1 (45) ß: |b ^ bfP+1akP • to1 (.5) for some to1,to1 € Ö2(H). We shall first prove the following claim. Claim. For any r > 2, aßr = bCrP+2IadrPTOr for some cr, dr € Zp and TOr € Ö2(H). Since t > 1, for any positive integer i0, by Lemma 3.2(1)&(3), we have abi0 = bi0a(1+pt)i0 = bi0a1+iopt = bioa • to0, (4.6) for some to0 € Ö2(H). Then by Equations (4.5) and (4.6), we have aß2 = (b/p+1afcp • to 1 )mp+l(bmp+lanp+1 • TO1)np+1 • to£ = b(2m+/i+ni)p+2ia(2n+fci)p ^ for some to2 € Ö2(H). Take c2, d2 € Zp such that 2m + f/ + n/ = c2 (mod p) and 2n + k/ = d2 (mod p). If r = 2, then Claim is clearly true. Now assume that r > 2 and Claim holds for any positive integer less than r. Then for some cr_1, dr-1 G Zp and TOr_ ; G Ö2(H), and then aßr = (6fp+1afcp • to;)1 p+2l(bmp+lanp+1 • TOi)dr-lp • to£ r_ 1 = b(cr-i + 2/i+idr-i)p+2ia(2fci+dr_i)p _ for some TOr G Ö2(H). Take cr, dr G Zp such that cr-1 + 2/1 + 1dr-1 = cr (mod p) and 2k1 + dr-1 = dr (mod p). By induction, we complete the proof of Claim. Now by our Claim, we have /p = bcPp+2Iadpp • TOp for some cp, G Zp and top g H2(H). It follows that cpp + 21 = 0 (mod p2), a contradiction. This completes the proof of our lemma. □ r 226 Ars Math. Contemp. 16 (2019) 141-155 Lemma 4.7. If H = (a,b | apt+l = bpt+l = 1, b-1ab = apt+1^ (t > 0), then p = 3. Proof. Suppose to the contrary that p > 3. We first define the following four maps. Let Y : a ^ a1+p, b ^ b, 6: a ^ a,b ^ b1+p, a : a ^ bpa, b ^ b, t : a ^ a,b ^ ba. Let x1 = a1+p, x2 = x4 = a, x3 = bpa, y1 = y3 = b, y2 = b1+p and y4 = ba. Since H is an inner-abelian metacyclic-p group, by Proposition 3.1 and a direct computation, we have o(xil) = o(a) = pt+1, o(yil) = o(b) = pf' and it is direct to check that xil and yil have the same relations as do a and b, where i1 G {1,2, 3,4}. Moreover, for any i1 G {1, 2, 3,4}, we have (xil,yil} = H. It follows that each of the above four maps induces an automorphism of H. Set P = (a, y, 6, t}. By a direct computation, we have o(y) = o(6) = pl, o(a) = pf' and o(t) = pt+1. Moreover, we have y6 = 6y, S-1aS = ap+1 and Y-1aY = ae with l(p +1) = 1 (mod pi+1). As both y and 6 fixes the subgroup (a} while a does not, one has (a,Y,6} = (a} x ((y} x (6}) = Zpt x (Zpt x Zpt). Observing that (a, y, 6} fixes the subgroup (b} setwise but t does not, it follows that (a,Y, 6} n (t} = 1, and hence |P| > p4i+1. In view of [13, Theorem 2.8], Aut(H) has a normal Sylow p-subgroup of order p4i+1. It follows that P = (a, y, 6, t} is the unique Sylow p-subgroup of Aut(H). In particular, we have P = (Y}(6}(a}(T}. Recall that S = {1, h, hha, ...,hha ■■■ haP-2} and o(a) = p. Assume that h = buav for some u G Zpt+l and v G Zpt+l. Since H = (S}, we obtain that o(h) = exp(H). It follows that (u,p) = 1. Then there exists u' G Z*t+l such that u = u'v (mod pi+1). Let p = au(6u)-1 (tv)-1. Then p G Aut(H) and h* = b. By Proposition 2.4(3), we have r = BiCay(H, 0, 0, S*). Let r' = BiCay(H, 0, 0, S*) and 3 = p-1ap. Then G Aut(r') cyclically permutates the elements in r'(10). It follows that bbf3 b$ ■■■ b$p =1 and s* = {1, b, bb$, bb$ b$2,..., bb$ b$2 ■ ■ ■ b$p-2}. Since o(3) = o(a) = p, we have 3 G P. Assume that 3 = Yi6jakt 1 for some i,j,k G Zpt and l G Zpt+l. Then by Lemma 3.2(2)-(3) and Proposition 3.1(1), we have fa ^ (ba1 )(1+p)ikpa(1+p)i = b^*^ a(1+p)i(1+klp) 3 : U ^ (ba1)(1+p)j = b(1+p)ja(1+p)j 1 (4.7) and then ß : ? ^ l\ Wt (4.8) \b^bal ■ w' for some w, w' G U1(H). Since r' = r is connected, we derive from Proposition 2.3 that H = (Sv). By Proposition 3.1(1), it follows that (l,p) = 1. We shall finish the proof by the following steps. Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 227 Step 1: t > 1. Suppose to the contrary that t = 1. Then H = (a, b | ap = bp =1, b-iab = ai+p). We shall first show that for any r > 1, = bi+(rJ +1 r(r-i)fci)pari+ 2 r(r+i)jp+ 2 r(r-i)(i+fci)ip+1 r(r-i)(r-2)fci2p. (4.9) By Equation (4.7), we have fa H bfcpai+(i+fcl)p ' jb H bi+jpal+jlp. Thus Equation (4.9) holds when r =1. Now assume that r > 1 and b^r-1 = bi + ((r-i)j+ 2 (r-i)(r-2)kl)p . a(r-i)l+ 2(r-i)rjlp+ 2(r-i)(r-2)(i+fci)ip+1 (r-i)(r-2)(r-3)fci2p By a direct computation, we have b^" = (bi+jPa'+j'P)i + ((r-i)j+ 2 (r-i)(r-2)kl)p • (bfcPai+(i+fc')P)(r-i)'+ 2 (r-i)rjp+ 2 (r-i)(r-2)(i+fci)ip+ i (r-i)(r-2)(r-3)fci2p = bi+(rj+1 (r-i)(r-2)fci + (r-i)fci)p _ ai+jp+[(r-i)ij + 2 (r-i)(r-2)fci2]p . a(r-i)i(i+(i+fci)p)+1 (r-i)rjp+ 2(r-i)(r-2)(i+fci)ip+ i (r-i)(r-2)(r-3)fci2p = bi+rjP+[ 1 (r-i)(r-2) + (r-i)]fcip . a[l+(r-i)l] + [i+(r-1)+1 (r-i)r]jp+1 r(r-i)(i+fci)ip+[ 1 +1 (r-3)](r-i)(r-2)fci2p = bi+(rj+1 r(r-i)fci)pari+ 2 r(r+i)jp+1 r(r-i)(i+fci)ip+ i r(r-i)(r-2)fci2p By induction, we have Equation (4.9). Then by Equation (4.9), we have = bi + (Pj+ 2 p(p-i)fc')pap+ 2 P(P+i)j'P+1 p(p-i)(i+fci)ip+ 6 p(p-i)(p-2)k 12p = bapl = b, a contradiction. Step 2: A final contradiction. 1 2 Let ^2(H) = {xp | x G H}. Then ^(H) < Z(H). By Equation (4.7), we have • w b^ = bjp+iajlp+l • w for some w, w' G U2(H). Let f = i + kl (mod p), n = j (mod p), m = jl (mod p) for some m, n, f G Zp. Then fa H bfcpafp+i • w' ^' H b"p+ia™p+(4.10) for some wi,wi G H2(H). We shall first prove the following claim. 228 Ars Math. Contemp. 16(2019)203-213 Claim. For any r > 1, bpr = brnp+klp+1armp+ ^r11 (n+f )lp+r(r ^ 2) kl2p+rl • with wr e U2(H). If r = 1, then by Equation (4.10), Claim is clearly true. Now assume that r > 1 and Claim holds for any positive integer less than r. Then }/T-1 = b(r-1)np+ (r-1)2(r-2) fc'P+1 • a(r-1)mp+ (r-1)2(r-2) (n+f )lp+ (r-1)(r;2)(r-3) fci2p+(r-1)i ^ ^ for some wr-1 e U2(H). Since t > 1, for any positive integer i0, by Lemma 3.2(1)&(3), we have abio = bi0a(1+pt)i0 = bi0a1+iopt = bioa • (4.11) for some w0 e U2 (H). Then by Equations (4.10) and (4.11), we have (r 1)(r 2) , = (b"P+1amP+' • ro1)(r-1)"P+ ^-2 -) kip+1 afp+1 • )(r-1)mp+ (r-12(r-2) (n+f)lp+ (r-1)(r;2)(r-3) fci2p+(r-1)i ^ b(r-1)np+ (r-1)2(r-2) fcip+np+1+fc(r-1)ip _ ^ _ a(r-1)nlp+ (r-1)2(r-2) fci2p i amp+l + (r-1)mp+ (r-1)2(r-2) (n+f )lp+ (r-1)(r^2)(r-3) fci2p+(r-1)i + (r-1)f = b™p+ r(r2-1) fcip+1 • armp+ r(r2-1) (n+f )lp+ r(r-16(r-2) fc;2p+ri ^ ^ for some e (H). By induction, we complete the proof of Claim. Now by our Claim and o(P) = p, we have bPP = bnp2 + ^ fcip2 + 1 • amp2 + ^ (n+f )lp2+ (p-1)6(p-2) kl2p2+ppl • ^ = b for some mp e U2(H). It follows thatpl = 0 (mod p2), a contradiction. This completes the proof of our lemma. □ 5 Proof of Theorem 1.3 We first prove a lemma. Lemma 5.1. Let p be an odd prime, and let H be a metacyclic p-group. If r is a connected edge-transitive bi-Cayley graph over H of valency p, then H is either abelian or inner-abelian. Proof. We may assume that H is non-abelian. By Proposition 3.1, the group H has the following presentation: H = (a,b | aPr+S+" = 1, bPr+S+' = aPr+S, ab = a^ ) , where r,s,t,u are non-negative integers with u < r and r > 1. Let r = BiCay(H, R, L, S) be a connected edge-transitive bi-p-Cayley graph over H of valency p. Let A = Aut(r), and let P be a Sylow p-subgroup of A such that Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 229 R(H) < P. From the proof of Lemma 4.4(1), we see that P is transitive on the edges of r. Since H' = (apr} = Zps+u, we have H/H' = ( a, b | op = b ^ i,ab — - Zpr X Zpr+s + t , where a = aH' and b = bH'. By Lemma 4.4(2), we have s + t = 0 or 1, and so (s,t) = (0,0), (1,0) or (0,1). Let n = 2r + 2s+u+1. We use induction on n. If n = 1 or 2, then H is clearly abelian, as desired. Assume n > 3. Let N be a minimal normal subgroup of P and N < R(H). Since H is metacyclic, we have N = Zp or Zp x Zp. Suppose that N = Zp x Zp. Note that R(H)' = Zps+u. Let Q be the subgroup of R(H)' of order p. Since Q is characteristic in r(h)' and R(H)' is characteristic in R(H), R(H)

1. If r = 1, then by Magma [5], there is no cubic edge-transitive bi-Cayley graph over H, a contradiction. If r > 2, then by Lemma 4.4(1), we have R = L = 0. Assume that S = {1, g, h}. Since r is connected, by Proposition 2.3(1), we have H = (S} = (g, h}. It follows that o(g) = o(h) = exp(H) = 3r+2, and so H' = (x3r} = (g3r} = (h3r}. Moreover, by Lemma 4.4(1), there exists a G Aut(H) such that ga = g-1h, ha = g-1 and o(a) | 3. Suppose that a is trivial. Then h = g-1, and then H = (g}, a contradiction. Thus, a has order 3. Assume that (g3r )a = gA• 3r for some A G Zg. Then (h3r )a = hA •3r. Since ga = g-1h and ha = g-1, we have gA •3r = g-3r h3r and hA •3r = g-3r. Then gA2 = (gA •3r ) A = (g-3r h3r ) A = g-A•3r hA •3r = g-A •3rg-3r = g(-A-1) • 3r . It follows that g(A2+A+1) • 3r = 1, and so 9 | A2 + A + 1, a contradiction. Case 2: (s,t) = (1,0). In this case, we have TT /71 3r+u+1 ,3r+1 3r+1 b 1+3r\ H =( a, b | a3 =1,b3 = a3 , « = a > 1 Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 231 Let x = a and y = ba 1. Since b3' = a3' , by Proposition 3.1(2), we obtain that y3r+1 = (ba-1)3r+1 = 1 and x" = ab°-1 = (ab)°-1 = (a1+3r )°-1 = a1+3' = x1+3'. Then R(H) = H = (x, y | x3r+"+1 = y3r+1 = 1, x" = x1+3^ , Recall that N = Z3 and N < R(H). By Lemma 3.2(4), N is one of the following four groups: (x3 ), (y3 ), (y3 x3 ), (y3 x2 3 ). Suppose first that N = (x3 ). Then x has order 3r+"+1. We shall show that H/N has the following presentation: H/N = (x, h | x3r+"+1 = h3' = I, xh = x1+3r Actually, if N = (y3' ), then we may take h = y. If N = (y3'x3'+" ), then take h = yx3", and then by Lemma 3.2(2)- (3), we have / 3u\3r : (yx ) = y 3 r x3 "[1+(] 3r•(3r-1) = y3r x3"[1+(1+3r ) + (1+2-3r)H-----+(1+(3r-1)3r )] = y3"X3"[3r + 3 2 ] or q " + r = y3 x3 + € N. If N = (y3'x2'3'+"}, then take h = yx2'3", and then by Lemma 3.2(2)-(3), we have (yX2'3" )3r = y3rx2-3"[1 + (1+3r) + (1+3r)2 + --- + (1+3r)3r-1] = y3r x2-3"[1 + (1+3r ) + (1+2-3r ) + --- + (1 + (3r-1)-3r)] = y3rx2 3"[3r + 3"•(32''-1) 3r] = y3' x2 3"+' € N. Clearly, in each case, we have xh = x1+3'. So H/N always has the above presentation. Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 0. Then 3r+1 3r+1 - y 1+3'^ H = ^x, y | x3r+1 = y3r+1 = 1, x" = x1+3^ , where r > 1. By [20] or [3, Lemma 65.2], H is inner-abelian, as required. Suppose now N = (x3r+"}. Then R(H)/N = (x,y | x3r+" = y3r+1 = T,xy = x1+3r) . Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 1. Then H = (x, y | x3r+2 = y3r+1 = 1, x" = x1+3^ ) , 232 Ars Math. Contemp. 16(2019)203-213 where r > 1. If r = 1, then by Magma [5], there is no cubic edge-transitive bi-Cayley graph over H, a contradiction. If r > 2, then by Lemma 4.4(1), we have R = L = 0. Assume that S = {l,g, h}. Since r is connected, by Proposition 2.3(1), we have H = {S} = {g, h}. It follows that o(g) = o(h) = exp(H) = 3r+2. By Lemma 4.4(1), there exists a G Aut(H) such that ga = g-1h, ha = g-1 and o(a) | 3. Suppose that a is trivial. Then h = g-1, and then H = {g}, a contradiction. Thus, a has order 3. Note that Qr(H) = (z3 | z G H^ = (x3^ x (y3^ = Z9 x Z3 and g3 ,h3 G fy (H). If {g3r} = {h3}, then we may assume that (g3)a = gx'3r for some A G Zg. Then (h3)a = hx'3r. Since ga = g-1h and ha = g-1, we have gx3 = g-3h3 and hx'3r = g-3. Then gx2-3r = (gX • ^ )A = (g-3 h3 )x = g-X • 3r hx • 3r = g-X • 3r g-3 = g(-X-1) • . It follows that g(x2+x+1)•3r = 1, and so 9 | A2 + A + 1, a contradiction. Suppose {g3r} = {h3}. Then Qr(H) = {g3,h3r} and H' = {x3} = Z9. Assume that x3 = gl•3hj•3 for some i,j G Z9. Then either (i, 3) = 1 or (j, 3) = 1. Since H' = {x3}, we have {x3}a = {x3}. So (gi•3hj•3)a = (gi•3hj•3)k for some k G Z9. Then gik •3 hjk •3 = (gi •3 hj •3r )a = (ga)i •3 (ha)j •3r = g-i •3 v 3r g-j •3r = g-(i+j) •3r hi •3r. It follows that -(i + j) = ik (mod 9) and i = jk (mod 9). Then -(jk + j) = jk2 (mod 9), and so j (1 + k + k2 ) = 0 (mod 9), forcing that 3 | j. Furthermore, since i = jk (mod 9), we have 3 | i, a contradiction. Case 3: (s,t) = (0,1). In this case, we have H = (a,b | a3r+u =1,b3r+1 = a3 ,ab = a^ ) . Let x = b,y = b3a-1. Since ab = a1+3r, we have b-1 aba-1 = a3, and then aba-1 = ba3r = bbT+1 = b1+3r+1. Since b3r = a3, by Proposition 3.1(2), we have x3r+U+1 = b3r+u+1 = a3r+u = 1, y3r = (b3a-1)3r = 1, xy = bb3a-1 = (b)a-1 = aba-1 = b1+3r+1 = x1+3r+1. Then R(H) = H = (x,y | x3r+u+1 = y3r = 1^ = x. Recall that N = Z3 and N < R(H). By Lemma 3.2(4), N is one of the following four groups: {x3r+u}, {y3 1}, {y3 1 x3r+u}, {y3 1 x2•3r+u}. Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 233 Suppose first that N = (x3 }. Then x has order 3r+"+1. We shall show that H/N has the following presentation: H/N = / X, h | x3r+U+1 = h3r-1 = 1, Xh = x1+3r+1 Actually, if N = (y3r }, then we may take h = y. If N = (y3r x3r "}, then take h = yx3"+1, and then by Lemma 3.2(2) -(3), we have (yx3" + 1 )3r-1 = y3r-1 x3u + 1[1 + (1+3r+1) + (1+3r+1)2H-----+(1+3r+1 1 -1 ] = y3r-1 x3" + 1[1 + (1+3r+1) + (1+2-3r+1)H-----h(1 + (3r-1-1)-3r+1)] or —1 ou + 1 To r — 1 i 3r 1-(3r 1 — 1) or+1i = y3 x3 [3 + 2 3 ) or—1 ou+r = y3 x3 + G N. If N = (y3r 1 x2 ^r+"}, then take h = yx2 3"+1, and then by Lemma 3.2(2)-(3), we have (yX2-3" + 1 )3r —1 = y3r —1 x2-3u + 1[1+(1+3r+1) + (1+3r+1)2 + --- + (1+3r+1)3r —1 —1] = y3r —1 x2*3u + 1 [1+(1+3r+1) + (1+2-3r+1 )+-----+(1+(3r —1 — 1) • 3r+1 )] = y3r —1 x2 3u + 1[3r —1 + 3r —1-(32r—1 —1) •3r+1] or — 1 o ou + r = y3 x2 3 + G N. Clearly, in each case, we have xh = x1+3r. So H/N always has the above presentation. Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 1. However, by Proposition 4.1, there is no cubic edge-transitive bi-Cayley graph over R(H)/N, a contradiction. Suppose now that N = (x3r+u}. Then R(H)/N = (x, y | x3r+u = y3r = T, x^ = x1+3r+1 ) . Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 2. However, by Proposition 4.1, there is no cubic edge-transitive bi-Cayley graph over R(H)/N, a contradiction. □ Now we are ready to finish the proof of Theorem 1.3. Proof of Theorem 1.3. By Lemma 5.1, if H is non-abelian, then H is inner-abelian. By Theorem 4.2, we have p = 3, and then by Proposition 4.1, r is isomorphic to either rr or Er, as desired. □ References [1] B. Alspach and T. D. Parsons, A construction for vertex-transitive graphs, Canad. J. Math. 34 (1982), 307-318, doi:10.4153/cjm-1982-020-8. [2] Y. Berkovich, Groups of Prime Power Order, Volume 1, volume 46 of De Gruyter Expositions in Mathematics, De Gruyter, Kammergericht, Berlin, 2008, doi:10.1515/9783110208221. 234 Ars Math. Contemp. 16(2019)203-213 [3] Y. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 2, volume 47 of De Gruyter Expositions in Mathematics, De Gruyter, Kammergericht, Berlin, 2008, doi:10.1515/ 9783110208238. [4] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., New York, 1976. [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235-265, doi:10.1006/jsco.1996.0125. [6] I. Z. Bouwer, An edge but not vertex transitive cubic graph, Canad. Math. Bull. 11 (1968), 533-535, doi:10.4153/cmb-1968-063-0. [7] M. Conder, J.-X. Zhou, Y.-Q. Feng and M.-M. Zhang, Edge-transitive bi-Cayley graphs, 2016, arXiv:1606.04625 [math.CO]. [8] Y.-Q. Feng, J. H. Kwak and M.-Y. Xu, Cubic s-regular graphs of order 2p3, J. Graph Theory 52 (2006), 341-352, doi:10.1002/jgt.20169. [9] B. Huppert, Endliche Gruppen I, volume 134 of Die Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1967. [10] H. Koike and I. Kovacs, Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs, Filomat 30 (2016), 321-331, doi:10.2298/fil1602321k. [11] C. H. Li, J. Pan, S. J. Song and D. Wang, A characterization of a family of edge-transitive metacirculant graphs, J. Comb. Theory Ser. B 107 (2014), 12-25, doi:10.1016/j.jctb.2014.02. 002. [12] C. H. Li and H.-S. Sim, Automorphisms of Cayley graphs of metacyclic groups of prime-power order, J. Aust. Math. Soc. 71 (2001), 223-231, doi:10.1017/s144678870000286x. [13] C. H. Li and H.-S. Sim, On half-transitive metacirculant graphs of prime-power order, J. Comb. Theory Ser. B 81 (2001), 45-57, doi:10.1006/jctb.2000.1992. [14] C. H. Li, S. J. Song and D. J. Wang, A characterization of metacirculants, J. Comb. Theory Ser. A 120 (2013), 39-48, doi:10.1016/j.jcta.2012.06.010. [15] P. Lorimer, Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph Theory 8 (1984), 55-68, doi:10.1002/jgt.3190080107. [16] Z. Lu, C. Wang and M. Xu, On semisymmetric cubic graphs of order 6p2, Sci. China Ser. A 47 (2004), 1-17, doi:10.1360/02ys0241. [17] A. Malnic, D. Marusic and C. Wang, Cubic edge-transitive graphs of order 2p3, Discrete Math. 274 (2004), 187-198, doi:10.1016/s0012-365x(03)00088-8. [18] D. Marusic and P. Sparl, On quartic half-arc-transitive metacirculants, J. Algebraic Combin. 28 (2008), 365-395, doi:10.1007/s10801-007-0107-y. [19] Y.-L. Qin and J.-X. Zhou, Cubic edge-transitive bi-p-metacirculant, Electron. J. Combin. 25 (2018), #P3.28, http://www.combinatorics.org/ojs/index.php/eljc/ article/view/v25i3p2 8. [20] L. Redei, Das "schiefe Produkt" in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehören, Comment. Math. Helv. 20 (1947), 225-264, doi:10.1007/bf02568131. [21] H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964, translated from the German by R. Bercov. [22] M. Xu and Q. Zhang, A classification of metacyclic 2-groups, Algebra Colloq. 13 (2006), 2534, doi:10.1142/s1005386706000058. Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 235 [23] J.-X. Zhou and Y.-Q. Feng, Cubic bi-Cayley graphs over abelian groups, European J. Combin. 36 (2014), 679-693, doi:10.1016/j.ejc.2013.10.005. [24] J.-X. Zhou and Y.-Q. Feng, The automorphisms of bi-Cayley graphs, J. Comb. Theory Ser. B 116 (2016), 504-532, doi:10.1016/j.jctb.2015.10.004. ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ars mathematica contemporanea 16 (2019) 237-244 https://doi.org/10.26493/1855-3974.1664.4b6 (Also available at http://amc-journal.eu) Comparing the expected number of random elements from the symmetric and the alternating groups needed to generate a transitive subgroup Andrea Lucchini, Mariapia Moscatiello Université degli Studi di Padova, Dipartimento di Matematica "Tullio Levi-Civita" Received 5 April 2018, accepted 29 May 2018, published online 23 December 2018 Given a transitive permutation group of degree n, we denote by ey(G) the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators of a transitive subgroup of G is found. We compare ey(Sym(n)) and Keywords: Transitive groups, generation, expectation. Math. Subj. Class.: 20B30, 20P05 1 Introduction Let n e N and suppose that we are in the following situation. There are two boxes, one is blue and one is red. The balls in the blue box correspond to the elements of Sym(n), the balls in the red box correspond to the elements of Alt(n). We choose one of the boxes, and then we extract balls from the chosen box, with replacement, until a transitive permutation group of degree n is generated. In order to minimize the number of extractions, is it better to choose the red box or the blue one? We are going to prove that the answer depends on the parity of n. If n is even the best choice is the blue box, if n is odd the red one. In order to formulate and discuss this problem in an appropriate way, we need to introduce some definitions. Let G be a transitive permutation group of degree n and x = (xm)meN be a sequence of independent, uniformly distributed G-valued random variables. We may define a random variable tg by setting tg = min{t > 1 | (xi,..., xt) is a transitive subgroup of Sym(n)} e [1, E-mail addresses: lucchini@math.unipd.it (Andrea Lucchini), moscatie@math.unipd.it (Mariapia Moscatiello) Abstract eT (Alt(n)). ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 238 Ars Math. Contemp. 16(2019)203-213 We denote with ex (G) = J21> i tP (tg = t) the expectation of the random variable tg. Thus ex(G) is the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators of a transitive subgroup of G is found. The case when G = Sym(n) has been studied in [2, Section 5]. Denote by nn the set of partitions of n, i.e. nondecreasing sequences of natural numbers whose sum is n. Given w = (ni,..., nfc) G n„ with ni — • • • — nk1 > nfcj+i — • • • — nkl+k2 > • • • > nfci +-----hkr-1 + i — • • • — nkt+-----+kr define — ( —1)k-1(k - 1)!, n! l(w) — —i—i-1' ni!n2! • • • nk! v(w) — ki!k§! • • • kr!. It turns out (see [2, Theorem 9]) that for every n > 2, ^(w)t(w)2 ex (Sym(n)) = - in. V(w)(i(w) - 1)J where n^ is the set of partitions of n into at least two subsets. The aim of this paper is to consider the case G — Alt(n). Our main result is the following. Theorem 1.1. For every natural number n > 3 (-1)n+in!(n - 1)! eT(Sym(n)) - ex(Alt(n)) = (n! - 1)(n! - 2) ' So the difference ex(Sym(n)) - ex(Alt(n)) tends to zero when n tends to infinity, but it is positive if n is odd and negative otherwise. To explain this behaviour notice that, if G < Sym(n), then P(tg — 1) concides with the probability Px(G, 1) that one randomly chosen element g in G generates a transitive subgroup of Sym(n), i.e. that g is an n-cycle: in particular Px(Sym(n), 1) — 1/n, Px(Alt(n), 1) — 2/n if n is odd and Px(Alt(n), 1) — 0 if n is even. In [2, Section 5], it is proved that ex(Sym(n)) — 2 and 7982 2 — ex(Sym(2)) < ex(Sym(n)) < ex(Sym(4)) — — ~ 2.1033. 3795 A similar result can be obtained in the alternating case. Theorem 1.2. Assume n > 3. 1. If n is odd, then § — ex(Alt(3)) < ex(Alt(n)) < 2. 2. If n is even, then 2 < ex(Alt(n)) < ex(Alt(4)) — 3ff - 2.3879. Moreover lim„^TO ex(Alt(n)) — 2. A. Lucchini andM. Moscatiello: Comparing the expected number of random elements from ... 239 2 Proof of Theorem 1.1 Let A = (X, <) be a finite poset. Recall that the Möbius function on the poset A is the unique function : X x X ^ Z, satisfying ^(x, y) = 0 unless x < y and the recursion formula y^ ^ I 1 if X = Z, y^L ' 10 otherwise. x t if and only if (xi,..., xt) is not a transitive subgroup of G, so we have P(rG >t) = 1 - Pq(G, t). We get that eT(G) = £ tP(rG = t) = £ I £ P(tg = m) t>1 t>1 \m>t (2.1) = EP(tg ^ t) = £P(tg > t) = £(1 - Pt(G, t)). t> 1 t>0 t>0 Consider the poset XG of the intransitive subgroups of G, let IG be the set of subgroups of G than can be obtained as intersection of maximal elements of the the poset XG, and let JG = IG U {G}. From [1, Section 2] we have that d triv^ ^t,G(h,g) ^T,G(H, g) Pt(G,t)= E |G : H|t = E |G: H|t , HeLr(G) 1 1 Hejg 1 1 where ^t,g denotes the Mobius function on the lattice Lt(G) = XG U {G}. So in order to compute the function Pt(G, t) we need information about the subgroups in JG. Let Pn be the poset of partitions of {1,..., n}, ordered by refinement. The maximum i of Pn is {{1,..., n}} (the partition into only one part), while the minimum 0 is {{1}, {2},..., {n}} (the partition into n parts of size 1). The orbit lattice of G is defined as Pn(G) = {a € Pn | the orbits of some H < G are the parts of a}. If a = {Q1,..., Qfc} € Pn, then we define G(a) = (Sym(Q1) x • • • x Sym(Qfc)) n G. If a € Pn(G), then G(a) is the maximal element in the lattice of those subgroups of G whose orbits are precisely the parts of a. Notice that H € JG if and only if there exists a € Pn(G) with H = G(a); moreover ^t,G(G(a), G) = ^pn(G)(a, 1) so P (G t) V^ Mpn(G)(a, 1) (2 2) Pt(G,t)= £ |Gi:G?(a)|t . (2.2) ^epn(G) | v y| We want now to use (2.2) in order to compute Pt(Sym(n), t) - Pt(Alt(n), t). Let P2,n be the subset of Pn consisting of the partitions of {1,..., n} into n - 1 parts (one of size 2, the others of size 1) and let = P2,n U {°}. The following two lemmas are immediate but crucial in our computation. 240 Ars Math. Contemp. 16(2019)203-213 Lemma 2.1. P„(Sym(n)) = P„ andP„(Alt(n)) = P„ \ P2,n. Lemma 2.2. If a eVn \ P|,„, then 1 Mp„(Sym(n))(a, 1) = Mp„(Alt(n)) (a, 1) = Mpn (a> 2. | Sym(n) : Sym(n)(a)| = | Alt(n) : Alt(n)(a)|. Lemma 2.3. We have 1. Mp„(Sym(n))(Ö, i) = (-1)n-1(n - 1)!; 2. Mp„(Ait(n))(Ö, i) = (-1)n-1(n - 1)! + ^^. Proof. We use the following known result (see for example [3, p. 128]): ({fii,..., Qfc}, 1) = (-1)k-1(k - 1)!. (2.3) This immediately implies Mpn(Sym(n))(Ö, 1) = Mpn (Ö, 1) = (-1)n-1(n - 1)!. Moreover Mp„(Alt(n)) (Ö, Ö) = ^pn(Alt(n)) (a? Ö) ff£p„(Alt(n))\{0> = - ^pn (a> Ö) = - Y^ ^Pn (°"> Ö)+ ^pn (°"> Ö) = Mpn (0, 1)+ ^pn (a' Ö) = (-1)n-1 (n - 1)! + (n 2) (-1)n-2(n - 2)! = (-1)n-1 (n - 1)! + ( 1)n n!. □ Theorem 2.4. For every natural number n > 2 Pt(Sym(n), t) - PT(Alt(n),t) = (-1)"+1(n -^^ ~ ^. Proof. For every t G N (and using (2.3) and Lemma 2.3) let ni(n t)= V MPn(Sym("))(a1) n1( ' ) ^ | Sym(n) : Sym(n)(a)|4 (-1)n-2(n - 2)!24 _ (-1)n-2(n!)2t (n!)4 2(n!)4 Mpn(Sym(n))(Ö , Ö) _(-1)n-1(n - 1)! n2(n ,t) n3(n,t) = r'n(Alt(")^ = (-1)n-1(n - 1)! + m | Alt(n) : Alt(n)(Ö)|* ^ ; v ; | Sym(n) : Sym(n)(Ö)|4 (n!)4 Mpn(Alt(„)) (Ö,1) _(. u ,, (-1)n-2n! N( 2 2 t 2 A. Lucchini andM. Moscatiello: Comparing the expected number of random elements from ... 241 From (2.2), Lemma 2.1 and Lemma 2.2, we deduce that MP„(Sym(n))(^, 1) Pt (Sym(n),t) = ]T CT£P„(Sym(n)) E | Sym(n) : Sym(n)(a)|4 MP„(Alt(n))(^, 1) ff£P„(Alt(n)) | Alt(n) : Alt(n)(a)|i + E 1) + ff£?2,„(Sym(n)) MP„(Sym(n))(°, 1) | Sym(n) : Sym(n)(a)|4 MP„(Alt(n))(°, 1) | Sym(n) : Sym(n)(0)|4 | Alt(n) : Alt(n)(0)|t Pt(Alt(n), t) + ni(n,t) + %(n,t) - %(n,t) = Pt (Alt(n), t) + (-1)n(n - 1)!(2t - 1) (n) □ Proof of Theorem 1.1. Using equation (2.1) we obtain that eT(Sym(n)) - eT(Alt(n)) = E (Pt(Alt(n),t) - Pt(Sym(n),t)) t> 0 E t0 (-1)"+1(n - 1)!(2t - 1) (n) (-i)"+1(« -1)! |£ (£)t - E t = (-1)"+1(n - 1)! . n! . ,t>0 v 7 t>0 n! n! n! 2 n! 1 (-1)n+1n!(n - 1)! (n! - 1)(n! - 2) ' □ 3 Examples In this section we want to verify Theorem 1.1 in the particular case when n G {3,4} using some direct, elementary arguments to compute et(Sym(n)) and et(Alt(n)). First assume n = 3. Notice that rAlt(3) is a geometric random variable with parameter 2, so et(Alt(3)) = 2' To generate a transitive subgroup of Sym(3) first of all we have to search for a nontrivial element of Sym(3). The numbers of trials needed to obtain a nontrivial element x of Sym(3) is a geometric random variable of parameter |: its expectation is equal to E0 = | .If this element has order 3, we have already obtained a transitive subgroup. However, with probability p1 = §, the nontrivial element x is a transposition: in this case in order to generate a transitive subgroup we need to find an element y G (x) and the number of trials needed to find y G (x) is a geometric random variable with parameter 2 and expectation E1 = 2. Definitely 6 3 3 21 eT(Sym(3)) = E + P1E = - + - • - = -. 5 5 2 10 242 Ars Math. Contemp. 16(2019)203-213 In particular 913 3 eT(Sym(3)) - eT(Alt(3)) = - - - = -, according with Theorem 1.1. Now assume n = 4. The transitive subgroups of Alt(4) are the noncyclic subgroups. Thus the subgroup (xi,..., xt) of Alt(4) is transitive if and only if there exist 1 < i < j < t such that xj = 1 and xj G (xj). The numbers of trials needed to obtain a nontrivial element x of Alt(4) is a geometric random variable of parameter 12 and expectation E0 = 11. With probability p1 = H the nontrivial element x has order 2: in this case the number of trials needed to find an element y G (x) is a geometric random variable of parameter 10 and expectation E1 = 12 .On the other hand, with probability p2 = ii the nontrivial element x has order 3: in this second case the number of trials needed to find an element y G (x) is a geometric random variable of parameter 12 and expectation E2 = 12. Thus 394 eT (Alt(4)) = Eo + P1E1 + p2E2 = —. 165 The case of Sym(4) is more complicated. To generate a transitive subgroup of Sym(4) first of all we have to search for a nontrivial element of Sym(4). The numbers of trials needed to obtain a nontrivial element x of Sym(4) is a geometric random variable of parameter 23: its expectation is equal to E0 = 23 .If x is a 4-cycle, then we have already generated a transitive subgroup. With probability p1 = 23, x is a product of two disjoint transposition: in this case to generate a transitive subgroup it is sufficient to find an element y G (x) and the number of trials needed to find such an element is a geometric random variable of parameter 20 and expectation E1 = 24. With probability p2 = , x is a 3-cycle: to generate a transitive subgroup we need an elements y which does not normalizes (x): the number of trials needed to find such an element is a geometric random variable of parameter 14 and expectation E2 = 11. Finally, with probability p3 = 23, x is a transposition. To find an element y G (x) we need E3 = 22 trials. If y is a 4-cycle or a 3-cycle with | supp(y) n supp(x)| = 1 or a product of two disjoint transpositions (a, b) (c, d) with x G {(a, b), (c, d)}, then we have already generated a transitive subgroup. With probability q1 = 22, (x, y) is an intransitive subgroup of order 4: to generate a transitive subgroup we need an elements z G (x, y). The number of trials needed to find such an element is a geometric random variable of parameter 20 and expectation E2 = 20. With probability q2 = 22, (x, y) = Sym(3) and to generate a transitive subgroup we need other E| = 28 trials. Definitely eT (Sym(4)) = Eo + P1E1 + P2E2 + p3(E3 + q^2 + q2E2) 24 3 24 8 24 6 (24 2 24 8 24 A 7982 + ™ • ™ + ™ • ttt + ™ ™ + ™ • ™ + 23 23 20 23 18 23 122 22 20 22 18 / 3795 In particular 7982 394 72 eT(Sym(4)) - eT(Alt(4)) = 3795 165 253 according with Theorem 1.1. A. Lucchini andM. Moscatiello: Comparing the expected number of random elements from ... 243 4 Proof of Theorem 1.2 Lemma 4.1. Let e = 0 if n is even, e =1 if n is odd. Then ,, 2e 1 2 3n eT(Alt(n)) < 2 - 2_ +-_ + _-+ _-^-. n n — 1 n(n — 1) — 2 n(n — 1)(n — 2) — 6 Proof. Since an element of Alt(n) generates a transitive subgroup if and only if it is a cycle of length n, we have that P^(Alt(n), 1) = 2e/n. Let now t > 2 and let X\ , . . . , Xt £ Alt(n) and Y = (xi,... ,xt) < Alt(n). If Y is contained in an intransitive maximal subgroup, then Y is contained in a subgroup conjugate to Sym(k) x Sym(n — k) for some 1 < k < LJ. Let k g {1,..., n — 1}. The probability that Y is contained in a subgroup conjugate to Sym(k) x Sym(n — k) is bounded by (k) t. So , ) i-t 1 — Pt(Alt(n),t) < ]T Notice that Ef n\ 1 t n /n 1 t 1 \k) < 2V3, 30 = (1 — Pt (Alt(n), 0)) + (1 — Pt (Alt(n), 1)) + £(1 — Pt (Alt(n), t)) t> 2 < 2—i+§( n1-•+(n)1-'+n (3)1-n 2e 1 1 n 1 = 2 — - + -r + 7n-7 + n - 1 (2) - 1 2 (3) - 1 _ 2e 1 2 3n n n — 1 n(n — 1) — 2 n(n — 1)(n — 2) — 6 Proof of Theorem 1.2. Let = ( — 1)"+1n!(n — 1)! f(n)= (n! — 1)(n! — 2) ' In [2, Section 5] it has been proved that limn^w (Sym(n)) = 2. This implies lim ex(Alt(n)) = lim (ex(Sym(n)) — f(n)) = lim ex(Sym(n)) — lim f (n) = 2. Moreover, again by [2, Section 5], if n > 2, then 2 < eT(Sym(n)) < eT(Sym(4)) - 2.1033. (4.1) 1 5. Notice that |f (n)| is a decreasing function and that f (n) < 0 if n is even, f (n) > 0 otherwise. Assume that n is even: ex(Alt(n)) = ex(Sym(n)) - f (n) > 2 - f (n) > 2, ex(Alt(n)) = ex(Sym(n)) - f (n) < ex(Sym(4)) - f (4) = ex(Alt(4)). Assume that n is odd: it follows immediately from Lemma 4.1, that ex(Alt(n)) < 2 if n > 9. Moreover eT (Alt(5)) eT (Alt(7)) 2205085 1.8842, 1170324 1493015628619946854486 779316363245447358045 1.9158. Finally 1440 3 eT(Alt(n)) = eT(Sym(n)) - f (n) > 2 - f (5) > 2 - — > 3 = eT(Alt(3)). □ References [1] E. Detomi and A. Lucchini, Some generalizations of the probabilistic zeta function, in: T. Hawkes, P. Longobardi and M. Maj (eds.), Ischia Group Theory 2006, World Scientific Publishing, Hackensack, NJ, 2007 pp. 56-72, doi:10.1142/9789812708670_0007, proceedings of the conference in honor of Akbar Rhemtulla held in Ischia, March 29 - April 1, 2006. [2] A. Lucchini, The expected number of random elements to generate a finite group, Monatsh. Math. 181 (2016), 123-142, doi:10.1007/s00605-015-0789-5. [3] R. P. Stanley, Enumerative Combinatorics, Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1997, doi:10.1017/ cbo9780511805967. /^creative ^commor ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 245-255 https://doi.org/10.26493/1855-3974.1591.92e (Also available at http://amc-journal.eu) ARS MATHEMATICA CONTEMPORANEA On the domination number and the total domination number of Fibonacci cubes Elif Saygi * Department of Mathematics and Science Education, Hacettepe University, 06800, Beytepe, Ankara, Turkey Received 2 February 2018, accepted 9 October 2018, published online 4 January 2019 Abstract Fibonacci cubes are special subgraphs of the hypercube graphs. Their domination numbers and total domination numbers are obtained for some small dimensions by integer linear programming. For larger dimensions upper and lower bounds on these numbers are given. In this paper, we present the up-down degree polynomials for Fibonacci cubes containing the degree information of all vertices in more detail. Using these polynomials we define optimization problems whose solutions give better lower bounds on the domination numbers and total domination numbers of Fibonacci cubes. Furthermore, we present better upper bounds on these numbers. Keywords: Fibonacci cubes, domination number, total domination number, integer linear programming. Math. Subj. Class.: 05C69, 68R10, 11B39 1 Introduction Let G = (V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). D C V(G) is a dominating set of G if every vertex in V(G) either belongs to D or is adjacent to some vertex in D. The domination number y(G) is defined as the minimum cardinality of a dominating set of the graph G. Similarly, D C V(G) is a total dominating set if every vertex in V(G) is adjacent to some vertex in D and the total domination number jt(G) is defined as the minimum cardinality of a total dominating set of G. Note that the total domination number is defined for isolate-free graphs and it is not defined for the graphs that contain isolated vertices. The domination number of the Fibonacci cubes rn is first given * Supported by TUBiTAK under Grant No. 117R032. The author would like to thank the anonymous reviewers for their valuable comments and would like to thank Prof. Dr. Omer Egecioglu for useful discussions and suggestions. E-mail address: esaygi@hacettepe.edu.tr (Elif Saygi) ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 246 Ars Math. Contemp. 16 (2019) 245-255 in [14] and [2]. These results are extended in [8] by using integer linear programming for some cases. Total domination number of rn is considered in [1], in which an upper bound and a lower bound on Yt(rn) are obtained. The exact values of y(T„) and Yt(rn) are also considered by using integer programming in [1]. The upper bound on Yt(rn) given in [1] is improved in [15]. We summarize these results in Section 2. The aim of this work is to improve some of the results given in [1] and [15]. The hypercube Qn of dimension n > 1 is the graph with vertex set V(Qn) = {0,1}n, in which two vertices are adjacent if they differ in one coordinate. For convenience Q0 = K1. All the vertices of Qn are labeled by the binary strings of length n. The Fibonacci cubes rn are special subgraphs of Qn and they were introduced by Hsu [7] as a model of interconnection networks. In literature, many interesting properties of the Fibonacci cubes have been investigated, see survey [9] for details. In recent years results on disjoint hypercubes in rn are presented in [5, 13, 16] and the cube enumerator polynomial of rn is considered in [10, 11, 17] and many combinatorial results are given. The domination-type invariants of rn are considered in [1, 2, 8, 14, 15] and some numerical results and bounds are presented. It is known that Fibonacci strings of length n are the binary strings of length n that contain no consecutive ones. For this reason we can write V (rn) = {b1b2 ••• bn | bi e {0,1}, 1 < i < n, and b, • bi+1 =0 for 1 < i < n} and E (rn) = {(u,v) | U,v e V (rn), dH (u, v) = 1}, where dH(u, v) denotes the Hamming distance between u and v, that is, the number of different coordinates in u and v. The number of vertices of the Fibonacci cubes rn is Fn+2, where Fn are the Fibonacci numbers defined as F0 =0, F1 = 1 and Fn = Fn—1+Fn—2 for n > 2. For n > 2 we will use the following formulation for the fundamental decomposition of rn (see, [9]): rn orn_ i + iorn_ 2. (1.1) Here note that 0rn—1 is the subgraph of rn induced by the vertices that start with 0 and 10rn-2 is the subgraph of rn induced by the vertices that start with 10. Furthermore, 0rn-1 has a subgraph isomorphic to 00rn—2, and there is a matching between 00rn-2 and 10rn —2 (see Figure 1). Figure 1: Fundamental decompositions of the Fibonacci cube rn, n > 3. In this paper, we present upper bounds on Y(Tn) and Yî(T„). Furthermore, we introduce the up-down degree polynomials for rn containing the degree information of all vertices V (T„) in more detail. Using these polynomials we define optimization problems whose solutions give lower bound on y (rn) and 7t(rn). E. Saygi: On the domination number and the total domination number of Fibonacci cubes 247 2 Known results and new upper bounds on 7(rn) and Yt(rn) In this section, first we summarize some known results on the domination number and the total domination number of Fibonacci cubes and then we present new uppper bounds for these numbers. We start with Figure 2 and Figure 3 showing a dominating set and a total dominating set for small dimensional rn's. Figure 2: r0,..., r5 and their dominating sets. Figure 3: ri,..., r5 and their total dominating sets. We collect the known values of y(T„) and Yi(T„) in Table 1. The values of y(rn) for n < 8 are obtained in [14]. The other values of Y(rn) are obtained by integer programming. For n = 9 and n = 10 they are obtained in [8] and for n = 11 and n =12 they are obtained in [1]. Similarly, all the values of Yt(rn) given in Table 1 are obtained by computer using integer programming in [1]. 248 Ars Math. Contemp. 16(2019)203-213 Table 1: Known values of 7(T„) and 7t(rn). n 1 2 3 4 5 6 7 8 9 10 11 12 13 |V (r„)| 2 3 5 8 13 21 34 55 89 144 233 377 610 Y(r„) 1 1 2 3 4 5 8 12 17 25 39 54 - 61 7t(r„) 2 2 2 3 5 7 10 13 20 30 44 65 97 -101 Now we describe the integer linear programming used in [8] and [1]. Suppose each vertex v e V(rn) is associated with a binary variable xv. Let N(v) be the set of vertices adjacent to v and N[v] = N(v) U {v}. The problems of determining Y(rn) and Yt(rn) can be expressed as a problem of minimizing the objective function E ^ vev (r„) (2.1) subject to the condition that for every v e V(rn) we have xa > 1 (for domination number), aeN [v] xa > 1 (for total domination number). aeN (v) The value of the objective function is then 7(rn) and Yt(rn) respectively. Note that this problem has Fn+2 variables and Fn+2 constraints. In [1], it is stated that Y(r12) and 7i(r13) were not computed in real time using the above optimization problem. They got the estimates 54 < Y(r12) < 61 and 97 < 7t(r13) < 101. Here, the main difficulty is the order of rn which equals to the number of variables and the number of constraints. By using the degree information of the vertices in rn the following lower bound on 7 (rn) is presented in [14]. Theorem 2.1 ([14]). If n > 9, then Y(r„) > Fn+2 — 2 n - 2 By using a similar technique the following lower bound on jt (rn) is obtained in [1]. Theorem 2.2 ([1]). If n > 9, then ~ Fn+2 - 11" 7t(r„) > n — 3 - 1. In Section 3 we propose an optimization problem having less number of variables and constraints to estimate lower bounds on Y(rn) and Yt(rn). Our results improve the lower v E. Saygi: On the domination number and the total domination number of Fibonacci cubes 249 bounds given in Theorem 2.1 and Theorem 2.2 and we present some numerical values in Table 2 and Table 3. By using the exact values in Table 1 and the fundamental decomposition (1.1) of rn, the following upper bound on Yi(Tn) is obtained in [1]. Theorem 2.3 ([1]). If n > 11, then Yt(T„) < 21F„_g + 2F„_io. In [1], using the computer result Yt(r13) < 101 the upper bound in Theorem 2.3 improved to Yt(rn) < 601K-1 - 371 F„, n > 12. These two upper bounds further improved in [15] by using the values of y(rn) and the fundamental decomposition (1.1) of rn more than once. Theorem 2.4 ([15]). If n > 15, then Y(r„) < Yt(rn) < 3Y(r„-3) + 2Y(r„_4) < 116F„ - 187K-1 = 21F„-s - (2K-10 + Fn_12). Furthermore, Yt(r14) < 166 . We implemented the same integer linear programming problem (2.1) using CPLEX in NEOS Server [3, 4, 6] for n =13 and obtain the estimates (takes approximately 2 hours) 78 < Y(r13) < 93. Using this result with y (r12) < 61 we obtain the following bound on the domination number of rn. Theorem 2.5. If n > 12, then Y(rn) < 21Fn-8 - (2Fn-10 + 8Fn-12). Proof. The proof mimics the proof of [1, Theorem 2.1]. By the fundamental decomposition (1.1) of r„ we have y(r„) < y(r„-1) + Y(r„-2). We know that y(^2) < 61 and Y(r13) < 93. For n > 12 define the sequence (bn) with bn = 6n-1 + 6„_2 where b12 = 61 and b13 = 93. Then by induction we have bn = 21Fn-8 - 2Fn-10 - 8Fn-12 for any n > 12. We complete the proof since y(rn) < bn for n > 12. □ Combining the results in Theorem 2.5 and Theorem 2.4 we get the following result which improves Theorem 2.3 and Theorem 2.4. Theorem 2.6. If n > 16, then Yt(rn) < 21F„_8 - (2K-10 + 8F„-12). 3 Up-down degree enumerator polynomial In this section we present the up-down degree enumerator polynomial for rn. It contains the degree information of all vertices V (T„) in more detail. Using this polynomial we write optimization problems whose solutions are lower bounds on Y(rn) and 7t(rn). For each fixed v G V(rn) we write a monomial where d = w(v) is the Hamming weight of v and u is deg(v) - d (that is, deg(v) = u + d). Recall that, by the definition of rn, (v, v') G E(rn) if and only if dH (v, v') = 1. Therefore, the number of neighbors of v 250 Ars Math. Contemp. 16(2019)203-213 whose weight is one more than the weight of v (say up neighbors of v, w(v') = w(v) + 1) is u and the number of neighbors of v whose weight is one less than the weight of v (say down neighbors of v, w(v') = w(v) — 1) is d. For this reason we call the polynomial Pn(x,y) = ^ xdeg(v)-w(v)yw(v) = ^ xuyd vev (r„) vev (r„) as the up-down degree enumerator polynomial of rn. By using the fundamental decomposition (1.1) of rn we obtain the following recursive relation which will be useful to calculate Pn(x, y). Theorem 3.1. Let Pn(x,y) be the up-down degree enumerator polynomial of rn. Then for n > 3 we have Pn(x, y) = xP„_i(x, y) + yPn-2(x,y) + yP„_3(x,y) — xyP„_3(x,y), where Po(x,y) = 1, Pi(x,y)= x + y and P2(x, y) = x2 + 2y. Proof. P0, P1 and P2 are clear from Figure 2. Assume that n > 3. We know that the up-down degree enumerator polynomials of rn-1, rn-2 and rn-3 are Pn-1(x, y), Pn-2(x, y) and Pn_3(x, y) respectively. By (1.1) we have r„ = or„_i + ior„_2 (3.1) = (oor„_2 + oior„_3) + iorn_2 (3.2) and there is a matching between oorn-2 and iorn-2 (see also Figure 1). From this decomposition we have the following three different cases: 1. Assume that v g iorn-2. These vertices are the ones in rn-2 whose weights d = w(v) increase by one in rn. Furthermore, their degrees increase by one due to the matching between oorn-2 and iorn-2, which means that u = deg(v) — w(v) remains the same in rn. Therefore, these vertices contribute yPn-2(x, y) to Pn(x, y). 2. Assume that v g oiorn-3. These vertices are the ones in rn-3 whose weights d = w(v) increase by one in rn and their degrees increase by one due to the matching between oiorn-3 and ooorn-3 c oorn-2, which means that u = deg(v)—w(v) remains the same in rn. Therefore, these vertices contribute yPn-3(x, y) to Pn(x, y). 3. Assume that v g oorn-2. These vertices are the ones in orn-1 that are not in oior„_3. In rn -i the up-down degree enumerator polynomial of these vertices becomes Pn-1(x, y) — yPn-3(x, y). The weights d = w(v) of all such vertices remain the same in rn but their degrees increase by one due to the matching between oorn-2 and iorn-2, that is, u = deg(v) — w(v) increase by one in rn. Therefore, these vertices contribute x(Pn-1(x, y) — yPn-3(x, y)) to Pn(x, y). By adding all of the above contributions we get the desired result. □ Now we describe an optimization problem using the up-down degree enumerator polynomial Pn(x, y). Let be a total dominating set of rn. Then by the definition of Fibonacci cubes for every vertex v g V(rn) with weight w(v) then there must exist a vertex E. Saygi: On the domination number and the total domination number of Fibonacci cubes 251 vD € N(v) n DT with w(vD) = w(v) T 1. Furthermore, assume that for any fixed vertex vD € Dt its corresponding monomial be xuyd in the Pn(x, y). This means that vD dominates u distinct vertices v € V(rn) with weight w(v) = w(vD) + 1 and d distinct vertices v € V(rn) with weight w(v) = w(vD) - 1. Note that for all vD € DT some of the dominated vertices may coincide. Now assume that P„(x,y) = £ xuyd = £ cUxuyd. (3.3) vev (r„) For each pair (u, d) in Pn(x, y) we associate an integer variable zU which counts the number of vertices in DT with weight d and degree u + d, that is, the number of vertices in DT having d down neighbors and u up neighbors. Clearly, we have the bounds 0 < zU < cU. Our aim is to minimize |DT |, that is, our objective function is to minimize £ zu. u,d Then by the above observation to dominate all the vertices having a fixed weight d such that 1 < d < [ n 1 - 1 we must have rd : £(u • zU_i + (d +1) • zU+1) > £ cu d since any vertex with weight d - 1 having u up neighbors can dominate u distinct vertices with weight d and any vertex with weight d + 1 (all have d + 1 down neighbors) can dominate d + 1 distinct vertices with weight d. By the same argument, for d = 0 we must have ro : £ zU >£ cU = 1 and for d = [ n 1 we must have rr21 : £u • zUfl-1 >£crf 1 ] n + 1 if n is even. 1 if n is odd, n 2 Now subject to these constraints r0,..., rp n 1 the value of the objective function will be a lower bound on 7t(rn). Similarly, to find a lower bound on Y(rn) we need to modify all of the constraints rd, 0 < d < [n 1. By the definition of the dominating set, for each fixed d we need to add all of the variables zU to the left side of the constraint rd. Remark 3.2. We remark that using [12, Theorem 4.6] we can easily obtain the coefficients cU in (3.3). By the definition of the up-down degree enumerator polynomial we know that cU is the number of vertices in rn whose number of up neighbors is u and weight is d. That is, cU equals to the number of vertices of rn having degree u + d and weight d. Therefore, [12, Theorem 4.6] gives d +1 \/n - 2d' n — 2d — u + 1/1 u 252 Ars Math. Contemp. 16(2019)203-213 Remark 3.3. We know that the number of vertices of rn with weight d is equal to the right hand side of the above constraints rd. By definition of rn this number is equal to the number of Fibonacci strings of length n and weight d. Therefore we have n — d +1 "V d u v Remark 3.4. To find the number of variables zU we need to find the number of monomials in Pn(x, y). Assume that n is even. Then by the structure of the vertices in Fibonacci cubes (it can also be seen from the structure of Fibonacci strings) n - 3d < u < n - 2d. Therefore the number of variables zU becomes L fJ f ^(d +1)+ ^ (n - 2d +1) d=0 d=LfJ+1 which is equal to s2 - 2sr +3r(r + 1) +1 where r = [f J and s = n/2. Similarly, if n is odd we obtain that the number of variables is s2 — s(2r + 1) + r(3r2+-5) + 2 where r = [ f J and s = [ f ]. Now we illustrate our optimization problem for n = 14. We have the following polynomial by Theorem 3.1. PM(x,y) = 8y7 + 7y6x2 + 42y6x + 35y6 + 6y5x4 + 60y5x3 + 120y5x2 + 60y5x + 6y5 + 5y4x6 + 60y4x5 + 150y4x4 + 100y4x3 + 15y4x2 + 4y3x8 + 48y3x7 + 112y3x6 + 56y3x5 + 3y2x10 + 30y2 x9 + 45y2x8 + 2yx12 + 12yx11 + „14 and this polynomial corresponds to the following optimization problem: Objective function: min : z24 + z12 + z" + z210 + zf + zf + zf + zf + zf + zf + z46 + z4 + z4 + z4 + z2 + z54 + zf + z52 + z2 + z50 + z2 + z2 + z0 + z0 E. Saygi: On the domination number and the total domination number of Fibonacci cubes 253 Constraints for jt (r 14): rr : 2z 2 + z1 6 + z6 >8 r6 : 4zf + 3zf + 2z2 + zf + 7z0 > 84 r5: 6z46 + 5z| + 4z| + 3z| + 2z| + 6z| + 6z1 + 6z0 > 252 r4 : 8zf + 7z7 + 6zf + 5zf + 5zf + 5zf + 5zf + 5zf + 5zf > 330 r3: 10z10 + 9z2 + 8zf + 4z| + 4zf + 4z| + 4z| + 4z| > 220 r2 : 12z12 + Hz111 + 3zf + 3zf + 3z| + 3zf > 78 r1 : 14z14 + 2z10 + 2z9 + 2zf > 14 r0 : z12 z1 + z1" >1 Constraints for 7 (r 14): rr: r6: r5 : r4 : r3: r2 : r1 : ro: 2z2 + + z° > 8 4zf + 3zf + 2zf + Z51 + z2 + + z0 + 7z0 > 84 6z6 + 5z4 + 4z4 + 3z4 + 2z4 + zf + zf + z5 + z1 + zf + 6z2 + 6z1 + 6z0 > 252 8zf + 7zr + 6z6 + 5zf + z6 + z4 + z4 + z4 + z2 + 5z4 + 5zf + 5zf + 5z1 + 5z0 > 330 10z210 + 9z9 + 8zf + zf + z7 + zf + zf + 4z6 + 4z4 + 4z4 + 4zf + 4z4 > 220 12z12 + llz!1 + z10 + z29 + zf + 3zf + 3zf + 3z6 + 3zf > 78 14z14 + z12 + z!1 + 2z10 + 2z9 + 2zf > 14 z14 + z112 + z111 > 1 Bounds: z14 < 1 z112 <2 z111 < 12 z210 <3 z29 < 30 zf < 45 zf <4 zr < 48 z6 < 112 zf < 56 z46 <5 z4 < 60 z44 < 150 zf < 100 z42 < 15 z4 <6 zf < 60 z52 < 120 zf < 60 zff <6 z62 <7 z61 < 42 z0 < 35 z0 <8 The value of the objective function gives a lower bound on Y(r14) and Yt(r14) respectively. Note that the above problem have only 24 variables and 8 constraints (instead of having 987 variables and 987 constraints, see Section 2). In general using the up-down degree enumerator polynomial Pn(x, y) of rn in Theorem 3.1 we can write an optimization problem having less number of variables z^ (see Remark 3.4) and [f 1 + 1 constraints rd. The solutions of these problem give lower bounds on 7 (rn) and Yt(rn). One can easily see that the number of variables and the number of constraints are very smaller than the ones in the optimization problem described in Section 2. 254 Ars Math. Contemp. 16(2019)203-213 For illustration we implemented the above integer linear programming problem using CPLEX in NEOS Server [3, 4, 6] for 13 < n < 26 and immediately obtain the lower bounds presented in Table 2 and Table 3. Note that for n = 26, the number of variables in our optimization problem is 70 by Remark 3.4 and it is F28 = 317811 for the general optimization problem (2.1). In addition, the upper bounds in these tables are obtained by Theorem 2.5 for n > 14 and Theorem 2.6 for n > 16. Note that the first bounds in both tables are obtained in [1] and the upper bounds on 74(^4) and 74(^5) comes from Theorem 2.4. Table 2: Current best bounds on Y(rn), 12 < n < 26. n 7(r„) n 7(r„) n 7(r„) 12 54 - 61 17 344 - 648 22 3060 - 7189 13 78 - 93 18 528 -1049 23 4748 -11632 14 98 -154 19 819 -1697 24 7381 -18821 15 148 - 247 20 1270 - 2746 25 11472 - 30453 16 224 - 401 21 1970 - 4443 26 17912 - 49274 Table 3: Current best bounds on 7t(rn), 13 < n < 26. n 7(r„) n 7(r„) n 7(r„) 13 97 -101 18 578 -1049 23 5075 -11632 14 110 -166 19 890 -1697 24 7865 -18821 15 164 - 261 20 1374 - 2746 25 12191 - 30453 16 246 - 401 21 2121 - 4443 26 19033 - 49274 17 376 - 648 22 3281 - 7189 Remark 3.5. For n = 12, Theorem 2.1 gives Y(r12) > 38 and Theorem 2.2 gives Yi(r12) > 40. The values of the objective function in our optimization problems having 19 variables and 7 constraints give lower bounds Y(r12) > 44 and 7t(r12) > 50. For the case n — 13, Theorem 2.1 gives 7^13) > 56 and Theorem 2.2 gives 7t(r13) > 59. The values of the objective function in our optimization problems having 22 variables and 8 constraints give lower bounds Y(r13) > 65 and 7t(r13) > 75. References [1] J. Azarija, S. KlavZar, Y. Rho and S. Sim, On domination-type invariants of Fibonacci cubes and hypercubes, Ars Math. Contemp. 14 (2018), 387-395, doi:10.26493/1855-3974.1172.bae. [2] A. Castro, S. KlavZar, M. Mollard and Y. Rho, On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes, Comput. Math. Appl. 61 (2011), 2655-2660, doi:10.1016/j.camwa.2011.03.012. [3] J. Czyzyk, M. P. Mesnier and J. J. More, The NEOS server, IEEE Comput. Sci. Eng. 5 (1998), 68-75, doi:10.1109/99.714603. E. Saygi: On the domination number and the total domination number of Fibonacci cubes 255 [4] E. D. Dolan, NEOS Server 4.0Administrative Guide, Technical Memorandum ANL/MCS-TM-250, Mathematics and Computer Science Division, Argonne National Laboratory, May 2001. [5] S. Gravier, M. Mollard, S. Spacapan and S. S. Zemljic, On disjoint hypercubes in Fibonacci cubes, Discrete Appl. Math. 190/191 (2015), 50-55, doi:10.1016/j.dam.2015.03.016. [6] W. Gropp and J. J. More, Optimization environments and the NEOS server, in: M. D. Buhmann and A. Iserles (eds.), Approximation Theory and Optimization, Cambridge University Press, Cambridge, pp. 167-182, 1997, selected papers from the Conference on Numerical Mathematics, in honor of M. J. D. Powell on the occasion of his 60th birthday, held in Cambridge, July 27-30, 1996. [7] W.-J. Hsu, Fibonacci cubes—a new interconnection topology, IEEE Trans. Parallel Distrib. Syst. 4 (1993), 3-12, doi:10.1109/71.205649. [8] A. Ilic and M. Milosevic, The parameters of Fibonacci and Lucas cubes, Ars Math. Contemp. 12 (2017), 25-29, doi:10.26493/1855-3974.915.f48. [9] S. KlavZar, Structure of Fibonacci cubes: a survey, J. Comb. Optim. 25 (2013), 505-522, doi: 10.1007/s10878-011-9433-z. [10] S. KlavZar and M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117 (2012), 93-105, doi:10.1007/s10440-011-9652-4. [11] S. KlavZar and M. Mollard, Daisy cubes and distance cube polynomial, European J. Combin. (2018), doi:10.1016/j.ejc.2018.02.019. [12] S. KlavZar, M. Mollard and M. Petkovsek, The degree sequence of Fibonacci and Lucas cubes, Discrete Math. 311 (2011), 1310-1322, doi:10.1016/j.disc.2011.03.019. [13] M. Mollard, Non covered vertices in Fibonacci cubes by amaximum set of disjoint hypercubes, Discrete Appl. Math 219 (2017), 219-221, doi:10.1016/j.dam.2016.10.029. [14] D. A. Pike and Y. Zou, The domination number of Fibonacci cubes, J. Combin. Math. Combin. Comput. 80 (2012), 433-444. [15] E. Saygi, Upper bounds on the domination and total domination number of fibonacci cubes, SDUJ. Nat. Appl. Sci. 21 (2017), 782-785, doi:10.19113/sdufbed.05851. [16] E. Saygi and O. Egecioglu, Counting disjoint hypercubes in Fibonacci cubes, Discrete Appl. Math. 215 (2016), 231-237, doi:10.1016/j.dam.2016.07.004. [17] E. Saygi and O. Egecioglu, q-cube enumerator polynomial of Fibonacci cubes, Discrete Appl. Math. 226 (2017), 127-137, doi:10.1016/j.dam.2017.04.026. /^creative ^commor ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 257-276 https://doi.org/10.26493/1855-3974.1619.a03 (Also available at http://amc-journal.eu) Pappus's Theorem in Grassmannian Gr (3, Cn) Sumire Sawada, Simona Settepanella *, So Yamagata Department of Mathematics, Hokkaido University, Japan Received 26 February 2018, accepted 13 August 2018, published online 6 January 2019 Abstract In this paper we study intersections of quadrics, components of the hypersurface in the Grassmannian Gr(3, Cn) introduced by S. Sawada, S. Settepanella and S. Yamagata in 2017. This lead to an alternative statement and proof of Pappus's Theorem retrieving Pappus's and Hesse configurations of lines as special points in the complex projective Grassmannian. This new connection is obtained through a third purely combinatorial object, the intersection lattice of Discriminantal arrangement. Keywords: Discriminantal arrangements, intersection lattice, Grassmannian, Pappus's Theorem. Math. Subj. Class.: 52C35, 05B35, 14M15 1 Introduction Pappus's hexagon Theorem, proved by Pappus of Alexandria in the fourth century A.D., began a long development in algebraic geometry. In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality. (D. Eisenbud, M. Green and J. Harris [4]) There are several knowns proofs of Pappus's Theorem including its generalizations such as Cayley Bacharach Theorem (see Chapter 1 of [9] for a collection of proofs of Pappus's Theorem and [4] for proofs and conjectures in higher dimension). In this paper, by mean of recent results in [6] and [10], we connect Pappus's hexagon configuration to intersections of well defined quadrics in the Grassmannian providing a new statement and proof of Pappus's Theorem as an original result on dependency conditions for defining polynomials of those quadrics. This result enlightens a new connection *The second named author was supported by JSPS Kakenhi Grant Number 26610001. E-mail addresses: b.lemon329@gmail.com (Sumire Sawada), s.settepanella@math.sci.hokudai.ac.jp (Simona Settepanella), so.yamagata.math@gmail.com (So Yamagata) ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 258 Ars Math. Contemp. 16 (2019) 245-255 between special configurations of points (lines) in the projective plane and hypersurfaces in the projective Grassmannian Gr(3, Cn). This connection is made through a third combinatorial object, the intersection lattice of the Discriminantal arrangement. Introduced by Manin and Schechtman in 1989, it is an arrangement of hyperplanes generalizing classical braid arrangement (cf. [7, p. 209]). Fixed a generic arrangement A = {H0,..., H} in Ck, the Discriminantal arrangement B(n, k, A), n, k G N for k > 2 (k = 1 corresponds to Braid arrangement), consists of parallel translates H^1,..., Htn, (ti,..., tn) G Cn, of A which fail to form a generic arrangement in Ck. The combinatorics of B(n, k, A) is known in the case of very generic arrangements, i.e. A belongs to an open Zariski set Z in the space of generic arrangements H0, i = 1,..., n (see [7], [1] and [2]), but still almost unknown for A G Z .In 2016, Libgober and Settepanella (cf. [6]) gave a sufficient geometric condition for an arrangement A not to be very generic, i.e. A G Z .In particular in the case k = 3, their result shows that multiplicity 3 codimension 2 intersections of hyperplanes in B(n, 3, A) appears if and only if collinearity conditions for points at infinity of lines, intersections of certain planes in A, are satisfied (Theorem 3.8 in [6]). More recently (see [10]) authors applied this result to show that points in a specific degree 2 hypersurface in the Grassmannian Gr(3, Cn) correspond to generic arrangements of n hyperplanes in C3 with associated discriminantal arrangement having intersections of multiplicity 3 in codi-mension 2 (Theorem 5.4 in [10]). In this paper we look at Pappus's configuration (see Figure 1) as a generic arrangement of 6 lines in P2 which intersection points satisfy certain collinearity conditions (see Figure 2). This allows us to apply results on [6] and [10] to restate and re-prove Pappus's Theorem. More in details, let A be a generic arrangement in C3 and the arrangement of lines in ~ P2 directions at infinity of planes in A. The space of generic arrangements of n lines in (P2)n is Zariski open set U in the space of all arrangements of n lines in (P2)n. On the other hand in Gr(3, Cn) there is open set U' consisting of 3-spaces intersecting each coordinate hyperplane transversally (i.e. having dimension of intersection equal 2). One has also one set U in Hom(C3, Cn) consisting of embeddings with image transversal to coordinate hyperplanes and U/ GL(3) = U' and U/(C*)n = U. Hence generic arrangements in C3 can be regarded as points in Gr(3, Cn). Let {si < • • • < s6} C {1,..., n} be a set of indices of a generic arrangement A = {H0,..., H} in C3, ai the normal vectors of H0's and fyi = det(a^ aj, a;). For any permutation a G S6 denote by [a] = {{ii,i2}, {i3,i4}, {i5,i6}}, ij = sCT(j), and by QCT the quadric in Gr(3,Cn) of equation fii2i5i6 - fii2i3i4A^ie = 0. The following theorem, equivalent to the Pappus's hexagon Theorem, holds. Theorem 5.3 (Pappus's Theorem). For any disjoint classes [ai] and [a2], there exists a unique class [a3] disjoint from [ai] and [a2] such that {QCT1, QCT2, QCT3} is a Pappus configuration, i.e. 3 Qai1 n Qffi2 = p| Qffi i=i for any {ii, i2} C [3]. In the rest of the paper, we retrieve the Hesse configuration of lines studying intersections of six quadrics of the form Qa for opportunely chosen [a]. This lead to a better understanding of differences in the combinatorics of Discriminantal arrangement in the complex and real case. Indeed it turns out that this difference is connected with existence of the Hesse arrangement (see [8]) in P2 (C), but not in P2 (R). S. Sawada et al.: Pappus's Theorem in Grassmannian Gr(3, Cn) 259 From above results it seems very likely that a deeper understanding of combinatorics of Discriminantal arrangements arising from non very generic arrangements of hyperplanes in Ck (i.e. A 2 and n > 3s, a good 3s-partition (see [10]) is a set T = {L i, L2, L3}, with Li subsets of [n] such that |Li| = 2s, |Li n Lj | = s (i = j), L i n L2 n L3 = 0 (in particular | U Li| = 3s), i.e. L i = {i i,..., i2S}, L2 = {i i,..., is,i2S+i,... ,i3s}, L3 = 260 Ars Math. Contemp. 16 (2019) 245-255 {¿s + 1, . . . , ¿3s}. Notice that given a generic arrangement A in C2s-1, subsets L define hyperplanes DLi in the Discriminantal arrangement B(n, 2s - 1, ATO). In this paper we are mainly interested in the case s = 2 corresponding to generic arrangements in C3. 2.3 Matrices A () and At () Let a = (a^,..., ajk) be the normal vectors of hyperplanes ffj, 1 < i < n, in the generic arrangement A in Ck. Normal here is intended with respect to the usual dot product (ai, .. ., afc) • (vi,. .., vfc) = ajVj. j Then the normal vectors to hyperplanes DL, L = {s1 < ••• < sfc+1} c [n] in S ~ Cn are nonzero vectors of the form k+1 «L = 53(-1)j det(asi,. .. ,aSi,.. ., aSfc+i )eSi, (2.1) i=1 where {ej}1 if j = ii e I, 0 otherwise. Plucker relations, i.e. conditions for dim(ker y>x) = k, are vanishing conditions of all (n — k + 1) x (n — k +1) minors of Mx. It is well known (see for instance [5]) that Plucker relations are degree 2 relations and they can also be written as k E(—1)1 ..pk-iqi ftqo...qi...qk = 0 (2.2) 1=0 for any 2k-tuple (pi,... ,Pk-1, qo,..., qk). Remark 2.1. Notice that vectors in the equation (2.1) normal to hyperplanes correspond to rows indexed by L in the Plucker matrix Mx, that is A(ATO ) = Mx, up to permutation of rows. Notice that, in particular, det(asi,..., ofSi,..., aSfc+1) is the Plucker coordinate ,0/, I = {s1, s2,..., sk+1} \ {«»}. 2.5 Relation between intersections of lines in and quadrics in Gr (3, Cn) Let A = {H0,..., H} be a generic arrangement in C3. If there exist L1, L2, L3 c [n] subsets of indices of cardinality 4, such that codimension of DLl n DL2 n DL3 is 2 then A is non very generic arrangement (see [2]). Let T = {L1, L2, L3} be a good 6-partition of indices {s1,...,s6} C [n]. In [6], authors proved that the codimension of DLl n DL2 n DL3 is 2 if and only if points f1ieL1nL2 f1ieL1nL3 and PlieL2nL3 are collinear in ([6, Lemma 3.1]). Since is vector normal to , the codimension of DLl n DL2 n DL3 is 2 if and only if rank AT(ATO) = 2, i.e. all 3 x 3 minors of AT(ATO) vanish. In [10] authors proved the following Lemma. Lemma 2.2 ([10, Lemma 5.3]). Let A be an arrangement of n hyperplanes in C3 and O.T = {{i1, i2, i3, i4}, {i1,i2, i5,i6}, {i3, i4, i5, i6}} a good 6-partition of indices s1 < • • • < s6 e [n] such that j = sCT(j), o permutation in S6. Then rank ACT.T(ATO) = 2 if and only if A is a point in the quadric of Grassmannian Gr (3, Cn) of equation £¿2 ¿5 ¿6 ^¿2«3i4 ^¿1i5«6 0. (2.3) 262 Ars Math. Contemp. 16(2019)203-213 As consequence of above results, we obtain correspondence between points x = ^ Piej, pj = 0, IC[n] | I | =3 in the quadric of equation (2.3) and generic arrangements of n hyperplanes A in C3 such that n HOTji2, HOTjia n and HOTji5 n HOTji6 are collinear in HOT. Notice that condition pI = 0 is direct consequence of A being generic arrangement. 3 Motivating example of Pappus's Theorem for quadrics in Gr (3, Cn) In classical projective geometry the following theorem is known as Pappus's theorem or Pappus's hexagon theorem. Theorem 3.1 (Pappus). On a projective plane, consider two lines l1 and l2, and a couple of triple points A, B, C and A', B', C' which are on 11 and l2 respectively. Let X, Y, Z be points of AB' n A'B, AC' n A'C and BC' n B'C respectively. Then there exists a line l3 passing through the three points X, Y, Z (see Figure 1). Figure 1: Original Pappus's Theorem. This theorem was originally stated by Pappus of Alexandria around 290-350 A.D. In this section, we restate this classical theorem in terms of quadrics in the Grassman-nian. Indeed the six lines AB', A'B, BC', B'C, AC', A'C e P2 (C) correspond to lines in the trace at infinity of a generic arrangement A in C3 and lines l1, l2 and l3 correspond to collinearity conditions for intersection points of lines in AOT. Consider a generic arrangement A = {H1,..., H6} of 6 hyperplanes in C3, its trace at infinity and T = {L1,L2,L3} the good 6-partition defined by L1 = {1,2,3,4}, L2 = {1,2,5,6}, L3 = {3,4,5,6}. By Lemma 2.2 we get that the triple points nieLinL2 Hi n H~, fWLa Hi n H~, nieL2nL3 Hi n H~ S. Sawada et al.: Pappus's Theorem in Grassmannian Gr(3, Cn) 263 are collinear if and only if A is a point of the quadric Ql : ^134^256 — ^234^156 = 0 in Gr(3, C6). Analogously if T' = {Ll, L2, ¿3}, Ll = {4, 6, 2, 5}, ¿2 = I4, 6,1, 3}, ¿3 = {2, 5,1, 3} and T'' = {¿1', ¿2', ¿3'}, ¿1' = {2,4,1, 6}, ¿2' = {2, 4, 3, 5}, ¿3' = {1, 6, 3, 5} are different good 6-partitions then triple points fliGLJnLi. Hi n nieiinL3 Hi n HieL^nL^ ^ n and PlieL'/nL^' H n PlieL'/nL^' Hi n PlieL2'nL3' Hi n are collinear if and only if A is, respectively, a point of quadrics Q2: ^425^613 — ^625^413 = 0 and Q3 : ^216^435 — ^416^235 = 0. With above remarks and notations we can restate Pappus's Theorem as follows (see Figure 2). Theorem 3.2 (Pappus's Theorem). Let A = {H1,..., H6} be a generic arrangement of hyperplanes in C3. If A is a point of two of three quadrics Q1, Q2 and Q3 in the Grassmannian Gr (3, C6), then A is also a point of the third. In other words 3 Qi ' n Qi2 = ff Qi, {¿1,i2}c [3]. i=1 We develop this argument in the following sections providing in Theorem 5.3 a general statement on quadrics in the Grassmannian which implies Pappus hexagon Theorem in the projective plane. 4 Pappus Variety In this section, we consider a generic arrangement {H1,..., Hn} in C3 (n > 6). Let's introduce basic notations that we will use in the rest of the paper. Notation. Let {s1,..., s6} be a subset of indices {1,..., n} and T = {¿1, ¿2, ¿3} be the good 6-partition given by ¿1 = {s1, S2, S3, S4}, ¿2 = {s1, S2, S5, S6} and ¿3 = {S3, S4, S5, S6}. Then for any permutation a G S6 we denote by a.T = , aX3} the good 6- partition given by subsets = {¿1, ¿2, ¿3, ¿4}, a^2 = {¿1, ¿2, ¿5, ¿6} and 0X3 = {¿3, ¿4, ¿5, ¿6} with j = sCT(j). Accordingly, we denote by QCT the quadric in Gr (3, Cn) of equation Q : A 1 i3 i4 ^i2i5i6 ^i2i3i4 Ali5i6 0. 264 Ars Math. Contemp. 16(2019)203-213 The following lemma holds. Lemma 4.1. Let ct, ct' g S6 be distinct permutations, then QCT = if and only if there exists t g S3 such that ct.Lj n CT.Lj = ct'.Lt(j) n ct'.Ltj (1 < i < j < 3). Proof. By definition of good 6-partition we have that Li = (Li n L2) U (Li n L3), L2 = (L2 n Li) u (L2 n L3), L3 = (L3 n Li) u (L3 n L2). Then there exists t g S3 such that ct and ct' satisfy a.Lj n a.Lj = ct'.Lt(^ n ct'.Ltj (1 < i < j < 3) if and only if ct.L; = ct'.Lt(() for l = 1,2,3, that is ACT/.T(AOT) is obtained by permuting rows of ACT.T(AOT). It follows that rank ACT.T(AOT) = 2 if and only if rank ACT/.T(AOT) = 2 and hence by Lemma 2.2 this is equivalent to QCT n NS1... S6 = Qct' n NS1i...jS6, where NS1I...IS6 = {x = £ ft ei | = 0 for any I c {si,... ,se}}. IC[n] |I|=3 Since NS1 i...jS6 is dense open set in 7(Gr(3,Cn)), Q- n NS1i...iS6 = Q- n NS1i...jS6 if and only if QCT = . Vice versa if QCT n Ns1... s6 = n Ns1... s6, then any generic arrangement A corresponding to a point in QCT n Ns1. . . s6 corresponds to a point in n NS1 i...jS6, that is rank ACT.T(AOT) = 2 if and only if rank ACT/.T(AOT) = 2. It follows that Act T(Aot) and ACT/.T(AOT) are submatrices of A(AOT) defined by the same three rows, i.e. ct.L; = ct'.Ltfor l = 1,2,3. □ Definition 4.2. For any 6 fixed indices T = {si,..., s6} C [n] the Pappus Variety is the hypersurface in Gr(3, Cn) given by pt = U Q-. CTGS6 S. Sawada et al.: Pappus's Theorem in Grassmannian Gr(3, Cn) 265 Notice that all the content of this section and the following section is based on the choice of six indices {si < • • • < s6} C [n]. This is related to result in Theorem 3.8 in [6] and, consequently, Lemma 5.3 in [10] (Lemma 2.2 in this paper). Indeed Theorem 3.8 in [6] states that in order to study special configurations of n lines in P2, that is non very generic arrangements of n lines in P2, it is sufficient to study subsets of six lines out of n. On the other hand since Pappus Variety can be defined inside Gr(3, Cn), we decided to keep the discussion more general picking six indices {s1 < ••• as following: a.T - a'.T t g S3 such that a.L, n a.Lj = a'.LT(i) n a'.LT(1 < i < j < 3). We denote by [a] the equivalence class containing a.T and by QCT the corresponding quadric (notice that a in the notation QCT can be any representative of [a]). By Lemma 4.1 [a] only depends on couples Lj n Lj hence for each class [a] we can choice a representative a.To = {{ji, j2, j3, j'4}, {ji, j2, j5, je}, {j3,j4,j5, je}} such that j < j2, j3 < j4, j5 < j6 and j < j'3 < j5 and we can equivalently define [a] = {{j1, j2}, {j3, j4}, {j5,j6}}. (6)(4)(2) Since the number of choices of [a] is V2y v32/V2y =15, Pappus Variety is composed by 15 quadrics. Finally remark that [a] = {{ji, j'2}, {j'3, j'4 }, {j5, j'e}} and [a'] = {{ji,j2}, {j'3,j'4}, {j'5,j'6}} are disjoint, i.e. [a] n [a'] = 0, if and only if {j2;-i, j'2;} = {j'2i'-i,j'2v} for any 1 < /,/' < 3. Definition 4.3 (Pappus configuration). Let [ai], [a2] and [a3] be disjoint classes, a Pappus configuration is a set {QCT1, QCT2, QCT3} of quadrics in Gr (3, Cn) such that 3 Q4 It follows that in the real case, for any choice of indices {s1,..., s6} C [n], we have at most 4 collinearity conditions (see Figure 7) corresponding to 15 hyperplanes in the Dis-criminantal arrangement with 4 multiplicity 3 intersections in codimension 2 (see Figure 8). While in the complex case Hesse configuration (see Figure 6) gives rise to a Discriminantal arrangement containing 15 hyperplanes intersecting in 6 multiplicity 3 spaces in codimen-sion 2. This remark allows a better understanding of differences in the combinatorics of Dis-criminantal arrangement in the real and complex cases. Indeed the existence of a discrimi-nantal arrangement of 15 hyperplanes intersecting in 6 multiplicity 3 spaces in codimension 2 in C but not in R implies that there exist combinatorics of Discriminantal arrangements that cannot be realised in any field. This is especially interesting since in the case known until now, i.e. in the case of very generic arrangements A, the combinatorics of Discriminantal arrangement B(n, k, A) is independent from the field (see [1]). S. Sawada et al.: Pappus's Theorem in Grassmannian Gr(3, Cn) 275 Q, Figure 7: Generic arrangement A in R3 containing 6 lines satisfying 4 collinearity conditions. A " " Figure 8: Codimension 2 intersections of 15 hyperplanes in B(n, 3, ATO) indexed in {s1,...,s6| c [n] with 4 multiplicity 3 points ▲ corresponding to intersections fli=1 D