ARS MATHEMATICA CONTEMPORANEA Volume 10, Number 2, Spring/Summer 2016, Pages 211-437 Covered by: Mathematical Reviews Zentralblatt MATH COBISS SCOPUS Science Citation Index-Expanded (SCIE) Web of Science ISI Alerting Service Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES) The University of Primorska The Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The publication is partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications. ARS MATHEMATICA CONTEMPORANEA Q1 We just learned that Ars Mathematica Contemporánea has been ranked 60th among the 312 mathematical journals in the ISI's Journal Citation Report for the year 2015. This makes it the first ever scientific journal published in Slovenia that has been placed in the upper quartile: Q1. We understand very well that a high score is only a necessary condition for excellence. 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Dragan Marušic and Tomaž Pisanski Editors In Chief viii ARS MATHEMATICA CONTEMPORANEA GEMS 2013 This issue of Ars Mathematica Contemporánea offers a collection of papers presented at the Sixth Workshop 'Graph Embeddings and Maps on Surfaces' (GEMS), which took place in Smolenice, Slovakia, the week 14-19 July 2013. The GEMS workshop series began with the idea of a small conference in Slovakia that would bring together researchers interested in various aspects of graphs embedded in surfaces. The first GEMS workshop was held in Donovaly the week 21-26 August 1994, and was attended by 33 participants from 14 countries. The topics covered by the workshop included combinatorial and topological properties of embedded graphs, construction of graph embeddings in surfaces, symmetries of embedded graphs, regular maps and hy-permaps, group actions on graphs and surfaces, and convex polytopes. The Donovaly workshop was followed by similar workshops in Banská Bystrica (1997), Bratislava (2001), Stará Lesná (2005), Tále (2009) and Smolenice (2013). The GEMS workshop is now held regularly every four years, organised by the leaders of the Slovak topological graph theory school: Roman Nedela, Jozef Siráft, and Martin Skoviera. These workshops have become very well known for their informal atmosphere, allowing time for discussion of research problems and exchange of information between both individual researchers and international research teams. The venue for the most recent GEMS workshop was Smolenice Castle, the very same place where an event considered the world's first truly international graph theory meeting was held fifty years earlier. Together with the the Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications (held in Kosice a week before the GEMS 2013 workshop), it constituted one of the highlights of celebrations to commemorate the 50th anniversary of this unique scientific event. We believe that readers will find the selected papers from GEMS 2013 both interesting and inspirational for further research. We also hope that after a similar special issue devoted to GEMS 2009, this issue will be followed by subsequent collections of papers, based on lectures delivered at GEMS workshops in 2017 and beyond. Jozef Siráft and Martin Skoviera Guest Editors viii ARS MATHEMATICA CONTEMPORANEA Contents Algorithmic enumeration of regular maps Thomas Connor, Dimitri Leemans . . 211 Isospectral genus two graphs are isomorphic Alexander Mednykh, Ilya Mednykh . . . 223 Combinatorial categories and permutation groups Gareth Jones................... 237 Iterated claws have real-rooted genus polynomials Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, David G. L. Wang 255 2-Arc-Transitive regular covers of Kn,n — nK2 with the covering transformation group Zp Wenqin Xu, Yanhong Zhu, Shaofei Du....................269 Cube-contractions in 3-connected quadrangulations Yusuke Suzuki ................................281 One-point extensions in n3 configurations William L. Kocay...............................291 The number of edges of the edge polytope of a finite simple graph Takayuki Hibi, Aki Mori, Hidefumi Ohsugi, Akihiro Shikama.......323 Equitable coloring of corona products of cubic graphs is harder than ordinary coloring Hanna Furmanczyk, Marek Kubale......................333 Petrie polygons, Fibonacci sequences and Farey maps David Singerman, James Strudwick.....................349 Odd edge-colorability of subcubic graphs Risto Atanasov, Mirko Petrusevski, Riste Skrekovski............359 The spectrum of a-resolvable A-fold (K4 — e)-designs Mario Gionfriddo, Giovanni Lo Faro, Salvatore Milici, Antoinette Tripodi . 371 The endomorphisms of Grassmann graphs Li-Ping Huang, Benjian Lv, Kaishun Wang .................383 On convergence of binomial means, and an application to finite Markov chains David Gajser 393 An infinite class of movable 5-configurations Leah Wrenn Berman, Elliott Jacksch, Lander Ver Hoef 411 Odd automorphisms in vertex-transitive graphs Ademir Hujdurovic, Klavdija Kutnar, Dragan Marusic 427 Volume 10, Number 2, Spring/Summer 2016, Pages 211-437 xvii ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 211-222 Algorithmic enumeration of regular maps * Thomas Connor t Université Libre de Bruxelles, Département de Mathématiques - C.P.216, Boulevard du Triomphe, B-1050 Bruxelles Dimitri Leemans * University of Auckland, Department of Mathematics, Private Bag 92019, Auckland, New Zealand Received 23 September 2013, accepted 22 July 2015, published online 24 September 2015 Given a finite group G, we describe an algorithm that enumerates the regular maps having G as rotational subgroup, using the knowledge of its table of ordinary characters and its subgroup lattice. To show the efficiency of our algorithm, we use it to compute that, up to isomorphism, there are 796,772 regular maps whose rotational subgroup is the sporadic simple group of O'Nan and Sims. Keywords: Regular map, O'Nan sporadic simple group, subgroup lattice, character table. Math. Subj. Class.: 05E18, 52B10, 20D08 1 Introduction According to Coxeter (see [9], Chapter 8), systematic enumeration of orientable regular maps began in the 1920s by fixing a genus g and enumerating all maps embeddable on surfaces of genus g. Genus 2 was the first case considered by Errera and finished by Threlfall. Since then, a lot of work has been done on the subject, culminating in the enumeration of all orientable maps on surfaces of genus up to 301 by Conder (see [5, 4] and Conder's website for the latest results1). * Research is supported by Marsden Grant UOA1218 of the Royal Society of New Zealand. t Boursier FRIA. i Corresponding author. E-mail addresses: tconnor@ulb.ac.be (Thomas Connor), d.leemans@auckland.ac.nz (Dimitri Leemans) 1http://www.math.auckland.ac.nz/~ conder/ Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 212 ArsMath. Contemp. 10(2016)211-222 Another way to enumerate orientable maps is to fix a group G (or a family of groups) and count how many regular maps have G acting as rotational subgroup of the full automorphism group. In other words, we want to determine, for a given group G, the number of pairs of elements [R, S] G G2 such that o(RS) = 2, o(R) = p, o(S) = q and (R, S) = G (1.1) where p and q are arbitrary orders of elements in G. The second type of enumeration can be done using a formula due to Frobenius [12] (see Section 2.2) based on character theory. Frobenius' formula has been used by Sah (see [20], Section 2) to obtain some enumeration results for the first group of Janko and the small Ree groups 2G2 (q) with q = 32e+1, among other things. Conder et al. [6] extracted an enumeration result for all regular hypermaps of a given type with automorphism group isomorphic to PSL(2, q) and PGL(2, q) from the latter reference. Their result does not make use of character theory. Jones and Singerman [16] set up the theoretical framework that links the study of maps to that of Riemann surfaces, showing among others that every map M is isomorphic to some canonical map M on a Riemann surface. In [11], Downs and Jones set up the theoretical framework to determine the number of orientable maps of type {3,p} with automorphism group a group PSL(2, q) or PGL(2, q). Jones and Silver showed in [15] that the Suzuki groups Sz(q) are automorphism groups of regular maps of type {4, 5}. They also enumerated these maps: they used character theory and techniques developed by Philip Hall in [13] using Mobius inversion to show that there is at least one pair [R, S] as above in each Sz(q). Then they used the fact that each element of order 4 is not conjugate to its inverse in Aut(Sz(q)) to conclude that every such map has to be chiral. For more results of that kind, we refer to [15, 14]. Mazurov and Timofeenko also used similar techniques to find those sporadic groups that can be generated by triples of involutions, two of which commute (see [18, 21]), therefore determining which sporadic groups are full automorphism groups of non-orientable regular maps. Given a pair [R, S] G G2 satisfying (1.1), we can construct a regular map M of type {p, q} from it with G being the orientation-preserving subgroup of the full automorphism group of M. Frobenius' formula therefore gives us the number of regular maps that have G as such subgroup. The idea of the present paper is to use this formula in a systematic way to determine for a given group G what are the possible types for a map M with G being either the orientation-preserving subgroup of Aut(M) or G being the full automorphism group of M in the non-orientable case. In this paper, we design an algorithm to compute up to isomorphism the number of regular maps (reflexible or chiral) having a given group G as group of orientation-preserving automorphisms, based on the character tables of G and its subgroups and on the subgroup lattice of G. To show the efficiency of our algorithm, we implemented it in MAGMA [2] and used it on the O'Nan sporadic simple group O'N. The choice of O'N is motivated by the fact that this is one of the most mysterious sporadic groups. Its smallest permutation representation is on 122,760 points and its subgroup lattice is relatively small. The motivation of the paper first came from abstract regular polytopes. A recent paper by the authors and Mark Mixer [8] classifies all abstract regular polytopes of rank at least four for the O'Nan group. Hence rank three remains open. For a simple group G, a non-orientable regular map M whose full automorphism group is G is also an abstract regular polyhedron while a chiral map is a chiral polyhedron. Hence, getting to know which types are possible for G is also interesting in the study of abstract polyhedra whose automorphism T. Connor and D. Leemans: Algorithmic enumeration of regular maps 213 group is G. There is most likely a very large number of pairwise non-isomorphic abstract polyhe-dra having the O'Nan group as automorphism group. For instance, as shown in [17], the third Conway group, whose order is comparable, has 21,118 abstract regular polyhedra up to isomorphism. Here, we derive the possible types {p, q} for maps having O'N as automorphism group. Our results for the O'Nan group may be summarized as follows. Theorem 1.1. Let G be the O'Nan sporadic simple group and let P := {3, 4, 5, 6, 7, 8,10,11,12,14,15,16,19, 20, 28, 31}. 1. There exist two elements ñ, S G G such o(ñ) = p, o(S) = q, o(ñS) = 2, (ñ, S) = G for every p < q G P except for {p,q} = {3, 3}, {3,4}, {3, 5}, {3, 6}, {3, 7}, {3,12} and {4,4}. 2. There are 796,772 orbits of such pairs {ñ, S} under the action of Aut(O'N) = O'N : C2. 3. Orientably-regular but chiral maps M with Aut(M) = G exist for all pairs {p, q} of(1) except {3,15} (that is 128possible types). 4. Non-orientable regular maps M with Aut(M) = G exist for all pairs {p, q} of(1) except {20, q}, {31, q} (with q G P), {3,10}, {4, 5} and {4, 6} (that is 95 possible types). 5. Reflexible maps M with Aut(M) = Aut(G) exist for all pairs {p, q} of(1) except {8, q}, {16, q} (with q G P) (that is 98possible types). The paper is organized as follows. In Section 2, we introduce the theoretical background needed to understand this paper. In Section 3, we describe our algorithm. In Section 4, we summarize the results obtained on the O'Nan sporadic simple group and obtain (1) and (2) of Theorem 1.1. In Section 5, we determine the types of maps that exist for the O'Nan group, deriving (3), (4) and (5) of Theorem 1.1. In Section 6, we give an algorithm to generate efficiently all maps of type {p, q} for a fixed p. Finally, in Section 7, we conclude our paper with some remarks. 2 Theoretical background 2.1 Regular maps In this paper, a map is a 2-cell embedding of a connected graph into a closed surface without boundary. Such a map M has a vertex-set V := V(M), an edge-set E := E(M) and a set of faces F := F(M). We call V U E U F the set of elements of M. A triple T := {v,e,f} where v G V, e G E and f G F is called a flag if each element of T is incident with the other elements of T. The map is called orientable if the underlying surface on which the graph is embedded is orientable. Otherwise, it is called non-orientable. Faces of M are simply-connected components of the space obtained by removing the embedded graph from the surface. An automorphism of a map is a permutation of its elements preserving the sets V, E and F and incidence between the elements. Automorphisms form a group under composition called the automorphism group of the map and denoted by Aut(M). 212 ArsMath. Contemp. 10(2016)211-222 If there exist a face f and two automorphisms R and S such that R cyclically permutes the consecutive edges of f and S cyclically permutes the consecutive edges incident to some vertex v of f, then M is called a regular map in the sense of Brahana [3]. In this case, the group Aut(M) acts transitively on the vertices, on the edges and on the faces. All faces are thus bordered by the same number of edges, say p and all the vertices have same degree, say q. The pair {p, q} is known as the type of M. Observe that the topological dual of M, denoted by M* is obtained by switching vertices and faces (that is V(M*) := F(M), E(M*) := E(M), F(M*) := V(M)). It is also regular and its type is {q,p}. Note that R and S may be assumed to be such that RS interchanges v with one of its neighbors along an edge e on the border of f, interchanging f with the other face containing e. The three automorphisms R, S and RS then satisfy the following relations. If a regular map M also has an automorphism a which flips the edge e but preserves f, then we say that M is reflexible. In that case, Aut(M) has a unique orbit on the set of flags. Moreover, Aut(M) is generated by the three automorphisms a, b := aR and c := bS that satisfy the following relations: a2 = b2 = c2 = (ab)p = (ac)2 = (bc)q. If the map M is orientable, then the elements R = ab and S = bc generate a normal subgroup of Aut(M) of index 2, consisting of all elements expressible as words of even length in {a, b, c}. This subgroup is called the rotational subgroup and denoted by Aut+(M). All elements of Aut+(M) are precisely those preserving the orientation of the underlying surface while all other elements of Aut(M) reverse the orientation. In the non-orientable case, each of a, b and c can be expressed as a word in {R, S} and hence, Aut(M) = (R,S). If there is no automorphism a which flips the edge e but preserves f, then we say that the map M is chiral. Its automorphism group can be generated by the rotations R and S and M is necessarily orientable. Moreover, chiral maps occur in opposite pairs, each member of which is obtainable from the other by reflection. 2.2 Frobenius' formula The search for maps having G := (R, S) as an automorphism group is equivalent to the search for triples of elements x, y, z G G satisfying (1.1) by posing x = (RS)-1 = RS, y = R and z = S. Let G be a finite group and let Hc({p, q}) := {[x, y, z] G G3|o(x) = 2, o(y) = p,o(z) = q, o(xyz) = 1}. In order to determine the cardinality nG ({p, q}) of nG ({p, q}), we use the following result, due to Frobenius (see [12], section 4, equation 2). Theorem 2.1. If Ci, Cj and Ck denote conjugacy classes of elements in a finite group G, the number of solutions of gi gj gk = 1 in G, with each gx G Cx is Rp = Sq = (RS)2 = 1 2 (2.1) (2.2) where Irr(G) is the set of irreducible characters of G. This theorem gives us an easy way to compute nG({p, q}). T. Connor and D. Leemans: Algorithmic enumeration of regular maps 213 Corollary 2.2. Let G be a group. Let Cl,...,Cr be the conjugacy classes of elements of G. Let Kn := {i G {1,..., r} | o(x) = n for some x G Cj}. Then nG({P,q})= EEE |C 1, we have YG({p, q}) = nGq}) - E yh({p, q}) H p, we construct a permutation representation of O'N on its involutions. This is done by constructing the coset space of O'N on C0'n(p) for an arbitrary involution p G O'N. Let P be a sequence. We will use P to store pairs of elements of O'N. Let G be the permutation representation on the cosets of CO'N(p) and let ^ : O'N ^ G be an isomorphism between O'N in its natural permutation representation and G. Let S be a sequence containing one representative of each conjugacy class of elements of order p in O'N. For s g S, let O be the set of orbits of ^(s). For each o g O, let x be a representative of o and let ^-1(Gx) be the centralizer of an involution in O'N that correspond to the fixed point x. Let t be the involution centralized by ^-1(Gx). Let R := t * S-1. Then {R, S} is a pair with RS = t an involution. If (R, S) = O'N and there is no pair {R', S'} in P isomorphic to {R, S}, append {R, S} to P. When a new pair {R, S} is found, we can determine whether it gives an orientably-regular but chiral map or a non-orientable map whose full automorphism group is O'N. In the process, we use the results of Section 5 to shorten the computations: we keep track of how many pairs of each type have been generated so that, once we get the total number for a given type, we do not have to consider that type anymore. Each chiral map (respectively non-orientable map) whose full automorphism group is O'N is also an abstract chiral polyhedron (respectively abstract regular polyhedron). Therefore, the algorithm described above permits in theory to construct all chiral and regular polyhedra for the O'Nan group. 7 Concluding remarks In practice, to generate all the 284 pairs of type {3, q}, it took less than 4 hours on a computer with a processor running at 2.9Ghz. We needed 11 days to generate all 5176 pairs of type {4, q} and 28 days for the 7738 pairs of type {5, q}. Experiments with other types gave an average time of more than five minutes per map. Out of the 284 pairs of type {3, q}, 230 give a chiral map and 39 a non-orientable map with full automorphism group O'N. Out of the 5176 pairs of type {4, q}, 4906 give a chiral map and 114 a non-orientable map with full automorphism group O'N. Out of the 7738 pairs of type {5, q}, 7340 give a chiral map and 188 a non-orientable map with 212 ArsMath. Contemp. 10(2016)211-222 full automorphism group O'N. The tendency of maps of chiral type being more prevalent seems confirmed by the partial results we obtained on maps of type {p, q} with q > p > 6. For all these maps, answering questions like "what are the exponents3 of M, is it self-dual, etc." is possible. 8 Acknowledgements Part of this research was done while the first author was visiting the second author at the University of Auckland. He therefore gratefully acknowledges the University of Auckland for its hospitality. He also acknowledges the Fonds pour la formation a la Recherche dans I'Industrie et I'Agriculture, (F.R.I.A.) and the Fonds National pour la Recherche Scientifique (F.N.R.S.) for financial support. The second author acknowledges financial support of the Royal Society of New Zealand Marsden Fund (Grant UOA1218). The authors also thank Marston Conder, Gareth Jones and Jozef Siran for interesting discussions while writing this paper. 3See [19] for a definition. 3 4 5 6 7 8 10 11 12 14 15 16 19 20 28 31 3 0 0 0 0 0 10 7 37 0 10 6 68 57 44 20 25 4 0 18 43 102 284 285 503 120 166 234 1292 846 554 370 359 5 26 98 150 470 365 718 211 290 340 1966 1242 800 560 502 6 165 354 1122 953 1776 474 597 874 4700 2943 1948 1370 1268 7 648 1848 1506 2687 815 1054 1284 7448 4725 2916 2214 1995 8 5424 4460 8096 2370 3056 3960 22040 13926 8880 6276 5752 10 3613 6532 1969 2526 3072 17822 11262 7224 4992 4632 11 11839 3583 4601 5814 32488 20493 13094 9202 8325 12 1072 1391 1710 9764 6126 3984 2796 2577 14 1796 2330 12504 7899 5024 3616 3266 15 2834 15808 10020 6424 4496 4062 16 88784 56052 35644 25316 23048 19 35442 22572 15978 14553 20 14238 10292 7246 28 7246 6658 31 5999 Table 1: Values of nG({p, q}) with G ^ O'N 212 ArsMath. Contemp. 10(2016)211-222 References [1] M. A. Al-Kadhi. On the generation of sporadic simple group O'N by (2, 3,t) generators. International J. Alg., 7 (2013), 167-176. [2] W. Bosma, J. Cannon, and C. Playoust. The Magma Algebra System. I. The User Language. J. Symbolic Comput. 24 (1997), 235-265. [3] H. R. Brahana. Regular maps and their groups. Amer. J. Math 49 (1927), 268-284. [4] M. Conder. Regular maps and hypermaps of Euler characteristic —1 to -200. J. Combin. Theory Ser. B 99 (2009), 455-459. [5] M. Conder and P. Dobcsanyi. Determination of all regular maps of small genus. J. Combin. Theory Ser. B 81 (2001), 224-242. [6] M. Conder, P. Potocnik, and J. Siran. Regular hypermaps over projective linear groups. J. Aust. Math. Soc. 85 (2008), 155-175. [7] T. Connor and D. Leemans. An atlas of subgroup lattices of finite almost simple groups. Ars Math. Contemp. 8 (2015), 259-266. [8] T. Connor, D. Leemans, and M. Mixer. Abstract regular polytopes for the O'Nan group. Int. J. Alg. Comput. 24 (2014), 59-68. [9] H. S. M. Coxeter and W. O. J. Moser. Generators and relations for discrete groups, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin, fourth edition, 1980. [10] M. R. Darafsheh, A. R. Ashrafi, and G. A. Moghani. (p, q, r)-generations of the sporadic group O'N. In Groups St. Andrews 2001 in Oxford. Vol. I, volume 304 of London Math. Soc. Lecture Note Ser., pages 101-109. Cambridge Univ. Press, Cambridge, 2003. [11] M. L. N. Downs and G. A. Jones. Enumerating regular objects with a given automorphism group. Discrete Math. 64 (1987), 299-302. [12] G. Frobenius. Ueber Gruppencharaktere. Berl. Ber. 1896, 985-1021. [13] P. Hall. The Eulerian functions of agroup. Q. J. Math., Oxf. Ser. 7 (1936), 134-151. [14] G. A. Jones. Ree groups and Riemann surfaces. J. Algebra 165 (1994), 41-62. [15] G. A. Jones and S. A. Silver. Suzuki groups and surfaces. J. London Math. Soc. (2) 48 (1993), 117-125. [16] G. A. Jones and D. Singerman. Theory of maps on orientable surfaces. Proc. London Math. Soc. (3) 37 (1978), 273-307. [17] D. Leemans and M. Mixer. Algorithms for classifying regular polytopes with a fixed automorphism group. Contr. Discrete Math. 7 (2012), 105-118. [18] V. D. Mazurov. On the generation of sporadic simple groups by three involutions, two of which commute. Sibirsk. Mat. Zh. 44 (2003), 193-198. [19] R. Nedela and M. Skoviera. Exponents of orientable maps. Proc.London Math. Soc. 75 (1997), 1-31. [20] C.-H. Sah. Groups related to compact Riemann surfaces. Acta Math. 123 (1969), 13-42. [21] A. V. Timofeenko. On generating triples of involutions of large sporadic groups. Diskret. Mat. 15 (2003), 103-112. [22] A. J. Woldar. On Hurwitz generation and genus actions of sporadic groups. Illinois J. Math. 33 (1989), 416-437. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 223-235 Isospectral genus two graphs are isomorphic Alexander Mednykh , Ilya Mednykh Sobolev Institute of Mathematics, 630090, Novosibirsk, Russia Novosibirsk State University, 630090, Novosibirsk, Russia Siberian Federal University, 660041, Krasnoyarsk, Russia Received 10 March 2013, accepted 24 July 2015, published online 3 October 2015 By a graph we mean a finite connected multigraph without bridges. The genus of a graph is the dimension of its homology group. Two graphs are isospectral is they share the same Laplacian spectrum. We prove that two genus two graphs are isospectral if and only if they are isomorphic. Also, we present two isospectral bridgeless genus three graphs that are not isomorphic. The paper is motivated by the following open problem posed by Peter Buser: are isospectral Riemann surfaces of genus two isometric? Keywords: Graph, Laplacian spectrum, isospectral graphs, Laplacian polynomial, spanning tree. Math. Subj. Class.: 05C50, 15A18, 58J53 1 Introduction Over the last decade, a few discrete versions of the theory of Riemann surfaces were created ([1, 18, 2, 8, 11]). In these theories, the role of Riemann surfaces is played by graphs. The genus of a graph is the dimension of its homology group. Under these assumptions, the theory of Jacobi manifolds is constructed and analogues of the Riemann-Hurwitz and Riemann-Roch theorems were proved. Counterparts of many other theorems from the classical theory of Riemann surfaces were derived in the discrete case ([9, 10, 16]). Since the classical paper by Mark Kac [14], the question of what geometric properties of a manifold are determined by its Laplace operator has inspired many intriguing results. One class of manifolds whose spectral theory has been studied with many beautiful results is the class of compact Riemann surfaces with the canonical constant curvature metric. Wolpert [19] showed that a generic Riemann surface is determined by its Laplace spectrum. Nevertheless, pairs of isospectral non-isometric Riemann surfaces in every genus > 4 are known. See papers by Buser [7], Brooks and Tse [5], and others. There are also examples of E-mail addresses: smedn@mail.ru (Alexander Mednykh), ilyamednykh@mail.ru (Ilya Mednykh) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 235 Ars Math. Contemp. 10 (2016) 183-192 isospectral non-isometric surfaces of genus two and three with variable curvature ([5, 3]). At the same time, isospectral genus one Riemann surfaces (flat tori) are isometric [4]. Similar results are also known for graphs ([12, 13]). Peter Buser [6] posed an interesting problem: are two isospectral Riemann surfaces of genus two isometric? Up to our knowledge the problem is still open but, quite likely, can be solved positively. The aim of this paper is to give a positive solution of an analogous problem for bridgeless graphs of genus two (Theorem 3.1). Also, we show that there are two isospectral bridgeless graphs of genus three that are not isomorphic (Figure 5). Because of the intrinsic link between Riemann surfaces and graphs we hope that our result will be helpful to make a progress in solution of the Buser problem. 2 Preliminary results 2.1 Laplacian matrix and Laplacian spectrum The Laplacian matrix of a graph and its eigenvalues can be used in several areas of mathematical research and have a physical interpretation in various physical and chemical theories. The related adjacency matrix of a graph and its eigenvalues were much more investigated in the past than the Laplacian matrix. At the same time, the Laplacian spectrum is much more natural and more important than the adjacency matrix spectrum because of it numerous application in mathematical physics, chemistry and financial mathematics. Graphs in this paper are finite and undirected, but they may have loops and multiple edges. Denote by V(G) and E(G), respectively, the number of vertices and edges of a graph G. Following [2] we denote by g(G) = E(G) — V(G) + 1 the genus of G. This is the dimension of the first homology group of G. In graph theory, the term "genus" is traditionally used for a different concept, namely, the smallest genus of any surface in which the graph can be embedded, and the integer g = g(G) is called the cyclomatic or the Betti number of G. We call g the genus of G in order to highlight the analogy with Riemann surfaces. A bridge is an edge of a graph G whose deletion increases the number of connected components. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. A graph is said to be bridgeless if it contains no bridges. Let G be a graph. Denote by V(G) and E(G) the set of vertices and edges of a graph G respectively. For each u, v e V(G), we set auv to be equal to the number of edges between u and v. The matrix A = A(G) = [auv]u,vey(G), is called the adjacency matrix of the graph G. Let d(v) denote the valency of v e V(G), d(v) = J2u auv, and let D = D(G) be the diagonal matrix indexed by V(G) and with dvv = d(v). The matrix L = L(G) = D(G) — A(G) is called the Laplacian matrix of G. It should be noted that loops have no influence on L(G). Throughout the paper we shall denote by ^(G, x) the characteristic polynomial of L(G). For brevity, we will call ^(G, x) the Laplacian polynomial of G. Its roots will be called the Laplacian eigenvalues (or sometimes just eigenvalues) of G. They will be denoted by ^i(G) < ^2(G) < ... < ^n(G), (n = V(G)), always enumerated in increasing order and repeated according to their multiplicity. Recall [17] that for connected graph G we always have ^1(G) = 0 and ^2(G) > 0. Two graphs G and H are called Laplacian isospectral (or isospectral) if their Laplacian polynomials coincide: ^(G, x) = ^(H, x). The matrix L(G) is sometimes called the Kirchhoff matrix of G due to its role in the A. Mednykh and I. Mednykh: Isospectral genus two graphs are isomorphic 225 well-known Matrix-Tree Theorem which is usually attributed to Kirchhoff. A generalization of the Matrix-Tree-Theorem was obtained in 1967 by A. K. Kel'mans who gave a combinatorial interpretation to all the coefficients of ^(X, x) in terms of the numbers of certain subforests of a graph X; see [15] and [17] for references and history of question. We present the result by Kel'mans in the following form. Theorem2.1. [15]If^(X,x) = xn-cix"-1 +... + (-1)icixn-i +... + (-1)n-1cn-1x then ci = £ T (Xs), SCV, |S|=n-i where T (H) is the number of spanning trees of H, and XS is obtained from X by identifying all points of S to a single point. 2.2 Theta graphs Let u and v are two (not necessary distinct) vertices. Denote by ©(k, l, m) the graph consisting of three internally disjoint paths joining u to v with lengths k, l, m > 0 (see Fig. 1). We set <7i = 0, then ©(k, l, m) is a graph of genus two. In this case at least two of numbers {k, l, m} are positive. (ii) If 01 > 0,02 = 0, then ©(k, l, m) is a graph of genus one. Then exactly one of numbers {k, l, m} is positive and the other two are zero. Moreover, ©(k, l, m) = Ck+i+m is a cyclic graph with k + l + m edges. 226 Ars Math. Contemp. 10 (2016) 183-192 (iii) If <7i = 0, then k = 1 = m = 0 and ©(k, 1, m) is a graph of genus zero. More precisely, ©(k, 1, m) = ©(0,0,0) consists of one vertex. Lemma 2.2. Let G be an arbitrary bridgeless graph of genus two. Then G is isomorphic to ©(k, 1, m) for some k, 1, m with ct2 = k 1 + 1m + km> 0. Proof. Since the graph G is bridgeless it has no vertices of valency one. Denote by H the graph obtained from G by deleting of all vertices of valency two. Suppose that H has V vertices of valences ni, n2,..., nv and E edges. Since the valency of each vertex of H is at least three we have n > 3, i = 1,2,..., V. Note that deleting of a vertex of valency two decreases the number of vertices and the number of edges of a graph by one. So, it does not affect the genus and H is still a graph of genus two. Thus g(H) = 1 - V + E = 2 and E = V + 1. Counting the sum of valences of H through vertices and through edges we obtain ni + n2 + ... + nv = 2E. Hence 3V < n1 + n2 + ... + nv = 2E = 2V + 2, or V < 2. If V =1 then n1 =4 and H is the figure eight graph consisting of one vertex and two loops. Putting back the vertices of valency two on the graph H we obtain the graph G isomorphic to ©(k, 1,0) for some positive k and 1. In particular, a2 = k 1 > 0. If V = 2 then n1 = n2 =3 and H is the theta graph consisting of two vertices and three edges. The graph G is obtained from H by adding the vertices of valency two. Hence, G is isomorphic to ©(k, 1, m) for some positive k, 1, m. □ 3 Main results 3.1 The main theorem and lemmas The main result of the paper is the following theorem. Theorem 3.1. Two genus two bridgeless graphs are Laplacian isospectral if and only if they are isomorphic. The proof of the theorem is based on the following three lemmas. Lemma 3.2. Let G = ©(k, 1, m) be a theta graph and let ^(G, x) = xn — c1xn-i + ... + ( — 1)n-1cn-1x be its Laplacian polynomial. Then n = k + 1 + m — 1, c1 = 2(k + 1 + m) and cn-1 = (k 1 + 1m + k m)(k + 1 + m — 1). Proof. The number of vertices, edges and spanning trees of graph G are given by V(G) = k + 1 + m — 1, E(G) = k + 1 + m, T(G) = k 1 + 1m + km. Then by ([15], formulas 2.15 and 2.16) we have n = V(G) = k + 1 + m — 1, c1 = 2E(G) = 2(k +1 + m) and cn-1 = V(G) • T(G) = (k1 + 1m + km)(k +1 + m — 1). □ A. Mednykh and I. Mednykh: Isospectral genus two graphs are isomorphic 227 Lemma 3.3. Let G = Q(k, l, m) be a theta graph and let G, x) = xn — c1xn 1 + ... + ( —1)n-1cn-1x be its Laplacianpolynomial. Then Cn-2 = A((J1, (J2) + B(ai, 02)03 where A(s,t) = (4t — 3st — 2s2t + s3t + 4t2 — st2)/12, B(s,t) = (3 — 4s + s2 — 3t)/12, 01 = k + l + m, 02 = kl + l m + km, and (J3 = klm. Proof. By Theorem 2.1 Cn-2 = E T (Xs ), (3.1) SCV, |S|=2 where XS runs through all graphs obtained from G = Q(k, l, m) by gluing two vertices. There are exactly four types of such graphs G1,G2,G3, and G4 shown in the Fig. 2. We will enumerate the spanning trees of each type separately. Type G1. Glue two 3-valent vertices of graph G. As a result we obtain the graph G1 shown on Fig. 2. The number of spanning trees of this graph is T1 = T(Ck) ■ T(Ci) ■ T (Cm) = klm. Type G2. Glue one 3-valent and one 2-valent vertices of graph G. The graph of type G2 shown in Fig. 2 is obtained by gluing the upper 3-valent of graph G and a 2-valent vertex on the path of G labelled by k. For given i, 1 < i < k — 1 the number of spanning trees for a graph of type G2 is equal to T (Ci) ■ T (Q(k — i,l,m)) = ia2(k — i, l, m). We k-1 set F(k, l, m) = J2 ia2(k — i, l, m). Then the total number of spanning trees for graphs i=1 of type G2 is T2 = 2(F(k, l, m) + F(l, m, k) + F(m, k, l)). The multiple 2 is needed since the graph Q(k, l, m) has two 3-valent vertices. Type G3. Glue two 2-valent vertices of graph G lying on different paths. We choose one of them on the path labelled by k and the second on the path labbeled by l. Fix i, 1 < i < k — 1 and j, 1 < j < l — 1 and consider a graph of type G3 shown in Fig. 2. This is a graph of genus three. To create a spanning tree on this graph we have to delete three edges. There are two different ways to do this. Firstly, we delete edges on three of the four paths labeled by i, j,k — i and l — j. This be done in a3 (i, j,k — i,l — j) ways, where a3(x, y, z, t) = xyz + xyt+xzt+yzt. Secondly, if we delete an edge from the path labeled by m (in m possible ways) then we have to remove one edge from the pair of paths i, j and one edge from the pair k — i,k — j. Then we have m((i + j)(k — i +1 — j)) possibilities to obtain a tree. As the result graph under consideration has G3(i, j, k, l, m) = a3(i, j, k — i, l — j) + m((i + j)(k — i + l — j) spanning trees. We set k-1l-1 J(k, l, m) ^^ G3(i, j, k, l, m). i= 1 j=1 Then the total number of spanning trees for graphs of type G3 is T3 = J(k, l, m) + J(l, m, k) + J(m, k, l). Type G4. Glue two 2-valent vertices lying on the same path of graph G. Choose the path labelled by k. Let us fix i and j such that 1 < i < j < k — 1. Then the number 228 Ars Math. Contemp. 10 (2016) 183-192 of spanning trees for a given graph of type G4 is T(Cj-i)T(0(k + i - j,l,m)) = (j -i)a2(k + i - j,l,m). We set k-2 k-1 H(k, l, m) ^^ ^^ (j - i)&2 (k + i - j, l, m). i=1 j=i+1 As a result, the number of spanning trees of the given type is T4 = H(k, l, m) + H(l, m, k) + H(m, k, l). Putting the obtained formulas in Mathematica 8 by (3.1) we get Cn-2 = Ti + T2 + T3 + T4 = A(ai, <72) + B(ai, 0-2)0-3. k-i I m Figure 2: The graphs obtained from 0(k, l, m) by gluing two vertices Lemma 3.4. Let G = Q(k, l, m) be a theta graph and let p(G, x)= xn - c1xn-1 + ... + (-l)n-1cn-1x be its Laplacian polynomial. Then Cn-3 = C (01,02) + D(<1,02)03 + E (01,02)03, where C(s,t) = (-34t + 21st + 25s2t - 10s3t - 3s4t + s5t - 50t2 + 10st2 + 12s2t2 - 2s3t2 - 16t3 + st3)/360, D(s,t) = (-45 + 50s + 5s2 - 12s3 + 2s4 + 24st - 9s2t + 15t2)/360, E(s,t) = -3(-8 + 3s)/360. A. Mednykh and I. Mednykh: Isospectral genus two graphs are isomorphic 229 Proof. By Theorem 2.1 cn-3 = E T (Xs ), (3.2) SCV, |S|=3 where XS runs through all graphs obtained from G = Q(k, l, m) by gluing three vertices. There are six types of such graphs Wi,W2, W3, W4, W5, and W6 shown on the Fig. 3. We examine the spanning trees of each type separately. Type Wi. To create a graph of type Wi we identify two 3-valent vertices of graph G and one 2-valent vertex of G (say on the path labelled by k). The obtained graph is shown in the k-i Fig. 3, has i(k — i)l m spanning trees. Consider the sum Fw (k, l, m) = J2 i(k — i)l m. i= 1 Find the total number of spanning trees for graphs of type Wi by the formula Tw = fw(k, l, m) + Fw (l, m, k) + Fw (m, k, l). Type W2. Glue one 3-valent vertices of graph G and two 2-valent vertices lying on different paths of G (say on the paths labelled by k an l), obtaining a graph in Fig. 3. For given i and j, 1 < i < k — 1, 1 < j < l — 1, the number of spanning trees for graph of k-ii-i type W2 is ija2(k — i,l — j, m). We set Hw(k, l, m) = J2 J2 ija2(k — i,l — j, m). i=ij=i Taking into account that graph Q(k,l, m) has two 3-valent vertices we obtain the following formula the number of spanning trees for graphs of type W2 : T2w = 2(Hw(k, l, m) + Hw(l, m, k) + Hw(m, k, l)). Type W3. Glue one 3-valent vertices and two 2-valent vertices lying on the same path of G. For fixed i and j, 1 < i < j < k — 1, we have i(j — i)a2(k — j, l, m) spanning trees for graph of type W3. Summing over i and j we get k-2 k-i Jw(k, l,m) = E E i(j — i)a2(k — j, l, m). i=i j=i+i Finally, the number of spanning trees for graphs of type W3 is given by T3? = 2(Jw(k, l, m) + Jw(l, m, k) + Jw(m, k, l)). Type W4. Glue three 2-valent vertices all lying on different paths of G. Fix i, j and s, 1 < i < k — 1, 1 < j < l — 1, 1 < s < m — 1. Then the number of spanning trees for a given graph of type W4 is equal to a2 (i, j, s)a2(k — i,l — j,m — s). Summing over i, j and s we obtain the total number of spanning trees for graphs of type W4 : k-i i-i m-i T4 = J2J2 J2 a2(i,j, s)°2(k — iJ — j,m — s). i=i j=i s=i Type W5. Glue two 2-valent vertices lying on a path and one 2-valent vertex lying on the other path of G. Denote by G3(i,j,k,l,m) the graph of type G3 shown in Fig. 2. From the proof of previous Lemma we have T (G3(i, j, k, l, m)) = a3(i, j, k — i,l — j) + m((i + 230 Ars Math. Contemp. 10 (2016) 183-192 j)(k — i + l — j)). Fix i, j and s, 1 < i < j < k — 1, 1 < s < l — 1. Then the number of spanning trees for a graph of type W5 in Fig. 3 is equal to T(Cj-i)T(G3(i, s,k + i — j, l, m)) = (j — i)T(G3(i, s,k + i — j, l, m)). Consider the sum k-2 k-1 l-l Kw(k,l, m) = ££ Y,(j — i)T(Gs(i,s,k + i — j, l, m)). i=l j=i+l s = l Then the number of spanning trees for graphs of type W3 is given by Tw = kw (k, l, m) + Kw (l, m, k) + Kw (m, k, l) + Kw(k, m, l) + Kw(l, k, m) + Kw(m, l, k). (3.3) Type W6. Glue three 2-valent vertices on the same path of G. Fixed i,j and s such that 1 < s < i < j < k — 1. Then the number of spanning trees for a given graph of type W6 is equal to T(Ci-a)T(Cj-i)T(G(k — j + s, l, m)) = (i — s)(j — i)a2(k — j + s, l, m). Summing over i, j and s we obtain Lw (k,l,m)= E (i — s)(j — i)&2(k — j + s,l,m). l 1 and o' > 1. Then the system of equations (3.4) gives o1 = o' and o2 = o2. The theorem will be proved if we show that 03 = 03. We will do this in two steps. First of all, we note that by [13] isospectral graphs with n < 5 vertices are isomorphic. So, we can assume that n = k + l + m - 1 > 5, that is, 01 = k + l + m > 6. By Lemma 3, Cn-2 = A(oi,02)+ B(oi, 02)03, (3.5) where A(s,t) = (4t - 3st - 2s2t + s3t + 4t2 - st2)/12 and B(s,t) = (3 - 4s + s2 - 3t)/12. Step 1. B(oi,02) = 0. Since c^-2 = cn-2,oi = oi ando2 = o2 from (3.5) we obtain B(oi, 02)03 = B(oi, 02)03. (3.6) Hence 03 = 03' and the theorem is proved. Step 2. B(0i, 02) = 0. Then by Lemma 3 cn-3 = C (0i,02) + D(0i ,02)03 + E (0i, 02)03, (3.7) where C(s, t) = (-34t + 21st + 25s2t - 10s3t - 3s4t + s5t - 50t2 + 10st2 + 12s2t2 - 2s3t2 - 16t3 + st3)/360, D(s,t) = (-45 +50s + 5s2 - 12s3 + 2s4 + 24st - 9s2t + 15t2)/360, E (s,t) = -3(-8 + 3s)/360. 232 Ars Math. Contemp. 10 (2016) 183-192 Since = cn-3, a1 = aj and a2 = a2 from (3-7) we obtain D(CTI,CT2)^3 + E(ai, a2)a?2 = D(aj, a2)a? + E(ai,a2)a|. (3-8) We note that E(a1,a2) = 0 for any integer a1. Then the above equation has two solutions with respect to a?. The first solution is a? = a3 and the second one is , D(a1,a2) a (3 9) a3 = -V - a3. (3-9) E(aj,a2) In the first case the theorem is proved- So we assume that a' is given by equation (3.9). Recall that B(a1,a2) = 0. Then a2 = (3-4a1 + a2)/3 and equation (3.9) can be rewritten in the form a3 = 7lg(2(425 " 357a1 - 144a2 + 27a?) - ) " as. (3-10) Since as and a? are integers the number N = 2(425 - 357a 1 - 144a2 + 27a?) - 490 -8 + 3a1 is an integer divisible by 729. Moreover, -8 + 3a1 is a divisor of 490 and the number a2 = (3 - 4a1 + a2)/3 is a positive integer. There are a finite number possibilities of a positive integer a1 to satisfy these three conditions, namely, a1 G {6,19,166}. The case a1 = 6 can be excluded since we suggested that a1 > 6. Another way to exclude a1 = 6 is to check that in this case a? = -3 - a? is negative. Consider the remaining cases a1 = 19 and a1 = 166. By (3.10) in these cases we have a? = 348 - a? and a? = 327789 - a? respectively. The respective values of a2 are 96 and 8965. Let a1 = 19. We have the following system of equations to find positive integer parameters k, l, m, a? of the graph G = ©(k, l, m) : k + l + m =19, kl + Im + mk = 96, klm = a?. This system has only one solution {k, l, m} = {3, 4, 12}, a3 = 144. Now we are able to find parameters k', l', m', a' of the graph G' = ©(k', l', m'). First of all, a' = 348 - a? = 204. Then we have k' + l' + m' = 19, k'l' + l'm' + m'k = 96, k'l'm' = 204. The latter system has no integer solutions. So the case a1 = 19 is impossible. Let a1 = 166. We have the following system k, l, m, a?. k + l + m = 166, k l + l m + m k = 8965, klm = a?. This system has only one solution {k, l, m} = {39,59,68}, a? = 39 • 59 • 68. Find parameters k', l', m', a' of the graph G' = ©(k', l', m'). Now, a' = 327789 -a? = 171321. Then we have k' + l' + m' = 166, k'l' + l'm' + m'k' = 8965, k'l'm' = 171321. The system has no integer solutions. The case a1 = 166 is also impossible. This completes the proof. □ A. Mednykh and I. Mednykh: Isospectral genus two graphs are isomorphic 233 4 Final remarks 1. The main Theorem 3.1 is not valid for genus two graphs with bridges. Indeed, the following two graphs (see Fig. 4) constructed in [12] are isospectral. They share the Laplacian polynomial —72x + 192x2 - 176x3 + 73x4 - 14x5 + x6. The first of these graphs is bridgeless, while the second one has a bridge. 2. There are isospectral bridgeless graphs of genus three which are not isomorphic (see Fig. 5). These two graphs were constructed in [13].They share the Laplacian polynomial —384x + 1520x2 - 2288x3 + 1715x4 - 708x5 + 164x6 - 20x7 + x8. 3. Any bridgeless graph of genus one is isomorphic to a cyclic graph Cn for some n > 1. If two cyclic graphs Cm and Cn are isospectral then their Laplace polynomials are of the same degree m = n. Hence, the graphs are isomorphic. 234 Ars Math. Contemp. 10 (2016) 183-192 At the same time, there are isospectral unicycle graphs [20]. For example, the two genus one graphs shown on Fig. 6 share the Laplacian polynomial 28x - 146x2 + 250x3 - 194x4 + 75x5 - 14x6 + x7. 4. One can hear the genus of a graph. That is, the genus of a graph G is completely determined by its Laplace spectrum. Indeed, g(G) = 1 - V(G) + E(G). Let p(G,x) = xn-c1xn-1 + ...+(- 1)n-1cn-1x be the Laplacian polynomial of G. By the arguments from the proof of Lemma 3.2 we have n = V(G) and c1 = 2E(G). Thus V(G) and E(G), as well as the genus, are uniquely determined by the Lapla-cian polynomial. It follows from this observation, the previous remark, and the main result of the paper that the bridgeless graphs of genera one and two are recognisable by their Laplacian spectra among all bridgeless graphs. 5. One cannot hear a bridge of a graph. Indeed, the two graphs in Fig. 4 are isospectral. We are not able to recognise the existence of a bridge of the second graph by its spectrum. 5 Acknowledgments The authors are thankful to the following grants for partial support of this investigation: the Russian Foundation for Basic Research (project no. 15-01-07906), the Grant of the Russian Federation Government at Siberian Federal University (grant 14.Y26.31.0006), the Program "Leading Scientific Schools" (project no. NSh-921.2012.1), the Dynasty Foundation and the Project: Mobility - enhancing research, science and education at the Matej Bel University, ITMS code: 26110230082, under the Operational Program Education cofi-nanced by the European Social Fund and by Slovenian-Russian grant (2014-2015). A. Mednykh and I. Mednykh: Isospectral genus two graphs are isomorphic 235 References [1] R. Bacher, P. de la Harpe, and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France, 125 (1997), 167-198. [2] M. Baker, S. 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ARS MATHEMATICA CONTEMPORANEA 10 (2016) 237-254 Combinatorial categories and permutation groups Gareth A. Jones * School of Mathematics, University of Southampton, Southampton SO171BJ, U.K. Received 24 September 2013, accepted 22 July 2015, published online 20 October 2015 The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group r, with automorphism group isomorphic to r/N. It is shown how to enumerate such objects with a given finite automorphism group G, how to represent them all as quotients of a single regular object U(G), and how the outer automorphism group of r acts on them. Examples constructed include kaleidoscopic maps with trinity symmetry. Keywords: Regular map, regular hypermap, covering space, permutation group, category. Math. Subj. Class.: 20B25, 05A15, 05C10, 05E18, 20J99, 57M10 1 Introduction In certain categories C, the objects O can be identified with the permutation representations of a particular group r = rC on sets $ = , and the morphisms O ^ O' correspond to the functions ^ $o> commuting with the actions of r. In the case of maps on surfaces one takes r to be the free product V4 * C2 acting on flags, or * C2 acting on directed edges of oriented maps. The corresponding groups for hypermaps are C2 * C2 * C2 and the free group F2 = * of rank 2. For abstract polytopes of a given type one can use the corresponding string Coxeter group, again acting on flags, though here one has to restrict attention to quotient groups satisfying the intersection property. In the case of coverings of a path-connected space X one uses the fundamental group n^X, acting on sheets or more precisely on the fibre over a base-point. *The author thanks the organisers of GEMS 2013 for inviting and supporting him to give a talk summarising this paper. It is based upon work supported by the project: Mobility — enhancing research, science and education at the Matej Bel University, ITMS code: 26110230082, under the Operational Program Education cofinanced by the European Social Fund. E-mail address: G.A.Jones@maths.soton.ac.uk (Gareth A. Jones) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 238 Ars Math. Contemp. 10 (2016) 183-192 In such a case we will call C a permutational category, with parent group r. Each object O in such a category C is a disjoint union of connected subobjects, corresponding to the orbits of r on $; one usually restricts attention to the connected objects, as we shall here, so that $ can be identified with the set of cosets in r of a point-stabiliser M = T^, where 0 € $. The permutation group induced by G on $ is the monodromy group G = Mon O = MonCO of O, a subgroup of the symmetric group Sym $ on $. The automorphism group A = Aut O = AutCO of O, regarded as an object in C, is the centraliser of G in Sym $; since G is transitive on $, A acts semiregularly on $, and A = Nr(M )/M = Ng(G0 )/G0. The most symmetric objects in C are the regular objects, those for which A acts transitively (and hence regularly) on $. This is equivalent to M being a normal subgroup of r, in which case A = r/M = G. Indeed, in this case A and G can be identified with the left and right regular representation of the same group. In principle, understanding regular objects is sufficient for an understanding of all objects in C, since each object O € C is the quotient of some regular object O € C, corresponding to the core N of M in r, by a group M/N of automorphisms of O; moreover, O is finite if and only if O is finite, since N has finite index in r if and only if M has finite index. We shall therefore concentrate, for the remainder of this paper, on the regular objects in various categories C. In particular, we will study the set R(G) = RC(G) of regular objects O € C with Aut O isomorphic to a given group G. If r is finitely generated and G is finite then r(G) := |R(G)| is finite. We will consider how to calculate r(G) in this case, how to represent the objects in R(G) as quotients of a single regular object U(G) in C, and how the outer automorphism group Out r of r acts on R(G). Examples will be given, in which the objects are maps, hypermaps or surface coverings, some of them relevant to recent work by Archdeacon, Conder and Siran on kaleidoscopic maps with trinity symmetry [1]. 2 Examples of permutational categories Let us call a category C a permutational category if it is equivalent to the category of permutation representations of some group r, called the parent group of C. This means that there are functors from each category to the other, so that their composition, in either order, is naturally equivalent to the identity. There are some well-known examples in the literature, though the equivalences are rarely expressed in terms of categories. We will summarise them briefly here; for further details, see, for example, [33] for maps, and [26] for hypermaps. The category M of maps on surfaces, with branched coverings of maps as its mor-phisms, is a permutational category, with parent group r = = (Rq| R? = (R0R2)2 = 1). (2.1) Here each R acts on the set $ of vertex-edge-face flags of a map O € M by changing, in the only way possible, the ¿-dimensional component of each flag while preserving its j-dimensional component for each j = ¿. (A boundary flag is fixed by R if no such change is possible.) This group, which is a free product (rq,r?)*(ri) = v4 * c? G. A. Jones: Combinatorial categories and permutation groups 239 of a Klein four-group and a cyclic group of order 2, can be regarded as the extended triangle group A[to, 2, to] of type (to, 2, to), generated by reflections in the sides of a hyperbolic triangle with angles 0, n/2,0. This gives a functor from maps to permutation representations of r. Conversely, given a permutation representation of r on a set $, one can take a set of triangles in bijective correspondence with $, each with vertices labelled 0,1,2, and use the cycles of Rj on $ to join pairs of triangles across edges jk (j, k = i); the result is the barycentric subdivision of a map O g M, with the vertices of O labelled 0 and its edges formed by edges of triangles labelled 01, so that midpoints of edges and faces of O are labelled 1 and 2. Branched coverings between maps O correspond to T-equivariant functions between sets $, so we obtain functors O ^ $ and $ ^ O which give the required equivalence of categories. Other triangle groups act as parent groups for related categories. For the category Mk of maps with all vertex-valencies dividing k we add the relation (RiR2)fc = 1 to the presentation (2.1), giving the parent group rOTfc = (Rc,Ri,R2 | R? = (R0R2)2 = (RiR?)k = 1) = A[k,2,to]. (2.2) Similarly, the isomorphic group A[to, 2, k] is the parent group for the dual maps, with all face-valencies dividing k. For the category H of hypermaps, where hyperedges may be incident with any number of hypervertices and hyperfaces, we delete the relation (R0R2 )2 = 1 from (2.1), giving the parent group rH = (R0, R1, R? | R? = 1) = A[to, to, to] = C? * C? * C? (2.3) again permuting flags. Similarly, the extended triangle group A[/, m, n] = (Ro, Ri, R? | R? = (RiR?)1 = (RoR?)m = (RoRi)n = 1) is the parent group for hypermaps of type dividing (/, m, n), that is, of type (/', m', n') where /', m' and n' divide /, m and n. For the corresponding categories M+, M+ and H+ of oriented maps and hypermaps we take the orientation-preserving subgroups of index 2 in these groups, generated by the elements X = RiR0, Y = R0R? and Z = R?Ri satisfying XYZ = 1. These are the triangle groups rM+ = (X, Y, Z | Y? = XYZ =1) = A(to, 2, to) = * C?, (2.4) rM+ = (X, Y, Z | Xk = Y? = XYZ =1) = A(k, 2, to) = Ck * C? (2.5) and rH+ = (X, Y, Z | XYZ =1) = A(to, to, to) = * = F?. (2.6) Similarly, the triangle group A(/, m, n) = (X, Y, Z | X1 = Ym = Zn = XYZ = 1) is the parent group for oriented hypermaps of type dividing (/, m, n). In the case of oriented hypermaps, the Walsh map [51] represents a hypermap as a bipartite map, with black and white vertices representing hypervertices and hyperedges, and edges representing their incidence; then X and Y permute the set $ of edges by following 240 Ars Math. Contemp. 10 (2016) 183-192 the local orientation around their incident black and white vertices. For oriented maps, X rotates directed edges around their target vertices, while Y reverses them; equivalently, one can convert a map into the Walsh map of a hypermap by adding a white vertex at the centre of each edge, so that new edges correspond to directed edges of the original map. If X is a path connected, locally path connected, and semilocally simply connected topological space [43, Ch. 13]), the unbranched coverings fi : Y ^ X of X form a permutational category C with the fundamental group r = niX as parent group, using unique path-lifting to permute the fibre $ = fi-1 (x0) C Y of fi over a chosen basepoint x0 € X. The regular coverings fi correspond to the normal subgroups N of r, with covering group Aut fi = r/N. If X is also a compact Hausdorff space (for instance, a compact manifold or orbifold), then r is finitely generated [43, p. 500]. The categories of maps and hypermaps described above can be regarded as obtained in the above way from suitable orbifolds X, such as a triangle with angles n/l, n/m, n/n for hypermaps of type dividing (l, m, n), or a sphere with three cone-points of orders l, m, n in the oriented case. Similarly, Grothendieck's dessins d'enfants [21, 22] are the finite coverings of a sphere minus three points, so their parent group is its fundamental group r = F2, with generators X, Y and Z inducing the monodromy permutations at the three punctures. For the rest of this paper, C will denote a permutations category with a finitely generated parent group r. 3 Counting regular objects For each group G, there is a natural bijection between the set R(G) = RC(G) of (isomorphism classes of) regular objects O € C with Aut O = G and the set N(G) = Nr(G) of normal subgroups N of r with r/N = G. These normal subgroups are the kernels of the epimorphisms r ^ G. Two such epimorphisms have the same kernel if and only if they differ by an automorphism of G, so there is a bijection between N(G) and the set of orbits of Aut G, acting by composition on the set Epi(r, G) of epimorphisms r ^ G. This action of Aut G is semiregular, since only the identity automorphism of G fixes an epimorphism. If G is finite then so is Epi(r, G), since each epimorphism r ^ G is uniquely determined by the images in G of a finite set of generators of r. In this case the sets R(G) and N(G) have the same finite cardinality r(G) = rc(G) = |R(G)| = n(G) = nr(G) = |N(G)| = ^q^ . (3.1) In [24], Hall developed a method for counting epimorphisms onto G by first counting homomorphisms (generally an easier task) to subgroups of G, and then using Mobius inversion in the lattice A(G) of subgroups of G. Let a and ^ be functions from isomorphism classes of finite groups to C such that a(G) = £ ¿(H) (3.2) H H with 6H,a = 1 if H = G and 0 otherwise. (One can view this as a group-theoretic analogue of the inclusion-exclusion principle, which applies to the lattice of all subsets of G; in that situation, by replacing the condition K > H in (3.4) with K D H one assigns the value (-1)|a\H| to ya(H) for each subset H of G.) Each homomorphism r ^ G is an epimorphism onto a unique subgroup H < G, so one can take 5. Equation (3.3) takes the form ¿(G) = a(G) — (p + 1)a(GTO) — a(Dp+i) — p(p + 1) dp-i + p(p +1)CT(Cp_i) + |G|S, where is the subgroup of index p +1 fixing to, and S depends on the congruence classes of p mod (5) and mod (8), which determine the existence of proper subgroups H = or S4. For example, if p = 5, or if p = ±2 mod (5) andp = ±3 mod (8), so that there are no such subgroups, then S = —12 ) + 4 ct(v4) + 1 a(Cs) + 1 a(C2) — a(1); G. A. Jones: Combinatorial categories and permutation groups 243 there are similar formulae in the other cases. In [10] Downs extended Hall's calculation of to L2(q) and PGL2(q) for all prime powers q; see [11] for a proof for L2(2e) and a statement of results for L2 (q) where q is odd, and [13] for some combinatorial applications by Downs and the author. Example 4.7 The Suzuki groups G = Sz(q) are a family of non-abelian finite simple groups, with q = 2e for some odd e > 1; see [5, 50, 52] for their properties, which are similar to those of the groups L2(2e). Downs calculated for these groups in [12]; see [14] for a statement of the results and some applications. 5 Counting homomorphisms In order to apply equation (3.7) to a group G, one needs to evaluate |Hom(r, H) | for those subgroups H < G with pa(H) = 0. If r has a presentation with generators X; and defining relations Rj, this is equivalent to counting the solutions (x;) in H of the equations Rj(xj) = 1. Example 5.1 If r is a free product Cmi *• • *Cmfc of cyclic groups of orders m; G NU{to}, then k |Hom(r, H )| = HE |H |m i=1 m| mi where |H |m denotes the number of elements of H of order m, and we regard all orders m as dividing to, so that J2m|TO |H|m = |H|. For instance, if r is a free group Fk of rank k then |Hom(r, H)| = |H|k. Similarly, the torsion theorem for free products [36, Theorem IV.1.6] implies that a homomorphism r ^ H is smooth if and only if it embeds each finite factor Cmi, so the number of such homomorphisms can be found by multiplying k factors equal to |H |mi or |H | as m; is finite or infinite. For certain groups r, the character table of H gives |Hom(r, H) |. Example 5.2 If r is a polygonal group A(mi,...,mk) = {Xl,...,Xk | X^"1 = ... = Xfcmfc = Xi ...Xk = 1) of type (m1,..., mk) for some integers m;, then |Hom(r, H)| can be found by summing the following formula (5.1) of Frobenius [18] over all choices of k-tuples of conjugacy classes of elements of orders dividing m;. Theorem 5.1. Let (i = 1,..., k) be conjugacy classes in a finite group H. Then the number of solutions of the equation x1... xk = 1 in H, with x; G for i = 1,..., k, is lCi| ... X(xi) . .. X(xk) (51) |H| ^ x(1)k-2 ) where x; G C; and x ranges over the irreducible complex characters of H. Similarly, the number of smooth homomorphisms r ^ H can be found by restricting the summation to classes of elements of order equal to m;. The case k = 3 of this theorem, where r is a triangle group, has often been used in connection with oriented maps and hypermaps: see [27] and [32], for instance. 244 Ars Math. Contemp. 10 (2016) 183-192 Example 5.3 If r is an orientable surface group ng, that is, the fundamental group g ns = nlSg = (Ai, Bi (i = 1,..., g) | fl[Ai, B] = 1) i= 1 of a compact orientable surface Sg of genus g > 1, one can use the following theorem of Frobenius [18] andMednykh [42], which counts solutions of the equation f]g=1[ai, bi] = 1: i=1 Theorem 5.2. If H is any finite group then |Hom(ng, H)| = |H|2g-1 £x(1)2-2g, (5.2) x where x ranges over the irreducible complex characters of H. Example 5.4 If r is a non-orientable surface group g n- = (Ai (i = 1,..., g) ^ A2 = 1) A2 ¿=i of genus g > 1, one can use the following result of Frobenius and Schur [19]: Theorem 5.3. If H is a finite group then |Hom(n- H)| = |H|g-1 Y 4x(1)2-s, (5.3) x where x ranges over the irreducible complex characters of H. Here cx is the Frobenius-Schur indicator |H|-1 J2heH x(h2) of x, equal to 1, -1 or 0 as x is the character of a real representation, the real character of a non-real representation, or a non-real character. See [28] for applications of these two theorems, and [48, Ch. 7] for several generalisations of them. 6 Enumerations Using Theorem 3.1 one can now enumerate, for a given finite group G, the regular objects in C with automorphism group G, and also the objects in C with monodromy group G. Example 6.1 It follows from a result of Hall [24] that if G = L2 (p) for some prime p > 5 and C = H+, so that r = F2, then r(G) = 1(p +1)(p2 - 2p - 1) - e, where e = 49, 40, 11 or 2 as p = ±1 mod (5) and ±1 mod (8), or ±1 mod (5) and ±3 mod (8), or ±2 mod (5) and ±1 mod (8), or ±2 mod (5) and ±3 mod (8). We also take e = 2 when p = 5, so that r(G) = 19 in this case; the 19 regular oriented hypermaps associated with the icosahedral group G = L2 (5) = A5 have been described by Breda and the author in [2]. Since this group G has eight conjugacy classes of proper subgroups, all with trivial core since G is simple, it follows from equation (3.8) that there are 19 x 8 = 152 oriented G. A. Jones: Combinatorial categories and permutation groups 245 hypermaps with monodromy group G, namely the quotients O/H where O G R(G) and H < G. Example 6.2 In [10], Downs considered the categories H, H+, M, M+, M3 and M+, and gave formulae for r(G) where G = L2 (q) or PGL2 (q) for any prime power q. The results for G = L2(2e) are given in [13]. Typical results for odd q are: rm(L2(pe)) = E M ( eW(Pf - a) f |e VJ/ for all p > 2 and odd e > 1, where a = 2 or 4 as p = 1 or -1 mod (4), and rOT3(PGL2(pe)) = 4e EM f (Pf - 1) for p > 3 and e > 1, where the sum is over all factors f of e with e/f odd. Example 6.3 Using Downs's calculation of the Mobius function for G = Sz(2e) in [12], he and the author have enumerated various combinatorial objects with automorphism group G in [14]. Typical results are that r«+ (G) = 1 E M (7) 2f (24f - 23f - 9) and rM(G) = 1 E^7) (2f - 1)(2f - 2). 7 f|e The second formula, which also gives the number of reflexible maps in (G), has been obtained by more direct means by Hubard and Leemans in [25]. Example 6.4 If G is infinite then R(G) could be finite or infinite. For instance, if C = H+, so that r = F2, then r(Z2) = 1 whereas r(Z) = H0. 7 Universal covers For any group G, and any C, let K(G) = kc(g)= H N. (7.1) N eN (G) This is a normal subgroup of r, so it corresponds to a regular object U(G) = WC(G)= V O (7.2) OeR(G) which we will call the universal cover for G, the smallest object in C covering each O G R(G). This has automorphism group G := AutU(G) =r/K(G). (7.3) 246 Ars Math. Contemp. 10 (2016) 183-192 If r has generators X (i G I) then one can realise G as the subgroup of the cartesian power GR(G) of G generated by the elements (x^, xj2,...) for i g I, where xik is the image of X in G = Aut for some numbering O1,02,... of the objects Ok G R(G). In particular, G has the same number of generators as r, and it satisfies all the identical relations satisfied by G: for instance, if G is nilpotent of class c, is solvable of derived length d, or has exponent e, then the same applies to G. Finally, if G is finite, as we will assume from now on, then so are U(G) and G, with |G| dividing |G|r where r = r(G). Example 7.1 Let C = H+, so that r = F2. If G = Cn then K(G) = Frn, so g = r/r'rn = cn x cn. Represented as a bipartite map, the hypermap U (G) is a regular embedding of the complete bipartite graph Kn,n in a surface of genus (n - 1)(n - 2)/2. In fact, we obtain the same universal cover U(G) and group G whenever G is a 2-generator abelian group of exponent n. This example shows that G can be a rather small subgroup of Gr, since G = G2 whereas r > n. However, if G is a non-abelian finite simple group, then the following result shows that G = Gr for any category C; see [29] for a proof. Lemma 7.1. Let N1,..., Nr be distinct normal subgroups of a group r, with each Gj := r/N non-abelian and simple. If K = N1 n • • • n Nr then r/K = G1 x ••• x Gr. Taking {N1,..., Nr } = Nr(G), so Gj = G for i = 1,..., r, gives the result. Example 7.2 Let C = H+ again, and let G = L2(5) = A5. By Example 6.1 we have r(G) = 19, so G = G19, of order 6019 = 609359740010496 x 1017 « 6.1 x 1031. Guralnick and Kantor [23] have shown that if G is a non-abelian finite simple group then each non-identity element of G is a member of a generating pair. If such a group G has exponent e then it follows that (G) has type (e, e, e), so by the Riemann-Hurwitz formula it has genus g = 1 + fi3 iGir • In Example 7.2, for instance, G has exponent 30, so (G) has genus 1 + — x 6019 = 274218830047232000000000000000001 « 2.742 x 1031. 20 For any finite group G we have |Epi(F2, G)| < |G|2, so I G|2 |G|.|Z(G)| r«+ (G) < |Aut G| |Out G| where Out G is the outer automorphism group Aut G/Inn G of G. In particular, if G has trivial centre then r«+ 1, the automorphism group of Fn is generated by the elementary Nielsen transformations: permuting the free generators, inverting one of them, and multiplying one of them by another [40, Theorem 3.2]. When n = 2 one can identify Q = Out r with GL2 (Z) through its faithful induced action on the abelianisation rab = r/r Z2 of r [38, Ch. I, Prop. 4.5]. This group Q can be decomposed as a free product with amalgamation as follows (see [6, §7.2] for presentations of Q). If we take the images of X and Y as a basis for rab, then there is a subgroup E = S3 = D3 of Q, generated by the matrices E =(! 0) and (-1 -i of order 2 and 3; this group, which simply permutes the three vertex colours of an oriented hypermap, regarded as a tripartite map by stellating its Walsh map, was introduced by Machi in [39]. The central involution -I of Q reverses the orientation of each hypermap and, together with E, generates a subgroup Qi = E x (-I) = S3 x C2 = De of Q which preserves the genus of each hypermap and permutes the periods in its type. If a hypermap is represented as a bipartite map, then the matrices -1 0 ^ A ( 1 0 0 1) and [0 -1 reverse the cyclic order of edges around each black or white vertex, while preserving the order around those of the other colour; they are sometimes called Petrie operations, since they preserve the embedded bipartite graph but replace faces with Petrie polygons (closed zig-zag paths), so the genus may be changed. These two matrices, together with E, generate a subgroup Q2 = D4 such that Q0 := Q1 n Q2 = (E, -I) = V4 = D2 G. A. Jones: Combinatorial categories and permutation groups 249 and Q = Q1 *q0 Q2 = DQ *d2 D4. The torsion theorem for free products with amalgamation [36, Theorem IV.2.7] shows that the operations of finite order are the conjugates of the elements of Q1 U Q2, described by Pinto and the author in [30]. For any 2-generator group G, the orbits of Q on R(G) correspond to the T2-systems in G, that is, the orbits of Aut F2 x Aut G acting by composition on Epi(F2, G) and hence on generating pairs for G. It is known [15, 45] that this action is transitive if G is abelian, whereas Garion and Shalev [20] have shown that if G is a non-abelian finite simple group then the number of orbits tends to to as |G| —^ ^o. Example 8.1 It follows from work of Neumann and Neumann [45] that the 19 hypermaps in R(A5) form two orbits of lengths 9 and 10 under Q, which acts as S9 x S10 on them. Those hypermaps whose type is a permutation of (2,5,5), (3,3, 5) or (3, 5,5)- form the first orbit, while those of type a permutation of (2,3,5), (3, 5, 5)+ or (5, 5,5) form the other; here the superscript + or - indicates that the generators of order 5 are or are not conjugate in A5. This example illustrates a useful result of Nielsen [46], that when r = F2 the order of the commutator [x, y] is an invariant of the action of Q on R(G) for any group G: here the order is 3 or 5 for the hypermaps in the two orbits. 8.2 Operations on all hypermaps When C = H we have r = C2 *C2 *C2, containing F2 as a characteristic subgroup of index 2. As shown by James [26] there is again an action of GL2 (Z) on hypermaps, as described above, but now extended to all hypermaps. In this case -I, induced by conjugation by R1, is in the kernel of the action (since any orientation is now ignored), and there is a faithful action on H of the group Outr = gl2(z)/(-i) = pgl2(z) = s3 *C2 v4. 8.3 Operations on oriented maps When C = M+ we have r = * C2, with Q = Outr = V4. This group Q is generated by vertex-face duality, induced by the automorphism of r transposing X and Z, and orientation-reversal, induced by inverting X and fixing Y. These two involutions commute, modulo conjugation by Y. If we restrict to the category M+ of oriented maps of valency dividing k, then r = Ck *C2, with Q isomorphic to the multiplicative group Z*k of units mod (k) provided k > 2. The elements of Q are the operations Hj defined by Wilson in [53], raising the rotation of edges around each vertex to its jth power, and induced by automorphisms fixing Y and sending X to Xj for j e Z*k. These operations Hj, studied by Nedela and Skoviera in [44], preserve the embedded graph, but can change the surface. When k = 5, for instance, H2 transposes the icosahedron and the great dodecahedron. 8.4 Operations on all maps When C = M we have r = V4 * C2, with Q = Out r = S3 induced by the automorphism group of the free factor (R0, R2) = V4 permuting its three involutions R0, R2 and R0R2. 250 Ars Math. Contemp. 10 (2016) 183-192 As shown by Thornton and the author [33], this group Q is simply an algebraic reinterpretation of the group of operations on regular maps introduced by Wilson in [53] (see also [37]). It is generated by the classical duality of maps, which transposes vertices and faces by transposing R0 and R2, and the Petrie duality, which transposes faces and Petrie polygons by transposing R0 and R0R2; these two operations have a product of order 3 which acts as a triality operation, cyclically permuting the sets of vertices, faces and Petrie polygons of each map. As noted by Wilson, maps admitting trialities but not dualities seem to be rather rare: Poulton and the author have given some infinite families of examples in [31]. If we restrict to the category Mk of maps of valency dividing k, then r = A[k, 2, rc] = (Ro, Ri) * (Ro, R2> = Dk *a2 D2, where the amalgamated subgroup C2 is generated by a reflection R0 in each factor. If k > 2 the automorphisms of Dk fixing R0 form a group isomorphic to Z*k, sending R0R1 to (R0R1)j for any j e Z*k, while those of D2 fixing R0 simply permute R2 and R0R2. These extend to automorphisms of r which generate a subgroup Z*k x C2 of Aut r: the first factor induces Wilson's operations Hj, and the second factor induces Petrie duality. The structure theorems for free products with amalgamation [36, §7.2] show that this subgroup maps onto Out r. Since H_1 is induced by conjugation by R0 we find that Q - (Zk/{±1}) x C2. When k = 3, with Q = C2, we obtain the outer automorphism of the extended modular group r = PGL2(Z) studied by Thornton and the author in [34]. 8.5 Operations on surface coverings If Sg is an orientable surface of genus g > 1, and r = n1Sg, then by the Baer-Dehn-Nielsen Theorem the group Q = Out r is isomorphic to the extended mapping class group Mod±(Sg) of Sg, that is, the group of isotopy classes of self-homeomorphisms of Sg (see [17, Ch. 8]). The mapping class group Mod(Sg) is the subgroup of index 2 corresponding to the orientation-preserving self-homeomorphisms; both groups are finitely presented, with ModSg generated by the Dehn twists [17, Ch. 3]. The induced action of Mod±(Sg) on coverings of Sg corresponds to the action of Q on permutation representations of r. Example 8.2 If g = 1 thenr - Z2 and Q = Mod±(S1) = GL2(Z), with Mod(S1) corresponding to SL2 (Z). This is generated by the Dehn twists corresponding to the elementary matrices j D and n 1 9 Invariance under operations Although it is natural to regard the regular objects in C as its most symmetric objects, some of these may have additional 'external' symmetries, in the sense that they are invariant (up to isomorphism) under some or all of the operations in Q. Self-dual maps, such as the tetrahedron, are obvious examples. For any C and G the group K(G) defined in (7.1) is a characteristic subgroup of r, so the corresponding regular object U(G) is invariant under G. A. Jones: Combinatorial categories and permutation groups 251 Q. This shows that each object O g C, regular or not, is covered by an Q-invariant regular object U(G) g C, which is finite if and only if O is, and which has automorphism group G where G = Mon O. The smallest Q-invariant regular object covering O can be obtained by restricting the normal subgroups N in (7.1) to those in the appropriate orbit of Q on Nr (G). Richter, Siran and Wang [47] have shown that for infinitely many k there are regular k-valent maps which are invariant under the group of operations Qi := Qm = S3 (see also [33, Theorem 3]), while Archdeacon, Conder and Siran [1] have recently constructed infinite families of k-valent orientably regular maps invariant under both Qi and the group Q2 := Qm+ = Z*k. They call these 'kaleidoscopic maps with trinity symmetry'. In both cases, examples of such maps can be constructed as maps Um(G) for finite groups G: for instance, the map denoted by Mn in [1, Theorem 2.2] has this form where G is a dihedral group of order 4n, with k(G) = r"(r')n in r = rm ^ v4 * c2. The connection is as follows. For orientably regular maps, invariance under the operation H_1 g Q2 is equivalent to reflexibility, so one needs to find normal subgroups of r which are invariant under the actions of Q1 = Outr (i.e. which are characteristic subgroups of r) and (for kaleidoscopic maps) of Q2 = Z*k, where k is the valency of the corresponding map. For any quotient G of r, these two groups Qi act by permuting the subgroups in Nr(G), so they leave invariant their intersection K(G); the map U(G) corresponding to K(G) is therefore kaleidoscopic with trinity symmetry. Example 9.1 Let G = A5, so that the three maps Mi (i = 1, 2, 3) in R(G) are the antipodal quotients of the icosahedron, the dodecahedron and the great dodecahedron (see Example 7.4); these have types {3,5}5, {5,3}5 and {5,5}3 where the subscript denotes Petrie length, as in [6, §8.6]. Their join U(G) is a non-orientable regular map of type {15,15} 15 and genus 39602, with automorphism group G = A5. The groups Q1 and Q2 permute the three maps Mi (Q1 transitively, while Q2 = Z15 = C2 x C4 has orbits {M2} and {M1, M3}), so U(G) is kaleidoscopic with trinity symmetry. (This is the non-orientable example constructed by a different method in [1, §7].) Example 9.2 For an orientable example, we can take G = A5 x C2, so U(G) is the join of U(A5), described in the preceding example, and U(C2), a reflexible map of type {2,2}2 on the sphere corresponding to the derived group K(C2) = r' of r. This gives an orientable map of type {30,30}30 and genus 187201, which is kaleidoscopic with trinity symmetry and has automorphism group (A5 x C2)3. More generally, if G is a non-abelian finite simple group which is a quotient of r (the only ones which are not are ¿3(9), U(q), ¿4(2®), U4(2e), Ae, A7, Mn, M22, M23 and McL, according to [49, Theorem 4.16]), these constructions yield a pair of non-orientable and orientable kaleidoscopic maps which have trinity symmetry and have automorphism groups Gr and Gr x C3, where r = rm(G). Example 9.3 If G is the Suzuki group Sz(8), of order 2e.5.7.13 = 29120, then r = 14 by Example 6.3; the resulting maps have types {k, k}k and {2k, 2k}2k where k = 455, the 252 Ars Math. Contemp. 10 (2016) 183-192 least common multiple of the valencies 5, 7 and 13 of the vertices in the 14 maps in R(G) (see [14]). The orientable map has genus 2912014 x 23 453 13 1 +- x -= 1 + 29120 x 28992. 4 910 If only trinity symmetry is required, as in [47], then smaller examples of this type can generally be found, with r dividing 6, by replacing U(G) with the join of an orbit of Qi on RM(G). For instance, if G = L?(p) for some primep = ±1 mod (24) one can take r = 1. 10 Finiteness Throughout this paper, we have generally assumed that the group G is finite. If it is not, then not only can RC(G) be infinite, it can even split into infinitely many orbits under the action of QC. Example 10.1 Let C = H+, so that r = F?, and let G = (x,y | x3 = y?), the group ni(S3 \ K) of the trefoil knot K. This group, isomorphic to the three-string braid group B3 = (a, b | aba = bab) with x = ab and y = ab?, has centre Z(G) = (x3) = CTO, with G/Z(G) = C3 * C = PSF^Z). Dunwoody and Pietrowski [16] have shown that the pairs xi = x3i+i, yi = y?i+i (i G Z) all generate G and lie in different T?-systems. The corresponding normal subgroups N G Nr(G), the kernels of the epimorphisms r ^ G given by X ^ xi, Y ^ yi, therefore all lie in different orbits of the group Q = Out r = GL?(Z), as do the corresponding hypermaps in RC(G). References [1] D. Archdeacon, M. Conder and J. 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Gross * Department of Computer Science, Columbia University, New York, NY 10027, USA Toufik Mansour Department of Mathematics, University of Haifa, 3498838 Haifa, Israel Thomas W. Tucker t Department of Mathematics, Colgate University, Hamilton, NY 13346, USA David G. L. Wang í School of Mathematics and Statistics, Beijing Institute of Technology, 102488 Beijing, P. R. China Received 10 September 2013, accepted 27 September 2015, published online 30 November 2015 Abstract We prove that the genus polynomials of the graphs called iterated claws are real-rooted. This continues our work directed toward the 25-year-old conjecture that the genus distribution of every graph is log-concave. We have previously established log-concavity for sequences of graphs constructed by iterative vertex-amalgamation or iterative edge-amalgamation of graphs that satisfy a commonly observable condition on their partitioned genus distributions, even though it had been proved previously that iterative amalgamation does not always preserve real-rootedness of the genus polynomial of the iterated graph. In this paper, the iterated topological operation is adding a claw, rather than vertex- or edge-amalgamation. Our analysis here illustrates some advantages of employing a matrix representation of the transposition of a set of productions. Keywords: Topological graph theory, graph genus polynomials, log-concavity, real-rootedness. Math. Subj. Class.: 05A15, 05A20, 05C10 ♦Supported by Simons Foundation Grant #315001. t Supported by Simons Foundation Grant #317689. * Supported by National Natural Science Foundation of China Grant No. 11101010 E-mail addresses: gross@cs.columbia.edu (Jonathan L. Gross), tmansour@univ.haifa.ac.il (Toufik Mansour), ttucker@colgate.edu (Thomas W. Tucker), david.combin@gmail.com; glw@bit.edu.cn (David G. L. Wang) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 256 Ars Math. Contemp. 10 (2016) 183-192 1 Introduction Graphs are implicitly taken to be connected. Our graph embeddings are cellular and orientable. For general background in topological graph theory, see [1, 9]. Prior acquaintance with the concepts of partitioned genus distribution (abbreviated here as pgd) and production (e.g., see [5, 11]) is prerequisite to reading this paper. Subject to this prerequisite, the exposition here is intended to be accessible both to graph theorists and to combinatorialists. The genus distribution of a graph G is the sequence g0(G), g\(G), g2(G), ..., where gi(G) is the number of combinatorially distinct embeddings of G in the orientable surface of genus i. A genus distribution contains only finitely many positive numbers, and there are no zeros between the first and last positive numbers. The genus polynomial is the polynomial rG(z) = go(G) + gi(G)z + g2(G)z2 + ... We say that a sequence A = (ak)n=0 is nonnegative if ak > 0 for all k. An element ak is said to be an internal zero of A if there exist indices i and j with i < k < j, such that aiaj = 0 and ak = 0. If ak-1ak+1 < akk for all k, then A is said to be log-concave. If there exists an index h with 0 < h < n such that ao < ai < • • • < ah-i < ah > ah+i > ••• > an, then A is said to be unimodal. It is well-known that any nonnegative log-concave sequence without internal zeros is unimodal, and that any nonnegative unimodal sequence has no internal zeros. A prior paper [7] by the present authors provides additional contextual information regarding log-concavity and genus distributions. 1.1 The LCGD Conjecture and Real-Rootedness Problems For convenience, we sometimes abbreviate the phrase "log-concave genus distribution" as LCGD. Proofs that closed-end ladders and doubled paths have LCGDs [2] were based on closed formulas for their genus distributions. Proof that bouquets have LCGDs [8] was based on a recursion. The following conjecture was formulated in [8]: LCGD Conjecture: Every graph has a log-concave genus distribution. Stahl [12] used the term "H-linear" to describe chains of graphs obtained by amalgamating copies of a fixed graph H. He conjectured that a number of "H-linear" families of graphs have genus polynomials with nonpositive real roots, which implies the log-concavity of their sequences of coefficients, by Newton's theorem. (Since all the coefficients of a genus polynomial are non-negative, it follows that all the roots are non-positive.) Although it was shown [14] that the genus polynomials of some such families do indeed have real roots, Stahl's conjecture of real-rootedness for W4-linear graphs (where W4 is the 4-wheel) was disproved by Liu and Wang [10]. Our previous paper [7] proves, nonetheless, that the genus distribution of every graph in the W4-linear sequence is log-concave. Thus, even though Stahl's proposed approach to log-concavity via roots of genus polynomials is sometimes infeasible, [7] does support Stahl's expectation that chains of copies of a graph are a relatively accessible aspect of the general LCGD problem. Moreover, Wagner [14] has proved the real-rootedness of the genus polynomials for a number of graph families for which Stahl made specific conjectures of real-rootedness. J. Gross et al.: Iterated claws have real-rooted genus polynomials 257 This leads to a couple of research problems that are subordinate to the LCGD Conjecture, as follows: Real-rootedness Problem: Characterize the graphs whose genus polynomials are not real-rooted. Real-rootedness Chain Problem: Characterize the graphs H whose genus polynomials are real-rooted but whose H-linear chains contain graphs whose genus polynomials are not real-rooted. Furthermore, we shall see here that Stahl's method of representing what we have elsewhere ([4, 6]) presented as a transposition of a production system for a surgical operation on graph embeddings as a matrix of polynomials can simplify a proof that a family of graphs has log-concave genus distributions. 1.2 Interlacing Roots in a Genus Polynomial Sequence The earliest proofs [2, 8] of the log-concavity of the genus polynomials for a sequence of graphs appealed directly to the condition aj_1aj+1 < a2. The need for more powerful techniques motivated the development of the linear combination techniques of [7]. Here, to prove the log-concavity of the genus polynomials for the sequence of iterated claws, we combine Newton's theorem that a real-rooted polynomial is log-concave (Theorem 4.1) with a focus on interlacing of roots of consecutive genus polynomials for the graphs in the sequence to prove their log-concavity. 2 The Sequence of Iterated Claws Let the rooted graph (Y0, u0) be isomorphic to the dipole D3, and let the root u0 be either vertex of D3. For n = 1, 2,..., we define the iterated claw (Yn,un) to be the graph obtained the following surgical operation: Newclaw: Subdivide each of the three edges incident on the root vertex un-1 of the iterated claw (Yn-1, un-1), and then join the three new vertices obtained thereby to a new root vertex un. Figure 1 illustrates the graph (Y3, u3). V2 Figure 1: The rooted graph (Y3,u3). The graph K13 is commonly called a claw graph, which accounts for our name iterated claw. The notation Yn reflects the fact that a claw graph looks like the letter Y. We observe x x 2 3 258 Ars Math. Contemp. 10 (2016) 183-192 that Y1 = K3 3. A recursion for the genus distribution of the iterated claw graphs is derived in [6]. We observe that, whereas all of Stahl's examples [12] of graphs with log-concave genus distributions are planar, the sequence of iterated claws has rising minimum genus. (Example 3.2 of [7] is another sequence of rising minimum genus. However, the graphs in that sequence have cutpoints, unlike the iterated claws.) We have seen in previous studies of genus distribution (especially [3]) that the number of productions and simultaneous recursions rises rapidly with the number of roots and the valences of the roots. The surgical operation newclaw is designed to circumvent this problem. For a single-rooted iterated claw (Yn, un), we can define three partial genus distributions, also called partials. Let an,i = the number of embeddings Yn ^ Si such that three different fb-walks are incident on the root un ; bni = the number of embeddings Yn ^ Si such that exactly two different fb-walks are incident on the root un; cn i = the number of embeddings Yn ^ Si such that one fb-walk is incident three times on the root un. We also define partial genus polynomials to be the generating functions œ An(z) ^ ] an,iZ i=0 Bn(z) = £ bn,iZi i=0 œ Cn(z) ^^ cn,izl. i=0 Clearly, the full genus distribution is the sum of the partials. That is, for i = 0,1,2,..., we have and gi(Yn) an,i + bn,i + cn,i Tyn (z) = An (z) + Bn(z) + Cri(z). We define gn,i = gi(Yn). Remark 2.1. Partitioned genus distributions and recursion systems for pgds were first used by Furst, Gross, and Statman [2]. Stahl [12] was first to employ a matrix equivalent of a production system to investigate log-concavity. Theorem 2.2. For n > 1, the effect on the pgd of applying the operation newclaw to the iterated claw (Yn-I,un-1) corresponds to the following system of three productions: ai —> I2bi+i +4ci+2 bi —> 2 ai +12bi+1 +2ci+1 Ci —> 8ai + 8ci+i (2.1) (2.2) (2.3) J. Gross et al.: Iterated claws have real-rooted genus polynomials 259 Proof. This is Theorem 4.5 of [6]. □ Corollary 2.3. For n > 1, the effect on the pgd of applying the operation newclaw to the iterated claw (Yn-i, un-i) corresponds to the following recurrence relations: an,i = 2bn-i,i + 8c„-i,j (2.4) bn,i = 12an-i,i-i + 12bn-i,i-i (2.5) Cn,i = 4a„-i,i-2 + 2b„-i,i-i + 8c„-i,i-i (2.6) Proof. The recurrence system (2.4), (2.5), (2.6) is induced by the production system (2.1), (2.2), (2.3). □ It is convenient to express such a recurrence system in matrix form: V(Yn) = M(z) • V(Yn-i) with the production matrix M(z) -- 0 2 8 12z 12z 0 4z2 2z 8z (2.7) (2.8) "Ao(z)" 2 V (Yo) = Bo(z) = 0 Co(z) 2z Since the initial graph Y0 in the sequence of iterated claws is isomorphic to the dipole D3, the initial column vector for the sequence V(Yn) is (2.9) Proposition 2.4. The column vector V (Yn) is the product of the matrix power M n(z) with the column vector V(Y0). Corollary 2.5. The column vector V(Yn) is the product of the matrix power Mn+1(z) with the (artificially labeled) column vector ( ° V (Y_i) = ( ° W4y Corollary 2.6. To prove that every iterated claw has an LCGD, it is sufficient to prove that the sum of the third column of the matrix Mn(z) is a log-concave polynomial. 3 Characterizing Genus Polynomials for Iterated Claws In this section, we investigate some properties of the genus polynomials of iterated claws. Corollary 2.6 leads us to focus on the sum of the third column of the matrix Mn(z), which is expressible as (1,1,1)Mn(z)(4V(Y_1)), which implies that it equals 4 times the genus polynomial of the iterated claw Yn-1. Theorem 3.1 formulates a generating function f (z,t) for this sequence of sums, and Theorem 3.2 uses the generating function to construct an expression for the genus polynomials from which we establish interlacing of roots in Section 4. 260 Ars Math. Contemp. 10 (2016) 183-192 Theorem 3.1. The generating function f (z,t) = £ „>0(1,1,1)M„(z)(4V (Y_i ))t„ for the sequence of sums of the third column of M n(z) has the closed form f(z,t) = _1 + (8 - 12z)t -^ 33 . (3.1) JK ' ' 1 - 20zt + 8z(8z - 3)t2 + 384z3t3 v y Proof. Let (p„, q„, r„) = (1,1,1)Mn(z) for all n > 0. Then (pn+i,q„+i,r„+i) = (p„ ,q„,r„)M (z) (3.2) = (12zq„ + 4z2r„, 2p„ + 12zq„ + 2zr„, 8p„ + 8zr„). The third coordinate of Equation (3.2) implies that P„ = 8(r„+i - 8zr„). (3.3) By combining (3.3) with the first coordinate of (3.2) we obtain q„ = 7^(r„+2 - 8zr„+i - 32z2r„). (3.4) 96z The second coordinate of (3.2) yields q„+i = 2p„ + 12zq„ + 2zr„ (3.5) Substituting (3.3) and (3.4) (twice) into (3.5) leads to the recurrence relation r„ = 20zr„_i + 8z(3 - 8z)r„_2 - 384z3r„_3 (3.6) with ro = 1, ri = 8 + 8z, (3.7) r2 = 160z + 96z2. By multiplying Recurrence (3.6) by tn and summing over all n > 0, we obtain Generating Function (3.1). □ It is easy to see that rYn (z) = rn+i/4, where r„ is defined in the proof of Theorem 3.1. In terms of rYn (z), the recurrence relation (3.6) becomes (z) = 20zrYn_1 (z) + 8z(3 - 8z)ry„-2(z) - 384z3ry„_3 (z). (3.8) Theorem 3.2 provides an explicit expression for the genus polynomial rYn (z), a result is of independent interest. It is not used here toward proof of log-concavity. Theorem 3.2. The genus polynomial of the iterated claw Y„ is given by (1,1,1)M„+i(z)V(Y_i) = 2„_i(h„+i(z)+2(2 - 3z)h„(z) - 6zh„_i(z)), where h„(z)= ^ fJ + ^ fJ + ^ fJ + ^ (1 + ^ (1 3j+ii (2z)„_j. J. Gross et al.: Iterated claws have real-rooted genus polynomials 261 Proof. By Theorem 3.1, we have / (z,t) = Dm, DM"*«*- ™r = n>0 Thus, 1 - 20 zt + 8z(8z - 3)t2 + 384z3t3 ' f (z/2, t/2) 1 + (4 - 3z)t - 3zt2 1 - 5zt + z(4z - 3)t2 + 6z3t3 1 + (4 - 3z)t - 3zt2 (1 - 2zt - 2z2t2)(1 - 3zt) - 3zt2 ^ (1 + (4 - 3z)t - 3zt2)3jzjt2j (1 - 3zt)j+1(1 + V'3zt)j+1(1 - V'3zt)j+1 ' /m-1+j\ ) — coefficient of tn, we derive the equation Using the combinatorial identity (1 - at) m = Xj>0 /+j) ajtj, and then finding the (1,1,1)M"(z/2)V(Yo) = 2" (h„(z) + 2(2 - 3z)h„_i(z) - 6zh„_2(z)), which, by Corollary 2.5, completes the proof. □ Now let g„,j be the coefficient of z® in rYn (z). The following table of values of g„,j for n < 4 is derived in [6]. gn,i i — 0 1 2 3 4 5 n — 0 2 2 0 0 0 0 1 0 40 24 0 0 0 2 0 48 720 256 0 0 3 0 0 1920 11648 2816 0 4 0 0 1152 52608 177664 30720 Denote by Ps,t the set of polynomials of the form J2fc=s afczk, where ak is a positive integer for any s < k < t. The above table suggests that rYn(z) G P|_(n+1)/2J,n+1 for n < 4. Theorem 3.3 shows that it holds true in general. Like Theorem 3.2, this enumerative result is of independent interest and is not used toward proof of log-concavity. Theorem 3.3. For all n > 0, the polynomial rYn(z) G P[(n+1)/2J,n+1. Moreover, we have the leading coefficient and, for any number i such that |_(n + 1)/2j +1 < i < n, we have 9n,i > 11gn-1,i-1- (3.10) 262 Ars Math. Contemp. 10 (2016) 183-192 Proof. We see in the table above, for n < 4, that Ymin(Yn) = |(n + 1)/2J and that 7max(Y„) = n + 1, or equivalently, that ^(z) € Pl(«+i)/2J,"+i• We see also, for n < 4, that Equation (3.9) and Inequality (3.10) are true. Now suppose that n > 5. For convenience, let =0 for all i < 0. We can also take =0 for i > k + 1, by induction using (3.8), for k < n. From Recurrence (3.8) and the induction hypothesis, we have = 20gn_i,i_i +24g„_2,i-i - 64gn-2,i-2 - 384gn-3,i-3, n > 3. (3.11) For i > n +1, the induction hypothesis implies that each of the four terms on the right side of Recurrence (3.11) is zero-valued. So the degree of rYn (z) is at most n +1. Let si = gi,i+1. Taking i = n +1 in (3.11), we get with the initial values so = 2, s1 = 24, s2 = 256. The above recurrence can be solved by a standard generating function method, see [15, p.8]. In practice, we use the command rsolve in the software Maple and get the explicit formula directly as It follows that gn,n+1 > 0. Hence the degree of rYn (z) is exactly n +1. Similarly, for i < |(n + 1)/2j, the four terms on the right side of (3.11) are zero-valued, so the minimum genus of Yn is at least |_(n + 1)/2j. Moreover, applying (3.11) with i = |_(n + 1)/2j and using the induction hypothesis gk,i = 0 for all i < |(k + 1)/2j with k < n, we find the first term is positive for n odd and zero for n even, the second term is always positive, and the third and fourth terms are always zero. In other words, gn,L(n+1)/2J = 2°gn-1,L(n+1)/2J-1 + 24gn-2, L(n+1)/2J_ 1 > 24gn-2, |_(n+1)/2J-1 > This confirms the minimum genus of Yn is exactly |(n + 1)/2j. Now consider i such that |(n + 1)/2j +1 < i < n. By (3.11), and using (3.10) inductively, we deduce gn,i = 11gn-1,i-1 + 24g„-2,i-1 + (9g„-1,i-1 - 64g„-2,i-2 - 384g„-3,i-3) > 11gn-1,i-1 + 24g„-2,i-1 + (35g„-2,i-2 - 384g n-3,i-3) > 11gn-1,i-1 + 24gn-2,i-1 + gn-3,i-3 > 11gn- 1,i-1. So Inequality (3.10) holds true. It follows that gn,i > 0. Hence Sn = 20Sn-1 - 64Sn-2 - 384Sn-3, (3.12) rY„ (z) € PL(n+1)/2J,n+1. This completes the proof. □ J. Gross et al.: Iterated claws have real-rooted genus polynomials 263 4 Genus Polynomials for Iterated Claws are Real-Rooted Our goal in this section is to establish in Theorem 4.3 the real-rootedness of the genus polynomials rYn (z) of the iterated claws, via an associated sequence Wn (z) of normalized polynomials. It follows from this real-rootedness that the genus polynomials for iterated claws are log-concave, by the following theorem of Newton. Theorem 4.1 (Newton's theorem). Let a0, ai, ..., an be real numbers and let all the roots of the polynomial n P(x) = ^^ ajx® j=o be real. Then a2 > aj_1aj+1 for j = 1,..., n — 1. Proof. For instance, see Theorem 2 of [13]. □ To proceed, we "normalize" the polynomials rYn (z) by defining Wn(z) = z-L(n+1)/2JrYn (z), (4.1) so that Wn(z) starts from a non-zero constant term, and has the same non-zero roots as rYn (z). We use the symbol dn to denote the degree of Wn(z), that is, dn = deg Wn(z) = (n + 1) — n + 1 "n + 1" _ 2 2 (4.2) By Theorem 3.3, we have Wn(z) G P0,dn. Substituting (4.1) into the recurrence relation (3.8), we derive W (z) = i20zWn-i(z)+8(3 — 8z)Wn-2(z) — 384z2Wn-3(z), if n is even, n(z) = |^20Wn_1(z) + 8(3 — 8z)Wn-2(z) — 384zWn-3(z), if n is odd, (4.3) with the initial polynomials Wo(z) = 2(1 + z), W1(z) = 8(5+ 3z), (4.4) W2(z) = 16(3+ 45z + 16z2). Let P denote the union Un>0P0,n = Un>0{^n=0 akzk | ak G Z+}. Lemma 4.2 is ultimately a consequence of the intermediate value theorem. Lemma 4.2. Let P(x), Q(x) G P. Suppose that P(x) has roots x1 < x2 < • • • < xdeg P, and that Q(x) has roots y1 < y2 < • • • < ydeg q. If deg Q — deg P G {0,1} and if the roots interlace so that x1 < y1 < x2 < y2 < • • • , then ( —1)i+deg PP(yi) > 0 for all 1 < i < deg Q, (4.5) ( — 1)j+deg QQ(xj) < 0 for all 1 < j < deg P. (4.6) 264 Ars Math. Contemp. 10 (2016) 183-192 Proof. Since P(x) is a polynomial with positive coefficients, we have (_1)deg Pp(_TO) > g. (4.7) We suppose first that degP(x) is odd, and we consider the curve P(x). We see that Inequality (4.7) reduces to P(-to) < 0. Thus, the curve P(x) starts in the lower half plane and intersects the x-axis at its first root, xi. From there, the curve P(x) proceeds without going below the x-axis, until it meets the second root, x2. Since x1 < y1 < x2, we recognize that (4.5) holds for i = 1, i.e., After passing through x2, the curve P (x) stays below the x-axis up to the third root, x3. It is clear that the curve P(x) continues going forward, intersecting the x-axis in this alternating way. It follows from this alternation that From (4.8) and (4.9), we conclude that (4.5) holds for all 1 < i < deg Q, when deg P(x) is odd. We next suppose that deg P(x) is even. In this case, we can draw the curve P(x) so that it starts in the upper half plane, first intersects the x-axis at x1, then goes below the axis up to x2, and continues alternatingly. Therefore the sign-alternating relation (4.9) still holds. Since P(y1) < 0 when deg P(x) is even, we have proved (4.5). It is obvious that Inequality (4.6) can be shown along the same line. This completes the proof of Lemma 4.2. □ Now we proceed with our main theorem on the genus polynomial of iterated claws. Beyond proving real-rootedness of the genus polynomials, we derive two interlacing relationships on their roots. Theorem 4.3. For every n > 0, the polynomial Wn(z) is real-rooted. Moreover, if the roots of Wk (z) are denoted by xk1 < xk 2 < • • •, then we have the following interlacing properties: (i) for every n > 2, the polynomial Wn(z) has one more root than Wn-2(z), and the roots interlace so that x„,1 < x„_2,1 < x„,2 < x„_2,2 < ••• < xn,d„-1 < x„_2,d„-1 < x„,dn; (ii) for every n > 1, the polynomial Wn (z) has either one more (when n is even) or the same number (when n is odd) of roots as Wn-1(z), and the roots interlace so that x„,1 < xn-1,1 < x„,2 < xn—1,2 < ••• < x„-1,dn-1 < xn,dn when n even; P(yi) > 0. (4.8) P(yfc)P(yfc+i) < 0 forall 1 < k < degQ - 1. (4.9) and Xn,i < xn-i,i < x„,2 < xn_i,2 < ••• < x„,dn < i„-i,d„ when n odd. J. Gross et al.: Iterated claws have real-rooted genus polynomials 265 Proof. From the initial polynomials (4.4), it is easy to verify Theorem 4.3 for n < 2. We suppose that n > 3 and proceed inductively. For every k < n - 1, we denote the roots of Wk (z) by < xk 2 < • • • < xkjdfc. For convenience, we define xfcj0 = -to and xkjdfc+1 = 0, for all k < n - 1. To clarify the interlacing properties, we now consider the signs of the function Wm(z) at -to and at the origin, for any m > 0. Since Wm(z) is a polynomial of degree dm, with all coefficients non-negative, we deduce that ( —1)dm Wm(-TO) > 0. (4.10) Having the constant term positive implies that Wm(0)= g„,0 > 0. (4.11) By the intermediate value theorem and Inequality (4.10), for the polynomial Wn(z) to have dn = deg Wn(z) distinct negative roots and for Part (i) of Theorem 4.3 to hold, it is necessary and sufficient that (-1)dn+j Wn(x„-2,j) > 0 for 1 < j < d„-2 + 1. (4.12) In fact, for j = dn-2 + 1, Inequality (4.12) becomes (_ 1)dn+dn-2+1w„(0) > 0. (4.13) By (4.11), Inequality (4.13) holds if and only if dn + dn-2 is odd, which is true since dn + 2 "n + 1" n - 1 =2 n - 1 + 2 2 2 + 1. Now consider any j such that 1 < j < dn-2. We are going to prove (4.12). We will use the particular indicator function Ieven, which is defined by J1, if n is even, Ieven(n) = \ 0, if n is odd. Note that x„_2j is a root of Wn-2(z). By Recurrence (4.3), we have W„(z„-2,j) = Xj20W„-i(x„-2,j) - 384x„_2,jW„_3(x„-2j)) . (4.14) Since x„_2jj < 0, the factor £^—2 jn) contributes (-1)n+1 to the sign of the right hand side of (4.14). On the other hand, it is clear that the sign of the parenthesized factor can be determined if both the summands 20Wn_1(xn_2 j) and -384x„_2j Wn_3(i„_2,j) have the same sign. Therefore, Inequality (4.12) holds if (-1)dn+j+n+1W„_i(x„_2,j) > 0, (4.15) (-1)dn+j+n+1W„_3(x„-2,j) > 0. (4.16) By the induction hypothesis on part (ii) of this theorem, we can substitute P = Wn-1 and Q = Wn-2 into Lemma 4.2. Then Inequality (4.5) gives (-1)dn-i+jW„-1(x„_2,j) > 0. (4.17) 266 Ars Math. Contemp. 10 (2016) 183-192 Thus, Inequality (4.15) holds if and only if the total power dn + j + n + 1 + d„_i + j = "n + 1" n + — 2 2 + n + 2 j + 1 of (-1) in (4.15) and (4.17) is even, which is clear by a simple parity argument. Moreover, again using the induction hypothesis on part (ii), we can make substitutions P(x) = Wn_2(x) and Q(x) = Wn_3(x) into Lemma 4.2. Then Inequality (4.6) gives (_1)dn-3+jW„-3(x„-2,j) < 0. Thus, Inequality (4.16) holds if and only if the total power dn + j + n + 1 + d„_3 + j n + 1 2 + n2 + n + 2j + 1 (4.18) (4.19) of (_1) in (4.16) and (4.18) is odd, which is also clear by a simple parity argument. This completes the proof of (4.12), and the proof of Part (i). The approach to proving Part (ii) is similar to that used to prove Part (i). By the intermediate value theorem and Inequality (4.10), Part (ii) holds if and only if (_1)dn+j Wn(xn-1,j ) > 0 for 1 < j < dn-1, (4.20) and also for j = dn-1 + 1 when n is even. In fact, when n is even and j = dn-1 + 1, we have (_1)dn+dn-1 + 1Wn(0) > 0. (4.21) By (4.11), Inequality (4.21) holds if and only if (_1)dn+dn-i + 1 = 1, which is clear since dn + dn 1 + 1 = "n + 1" n + — 2 2 + 1 = n + 2. For 1 < j < dn-1, we are now going to show (4.20). By setting x = xn-1,j, Recurrence (4.3) turns into Wn(xn-1,j) = 8(3 _ 8xn-1,j)W„_2(i„_1,j) _ 384xn+Iij(n)Wn-3(xn-1,j). (4.22) Since xn-1,j < 0, we see that 8(3_8xn_1,j-) > 0, and that the factor _384xn_1e'jen( ) contributes (_1)n+1 to the sign of the right-hand side of (4.22). Therefore, Inequality (4.20) holds if (_1)dn+j Wn_2(xn_1,j) > 0, (_1)dn+j+n+1Wn_3(xn_1,j) > 0. (4.23) (4.24) Substituting P(x) = Wn_1(x) and Q(x) = Wn_2(x) into Lemma 4.2, we find that Inequality (4.6) yields (_1)dn-2 + j Wn_2(xn_1,j) < 0 when 1 < j < dn_1. Thus, Inequality (4.23) holds if and only if the total power (4.25) dn + j + dn_2 + j "n + 1" n _ 1 + 2 2 + 2 j J. Gross et al.: Iterated claws have real-rooted genus polynomials 267 of (-1) in (4.23) and (4.25) is odd, which holds true, obviously, by parity. On the other hand, by the induction hypothesis on Part (i) and substituting P(x) = Wn-1(x) and Q(x) = Wn-3(x) into Lemma 4.2, Inequality (4.6) becomes (-1)dn-3+j Wn-3(xn-1,j ) < 0. (4.26) Therefore, Inequality (4.24) holds if and only if the total power dn + j + n + 1 + dn-3 + j of (-1) in (4.24) and (4.26) is odd, which coincides with (4.19). This completes the proof of (4.20), ergo the proof of Part (ii), and hence the entire theorem. □ Corollary 4.4. The sequence of coefficients for every genus polynomial rYn (z) is log-concave. Proof. Recalling Equation (4.1), we have ry„ (z) = zL(n+1)/2JWn(z). By Theorem 4.3, we know that the polynomial Wn(z) is real-rooted. It follows that the polynomial rYn (z) is real-rooted. Applying Theorem 4.1 (Newton's theorem), we know that the polynomial rYn (z) is log-concave. □ 5 On Real-Rootedness In the study of genus polynomials, the role of real-rootedness may rise beyond being a sufficient condition for log-concavity. The introductory section presents two basic research problems specifically on real-rootedness. One may reasonably anticipate that continuing study of the roots of genus polynomials will lead to new insights into the imbeddings of graphs. References [1] L. W. Beineke and R. J. Wilson (eds.), Topics in topological graph theory, volume 128 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2009, doi:10.1017/CBO9781139087223, with academic consultants J.L. Gross and T.W. Tucker, http://dx.doi.org/10.1017/CBO9781139087223. [2] M. L. Furst, J. L. Gross and R. Statman, Genus distributions for two first classes of graphs, J. Combin. Theory Ser.B 46 (1989), 22-36, doi:10.1016/0095-8956(89)90004-X, http://dx. doi.org/10.1016/0095-8956(89)900 04-X. [3] J. Gross, Embeddings of graphs of fixed treewidth and bounded degree, Ars Math. Contemp. 7 (2014), 379-403, presented at AMS Meeting in Boston, January 2012. [4] J.L. Gross, Genus distribution of graph amalgamations: self-pasting at root-vertices, Australas. J. Combin. 49 (2011), 19-38. [5] J. L. 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ARS MATHEMATICA CONTEMPORANEA 10 (2016) 269-280 2-Arc-Transitive regular covers of Kn,n — nK2 with the covering transformation group Zp Wenqin Xu, Yanhong Zhu , Shaofei Du * School of Mathematical Sciences,Capital Normal University, Beijing, 100048, P R China Received 27 January 2014, accepted 24 December 2015, published online 12 January 2016 Abstract In 2014, Xu and Du classified all regular covers of a complete bipartite graph Kn,n minus a matching, denoted by Kn n - nK2, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of Kn n - nK2, whose covering transformation group is isomorphic to Zp with p a prime and whose fibre-preserving automorphism group acts 2-arc-transitively. Actually, there are only few covers with these properties and it is shown that all of them are covers of K4,4 - 4K2. Keywords: Arc-transitive graph, covering graph, 2-transitive group. Math. Subj. Class.: 05C25, 20B25, 05E30 1 Introduction Throughout this paper graphs are finite, simple and undirected. For the group- and graph-theoretic terminology we refer the reader to [15, 17]. For a graph X, let V(X), E(X), A(X) and Aut X denote the vertex set, edge set, arc set and the full automorphism group of X respectively. An edge and an arc of X are denoted by {u, v} and (u, v), respectively. An s-arc of X is a sequence (v0, vi,..., vs) of s +1 vertices such that (vj, vj+1) G A(X) and vj = vi+2, and X is said to be 2-arc-transitive if Aut X acts transitively on the set of 2-arcs of X. Let X be a graph, and let P be a partition of V(X) into disjoint sets of equal size m. The quotient graph Y := X/P is the graph with the vertex set P and two vertices P1 and P2 of Y are adjacent if there is at least one edge between a vertex of P1 and a vertex of * corresponding author E-mail addresses: wenqinxu85@163.com (Wenqin Xu), zhuyanhong911@126.com (Yanhong Zhu), dushf@mail.cnu.edu.cn (Shaofei Du) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 270 Ars Math. Contemp. 10 (2016) 183-192 P2 in X. We say that X is an m-fold cover of Y if the edge set between Pi and P2 in X is a matching whenever P\P2 G E(Y). In this case Y is called the base graph of X and the sets Pi are called the fibres of X. An automorphism of X which maps a fibre to a fibre is said to be fibre-preserving. The subgroup K of all those automorphisms of X which fix each of the fibres setwise is called the covering transformation group. It is easy to see that if X is connected then the action of K on the fibres of X is necessarily semiregular, that is, Kv = 1 for each v G V (X). In particular, if this action is regular we say that X is a regular cover of Y. The main motivation for the present paper is to contribute toward the classification of finite 2-arc-transitive graphs. In [23, Theorem 4.1], Professor Praeger divided all the finite 2-arc-transitive graphs X into the following three subclasses: (1) Quasiprimitive type: every nontrivial normal subgroup of Aut X acts transitively on vertices; (2) Bipartite type: every nontrivial normal subgroup of Aut X has at most two orbits on vertices and at least one of them has two orbits on vertices; (3) Covering type: there exists a normal subgroup of Aut X having at least three orbits on vertices, and thus X is a regular cover of some graphs of types (1) or (2). During the past twenty years, a lot of results regarding the primitive, quasiprimitive and bipartite 2-arc-transitive graphs have appeared [11, 18, 19, 20, 23, 24]. However, very few results concerning the 2-arc-transitive covers are known, except for some covers of graphs with small valency and small order. The first meaningful class of graphs to be studied might be complete graphs. In [7], a classification of covers of complete graphs is given, whose fibre-preserving automorphism groups act 2-arc-transitively and whose covering transformation group is either cyclic or Zp. This classification is generalized in [8] to covering transformation group Zp. In [26], the same problem as in [7] and [8] is considered, but the covering transformation group considered is metacyclic. As for covers of bipartite type, in [25], all regular covers of complete bipartite graph minus a matching Kn n - nK2 were classified, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, we consider the same base graphs while the covering transformation group is Zp with p a prime. Remarkably, we shall show that all the regular covers with these properties are just covers of K4j4 - 4K2. Note that to classify regular covers of given graphs such as Kn and Kn,n, whose covering transformation group is an elementary group Zp and whose fibre-preserving automorphism group acts 2-arc-transitively is a very difficult task. Essentially, it is related to the group extension theory, the group representation theory and other specific branches of group theory. We believe that the classification of all such covers for all the values k is almost not feasible. Therefore, the first step might be to study the problem for small values k and to construct some new interesting covers. Except for the graph Kn n - nK2, another often considered graph is the complete bipartite graph Kn n. In further research, we shall focus on the 2-arc-transitive regular elementary abelian covers of this graph. For further reading on the topic of covers, see [4, 5, 9, 13, 14, 22]. A cover of a given graph can be derived through a voltage assignment, see Gross and Tucker [15, 16]. Let Y be a graph and K a finite group. A voltage assignment (or, K-voltage assignment) on the graph Y is a function f : A(Y) ^ K with the property that W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 271 f (u, v) = f (v, u)- for each (u, v) G A(Y). The values of f are called voltages, and K is called the voltage group. The derived graph Y Xf K from a voltage assignment f has for its vertex set V(Y) x K, and its edge set {{(u, g), (v, f (v, u)g)} I {u, v} G E(Y), g G K}. By the definition, the derived graph Y Xf K is a covering of the graph Y with the first coordinate projection p : Y x f K ^ Y, which is called the natural projection and with the covering transformation group isomorphic to K. Conversely, each connected regular cover X of Y with the covering transformation group K can be described by a derived graph Y x f K from some voltage assignment f. Moreover, the voltage assignment f naturally extends to walks in Y. For any walk W of Y, let fW denote the voltage of W. Finally, we say that an automorphism a of Y lifts to an automorphism a of X if ap = pa, where p is the covering projection from X to Y. Before stating the main result, we first introduce a family of derived graphs. Let Y = K4,4 - 4K2 with the bipartition V(Y) = {a, 6, c, d} U {w, x, y, z} as shown in Figure (a), and fix a spanning tree T of K4 4 - 4K2 as shown in Figure (b). Identify the elementary group Zp with the 2-dimensional linear vector space over Fp. Then we define a family of derived graphs X(p) := (K4,4 - 4K2) x^ Zp with voltage assignment ^ such that ^(6, y) = (1, 0), ^(c, w) = ^(d, w) = ^(d, x) = (0,1), ^(c, x) = (1,1) and ^(w,v) = 0 for any tree arc (u, v). Figure (a): the graph K4,4 — 4K2; (b): a spanning tree T of K4,4 — 4K2. a w a w The following theorem is the main result of this paper. Theorem 1.1. Let X be a connected regular cover of the complete bipartite graph minus a matching Kn,n — nK2 (n > 3), whose covering transformation group K is isomorphic to Zp with p a prime and whose fibre-preserving automorphism group acts 2-arc-transitively. Then n = 4 and X is isomorphic to X(p). 2 Preliminaries In this section we introduce some preliminary results needed in Section 3. The first result may be deduced from the classification of doubly transitive groups (see [2] and [3, Corollary 8.3]). Proposition 2.1. Let G be a 3-transitive permutation group of degree at least 4. Then one of the following occurs. 272 Ars Math. Contemp. 10 (2016) 183-192 (i) G S4; (ii) soc(G) is 4-transitive; (iii) soc(G) = M22 or A5, which are 3-transitive but not 4-transitive; (iv) PSL(2, q) < G < PrL(2, q), where the projective special linear group PSL(2, q) is the socle of G which does not act 3-transitively, and G acts on the projective geometry PG(1, q) in a natural way, having degree q +1, with q > 5 an odd prime power; (v) G = AGL(m, 2) with m > 3; (vi) G = Z4 x A7 < AGL(4, 2). Let G be a finite group and H be a proper subgroup of G, and let D = D-1 be inverse-closed union of some double cosets of H in G \ H. Then the coset graph X = X (G; H,D) is defined by taking V(X) = {Hg | g g G} as the vertex set and E(X) = {{Hg1,Hg2} | g2g-1 G D} as the edge set. By the definition, the size of V(X) is the number of right cosets of H in G and its valency is |D|/|H|. It follows that the group G in its coset action by right multiplication on V (X) is transitive, and the kernel of this representation of G is the intersection of all the conjugates of H in G. If this kernel is trivial, then we say the subgroup H is core-free. In particular, if H = 1, then we get a Cayley graph. Conversely, each vertex-transitive graph is isomorphic to a coset graph (see [21]). Let G be a group, let L and R be subgroups of G and let D be a union of double cosets of R and L in G, namely, D = |Ji RdjL. By [G : L] and [G : R], we denote the set of right cosets of G relative to L and R, respectively. Define a bipartite graph X = B(G, L, R; D) with bipartition V(X) = [G : L] U [G : R] and edge set E(X) = {{Lg, Rdg} | g G G, d G D}. This graph is called the bicoset graph of G with respect to L, R and D (see [10]). Proposition 2.2. ([10, Lemmas 2.3, 2.4]) (i) The bicoset graph X = B(G, L, R; D) is connected if and only if G is generated by elements of D-1D. (ii) Let Y be a bipartite graph with bipartition V (Y) = U (Y) U W (Y), let G be a subgroup of Aut (Y) acting transitively on both U and W, let u G U (Y) and w G W(Y), and set D = {g G G | wg G Y1(u)}, where Y1(u) is the neighborhood of u. Then D is a union of double cosets of Gw and Gu in G, and Y = B(G, Gu, Gw; D). In particular, if {u, w} G E(Y) and Gu acts transitively on its neighborhood, then D = Gw Gu. Proposition 2.3. ([17, Satz 4.5]) Let H be a subgroup of a group G. Then CG (H) is a normal subgroup of NG(H) and the quotient NG(H)/CG(H) is isomorphic with a subgroup of Aut H. Let G be a group and N a subgroup of G. If there exists a subgroup H of G such that G = NH and N n H = 1, then the subgroup H is called a complement of N in G. The following proposition is due to Gaschutz. Proposition 2.4. ([17, Satz 17.4]) Let G be a finite group. Let A and B be two subgroups of G such that A is abelian normal in G, A < B < G and (|A|, |G : B|) = 1. If A has a complement in B, then A has a complement in G. W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 273 Proposition 2.5. ([7, Lemma 2.7]) Ifp is a prime, then the general linear group GL(2, p) does not contain a nonabelian simple subgroup. A central extension of a group G is a pair (H, n) where H is a group and n : H ^ G is a surjective homomorphism with ker(n) < Z(H). A central extension (G, n) of G is universal if for each central extension (H, a) of G there exists the unique group homomorphism a : G ^ H with n = aa. If G is a perfect group, namely G' = G, then up to isomorphism, G has the unique universal central extension, say (G, n), (see [1, pp.166-167]). In this case, G is called the universal covering group of G and ker(n) the Schur multiplier of G. Proposition 2.6. ([6, page xv]) The Schur multiplier of the simple group PSL(2, q) is Z2 for q = 9, and Z6 for q = 9. The following proposition is quoted from [9]. Proposition 2.7. ([9, Lemma 2.5]) Let Y be a graph and let B be a set of cycles of Y spanning the cycle space CY of Y. If X is a cover of Y given by a voltage assignment f for which each C G B is trivial, then X is disconnected. 3 Proof of Theorem 1.1 Now we prove Theorem 1.1. Let U = {1,2, • • • ,n} and W = {1', 2', • • • ,n'}. Set Y = Kn,n - nK2 (n > 3) with the vertex set V(Y) = U U W and edge set E (Y) = {{i,j'} | i = j,i,j = 1,2 • • • , n}. Let X be a cover of Y with the covering projection $ : X ^ Y and the covering transformation group K = Z^, where p is a prime. Suppose that n = 3. Then Y is a 6-cycle and there is only one cotree arc. Since X is assumed to be connected, all the voltage assigned to the cotree arcs in Y should generate K. It means that K is a cyclic group, a contradiction. Suppose that n = 4. In [12, Theorem 4.1], all regular covers of K4i4 - 4K2 were classified, whose covering transformation group K is either cyclic or elementary abelian, and whose fibre-preserving automorphism group acts arc-transitively. Among them, X (p) is the unique cover when K = Zp and the fibre-preserving automorphism group acts 2-arc-transitively. In what follows, we will assume n > 5. Since our aim is to find the covers of Y whose fibre-preserving automorphism group acts 2-arc-transitively, this group module the covering transformation group K should be isomorphic to a 2-arc-transitive subgroup of Aut Y, in other word, there exists a 2-arc-transitive subgroup of Aut Y to be lifted. Now, let A < Aut Y be a 2-arc-transitive subgroup, and let^ < A be the corresponding index 2 subgroup of A fixing U and W setwise. Let A and G be the respective lifts of A and G. Clearly, Aut (Y) = Sn x {a), where a is the involution exchanging every pair i and i'. Now, we show that G has a faithful 3-transitive representation on the two biparts of Y. Take arbitrary two different triples {ui, v\,w\} and {u2, v2, w2} with ui, vi, wi e U and i e {1,2}. Since (ui,v',wi) and (u2,v2,w2) are both 2-arcs, and since A acts 2-arc-transitively on Y, there exists an element g e A such that (ui,v',wi)g = (u2,v2,w2), noting that v'f = v2 implying vg = v2. Moreover, it is obvious that g fixes two biparts setwise so that g e G. So G acts 3-transitively on U. By the symmetry, G acts 3-transitively on another bipart. Therefore, G should be one of the 3-transitive groups listed in Proposition 2.1. Since n > 5, we conclude the following four cases from Proposition 2.1: 274 Ars Math. Contemp. 10 (2016) 183-192 (1) either soc(G) is 4-transitive or soc(G) = M22; (2) n = 5 and soc(G) = A5; (3) soc(G) = PSL(2, q) with q > 5; (4) G is of affine type, that is the last two cases of Proposition 2.1. To prove the theorem, we shall prove the non-existence for the above four cases separately in the following subsections. 3.1 Either soc(G) is 4-transitive or soc(G) = M22 Lemma 3.1. There exist no regular covers X of Kn,n — nK2, whose fibre-preserving automorphism group acts 2-arc-transitively and whose covering transformation group is isomorphic to Zp with p a prime, such that either soc(G) acts 4-transitively on two biparts or soc(G) = Mp2. Proof. Suppose that G has a nonabelian simple socle T := soc(G) which is either 4-transitive or isomorphic to M22. Let T be the lift of T so that T/K = T. In view of Proposition 2.3, we have (T/K)/(Cf(K)/K) = T/Cf (K) < Aut (K) = GL(2,p). (3.1) Since Cf(K)/K > T/K and T/K is simple, we get Cf(K)/K = 1 or T/K. If the first case happens, then Eq(3.1) implies that GL(2,p) contains a nonabelian simple subgroup, which contradicts Proposition 2.5. Thus, C^,(K) = T, that is, K < Z(T). It was shown in [9, pp.1361-1364] that the voltages on all the 4-cycles and 6-cycles of the base graph Y are trivial, provided K < Z(T) and either T is 4-transitive or T = M22. Therefore, Proposition 2.7 implies that the covering graph X is disconnected. This completes the proof of the lemma. □ 3.2 n = 5 and soc(G) = A5 Lemma 3.2. Suppose that n = 5 and soc(G) = A5. Then, there are no connected graphs X arising as regular covers of Y whose covering transformation group K is isomorphic to Zp with p a prime, and whose fibre-preserving automorphism group acts 2-arc-transitively. Proof. Since G is isomorphic to either A5 or S5, it suffices to consider the case G = A5. Let G be a lift of G, that is, G/K = G. As in Lemma 3.1, a similar argument shows that K < Z(G). Set T := G'. In what follows, we divide our proof into four steps. Step 1: Show T n K =1 or Z2. Set T := G'. Since G' = G, we get Tt/T n K = TtK/K = (G/K)' = G' = G = Gt/K = A5, (3.2) which implies that G = TK. As K < Z(G), we have T = [G, G] = [TK, TK] = [T, T] = T'. W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 275 Thus, T n K < T' n Z(T) and Eq(3.2) implies that T is a proper central extension of T n K by G = A5. By Proposition 2.6, we know that the Schur Multiplier of A5 is Z2. Thus, T n K is either 1 or Z2. Let u G V(Y) be an arbitrary vertex, and take T G ¿-1(u), where ^ is the covering projection from X to Y. Step 2: Show D4 < Gs n T Now, we have Ga = G„ = A4 and so Gk/Gk n t 5 G5t/t < G/t = tk/t 5 k/k n t . (3.3) Since Ga n T > Gg = A4, it follows that Ga n T = 1, D4 or A4. If GK n T = 1, then Eq(3) implies that Ga = A4 is isomorphic to a quotient group of K = Zp, a contradiction. So, we get D4 < Ga n T. Step 3: Show T 5 A5 and G = T x K. By Step 1, we know that if n K =1 or Z2. If T n K = Z2, then Eq(3.2) implies that T = SL(2,5) which has the unique involution, contradicting the fact that D4 < Ga n T. Hence, it follows that T n K = 1, and so T = A5 and G = T x K. Step 4: Show the nonexistence of the covering graph X. Suppose that V(Y) = {1, 2, 3, 4, 5}U{1', 2', 3', 4', 5'} and E(Y) = {{i,j'} | i = j, 1 < i,j < 5}. Since T = A5, we may identify T with A5. In T, set x = (23)(45), y = (25)(34), z = (234), b = (15)(23). Then, GF = ((x,y) x (z)) x K, where F = ¿-1(1) is the fibre over the vertex 1 G V(Y). Take T G F. Since D4 < Ga n T, one may deduce that D4 = (x, y) < Gjj so that L := Ga = (x, y) x (zk1) for some k1 G K. Note that GF = GF/, where F' = ^_1(1') is the fibre over the vertex 1' g V(Y). Then, one may assume that R := Gw = (x, y) x (zk2) for some k2 G K and w G F'. By Proposition 2.2, the covering graph X should be isomorphic to a bicoset graph X' = B(G, L, R; D), where D = Rbk3L for some k3 G K with two biparts: U = {Lk | k G K} U {Lbx®yjk 1 i, j = 0,1,k G K}, W' = {Rk 1 k G K}U{Rbxy k|i,j =0,1,k G K}. Moreover, X' should satisfy the following two conditions. (i) d(X') = 4: 276 Ars Math. Contemp. 10 (2016) 183-192 Since the length of the orbit of L containing the vertex Rbk3L is 4, zk1 must fix the vertex Rbk3, that is, Rbk3 = Rbk3zk1 = Rbk3zk1(bk3)-1bk3 = Rzb k1bk3 = Rz-1k- 1k2kibk3 = Rbk 3k2ki, which implies that k2 = k-1. (3.4) (ii) Connectedness property: By Eq(4), we have (D-1D) = (LbRbL) = (L, Rb) = (x,y,zk1,xb,yb,zbk2)< T x (k1) = G. It follows from Proposition 2.2(i) that the bicoset graph X' is disconnected, which completes our proof. □ 3.3 G is of affine type Lemma 3.3. Suppose that either G = AGL(m, 2), where m > 3 or G = Z| x A7 < AGL(4,2). Then, there are no connected graphs X arising as regular covers of Y whose covering transformation group K is isomorphic to Z'2p with p a prime, and whose fibre-preserving automorphism group acts 2-arc-transitively. Proof. The arguments in both cases are exactly the same, and so here we just discuss the first case in details. Suppose that G = AGL(m, 2) = Zf x GL(m, 2), and let G be a lift of G, namely G/K = G. Since Gd(K)/K > G/K = Zf x GL(m, 2), it follows that Cq (K)/K=l, Zf or G/K. By Proposition 2.3, we have (G/K)/(Cq(K)/K) = G/Cq(K) < Aut (K) = GL(2,p). (3.5) If the first two cases happen, then Eq(3.5) implies that GL(2,p) contains a nonabelian simple subgroup, which contradicts Proposition 2.5. Thus, Cq (K) = G, that is K < Z(G). _ Let A be the group of fibre-preserving automorphism of X acting 2-arc-transitively. Let U and W be the two biparts of X. Take a fibre F in U and take a vertex u1 g F. Set M := ¿?Ul = GL(m, 2) and T/Kj= soc(G/K) = Zf. Then G = T x M. Let F' denote the unique corresponding fibre in W without edges leading to F and take a vertex w1 g F'. Then GF = GF>. Since M is the unique subgroup isomorphic to GL(m, 2) in K x M, it follows that GWl = M. First, suppose that p = 2. Now, Gf = K x M. Since (|G : Gf |, |K |) = (2f,p2) = 1, by Proposition 2.4, K has a complement in G. So, we may suppose that G = K x (L x M), where L = Zf. Since G is transitive on W, there exists an element x g G W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 277 such that (wi,wf) G E(X). By Proposition 2.2(ii), X is isomorphic to a bicoset graph B(G,M, Mx; D), where D = MMx. Since L x M > G, we get (D-1D) = (M, Mx) < L x M = G. It follows from Proposition 2.2(i) that X is disconnected. Next, suppose that p = 2, namely K = Z2 x Z2. Let F = |m1, T2, T3, w4} and F' = {w^ w2, w3, w4}. Clearly, M has four orbits on U \ F and W \ F', respectively, say Ai, A2, A3, A4; Al, A2, A3, A4. For i = 0,1,2, • • • , by Xi(tJ1) we denote the set of vertices of distance i from uj1. Without loss of generality, let X1 (UJ1) = A1. Since M acts 2-arc-transitively on the arcs initialed from jj1, it follows that X2(jj1) is an orbit of M, that is, X2(jj1) = Aj for some i G {1, 2,3,4}. Then X3(j1) = {Wj}, for some j G {1, 2, 3,4}. Clearly, X4(j1) = 0 and therefore X is disconnected. □ 3.4 soc(G) = PSL(2, q) for q > 5 In this subsection, identify V(Y) with two copies of the projective line PG(1, q). Lemma 3.4. Suppose that PSL(2, q) < G < PrL(2, q), where q = rl > 5 is an odd prime power. Then, there are no connected graphs X arising as regular covers of Y whose covering transformation group K is isomorphic to Z^ with p a prime, and whose fibre-preserving automorphism group acts 2-arc-transitively. Proof. Let G be the lift of G so that G/K = G. Since PrL(2, q)' = PSL(2, q) and PSL(2, q) < G < PrL(2, q), we have G' = PSL(2, q). Hence, G is insolvable and there exists a positive integer m such that G(m) = C?(m+1). Suppose that T = C?(m), it follows that T/T n K = TTK/K = C?(m)K/K = (G/K)(m) = G(m) = PSL(2, q). (3.6) Therefore, TTK/K is simple and so (TK/K) n (G^j(K)/K) = 1 or TK/K. Again, by Proposition 2.3 and 2.5, we have TK/K < Cg (K)/K, implying that T n K < Z(T). Thus, by Eq(3.6), TT is a proper central extension of T n K by PSL(2, q). In viewing of Proposition 2.6, the Schur Multiplier of PSL(2, q) is either Z2 for q = 9 or Z6 for q = 9. It is obvious that T n K_= 1 or Z2 for q = 9. Next, we show it is also true for q =9. Assume, the contrary, that T n K = Z3 for q =9. Since TK/K = PSL(2,9), we get (TTK)s = Z3 x Z4. Let = H < (TTK)5. As H n K =1 and (|!TK K|, |K|) = 1, it follows from Proposition 2.4 that Khas a complement in TK, say NV. Thus, TK = K x NV = Z2 x PSL(2, 9). Since [K, TV] = 1, one may get N = NV' = (TK)' = [TK, TK] = [T, TT] = T' = TT, contradicting TT n K = Z3. Therefore we have either TT n K = 1 or T n K = Z2. In what follows, we discuss these two cases respectively. Set M := TK so that M/K = PSL(2, q). Case 1: T n K = 1 278 Ars Math. Contemp. 10 (2016) 183-192 In this case, we have M = T x K and T = PSL(2, q), and we shall identify T with PSL(2, q). In PSL(2, q), set 1 M x -( ° M y - ° 1 0 1 ) ' x - ^ 0 e-1 ) ' y - ^ -1 ° where F* - (e) and i G Fg. Let Q - {U | i G Fg) = Z[ and Q < T be the lift of Q. Acting on PG(1 , q), set H :- (PSL(2 , q))œ - Q x (x) and the points i G PG(1 , q) \ {to} correspond to the cosets Hytj. Take ù G and set H :- Mïï. Since H is a lift of H, we may assume that H - Q1 x (xk1) for some k1 G K, and Q1 < Q x K. Actually, we are showing Q1 - Q below. Suppose that Q - Q^ it followsjthat p - r. Then, there exist two nontrivial elements c1 G Q and k G K such that c1k G Q^ Moreover, we have |¿ù 1 n Q| > r1-2. If l > 2, then there exists a nontrivial element c2 G Q1 n Q. Since (x) has two orbits both with length 2-11 on Q \ {1} by conjugacy action, (xk1) has the same property on Q1 \ {1}, whose jwo orbits should be Bi :- {(cik)} - {c1xfcl>k} and B2 :- {c2xkl>}. Therefore, Q1 - B1 U B2 U {1}. Noting r > 3, the inverse (c1k)-1 of c1k G Q1 is not contained in B1 U B2 U {1}, a contradiction. If l - 1, then we get Q1 n Q - 1. As q - r1 - r > 5, there exist two nontrivial elements C2 G Q and k' G K such that 02k' G QQi. Again, Qi - {c1xkl>k} U {c2xkl>k'} U {1}. Sincep - r > 5, take ks G K\{1, k, k'} for some integer s. Then, (c1k)s - c1sks G Q1 is neither contained in {c1xkl>k} nor in {c2xkl>k'}, a contradiction. If l - 2 and r > 5, we shall have the same discussion as in the case l - 1. Now, we only need to consider l - 2 and r - 3, that is, q - r1 - 9. Since c1k G Q^ it is easy to check that (xk1)-1(c1k)(xk1) - c^k - c-1k G {(c1)k} C (Q1. Hence, 1 - (c1k)(c-1k) - k2 G a contradiction again. By the above discussion, we may assume that L :- - Q x (xk1) and R :- My -Q x (xk2) for some k1, k2 G K and ù' G Then by Proposition 2.2, our graph X is isomorphic to a bicoset graph X' - B(M, L, R; D) for some double coset D with two biparts: | | U - {Lk | k G K} U {Lytjk | i G Fq, k G K}, W' - {Rk 1 k G K} U {Rytjk 1 i G Fq, k G K}. Since there is only one edge from L to the block {Ryk | k G K}, we may assume that the neighbor of L corresponds to the bicoset D - Ryk3L for some k3 G K. Then X' should satisfy the following two conditions. (i) d(X') - q: Since the length of the orbit of L containing the vertex Ryk3 L is q, we have xk1 should fix the vertex Ryk3, that is, Ryk3 - Ryk3xki - Ryk3xki(yk3)-1yk3 - Ryxyk-1kiyk3 - Rx-2k—1kiyk3 - Rk2kiyk3, W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 279 which implies that k2 = k-1. (3.7) (ii) Connectedness property: By Eq(3.7), we have (D-1D) = (L(yk3)-2R(yk3)L) = jL, Ry) _ _ = (Q, xk1, Qy, xyk2) = (Q, xk1, Qy, xyk-1) < T x (k1) = M. Again, Proposition 2.2(i) implies that the graph X' is disconnected. Case 2: T n K = Z2 and T = SL(2, q) In this case, we have K = Z2 x Z2 and identify T with SL(2, q). In SL(2, q), set e= ( -i -1 ), ti = ( 0 1), x= ( 0 ), y = (0 -1), where F* = (0) and i G Fq. Let Q = (ti | i G Fq) = Zlr. Take T g one may assume that M5 = Q1 x (xk) = Zlr x Z,-1, where Q1 < K x Q and k G K. Since Q = Zl and r is an odd prime, we get Q1 = Q. Moreover, as (xk)2-1 = 1 and K = Z2 x Z2, it follows that k= e, that is, k = e and is odd. Hence, we may assume that L := Ms = Q x (xe) and R := My = Q x (xe), where T G Finally, with the same discussion as Case 1, one may get the nonexistence of X. □ Combining the lemmas in Subsections 3.1, 3.2, 3.3 and 3.4, we complete our proof of Theorem 1.1. Acknowledgements The authors thank the referee for helpful comments and suggestions. This work is partially supported by the National Natural Science Foundation of China (11271267,11371259) and the National Research Foundation for the Doctoral Program of Higher Education of China (20121108110005). References [1] M. Aschbacher, Finite group theory, Cambridge University Press, Cambridge, 1986. [2] P.J. Cameron, Finite permutation groups and finite simple groups, Bull. London. Math. Soc. 13 (1981), 1-22. 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Xu, 2-arc-transitive metacyclic covers of complete graphs, J. Comb. Theory B 111 (2015), 54-74. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 281-290 Cube-contractions in 3-connected quadrangulations Yusuke Suzuki * Department of Mathematics, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan Received 4 October 2013, accepted 10 January 2016, published online 30 January 2016 Abstract A 3-connected quadrangulation of a closed surface is said to be K' -irreducible if no face- or cube-contraction preserves simplicity and 3-connectedness. In this paper, we prove that a K'-irreducible quadrangulation of a closed surface except the sphere and the projective plane is either (i) irreducible or (ii) obtained from an irreducible quadrangulation H by applying 4-cycle additions to F0 C F(H) where F(H) stands for the set of faces of H. We also determine K'-irreducible quadrangulations of the sphere and the projective plane. These results imply new generating theorems of 3-connected quadrangulations of closed surfaces. Keywords: Quadrangulation, closed surface, generating theorem. Math. Subj. Class.: 05C10 1 Introduction In this paper, we only consider simple graphs which have no loops and no multiple edges. We denote the vertex set and the edge set of a graph G by V(G) and E(G), respectively. We say that S c V(G) is a cut of G if G - S is disconnected. In particular, S is called a k-cut if S is a cut with |S| = k. A cycle C of G is said to be separating if V(C) is a cut. Similarly, a simple closed curve 7 on a closed surface F2 is said to be separating if F2 - 7 is disconnected. A quadrangulation G of a closed surface F2 is a simple graph cellularily embedded on the surface so that each face is quadrilateral; thus, a 2-path on the sphere is not a quadrangulation. We denote the set of faces of G by F(G) throughout the paper. For quadrangulations we consider applying three reductions, called a face-contraction, a 4-cycle removal *This work was supported by JSPS Grant-in-Aid for Young Scientists (B) No. 24740056. E-mail address: y-suzuki@math.sc.niigata-u.ac.jp (Yusuke Suzuki) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 282 Ars Math. Contemp. 10 (2016) 183-192 [voVi]!/ Avi face-contraction 4-cycle removal "t 3) into the sphere, put a vertex x on one side and a vertex y on Y. Suzuki: Cube-contractions in 3-connected quadrangulations 285 the other side and add edges xv and yu for i = 0,..., k —1. The resulting quadrangulation of the sphere with 2k + 2 vertices is said to be apseudo double wheel and denoted by W2k (see the left-hand side of Figure 3). The smallest pseudo double wheel is W6, which is isomorphic to a cube, when the graphs are assumed to be 3-connected. The cycle C of length 2k is called the rim of W2k. We call a quadrangulation of the sphere obtained from W6 by a single 4-cycle addition a double cube, which is isomorphic to C4 x P2. Secondly, embed a (2k -1)-cycle C = v0vi... v2k-2 (k > 2) into the projective plane so that the tubular neighborhood of C forms a Mobius band. Next, put a vertex x on the center of the unique face of the embedding and join x to v for all i so that the resulting graph is a quadrangulation. The resulting quadrangulation of the projective plane with 2k vertices is said to be a Mobius wheel and denoted by TW2fc-1 (see the right-hand side of Figure 3). 3 Lemmas to prove Theorem 1.4 The following lemma holds not only for quadrangulations but also for even embeddings of closed surfaces F2, that is, for graphs embedded on F2 with each face bounded by a cycle of even length. Taking a dual of an even embedding and using the odd point theorem, we can easily obtain this lemma. Lemma 3.1. An even embedding of a closed surface has no separating closed walk of odd length. Let G be a quadrangulation of a closed surface F2 and let f = v0v1 v2v3 be a face of G. Then a pair {vj, vi+2} is called a diagonal pair of f in G, where the subscripts are taken modulo 4. A closed curve 7 on F2 is said to be a diagonal k-curve for G if 7 passes only through distinct k faces f0,..., fk-1 and distinct k vertices x0,..., xk-1 of G such that for each i, f and fi+1 share xj, and that for each i, {xi-1, x4} forms a diagonal pair of f of G, where the subscripts are taken modulo k. Lemma 3.2. Let G be a quadrangulation of a closed surface F2 with a 2-cut {x, y}. Then there exists a separating diagonal 2-curve for G only through x and y. Proof. Observe that every quadrangulation of any closed surface F2 is 2-connected and admits no closed curve on F2 crossing G at most once. Thus there exists a surface separating simple closed curve 7 on F2 crossing only x and y, since {x, y} is a cut of G. We shall show that 7 is a diagonal 2-curve. Suppose that 7 passes through two faces f1 and f2 meeting at two vertices x and y. If 7 is not a diagonal 2-curve, then x and y are adjacent on df1 or df2. Since G has no multiple edges between x and y, and since {x, y} is a 2-cut of G, we may suppose that x and y are adjacent in df1, but not in df2. Here we can take a separating 3-cycle of G along 7. This contradicts Lemma 3.1. □ Lemma 3.3. Let G be a 3-connected quadrangulation of a closed surface F2, and let f = v0v1v2v3 be a face of G. If the face-contraction of f at {v0, v2} breaks 3-connectedness of the graph but preserves simplicity, then G has a separating diagonal 3-curve passing through v0,v2 and another vertex x G V(G) — {v0, v1, v2, v3}. Proof. Let G' be the quadrangulation of F2 obtained from G by the face-contraction of f at {v0, v2}. Since G' has connectivity 2, G' has a 2-cut. By Lemma 3.2, G' has a separating diagonal 2-curve 7' passing through two vertices of the 2-cut. Clearly, one of the two 286 Ars Math. Contemp. 10 (2016) 183-192 vertices must be [v0v2] of G', which is the image of v0 and v2 by the face-contraction of f. (Otherwise, G would not be 3-connected, a contradiction.) Let x be a vertex of G' on Y' other than [v0v2]. Note that x is not a neighbor of [v0v2] in G'. Now apply the vertex-splitting of [v0v2] to G' to recover G. Then a diagonal 3-curve for G passing through only v0,v2 and x arises from y' for G'. □ The next lemma plays an important role in a later argument. Lemma 3.4. Let G be a 3-connected quadrangulation on a closed surface F2. If G has a separating 4-cycle C = x0x1x2x3 and a face f of G such that (i) one of the diagonal pairs of f is {x^ xi+2} for some i, and (ii) f has a separating diagonal 3-curve y intersecting C only at xi and xi+2 transversely, then there exists a 3-contractible face in G. Proof. Suppose that G has a separating 4-cycle C = x0xix2x3 and a face f bounded by axicx3. Since C is separating, G has two subgraphs GR and GL such that GR U GL = G and Gr n Gl = C. Suppose that f is contained in GR. Furthermore, we assume that GR contains as few vertices of G as possible. Since C is separating, we have df = C. By (ii), f has a separating diagonal 3-curve Y through x1, x3 and some vertex x. Note that x G V (GL) - V (C) by the condition (ii) in the lemma. Now assume that f is not 3-contractible at {a, c}. Observe that y (or the 3-cut {x1, x, x3}) separates a from c. Further, G does not have both of edges ax and cx since df = C. Therefore, there is no path of G of length at most 2 joining a and c other than ax1c and ax3c. Moreover, if {a, c} n {x0, x2} = 0, then f has no separating diagonal 3-curve joining a and c. This contradicts our assumption by Lemma 3.3 and so we may suppose that a = x0 and c = x2, and f has a separating diagonal 3-curve, say y', through a (= x0 ) and c. Since y' separates x1 and x3 and since x2 is a common neighbors of x1 and x3, y' must pass through x2, and hence we can find a face f' of GR one of whose diagonal pair is {c, x2}. Let C' be the 4-cycle x1x2x3c of G. Since deg(c) > 3, we have df' = C', and hence C' is a separating 4-cycle in GR such that C' = C. Moreover, y' and C' cross transversely at x2 and c. Therefore, C' and f' are a 4-cycle and a face which satisfy the assumption of the lemma, and moreover, C' can cut a strictly smaller graph than GR from G. Therefore, this contradicts the choice of C. □ Lemma 3.5. Let G be a 3-connected quadrangulation of a closed surface F2. If G is K3-irreducible then G is K3-irreducible. Proof. Let G be a 3-connected quadrangulation of a closed surface. Assume that G is not K3-irreducible. Then, G has either a 3-contractible face or a contractible cube. If G has a 3-contractible face, then G is not K3-irreducible. Therefore, we suppose that G has no 3-contractible face but has a contractible cube Q with an inner 4-cycle C in the following argument. Now, we apply a 4-cycle removal of C to G and let G' be the resulting quadrangulation. Let f' = dQ be the new face of G' into which C was inserted. If G' is 3-connected, G is not K3-irreducible by the definition, and we are done. Therefore, we assume that G' is not 3-connected. By Lemma 3.2, there is a diagonal 2-curve y passing through f' and another Y. Suzuki: Cube-contractions in 3-connected quadrangulations 287 face f''; otherwise, G would have a 2-cut, contrary to our assumption. Note that f'' is also a face in G. Now dQ and f" satisfy the conditions of Lemma 3.4, and hence there exists a 3-contractible face in G. However, this contradicts the above assumption. Thus, the lemma follows. □ In the following argument, we denote the set of K3-irreducible (resp. K3-irreducible) quadrangulations of a closed surface F2 by K3I(F2) (resp. K31(F2)). Lemma 3.6. Let G be a 3-connected quadrangulation of F2. If G G K3I(F2) \K3I(F2), then G has an attached cube Q such that the graph obtained from G by applying a 4-cycle removal of Q is in K3I(F2). Proof. Let G be in K3I(F2) \ K3I(F2). By the definition, G has an attached cube Q with an inner 4-cycle C which is removable, but is not contractible. We apply a 4-cycle removal of C and let G- be the resulting quadrangulation. We denote the new face of G- by f-, where f- = dQ. First, we confirm that G- is 3-connected. Otherwise, G- has a 2-cut and has a separating diagonal 2-curve 7 on F2 by Lemma 3.2. If 7 does not pass through f- then 7 would also be a diagonal 2-curve in G, a contradiction. Let f0 be the other face passed by 7. Here, fo and dQ in G satisfy the conditions in Lemma 3.4 and there exists a 3-contractible face, contrary to G being K3-irreducible. By way of contradiction, assume that G is not in K3I(F2). That is, G has either (a) a 3-contractible face or (b) a contractible cube. First, we assume (a) and let f be a 3-contractible face in G-. If f- = f, the attached cube Q in G would be contractible, contrary to G being K3-irreducible. Thus, suppose f- = f. In this case, let G' be the resulting 3-connected quadrangulation after applying a face-contraction of f in G-. Since any 4-cycle addition doesn't break the 3-connectedness of a quadrangulation, the graph obtained from G' by a 4-cycle addition to f- is clearly 3-connected. This means that f is also 3-contractible in G, a contradiction. Next, suppose (b) and let Q' be such a contractible cube with dQ' = v0viv2v3. If Q' does not contain f- as one of its five faces, Q' is also contractible in G and G would not be K3-irreducible by the similar argument as above. Thus, we assume that Q' contains f-. Let C = m0m1m2m3 denotes the inner 4-cycle of Q' where WjVj G E(Q') for i = 0,1,2,3. We consider the following two cases up to symmetry; (b-1) f- = C and (b-2) f- = v0u0u1v1. At first, suppose (b-1). Here, we apply a face-contraction of f1 = v0u0u1v1 at {u0, v1} to G. If the above face-contraction breaks the 3-connectedness of G, there exists a face f2 = v1xv3y in the outside of Q' by Lemma 3.3; note that it clearly preserves the simplicity of the graph since v1 = v3. Now, a separating diagonal 3-curve passing through {v1, u0, v3} satisfies the conditions of Lemma 3.4 and hence G is not K3-irreducible, contrary to our assumption. In fact, an analogous proof is valid for (b-2) if we try to apply a face contraction at {v1, u2} to G. Therefore the lemma follows. □ Lemma 3.7. Let G be a 3-connected quadrangulation of a closed surface F2. If G G K3I (F2) \ K3I (F2), then G can be obtained from H G K3I (F2) by applying 4-cycle additions to F0 C F(H). Proof. Assume that G G K3I(F2) \ K3I(F2). By the previous lemma, there exists a sequence of K3-irreducible quadrangulations G = G0, G1,..., Gk such that Gi+1 is obtained from Gj by a single 4-cycle removal of Cj, where Gk G K3I(F2). (Since the 288 Ars Math. Contemp. 10 (2016) 183-192 number of vertices of G is finite, Gk € K3I(F2).) Let Qi denote an attached cube in Gi with an inner 4-cycle Ci. For a contradiction, we assume that there exists l € {0,..., k — 2} such that G; is obtained from G;+1 by a 4-cycle addition which is put on a face not of F(Gk); this l should be maximal. This implies that C; is put on a face of Q;+1 as one of its five faces. Then the same argument as the proof of Lemma 3.6 holds and hence G; would not be K'3-irreducible, contrary to our assumption. Thus for each i € {0,..., k — 1}, Gi is obtained from Gi+1 by a 4-cycle addition which is put on a face of F(Gk). □ Proof of Theorem 1.4. By Lemma 3.5, we have K3I (F2) C K3I(F2). Furthermore, by Theorem 1.3 and Lemma 3.7, we obtain (i) and (ii) in the statement. Thus, we have got a conclusion. □ 4 Spherical and projective-planar cases In this section, we discuss the spherical case and the projective-planar case. Proof of Theorem 1.5. Let G be a K3-irreducible quadrangulation of the sphere. We have K3I(S2) C K^I(S2) by Lemma 3.5, where S2 stands for the sphere. If G is K3 -irreducible, then G is isomorphic to a pseudo double wheel by Theorem 1.1. If G is in KgI(S2) \ K31(S2), G can be obtained from a pseudo double wheel W2k (k > 3) by some 4-cycle additions to faces of W2k by Lemma 3.7. However if k > 4, G has a 3-contractible face (or a contractible cube), as shown in the first operation in Figure 4. (For example, the entire Figure 4 presents a sequence of a face-contraction and a cube-contraction which deforms W8 with an attached cube Q into W6, preserving the 3-connectedness.) Figure 4: W8 with an attached cube Q deformed into W6. Therefore, we only consider the case of k = 3 in the following argument. Assume that G is obtained from W6 by at least two 4-cycle additions to faces of W6. Similarly to the above argument, G would have a 3-contractible face (or a contractible cube) , as in Figure 5, contrary to G being K3-irreducible; note that it suffices to discuss these two cases, up to symmetry. Therefore, we conclude that G is obtained from W6 by exactly one 4-cycle addition. This is nothing but a double cube; observe that a double cube has no 3-contractible face and no contractible cube. □ To conclude with, we prove the projective-planar case. Y. Suzuki: Cube-contractions in 3-connected quadrangulations 289 Proof of Theorem 1.6. In this case, we use Mobius wheels Wk (k > 3) and Qp as base graphs by Theorem 1.2. First we consider the former case. Similarly to the previous proof (and see Figure 6), we consider only a Mobius wheel W3 as a base to which we apply some 4-cycle additions. However, W3 (= Qp) is isomorphic to the complete graph with four vertices, and hence it is irreducible. This fact implies that every G obtained from W3 by applying at most three 4-cycle additions is fc3 -irreducible since any face-contraction and any cube-contraction to G destroys the simplicity of the graph, or results in a vertex of degree 2. From this case, we obtain exactly three quadrangulations in K'3I(P2) \ K3I(P2), up to homeomorphism, where P2 stands for the projective plane. Figure 6: W5 with an attached cube Q deformed into W3. Similarly, as the latter case, we obtain the other ten quadrangulations in K31(P2) \ K3I(P2) from Qp; consider all the way to put attached cubes into faces of Qp, up to symmetry. As aresult, we have \K'3I(P2) \ K3I(P2)| = 13 in total. □ 290 Ars Math. Contemp. 10 (2016) 183-192 References [1] V. Batagelj, An inductive definition of the class of 3-connected quadrangulations of the plane, Discrete Math. 78 (1989), 45-53, doi:10.1016/0012-365X(89)90159-3. [2] G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas and P. Wollan, Generation of simple quadrangulations of the sphere, Discrete Math. 305 (2005), 33-54, doi:10.1016/j.disc. 2005.10.005. [3] H. J. Broersma, A. J. W. Duijvestijn and F. Gobel, Generating all 3-connected 4-regular planar graphs from the octahedron graph, J. Graph Theory 17 (1993), 613-620, doi:10.1002/jgt. 3190170508. [4] M. Nagashima, A. Nakamoto, S. Negami and Y. Suzuki, Generating 3-connected quadrangulations on surfaces, Ars Combin. 116 (2014), 371-384. [5] A. Nakamoto, Irreducible quadrangulations of the Klein bottle, Yokohama Math. J. 43 (1995), 125-139. [6] A. Nakamoto, Irreducible quadrangulations of the torus, J. Combin. Theory Ser. B 67 (1996), 183-201, doi:10.1006/jctb.1996.0040. [7] A. Nakamoto, Generating quadrangulations of surfaces with minimum degree at least 3, J. Graph Theory 30 (1999), 223-234, doi:10.1002/(SICI)1097-0118(199903)30:3(223::AID-JGT7)3.3. CO;2-D. [8] A. Nakamoto and K. Ota, Note on irreducible triangulations of surfaces, J. Graph Theory 20 (1995), 227-233, doi:10.1002/jgt.3190200211. [9] S. Negami and A. Nakamoto, Diagonal transformations of graphs on closed surfaces, Sci. Rep. Yokohama Nat. Univ. Sect. I Math. Phys. Chem. (1993), 71-97. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 291-322 One-point extensions in n3 configurations William L. Kocay * Computer Science Department and St. Paul's College, University of Manitoba, Winnipeg, Manitoba, R3T 2N2 Canada Received 30 October 2014, accepted 18 December 2015, published online 30 January 2016 Given an n3 configuration, a 1-point extension is a technique that constructs an (n +1)3 configuration from it. It is proved that all (n + 1)3 configurations can be constructed from an n3 configuration using a 1-point extension, except for the Fano, Pappus, and Desargues configurations, and a family of Fano-type configurations. A 3-point extension is also described. A 3-point extension of the Fano configuration produces the Desargues and anti-Pappian configurations. The significance of the 1-point extension is that it can frequently be used to construct real and/or rational coordinatizations in the plane of an (n + 1)3 configuration, whenever it is geometric, and the corresponding n3 configuration is also geometric. Keywords: Fano configuration, Pappus, Desargues, (n, 3)-configuration. Math. Subj. Class.: 51E20, 51E30 1 Projective Configurations A projective configuration consists of a set S of points and lines, and an incidence relation n, such that two lines intersect in at most one point. We denote this by (S, n). For example, a triangle with points A, B,C and lines a, b, c can be represented by the pair ({A, B, C, a, b, c}, {Ab, Ac, Ba, Bc, Ca, Cb}). A configuration (S, n) can also be viewed as a bipartite incidence graph of points versus lines. We will always assume that the incidence graph of a configuration is connected. Excellent references on configurations are the recent books by Griinbaum [7], and by Pisanski and Servatius [11]. An n3-configuration is a projective configuration with n points and n lines such that every line is incident on 3 points, and every point is incident on 3 lines. There is a unique 73-configuration, the Fano configuration, and a unique 83-configuration, the Mobius-Kantor *This work is partially funded by a discovery grant from the Natural Sciences and Engineering Research Council of Canada. E-mail address: bkocay@cs.umanitoba.ca (William L. Kocay) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 292 Ars Math. Contemp. 10 (2016) 183-192 configuration. In 1887, Martinetti [10] presented a method to construct the (n +1)3 configurations from the n3 configurations. This is described in [7, 6]. Boben [1, 2] has analysed and extended Martinetti's construction significantly. Important related work has also been done by Carstens, Dinskiand Steffen [4]. See also [12]. A recent paper [13] by Stokes studies extensions of configurations in a very general setting. The 1-point extension presented here can be related to Stokes's construction, but does not follow directly from it. An n3 configuration which can be represented by a collection of points and straight lines in the real or rational plane, such that all incidences are respected, and no two points or two lines coincide, and no unwanted incidences occur, is termed a geometric n3 configuration. In order to show that an n3 configuration is geometric, the usual method is to assign suitable homogeneous coordinates to its points and lines. We call this a coordinatization of the configuration. Some n3 configurations are not geometric configurations, although it is currently an unsolved problem to determine which n3 configurations are geometric. The purpose of this paper is to present a theorem, the 1-point extension theorem, which describes another method to construct an (n +1)3-configuration from an n3-configuration; and to characterize which configurations can be obtained in this way. The significance of this construction is that if the n3 configuration is geometric, with a given coordinatization, then there is usually a simple method to extend the coordinatization to the (n + 1)3 configuration, that is, the (n + 1) 3 configuration will also be geometeric. This is too long to include here, it will be the subject of another paper, currently in preparation [8]. In particular the following theorem is proved. Theorem 1.1. Let (E, n) be an (n + 1)3-configuration. Then (E, n) can be constructed by a 1-point extension from an n3-configuration if and only if (E, n) is not one of the following configurations: a) the Fano configuration, b) the Pappus configuration, c) the Desargues configuration, d) a Fano-type configuration (to be described). We begin with the idea of a 1-point extension in an n3-configuration. Theorem 1.2. (1-Point Extension) Let (E, n) be an n3-configuration. Let a\,a2, a3 be 3 distinct points in E, and let t\,t2, ¿3 be 3 distinct lines in E such that a1 = ¿1 n t2, a2 = ¿2 n ¿3 and a3 G t3, where a3 G ¿1. We can represent this in tabular form as (E,n) ¿1 ¿2 ¿3 • •• ai ai a2 ■ ■ ■ ■ a2 a3 ••• where the dots indicate other points of the configuration. Let ¿' be the third line containing a1. Suppose further that if ¿' n ¿3 = 0, then ¿' n ¿3 = a3. Construct a new configuration (E', n') as follows. E' = E U ja0^0} where a0 is a new point and ¿0 is a new line. n = n — ja^, a^2, a^3} U ja^3, a^0, a^0, a^0, a^, a^2}. We can represent this in tabular form as W L. Kocay: One-point extensions in n3 configurations 293 (E', n') 4 «2 ao <33 • ao • Here the dots represent exactly the same points as in the previous table. Then (E', n') is an (n + 1)3-configuration. Proof. The only incidences in which (E', n') and (E,n) differ are those involving i0, i^ i2, i3. It is easy to verify from the tables that each of a1, a2 and a3 occurs in exactly 3 lines in both (E', n') and (E, n), and that a0 also occurs in exactly 3 lines. We must still verify that any two lines of (E', n') intersect in at most one point. Notice that i0 intersects and i2 in exactly one point, since a3 ^ ¿1, i2. Also, i0 intersects i3 in exactly one point. If i = ¿1, i2, i3 is any line of (E, n) intersecting ¿1, then in (E', n'), it intersects in either 0 or 1 point. If i intersects i2 in (E, n), then in (E', n'), it intersects i2 in either 0 or 1 point. If i = i', the third line of (E, n) containing a1, then in (E', n'), i intersects i3 in only a1, because of the condition concerning i'. If i = i' and i intersects i3 in (E, n), then then since a1 ^ i3 in (E, n), it follows that i intersects i3 in 0 or 1 point in (E', n'). Finally, if i is any line of (E, n) not intersecting i1, i2, then it does not intersect i1, i2 in (E', n'). If i does not intersect i3 in (E, n), it may intersect i3 in a1 in (E', n'). This completes the proof of the theorem. □ Example 1.3. The Fano configuration can be represented by the following table. Fano i1 i2 i3 i4 i5 i6 i7 1 2 3 4 5 6 7 2 3 4 5 6 7 1 4 5 6 7 1 2 3 Choose i1,i2,i3 as indicated, and choose a1 = 2, a2 = 3, a3 = 6, and let a0 = 8. Notice that the third line containing a1 is i' = i6, which intersects i3 in a3 = 6. Then by Theorem 1.2, the following table represents an 83-configuration, which is known to be unique. 83-config ¿0 ¿1 ¿2 ¿3 ¿4 ¿5 ¿6 ¿7 3 1 2 2 4 5 6 7 6 4 5 3 5 6 7 1 8 8 8 4 7 1 2 3 The 83 -configuration can be viewed as a double cover of the cube [9]. It is possible to apply a 1-point extension to this configuration in two possible ways, resulting in two distinct 93-configurations. The third 93-configuration, known as the Pappus configuration, cannot be obtained in this way. The 1-point extension theorem can be illustrated by the diagram of Figure 1. In (E, n), we have a substructure consisting of 3 points ai, a2, a3, and 3 lines, ¿1, ¿2,, sequentially incident, forming a self-dual substructure contained in the n3-configuration. After the extension, we find that (E', n') contains a triangle with vertices a1, a2, a0 and sides ¿2, ¿3,¿0, where the third point on ¿0 is a3, and the third line through a0 is ¿4. This is again a self-dual substructure in the configuration. ¿2 ¿3 ai ai ao a2 294 Ars Math. Contemp. 10 (2016) 183-192 Figure 1: A 1-point extension with 3 points Corollary 1.4. In (E', n'), the third line through ai does not intersect £i; the third point on £3 is not collinear with a3; and the third line through a2 does not intersect £2. Proof. If there were a line £ in (E', n') through ai which intersected £i in a point u, then in (E, n), £ would intersect £i in u and ai, which is impossible. If there were a point x in (E', n') on £3 collinear with a3, then the line £ containing a3 and x would also be a line in (E, n), where it would intersect £3 in two points. Finally, if there were a line £ in (E', n') through a2 which intersected £2 in a point u, then in (E, n), £ would intersect £2 in a2 and u, which is impossible. □ The purpose of this paper is to characterize the configurations that can be obtained using 1-point extensions. In practice, the 1-point extensions are very easy to find and apply, and can easily be done by computer. However, the characterization of which configurations can be obtained by them is very long and tedious. We shall refer to the Fano, Pappus, and Desargues configurations, illustrated in Figure 1.1. The conditions of Corollary 1.4 will be used frequently in the characterization. We state them here. We are concerned with an ordered triangle, denoted A(i, j, k), where i, j and k are the first, second, and third vertices, respectively, of the triangle. The line containing i and j is denoted ij, etc. Definition 1.5. Let (E, n) be a configuration containing an ordered triangle A(i, j, k). We define the following 3 conditions: A) The third line through k intersects ij ; W. L. Kocay: One-point extensions in n3 configurations 295 B) The third line through i intersects the third line through j; C) The third point on £ik is collinear with the third point on j. The definition is illustrated in Figure 3. Figure 3: Conditions A, B and C for triangle A(i, j, k) Theorem 1.6. Let (S', n' ) be an (n + l)3-configuration containing a triangle A. If conditions A, B and C do not apply to some ordering of the triangle, then (S', n') can be derived from an n3-configuration by a 1-point extension. Proof. Let the ordered triangle to which conditions A, B and C do not apply be A(a0, a1, a2), and let the sides of the triangle be l0, £2,£3, where a0 = l0 n £2, ai = £2 n £3, a2 = £3 n l0. Let a3 be the third point on £0, and let £1 be the third line through a0. Observe that a3 ^ £1. These incidences are characterized by the following table. (s, n) £0 £1 £2 £3 a2 a0 a1 a1 0,3 ■ a0 a2 a0 ■ ■ ■ We can then delete a0 and £0, and change the incidences to the following. (s', n') £1 £2 £3 a1 a1 a2 ■ a2 a3 Call the result (S', n'). If £ is the third line through a2 in (S, n), then since condition A does not apply, we know that in (S', n'), £ and £2 intersect in just one point. If £ is the third line through a1 in (S, n), then since condition B does not apply, we know that in (S', n'), £ and £1 intersect in just one point, a1. Since £ n £3 = a1 in (S, n), it follows that in (S', n'), if £ and £3 intersect, they intersect in a3. If £ is any line other than £0 through a3 in (S, n), then since condition C does not apply, we know that in (S', n'), £ and £3 intersect in just one point. The result is an n3-configuration to which Theorem 1.2 applies. □ 296 Ars Math. Contemp. 10 (2016) 183-192 Given an ordered triangle A(i, j, k), the dual is an ordered triangle whose sides are lines which can be denoted i', j', k'. The dual of condition A is that the third point on k' is collinear with i' n j'. But this is just condition A again applied to the triangle A(i' n k', j' n k', i' n j'). So condition A is self-dual. The dual of condition B is that the third point on i' is collinear with the third point on j'. This is just condition C applied to the triangle A(i' n k', j' n k', i' n j'). So B and C are dual conditions. Theorem 1.6 is the main tool which we will use to characterize the extensions. We will find all configurations such that at least one of conditions A, B, and C apply to every ordering of every triangle. We will also need longer cycles than triangles. 2 The General Extension Theorem Before beginning the characterization of the n3-configurations that can be obtained by 1-point extensions, we generalize Theorem 1.2 to m points and m lines, sequentially incident. Theorem2.1. (General 1-Point Extension) Let (E, n) be an n3-configuration. Let ai, a2, ..., am be m distinct points in E, where 3 < m < n, and let ¿^ ¿2,..., ¿m be m distinct lines in E such that ai = ¿1 n t2, a2 = ¿2 n ¿3, ..., am-i = n £m, and am G £m. Suppose that am-i, am G ¿^ ¿2, and that ai G ^i+3, where i =1, 2,..., m — 3. We can represent this in tabular form as (e, n) ¿i ¿2 ¿3 . . . ¿m—1 ¿ m ai ai a2 . . . am—2 am — 1 a2 a3 . . . am—1 am where the dots indicate other points of the configuration. Let ¿j be the third line containing ai, where 1 < i < m — 2. Suppose further that if ¿j n ¿i+2 = 0, then ¿j n ¿i+2 = ai+2. Construct a new configuration (E', n') as follows. E' = E U {ao,¿o} where a0 is a new point and ¿0 is a new line. n' = n — {a1¿1, a2¿2,..., am¿m} U {a^3, a2¿4,..., am—am—, a^0, a0¿0, a0¿1, a0¿2}. We can represent this in tabular form as (E', n') ¿0 ¿i ¿2 ¿3 . . . ¿ • • ^m—1 ¿ m am—1 a0 a0 ai . . . am—3 am—2 am ai a2 . . . am —2 am— 1 a0 Here the dots represent exactly the same points as in the previous table. Then (E', n') is an (n + 1)^-configuration. Proof. The only incidences in which (E', n') and (E, n) differ are those involving ¿0, ¿1, ¿2,..., ¿m. It is easy to verify from the tables that each of a1, a2,..., am occurs in exactly 3 lines in both (E', n') and (E, n), and that a0 also occurs in exactly 3 lines. We must still verify that any two lines of (E', n') intersect in at most one point. Notice that ¿0 intersects ¿4 and ¿2 in exactly one point, since am-1, am G ¿4, ¿2. It does not intersect ¿3,..., ¿m-1, and it intersects in exactly one point. Let I = ¿1, ¿2,..., be a line of (E, n). If I intersects in (E, n), then in (E', n'), it intersects in either 0 or 1 point. If I intersects ¿2 in (E, n), then in (E', n'), it intersects W. L. Kocay: One-point extensions in n3 configurations 297 12 in either 0 or 1 point. Suppose that i intersects i3 in (E, n). If i = i^, then ini3 = a3 in (E, n) according to the condition of the theorem concerning ij. It follows that i n i3 = a1 in (E', n'). If i = i1, then i intersects i3 in either 0 or 1 point in (E', n'). An identical argument holds if i intersects one of i4,..., im in (E, n). Suppose that i does not intersect i1 in (E, n). Then it also does not intersect i1 in (E', n'). Similarly, if i does not intersect i2 in (E, n), then it also does not intersect i2 in (E', n'). Suppose that i does not intersect i3 in (E, n). Then in (E', n'), it may intersect 13 only in a1. A similar argument holds if i does not intersect i4,..., im. Finally, let i, and ij, where 1 < i < j < m, be two lines of (E, n). If j = i + 1, then ij and ij intersect in one point in both (E, n) and (E', n'). Suppose that j = i + 2. If ii n ij = 0 in (E, n), then it is also 0 in (E', n'). Now i n ij = ai-1 in (E, n) (when i > 1), because of the hypothesis that ak & ik+3. Also, i, n ij = a,, because ii+1 contains a, and ai+1. It follows that |i, n ij| is the same in (E, n) and (E', n') when j = i + 2. Suppose now that j > i + 3. It is easy to see that |i, n ij | < 1 in (E', n'). This completes the proof of the theorem. □ Theorem 2.1 is illustrated in Figure 4, with m = 4. This general form of Theorem 2.1 is stated separately from Theorem 1.2, because the form with m = 3 is simpler, and because we shall mostly only require Theorems 1.2 and 1.6 when characterizing extensions. Figure 4: A 1-point extension with 4 points An ordered cycle in a configuration is a sequence of distinct points and lines which are cyclicly incident, for example C = (ai, ¿4, a2, ..., am, where a = £¿-1 n for i = 2,3,..., m, and a1 = £m n £1. Here m > 3. Each point of C is incident on two lines of C, and vice versa. Corollary 2.2. Let (E, n) and (E', n') be as in Theorem 2.1, so that C = (a0, £2, a1, £3, . . . , am—2 , £m, 1, £0) is an ordered cycle in (E', n'). Then in (E', n'): i) the third points of £m and £0 are not collinear; ii) the third point on £j is not contained in the third line through a^, for i = 2,..., m — 1; iii) the third lines through a0 and a1 do not intersect. 298 Ars Math. Contemp. 10 (2016) 183-192 Proof. The third point of i0 is am. If there were a line i in (E', n') containing am and the third point of im, then in (E, n), i and im would intersect in two points, which is impossible. Let i be the third line through aj in (E', n'), for some i = 2,..., m — 1, and let u be the third point on ij. Suppose that u G i. In (E', n'), aj is contained in ij+1 and ij+2, but in (E, n), aj is contained in ij and ij+1. We then find that in (E, n), i n ij = {u, a4}, which is impossible. The third line through ao is ii . Let i be the third line through ai . If i n ii = u in (E', n'), then in (E, n), i n i1 = {u, a1}, which is impossible. □ Observe that a triangle is a set of three distinct points and lines that are cyclically incident. Similarly, we define a quadrangle to be a set of four distinct points and lines that are cyclically incident. We will also need conditions similar to A, B, C for quadrangles. An ordered quadrangle with vertices i, j, k, m is denoted □ (i, j, k, m). In analogy with Definition 1.5 and Corollary 2.2, we make the following definition for a quadrangle. Definition 2.3. Let (E, n) be a configuration containing an ordered quadrangle □ (i, j, k, m). We define the following 4 conditions: D) The third point on ijm is collinear with the third point on ikm; E) The third line through m intersects ijk; F) The third line through k intersects ijj; G) The third line through j intersects the third line through i. These conditions are illustrated in Figure 5. D Figure 5: Conditions D, E, F, G for quadrangle □ (i, j, k, m) The analog of Theorem 1.6 for general 1-point extensions is the following. Theorem 2.4. Let (E', n') be an (n + 1)^-configuration containing an ordered cycle C = (ao,^2, ai,4, «2, 4,..., «m-2,^m, flm-i,^o), where m > 4; ao, ai,..., am_i are distinct points; and ... ,^m-1 are distinct lines. Let denote the third line containing a0 and let am denote the third point on t0. Suppose that is distinct from 4i, ^3,..., and that a2 G Let £i denote the third line containing ait for i = 1,2,..., m — 1. Suppose that £i does not not contain the third point of iit for i = 2,..., m — 1; that n = 0; and that am is not collinear with the third point of tm. Then (E', n') can be derived from an n3-configuration by a 1-point extension. Proof. The incidences of the ordered cycle can be represented by the following table. W. L. Kocay: One-point extensions in n3 configurations 299 (s, n) io io i1 i2 is . . . im_ 1 i m am_ 1 ao ao a1 . . . am_s am_2 am a1 a2 . . . am_2 am_ 1 ao We can then delete a0 and 4i, and change the incidences to the following. (£', n') h ai i2 is . . . im _ 1 i m a1 a2 . . . am_2 am_ 1 a2 as . . . am_1 am Call the result (S, n). It is clear that each point of (S, n) is contained in exactly three lines. We have to show that any two lines intersect in at most one point in (S, n), and that ¿4, i2,is,..., im are distinct lines in (S, n). Any two of i^ i2,..., im intersect in at most one point because we began with an ordered cycle of distinct points, and because a2 G ¿1. Let i be any line not in this set. Suppose that i intersects ij in two points, for some i = 2,..., m — 1. Now ij contains ai_1, aj and a third point z. If i contained a^ then i = ij, which does not intersect ij in (S', n'), by assumption. Therefore aj G i. Otherwise i must contain aj_1 and z. But these points are in ij in (S', n'), and i is unchanged. It follows that i intersects i2,..., im-1 in at most one point each. Suppose that i intersects i1 in two points in (S, n). Now i1 contains a1 and two other points u, v. As u and v are both on i1 in (S', n'), it follows that i does not contain both u and v. Therefore i = ii. But by assumption, ii n i1 = 0 in (S', n'). Suppose that i intersects im in two points in (S, n). The two points cannot be am-1, am, because these points occur on iQ in (S', n'). They cannot be am-1 and a third point w, because these points occur on im in (S', n'). And they cannot be am and the third point w, because by assumption, am is not collinear with the third point of im in (S', n'). We conclude that (S, n) is an n3-configuration to which the conditions of Theorem 2.1 apply. □ Corollary 2.5. Let (S', n') be an (n+1)3-configuration containing a quadrangle D(i, j, k, m). If conditions D, E, F and G do not apply to some ordering of the quadrangle, and if the third line through i does not contain k, then (S', n') can be derived from an n3-configuration by a 1-point extension. Proof. The conditions D, E, F, G, and a2 = k G i1 are the conditions of Theorem 2.4 applied to an ordered quadrangle. □ Theorem 2.6. Let (S', n') be an (n + 1)^-configuration. If (S', n') does not contain a triangle, then it can be derived by a 1-point extension from an n3-configuration. Proof. Choose a cycle of smallest possible length in (S', n'). Denote the cycle by (ao, i2, a1, i3, a2, i4,. .. , am_2, im, am_ 1, io), where m > 4. Let i1 be the third line containing aQ, and let am be the third point on iQ. This can be denoted in tabular from by 300 Ars Math. Contemp. 10 (2016) 183-192 (s, n) ¿q ¿i Om-1 aQ am * aQ • Let ¿i denote the third line containing ai, where i = 1, 2,..., m — 1. If ¿i were to intersect ¿i in a point z, where i = 2,..., m — 1, this would create a triangle A(ai-1, ai, z). If ¿1 were to intersect ¿1 in a point u, this would create a triangle A(aQ, a1, u). If am were collinear with the third point w of ¿m, this would create a triangle A(am-1, am, w). If ¿1 contained a2, this would create a triangle A(aQ, a1, a2). It follows that the conditions of Theorem 2.4 apply, so that (S', n') can be derived by a 1-point extension from an n3-configuration. □ 3 Fano-Type Configurations Let F denote the Fano configuration, the unique 73 configuration. We will use three subconfigurations to build a family of n3 configurations which cannot be obtained by 1-point extensions. Definition 3.1. Denote by F' the unique configuration obtained from F by removing a single incidence. Denote by Fi the unique configuration obtained from F by removing a line. Denote by Fp the unique configuration obtained from F by removing a point. Note that Fi and Fp are dual configurations. ¿2 ¿3 aQ ai a1 a2 . . ¿ m- 1 ¿m am-3 am-2 am - 2 am -1 Figure 6: The configurations Fi, Fp and F' The configurations Fe, Fp and F' are not n3-configurations. They can be used as building blocks of n3 configurations, which we call Fano-type configurations. F' has one point on only two lines, and one line containing only two points. Fp has three lines containing only two points. Every point is in three lines. Fe has three points in only two lines. Every line contains three points. These are illustrated schematically in Figure 7, where the points missing a line are indicated as black circles, and the lines missing a point are indicated as lines. These sub-configurations can be used as modules, which can be connected together like vertices of a graph, to create graphs representing n3 configurations. For example, two or more copies of F' can be connected into a cycle or path of arbitrary length. If only Fe and Fp are used, the resulting structure is a bipartite graph. W. L. Kocay: One-point extensions in n3 configurations 301 Theorem 3.2. Let G be a multigraph which is isomorphic to either a cycle of length > 2, or a subdivision of a 3-regular bipartite multigraph, with bipartition (X, Y ). Replace each vertex of X by a configuration Fp, replace each vertex of Y by a configuration Fi, and replace each vertex of degree two by a configuration F '. The result is an n3 configuration which can not be obtained by a 1-point extension. Proof. Refer to Figure 8, showing a cycle of length four, and a configuration constructed from the unique 3-regular bipartite multigraph on four vertices. Figure 8: Configurations constructed from F', Fi and Fp We must show that the n3 configurations constructed like this cannot be obtained by a 1-point extension. Observe first that the Fano configuration F is a projective plane, so that every two points are contained in a line, and every two lines intersect in a point. Consequently, every triangle contained in F', Fi or Fp has an ordering which satisfies one of conditions A, B or C .By Corollary 1.4, a Fano-type configuration cannot be obtained by a triangular 1-point extension (Theorem 1.2). Suppose that it can be obtained by a general 1-point extension (Theorem 2.1). By Corollary 2.2, there must be an ordered cycle C of length > 4 satisfying certain conditions. Let C = (a0, i2,a\,i3,..., am-2,im, am-l, £0) be as in Corollary 2.2, and let £i denote the third line containing a4, where i = 1,2,... ,m-1. Let l\ denote the third line containing a0, and let am denote the third point on l0. If C were contained within an F', Fi or Fp, then C would have length 4, because any 5 points of F necessarily contain three collinear points. But in F', Fi or Fp, every ordered quadrangle satisfies at least one of conditions D, E, F, G, since the Fano configuration is a projective plane. It follows that C is not contained within an F', Fi or Fp. Consider the portion of C contained within some F', Fi or Fp. It is a sequence of sequentially incident points and lines. Suppose first that it is contained within an F'. Referring to Figure 6 we see that the 302 Ars Math. Contemp. 10 (2016) 183-192 shortest possible portion of C contained within an F' is (ai; ¿i+2, ai+1, ¿i+3, ai+2, ¿¿+4), for some i = 0,1,..., m - 1 where subscripts are reduced modulo m. If ai+2 = a0, a1, then ¿¿+2 contains the third point of ¿i+2, which is in F'. If ai+2 = a0, then ai+1 = am-1 and ¿i+2 = ¿m, so that am is collinear in F' with the third point of ¿m. If ai+2 = a1, then ai+1 = a0, so that ¿1 and ¿1 are in F' and ¿1 n ¿1 = 0. Thus, the conditions of Corollary 2.2 are never satisfied if a portion of C is contained within an F'. Suppose next that a portion of C is contained within an F^. Referring to Figure 6 we see that the shortest possible portion of C contained within an Fe is (^¿i+2, ai+1^i+3, ai+2), for some i = 0,1,...,m - 1 where subscripts are reduced modulo m. If ai+2 = a0, a1, then ¿¿+2 contains the third point of ¿i+2, which is in F^. If ai+2 = a0, then ai+1 = am-1 and ¿i+2 = ¿m, so that am is collinear in Fe with the third point of ¿m. If ai+2 = a1, then ai+1 = a0, so that ¿1 and ¿1 are in Fe and ¿1 n ¿1 = 0. Thus, the conditions of Corollary 2.2 are never satisfied if a portion of C is contained within an F^. A similar result holds for Fp, which is the dual of F^. We conclude that the Fano-type configurations can not be obtained by a 1-point extension. □ 4 The Characterization Theorem In this section we will assume that (£, n) is an n3-configuration which cannot be derived by a 1-point extension. It follows from Theorem 2.6 that we can assume that (£, n) has a triangle. Let the points of (£, n) be numbered 1,2,..., n. Without loss of generality, we can assume that A(2, 3,1) is a triangle in (£, n). This is illustrated in Figure 9. It will be convenient to omit the commas and brackets from expressions like A(2,3,1), and write simply A231. We divide the analysis into two cases according to whether or not (£, n) has a triangle satisfying condition A. The theorem obtained will be the following. Theorem 4.1. If (£, n) is an n3-configuration which cannot be obtained from a 1-point extension, then either: i) (£, n) is one of the Fano, Pappus, or Desargues configurations; or ii) (£, n) is a Fano-type configuration. Proof. The proof of this theorem is very long, involving an analysis of many possible cases. Case A. (£, n) has a triangle satisfying condition A. W. L. Kocay: One-point extensions in n3 configurations 303 Let the ordered triangle be A231, as above. Condition A tells us that the third line through 1 intersects ^23. Call the point of intersection 4. This is shown in Figure 9. We will show that any n3 configuration that cannot be obtained by a 1-point extension, and which satisfies Condition A, is either a Fano-type configuration, or the Fano configuration. Now consider A142. It currently does not satisfy conditions A, B, or C. Since every triangle must satisfy at least one of these conditions, there are three possibilities, which we indicate by A142A, A142B, and A142C. These are shown in Figure 10. In A142A, the third line through 4 intersects £12 (in point 5). In A142B, the third lines through 1 and 4 intersect (in point 5). In A142C, the third points on ii2 (point 5) and t24 (point 3) are collinear. These three structures are easily seen to be isomorphic, by relabelling the points. Each structure is self-dual, having two points incident on 3 lines each, and two lines each containing 3 points. Thus, without loss of generality, we can assume that the subconfiguration A142A exists in (£, n) in Case A. Consider triangle A124. It currently does not satisfy condition A, B, or C. Since it must satisfy at least one of these conditions, there are three possibilities, which we indicate by A142AA124A, A142AA124B, and A142AA124C. These are shown in Figure 11. The structures A142AA124B and A142AA124C are duals of each other. The first has 6 points and 5 lines, while the other has 5 points and 6 lines. It can be verified by exhaustion that every ordered triangle in these structures satisfies at least one of conditions A, B, or C. Case A142AA124A. Consider the quadrangle D6431 in A142AA124A. It must satisfy at least one of 304 Ars Math. Contemp. 10 (2016) 183-192 conditions D, E, F, G (see Figure 5). Condition D is possible only if ¿25 intersects ¿43. Condition E is not possible. Condition F is possible only if the third line through 3 intersects ¿46. Condition G is possible only if there is a line ¿56. These cases are illustrated in Figure 12. Figure 12: A142AA124AD6431D, A142AA124AD6431F, A142AA124AD6431G Now the diagrams A142AA124AD6431D and A142AA124AD6431G are duals of each other, for the mapping which sends points 1,2,3,4, 5,6,7 of D to ¿15,¿16, ¿25, ¿24, ¿46, ¿13, ¿56 of G is an isomorphism. Therefore we need only consider cases D and F. Case A142AA124AD6431D. It can be verified that all triangles of the diagram satisfy one of conditions A, B, C. Consider the quadrangle D3164. Condition D is only possible if point 7 lies on line ¿46. Condition E is not possible. Condition F is only possible if there is a line ¿67. Condition G is only possible if there is a line ¿35. These cases are illustrated in Figure 13. Figure 13: A142AA124AD6431DD3164D, F, and G Case A142AA124AD6431DD3164D. It can be verified that every triangle satisfies at least one of conditions A, B, C, and every quadrangle satisfies at least one of conditions D, E, F, G. This configuration is isomorphic to the Fano configuration, with one line removed (¿356), which we denote as F^. The dual configuration is the Fano configuration, with one point removed, which we denote as Fp. Case A142AA124AD6431DD3164F. Consider the quadrangle D2376. Condition D requires that ¿15 intersects ¿67, which W. L. Kocay: One-point extensions in n3 configurations 305 is impossible. Condition E requires that ¿46 contains point 1, which is impossible. Condition F requires that ¿75 contains point 4, which is impossible. Condition G requires a line ¿35. The result is illustrated in Figure 14. \7 / (l \6/ 2 Figure 14: Case A142AA124AD6431DD3164FD2376G We then consider quadrangle 06237. Condition D requires that ¿45 intersects ¿67, which is impossible. Condition E requires that ¿75 contains point 4, which is impossible. Condition F requires that ¿35 contains point 1, which is impossible. Condition G requires that ¿46 and ¿25 intersect in point 5, which is impossible. We conclude that case A142AA124AD6431DD3164F is not possible. Case A142AA124AD6431DD3164G. Consider the quadrangle 104316. Condition D requires that ¿25 intersects ¿46. The point of intersection can only be 7. Condition E requires that ¿75 contains point 6, which is impossible. Condition F requires that ¿15 contains point 2, which is impossible. Condition G requires a line ¿356. These cases are illustrated in Figure 15. Figure 15: Cases A142AA124AD6431DD3164GD4316D and G These two configurations are easily seen to be isomorphic, by the permutation of the points given by (2, 3,4)(5,6, 7), mapping D onto G. They are both isomorphic to the Fano configuration, with one incidence removed, denoted by F'. Every triangle satisfies at least one of conditions A, B, C, and every quadrangle satisfies at least one of conditions D, E, F, G. 306 Ars Math. Contemp. 10 (2016) 183-192 Note that we can complete F' to the Fano configuration, which can not be constructed by a 1-point extension. We summarise Case A as follows: Consider an n3 configuration (E, n), where n > 7, which cannot be constructed by a 1-point extension. Every triangle satisfying condition A is contained in a unique sub-configuration isomorphic to one of Fg, Fp or F'. Case B. (E, n) has no triangle satisfying condition A. We begin with triangle A231. It must satisfy condition B or C. These two possibilities are shown in Figure 16. Figure 16: A231B and A231C These two structures are duals of each other. Hence we can assume without loss of generality that (E, n) contains the structure A231B. Consider the triangle A123. It must satisfy condition B or C. We must take these as two separate cases, Case BA123B and Case BA123C. They are shown in Figure 17. It will be necessary to examine a great many subcases. Figure 17: Cases BA123B and BA123C Case BA123B. Consider triangle A132. There are two possibilities, cases BA123BA132B and BA123BA132C, which must both be considered. They are shown in Figure 18. Case BA123BA132B. Consider triangle A243. There are two choices BA123BA132BA243B and W. L. Kocay: One-point extensions in n3 configurations 307 Figure 18: Cases BA123BA132B and BA123BA132C BA123BA132BA243C. They are shown in Figure 19. These structures both have 7 points {1, 2,..., 7}, so that a mapping from the first to the second can be denoted by a permutation. It is easy to see that the permutation (1,2, 3)(4, 6,5)(7) maps the first to the second. Thus, without loss of generality, we can suppose that (S, n) contains the structure BA123BA132BA243B. Figure 19: Isomorphic cases BA123BA132BA243 B and C Consider triangle A342. There are two possibilities, BA123BA132BA243B A342B and BA123BA132BA243BA342C. They are shown in Figure 20. We must consider both possibilities. Figure 20: Cases BA123BA132BA243BA342B and BA123BA132BA243BA342C This is beginning to look remarkably like the Pappus configuration. Case BA123BA132BA243BA342B. Consider the quadrangle D1248. At least one of conditions D, E, F, G must be satisfied. Of these, it is only possible to satisfy condition E, namely the third line 308 Ars Math. Contemp. 10 (2016) 183-192 through 8 must intersect ¿24. The point of intersection can only be 5. Therefore the left diagram of Figure 21 must exist in (£, n). Figure 21: Cases BD1248E and BD1248ED7238E Consider the quadrangle D7238. At least one of conditions D, E, F, G must be satisfied. Of these, it is only possible to satisfy condition E, namely the third line through 8 must intersect ¿23. Therefore the right diagram of Figure 21 must exist in (£, n). Consider the quadrangle D3159. It is only possible to satisfy condition E, namely the third line through 9 must intersect ¿15 in point 6. Therefore the following structure (Figure 22) must exist in (£, n). Figure 22: Case BD1248ED7238ED3159E Consider the quadrangle D1347. It is only possible to satisfy condition E, namely the third line through 7 must intersect ¿34. The point of intersection must be 6, so that point 7 is incident with ¿69. Therefore the diagram is completed to a 93-configuration, so that (£, n) can only be the Pappus configuration. Case BA123BA132BA243BA342C. This case is illustrated in Figure 20. Consider the triangle A274. There are two possibilities, A274B and A274C, shown in Figure 23. These are duals of each other. The mapping which sends the points 1,2,..., 8 of A274B to the lines ¿15, ¿25, ¿34, ¿32, ¿12, ¿13, ¿58, ¿47 of A274C is an isomorphism. Hence we only need to consider one of them, the first, say. Consider the quadrangle D1783. It is only possible to satisfy condition E, namely the third line through 3 must intersect ¿78. The point of intersection must be 6, so that ¿78 must be extended to include point 6. Consider next quadrangle D1745. It is W. L. Kocay: One-point extensions in n3 configurations 309 Figure 23: Case BA123BA132BA243BA342CA274, B and C only possible to satisfy condition E, namely the third line through 5 must intersect i47. The result is illustrated in Figure 24. \/l Figure 24: Case BA123BA132BA243BA342CA274BD1783D1745 Finally, consider quadrangle D7138. It is only possible to satisfy condition E, namely the third line through 8 must intersect £i3. The point of intersection must be 9, so that £i3 must be extended to include point 9. Once again we have the Pappus configuration. Case BA123BA132C. This case is illustrated in Figure 18. Consider the triangle A267. There are two possible ways to satisfy condition B, namely the third line through 6 could contain either 4 or 5. The first of these choices is illustrated in Figure 25. The second is not allowed, as it would create a triangle A125 satisfying condition A. There are two possible ways to satisfy condition C, namely £67 could intersect £i3 or £34. Call these two results Ci and C2, respectively, also shown in Figure 25. Case BA123BA132CA267B. Consider the quadrangle D1673. It is not possible to satisfy conditions D or F. Condition E can only be satisfied if £34 intersects £67. Condition G can only be satisfied if £i5 intersects £46. These cases are shown in Figure 26. Now case G (the right diagram) leads to a contradiction, for consider the quadrangle □3167. Conditions E, F, G are not possible. Condition D is only possible if 5 G £67. But this creates a triangle A156 satisfying condition A, a contradiction. Therefore we consider case E (the left diagram). Consider the quadrangle ^3761. Conditions D, F, G cannot be satisfied. Condition E can only be satisfied if £i5 intersects £67 in point 8, as shown in Figure 27. Consider next the quadrangle ^6137. Conditions 310 Ars Math. Contemp. 10 (2016) 183-192 Figure 25: Cases BA123BA132CA267 B, Cu and C2 Figure 26: Cases BA123BA132CA267BD1673 E and G D, F,G cannot be satisfied. Condition E can only be satisfied if the third line through 7 intersects l\3 in a point 9, also illustrated in Figure 27. Figure 27: Cases ED1673E and ED1673ED6137E Consider now the quadrangle 02685 in the right diagram of Figure 27. Conditions D, F, G cannot be satisfied. Condition E can only be satisfied if the third line through 5 contains point 7, which is only possible if 5 G £rg. The result is isomorphic to the diagram of Figure 24. Once again, we obtain the Pappus configuration. Case BA123BA132CA267Ci. Refer to Figure 25. Consider the quadrangle 02784. Conditions D and F cannot be satisfied. Condition E can only be satisfied if there is a line l46, which gives a result identical to the left diagram of Figure 26. Condition G can only be satisfied if the third line through 7 intersects ^26 in point 1, but this creates a triangle A127 W. L. Kocay: One-point extensions in n3 configurations 311 satisfying condition A, which is not allowed. This completes this case. Case BA123BA132CA267C2. Refer to Figure 25. Consider the quadrangle [II1376. Conditions D, E, F are not possible. Condition G is only possible if £15 and £34 intersect, shown in Figure 28. Consider now the quadrangle D1872. Conditions D, E, F are not possible. Condition G is possible if £15 intersects the third line through 8. The point of intersection can be either 5 or 9, resulting in G1 and G2, also shown in Figure 28. Figure 28: Cases C2D1376G, GD1872G1 and GD1872G2 Consider the quadrangle D7218 in diagram GD1872G1. Conditions D, E, F cannot be satisfied. Condition G can only be satisfied if the third line through 7 intersects £24. The point of intersection can be 4 or 5. But 4 creates a triangle A734 satisfying condition A, a contradiction. Therefore the intersection must be point 5, as shown in Figure 29. Then consider quadrangle D7812. Conditions D, E, F cannot be satisfied. Condition G can only be satisfied if £15 and £g9 intersect, also shown in Figure 29. Next, consider quadrangle D1572. Conditions D, E, F, G cannot be satisfied, a contradiction. This completes this case. Figure 29: Cases G1 : D7218G and D7218GD7812G Consider next GD1872G2, and quadrangle D7218. Conditions D,E, F cannot be satisfied. Condition G can only be satisfied if the third line through 7 intersects £24. The point of intersection must be 4. But this creates a triangle A734 satisfying condition A, a contradiction. This completes this case, and also case BA123BA132CA267C2, and case BA123BA132C and case BA123B. Case BA123C. 312 Ars Math. Contemp. 10 (2016) 183-192 Refer to Figure 17. Consider the triangle A132. Condition B can be satisfied if the third line through 1 intersects ¿34. There are two ways this can occur - the intersection can be point 4, or a new point. This gives Bi and B2, shown in Figure 30. Condition C can be satisfied if point 6 is collinear with the third point on ¿i2. There are two ways this can occur. The line through 6 intersecting ¿i2 can be 46 or a new line. This gives Ci and C2, shown in Figure 31. /TV A\' \5/ / \ \5/ \ \V l\ 6 \ 2 / I\ 6 \-V Figure 30: Case BA123CA132 Bi and B2 Figure 31: Case BA123CA132 Ci and C2 It can be observed that Ci is isomorphic to the dual of Bi. If we map points 1,2,3,4, 5, 6,7 of Ci to lines ^i2, ^23, ^i3, 46, ^i4, ^34, ^24, respectively, of Bi, we have an isomorphism. Similarly, C2 is isomorphic to the dual of B2. An isomorphism maps points 1, 2,3,4,5, 6, 7 of C2 to lines ¿i2, ¿i3, ^23, 46, ^24, ¿34, respectively, of B2. Consequently, we have only cases Bi and B2 to deal with. Case BA123CA132Bi. Consider the quadrangle D1562. Condition D can only be satisfied if the third point on ¿42 is collinear with point 3. But then triangle A123 would satisfy condition A, which is not allowed. Condition E can be satisfied if ^24 intersected 46. This is shown in Figure 32. Condition F can only be satisfied if the third line through 6 intersected ¿i5 in point 3. However, 6 and 3 are already collinear. Condition G can be satisfied if the third line through 5 intersected ¿i4. The third line through 5 cannot be ^24, for A124 would then satisfy condition A. Thus, the third line through 5 must be a new line, as shown also in Figure 32. Case BA123CA132B1D1562E. Consider the triangle A267. Condition B can be satisfied if the third line through 6 intersected ¿i2. The third line through 6 cannot be ¿i4, as the triangle A123 would then satisfy condition A. Hence, the third line through 6 must be a new line, as shown in Figure 33. Condition C can only be satisfied if points 4 and 5 are collinear. W. L. Kocay: One-point extensions in n3 configurations 313 \5 / y/i h\ \3/ ' 4 Figure 32: Case BA123CA132B1D1562 E and G The line containing 4 and 5 cannot be l14 and it cannot be Therefore Condition C is impossible, and we must have BA123CA132B1D1562EA267B, shown in Figure 33. Figure 33: Case BA123CA132B1D1562EA267B This structure is found to be isomorphic to the dual of BA123BA132CA267B □ 1673G, shown in Figure 26. The isomorphism maps points 1, 2,3,4,5, 6, 7,8 to lines ^24,^26,^56,^15,^34,^18,^68,^14. This completes case BA123CA132B1 □ 1562E. Case BA123CA132Bi□1562G. Consider the quadrangle ^2651. Condition D can only be satisfied if the third point on i23 is collinear with point 3. However triangle A132 would then satisfy condition A. Condition E can only be satisfied if l14 intersected l56. The point of intersection cannot be 7. If it were point 4, then A563 would then satisfy condition A. Hence condition E is not possible. Condition F can only be satisfied if intersected i26 in point 3. However 5 and 3 are already collinear. Condition G can be satisfied if the third line through 6 intersected ^24. The point of intersection cannot be 4. The only possibility is a new line through 6, as shown in Figure 34. Consider the quadrangle □4863. Condition D can only be satisfied if the third point on ^34 is collinear with point 2. The triangle A342 would then satisfy condition A, 314 Ars Math. Contemp. 10 (2016) 183-192 Figure 34: Cases BA123CA132B1D1562G : D2651G and □2651GD4863G which is not allowed. Condition E can only be satisfied if intersected ¿68 in either 1 or 5. However, 1 and 5 are already each on 3 lines. Condition F can only be satisfied if ¿56 intersected ¿4s in 2. However 6 and 2 are already collinear. Condition G can be satisfied if the third line through 8 intersected ¿14. The point of intersection can only be 7, shown in the right diagram of Figure 34. Consider the quadrangle ^6512. Condition D can only be satisfied if the third point on ¿42 were collinear with point 3. But triangle A123 would then satisfy condition A. Condition E can only be satisfied if ¿24 intersected ¿15 in 3. This is not possible. Condition F can only be satisfied if l14 intersected ¿56. This is not possible. Condition G can only be satisfied if ¿57 intersected ¿68. This is shown in Figure 35. Figure 35: Cases ^6512G and □6512Gœ743G Consider the quadrangle ^5743. Condition D can only be satisfied if the third point on ¿34 were collinear with point 1. But then triangle A341 would satisfy condition A. Condition E can only be satisfied if ¿23 intersected ¿47 in point 1. This is not possible. Condition F can only be satisfied if ¿24 intersected ¿57 in 9. This is not possible. Condition G can only be satisfied if ¿78 intersected ¿56 in a new point, also shown in Figure 35. Consider the triangle A157. Condition B can only be satisfied if ¿12 intersected W. L. Kocay: One-point extensions in n3 configurations 315 46. The point of intersection must be point 0. Condition C can only be satisfied if points 4 and 9 are collinear. The line of collinearity must be £34. The resulting two structures are both isomorphic to the Desargues configuration, with one incidence missing, as can be seen from Figure 1.1. If we then consider A268, the remaining incidence is forced. This completes case BA123CA132B1[H1562G and also case BA123C A132B1. Case BA123CA132B2. Refer to Figure 30. Consider the triangle A173. Condition B can be satisfied if the third line through 7 intersected £12. The point of intersection cannot be point 2. Therefore it is a new point, as shown in Figure 36. Condition C can be satisfied if points 4 and 5 are collinear. The line of collinearity cannot be £56, for triangle A453 would then satisfy condition A. Hence £45 is a new line, also shown in Figure 36. be satisfied if £57 intersected £68. This is shown in Figure 35. Figure 36: Cases BA123CA132B2 A173 B and C Now case BA123CA132B2 A173C is isomorphic to case BA123BA132CA267B, shown in Figure 25. As both diagrams have 7 points, the isomorphism can be given by a permutation, (1,5,6)(2, 3,4), which maps diagram BA123BA132CA267B to BA123CA132B2 A173C. Thus we need only consider case BA123CA132B2 A173B. Consider the triangle A781 in the left diagram of Figure 36. Condition B can be satisfied if the third line through 8 intersected £37. The point of intersection cannot be 3. Therefore there must be a line £48, as shown in Figure 37. Condition C can be satisfied if the third point on £17 is collinear with point 2. The line of collinearity cannot be £26, for if point 6 were on £17, triangle A173 would satisfy condition A. Hence £24 must intersect £17 in a new point. This is also shown in Figure 37. Case BA123CA132B2A173BA781B. Consider the triangle A365. Condition B can be satisfied if the third line through 6 intersected £37. The point of intersection cannot be 4, because £48 would then contain 6, causing a triangle A682 satisfying condition A. Line £17 cannot contain 6, for then triangle A136 would satisfy condition A. Therefore condition B requires that £78 contain 6, shown in Figure 38. Condition C can be satisfied if the third point on £56 were collinear with point 1. The line of collinearity must be £17, also shown in Figure 38. Figure 38: Case BA123CA132B2A173BA781BA365 B and C Case BA123CA132B2A173BA781BA365B. Refer to the left diagram of Figure 38. Consider the quadrangle □2176. Condition D can only be satisfied if points 3 and 8 were collinear. This is not possible as 3 and 8 are already incident on 3 lines each. Condition E can only be satisfied if 46 intersected ¿17, shown in Figure 39. Condition F can only be satisfied if ¿37 intersected ¿12 in 8. However, 7 and 8 are already collinear. Condition G can only be satisfied if ¿15 and ¿24 intersected. The point of intersection must be 5, making triangle A132 satisfy condition A. We conclude that only E is possible. Figure 39: Case BA123CA132B2A173BA781BA365BD2176E Consider the quadrangle D2156. Condition D can only be satisfied if points 3 and 9 were collinear, which is impossible. Condition E can only be satisfied if intersected ¿45 in point 3, which is impossible. Condition F can only be satisfied if W. L. Kocay: One-point extensions in n3 configurations 317 the third line through 5 intersected ¿12 in point 8, which is impossible. Condition G can only be satisfied if ¿17 and ¿24 intersected. The point of intersection must be point 9, also shown in Figure 39. As can be seen from the diagram, this is the Pappus configuration with one incidence missing. We conclude that this case results in the Pappus configuration. Case BA123CA132B2A173BA781BA365C. Refer to the right diagram of Figure 38. Consider the quadrangle D7123. Condition D can only be satisfied if points 4 and 6 are collinear, which is impossible. Condition E can only be satisfied if ¿13 contains 8, which is impossible. Condition F can only be satisfied if ¿24 contains point 9. Condition G can only be satisfied if ¿78 intersected ¿13. The point of intersection must be 5, creating a triangle A195 satisfying condition A, a contradiction. We conclude that only condition F is possible, shown in Figure 40. Figure 40: Cases BA123CA132B2 A173BA781BA365CD7123F and D2371F Consider the quadrangle D2371. Condition D can only be satisfied if points 8 and 9 are collinear, which is impossible. Condition E is only possible if ¿13 contains 4, which is impossible. Condition F is possible only if ¿78 contains 6. Condition G is only possible if ¿29 and ¿35 intersected, which is impossible. We conclude that condition F is necessary. We next consider quadrangle D4862. Condition D can only be satisfied if points 9 and 3 are collinear, which is impossible. Condition E can only be satisfied if ¿21 contains point 7, which is impossible. Condition F is possible only if ¿69 and ¿48 intersect in point 5. Condition G is only possible if ¿47 and ¿81 intersected, which is impossible. We conclude that condition F is necessary, giving the Pappus configuration. This completes case BA123CA132B2A173BA781B. Case BA123CA132B2A173BA781C. Refer to the right diagram of Figure 37. Consider triangle A243. Condition B can only be satisfied if the third line through 4 intersected ¿28. The point of intersection can only be 8, as shown in Figure 41. Condition C can only be satisfied if points 6 and 7 are collinear. The line of collinearity cannot be ¿17, as triangle A231 would then satisfy condition A. Hence, the line can only be ¿78, which must contain 6, as shown in Figure 41. Case C is isomorphic to the dual of BA123CA132B2A173BA178BA365B, shown in Figure 38. An isomorphism maps points 1, 2,..., 9 of C to lines ¿67, ¿34, ¿23, ¿24, ¿56, ¿13, ¿12, ¿17, ¿48, respectively, of B. Thus we only need consider case B. 318 Ars Math. Contemp. 10 (2016) 183-192 Figure 41: Case BA123CA132B2A173BA178CA243B and C Consider the quadrangle 08731. Condition D can only be satisfied if points 2 and 6 are collinear, which is impossible, as the line of collinearity could only be i24. Condition E cannot be satisfied. Condition F can only be satisfied if i36 intersects i87. The point of intersection must be 6, as shown in Figure 42. Condition G can only be satisfied if i84 and i79 intersect, which is impossible. Thus, only condition F is possible. But this diagram is isomorphic to case BA123BA132GA267BD1673ED6137E, shown in Figure 27. An isomorphism is given by (5,9)(6,7, 8). Figure 42: Case BA123CA132B2A173BA178CA243BD8731F We summarise Case B as follows: An n3 configuration (E, n), which cannot be constructed by a 1-point extension, and having no triangle satisfying condition A, is one of the Pappus or Desargues configurations. We still must show that the Fano, Pappus, and Desargues configurations cannot be obtained by 1-point extensions. This is clearly so for the Fano configuration, as there are no 63 configurations. Consider the Pappus configuration. One way to show that it cannot be obtained by a 1-point extension is to start with the unique 83 configuration and to show that the possible 1-point extensions do not produce the Pappus configuration. Another way is to show that every ordering of every triangle and quadrilateral in the Pappus configuration satisfies one of conditions A, B, C, D, E, F, G, so that the Pappus configuration does not W. L. Kocay: One-point extensions in n3 configurations 319 arise by a 1-point extension. The collineation group of the Pappus configuration has order 108. It is transitive on points, lines, triangles, and quadrangles, so that only one triangle and one quadrilateral need be tested. We omit the proof. 7 W 8 <> /. ' - ' 2 Figure 43: The Pappus configuration Consider next the Desargues configuration. Its collineation group has order 120. It is transitive on points, lines, triangles, quadrangles, and also on quadruples (a0, ¿2, ai, ¿3), where a0,ai G ¿2, a0 = ai, ai G ¿3, and ¿2 = ¿3. It is not transitive on pentagons, hexagons, etc. Refer to Figure 44. We look for a cycle beginning (a0, ¿2, ai, ¿3,..., ¿0) = (1, ¿i3,3, ¿34,...), satisfying the conditions of Theorem 2.4. Since ¿i n ¿i = 0, where ¿i = ¿37, and ¿i is the third line through a0 = 1, we must have ¿i = ¿i5, so that ¿0 = ¿i7. Since a2 G ¿4, by Theorem 2.4, we cannot have a2 = 5. Hence, a2 = 4. Figure 44: The Desargues configuration Then since ¿2 n ¿2 = 0, we cannot have ¿2 = ¿42, as ¿42 intersects ¿2 = ¿i3 in 2. Therefore ¿4 = ¿49, from which we have a3 = 9, and the cycle is (1, ¿i3,3, ¿34,4, ¿49, 9, ..., ¿i7). Since ¿3 n ¿3 = 0, we cannot have ¿3 = ¿59, as ¿59 intersects ¿3 = ¿34 in 5. It follows that ¿5 = ¿59. But then a4 must be either 1 or 5, both of which are impossible. We conclude that the Desargues configuration cannot be obtained by a 1-point extension. This completes the proof of Theorem 4.1. □ Observe that we have only used 1-point extensions based on triangles and quadrangles in the proof of Theorem 4.1. Hence we have proved that if an (n +1)3 configuration cannot be obtained using a 1-point extensions based on triangles or quadrangles, then it is the Fano, Pappus, Desargues, or a Fano-type configuration. Therefore we have the following corollary. 320 Ars Math. Contemp. 10 (2016) 183-192 Corollary 4.2. Every (n +1)3 configuration that can be obtained from an n3 configuration by a 1-point extension, can be obtained using a 1-point extension based on triangles or quadrangles. A consequence of this corollary is that the (n + 1)3 configurations can be constructed from the n3 configurations by constructing all sequences of sequentially incident points and lines of length at most 4, and testing whether they satisfy the conditions required for a 1-point extension. Isomorphism testing of the resulting (n + 1)3 configurations then gives all configurations that can be constructed by 1-point extensions. Those which cannot be constructed in this way are the Fano-type configurations, which can be constructed from cycles and subdivisions of bipartite 3-regular multigraphs, using Theorem 3.2. One of the central problems in the theory of n3 configurations is to determine whether they are geometric, that is, whether they can be coordinatized over the reals and/or rationals. See [3, 14, 15, 16]. This means to assign homogeneous coordinates in the real and/or rational projective plane, so that the lines are straight lines, and all incidences and non-incidences are respected. The application of 1-point extensions to geometric configurations will be described in another article (in preparation). 5 The 3-Point Extension Let (S, n) be an n3-configuration. Choose a line ¿, and let its points be ai, a2, a3. Construct a new configuration (S', n') as follows. S' = S U 62,63, ¿^ ¿2, ¿3}, where bi, b2, b3 are new points and ¿^ ¿2, ¿3 are new lines. The incidences n' are constructed as follows. ¿i contains the points ai, 62, 63. ¿2 contains the points bi, a2, b3, and ¿3 contains the points bi, b2, a3. Choose 3 lines ¿i, ¿2, ¿3 = ^ such that ¿i contains a4. Remove a from ¿i and place 6j on ¿i. This is illustrated in the following table. Then n' contains all remaining incidences of n, except for the incidences a^i, a^2, a^3. ¿ ¿i ¿2 ¿3 ¿'i ¿' ¿2 ¿' ¿3 ai ai 6i 6i 6i 62 63 a2 62 a2 62 a3 63 63 a3 Theorem 5.1. (S', n') is an (n + 3)3-configuration. Proof. Note that each 6j is incident on exactly 3 lines, and that each of ¿i, ¿2, ¿3 is incident on exactly 3 points. We must verify that any 2 lines of (S', n') intersect in at most one point. Clearly ¿, ¿^¿^¿3 intersect each other in at most one point. Similarly for ¿, ¿i, ¿2, ¿3. The same is true for all other lines of S', because it is true for (S, n). □ Example 5.2. The Fano configuration has 7 points and 7 lines, all of which are equivalent under automorphisms. There is one way to choose 3 points ai, a2, a3. The incidences of ¿, ¿i, ¿2, ¿3 are uniquely determined. The choice of ¿i, ¿2, ¿3 is not unique, as each is incident on two lines other than ¿. There results two possible 3-point extensions of the Fano configuration. One of these is the Desargues configuration. The other is known as the "anti-Pappian" configuration [5]. A complete quadrilateral in an n3 configuration is a set of four distinct lines intersecting in six distinct points. Notice that the extended configuration (S', n') always contains a complete quadrilateral ¿, ¿i, ¿2, ¿3, intersecting in the six points ai, a2, a3, bi, 62, 63. The W. L. Kocay: One-point extensions in n3 configurations 321 3-point extension can also be constructed from the dual point of view - rather than beginning with 3 collinear points ai,a2, a3, we begin with 3 concurrent lines, and so forth. This is equivalent to using the 3-point extension in the dual of (£, n), and then dualizing (£', n'). In this case, the 3-point extension will always contain a complete quadrangle, that is, the dual of a complete quadrilateral. Theorem 5.3. The Fano-type configurations cannot be obtained by a 3-point extension. Proof. Suppose that a Fano-type configuration (£, n) were obtained by a 3-point extension. It would then contain a complete quadrilateral I, li,l2,l3, intersecting in the six points ai, a2, a3,bi,b2,b3. These four lines and six points must all be part of a single F', Fp, or Fi. Refer to Figure 6. Now the points ai,a2, a3 must be collinear. Furthermore, there must be a line containing ai,b2, b3, and so forth. This determines the labelling of an F', Fp, or Fi. But we then find there is a line containing at least one of the pairs ai, bi; a2,b2; a3,b3, which is not possible in a 3-point extension. □ 6 Acknowledgement The author would like to thank an anonymous referee for very many helpful comments, and especially for bringing the articles [1, 2, 4, 12, 13] to his attention. References [1] M. Boben, Reductions of (V3) configurations, 2005, arXiv:math/0505136 [math.CO] . [2] M. Boben, Irreducible (v3) configurations and graphs, Discrete Math.. 307 (2007), 331-344, doi:10.1016/j.disc.2006.07.015. [3] J. Bokowski and B. Sturmfels, Computational synthetic geometry, Berlin etc.: Springer-Verlag, 1989, doi:10.1007/BFb0089253. [4] H. Carstens, T. Dinski and E. Steffen, Reduction of symmetric configurations n3, Discrete Appl. Math. 99 (2000), 401-411, doi:10.1016/S0166-218X(99)00147-X. [5] D. G. Glynn, On the anti-Pappian 103 and its construction, Geom. Dedicata 77 (1999), 71-75, doi:10.1023/A:1005167220050. [6] H. Gropp, Configurations between geometry and combinatorics, Discrete Appl. Math. 138 (2004), 79-88, doi:10.1016/S0166-218X(03)00271-3. [7] B. Grunbaum, Configurations of points and lines, Providence, RI: American Mathematical Society (AMS), 2009. [8] W. Kocay, Coordinatizing n3 configurations, in preparation. [9] W. Kocay and R. Szypowski, The application of determining sets to projective configurations, Ars Comb. 53 (1999), 193-207. [10] V. Martinetti, Sulle configurazioni piane Ann. Mat. Pura Appl. 15 (1887), 1-26. [11] T. Pisanski and B. Servatius, Configurations from a graphical viewpoint, New York, NY: Birkhauser, 2013, doi:10.1007/978-0-8176-8364-1. [12] E. Steffen, T. Pisanski, M. Boben and N. Ravnik, Erratum to: "Reduction of symmetric configurations n3" [Discrete Appl. Math. 99 (2000), no. 1-3, 401-411; by H. G. Carstens, T. Dinski and Steffen], Discrete Appl. Math. 154 (2006), 1645-1646, doi:10.1016/j.dam.2006.03.015, http://dx.doi.org/10.1016/j.dam.2006.03.015. [13] K. Stokes, Irreducibility of configurations, Ars Math. Contemp. 10 (2016), 169-181. 322 Ars Math. Contemp. 10 (2016) 183-192 [14] B. Sturmfels, Aspects of computational synthetic geometry. I. Algorithmic coordinatization of matroids, in: Computer aided geometric reasoning, Vol. I, II (Sophia-Antipolis, 1987), INRIA, Rocquencourt, pp. 57-86, 1987. [15] B. Sturmfels and N. White, Rational realizations of 113-and 123-configurations, in: Symbolic Computation in Geometry, H. Crapo et al. IMA preprint series, volume 389, 1988 . [16] B. Sturmfels and N. White, All 113 and 123-configurations are rational, Aequationes Math. 39 (1990), 254-260. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 323-332 The number of edges of the edge polytope of a finite simple graph Takayuki Hibi Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan Aki Mori Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan Hidefumi Ohsugi Department ofMathematical Sciences, School ofScience and Technology, Kwansei Gakuin University, Sanda, Hyogo, 669-1337, Japan Akihiro Shikama Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan Received 6 September 2014, accepted 8 October 2015, published online 5 February 2016 Let d > 3 be an integer. It is known that the number of edges of the edge polytope of the complete graph with d vertices is d(d - 1)(d - 2)/2. In this paper, we study the maximum possible number ^d of edges of the edge polytope arising from finite simple graphs with d vertices. We show that = d(d - 1)(d - 2)/2 if and only if 3 < d < 14. In addition, we study the asymptotic behavior of Tran-Ziegler gave a lower bound for ^d by constructing a random graph. We succeeded in improving this bound by constructing both a non-random graph and a random graph whose complement is bipartite. Keywords: Finite simple graph, edge polytope. Math. Subj. Class.: 52B05, 05C30 E-mail addresses: hibi@math.sci.osaka-u.ac.jp (Takayuki Hibi), a-mori@cr.math.sci.osaka-u.ac.jp (Aki Mori), ohsugi@kwansei.ac.jp (Hidefumi Ohsugi), a-shikama@cr.math.sci.osaka-u.ac.jp (Akihiro Shikama) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 324 Ars Math. Contemp. 10 (2016) 183-192 1 Introduction The number of ¿-dimensional faces of a convex polytope has been studied by many researchers for a long time. One of the most famous classical results is "Euler's formula." The extremal problem concerning the number of faces is an important topic in the study of convex polytopes. On the other hand, the study of edge polytopes of finite graphs has been conducted by many authors from viewpoints of commutative algebra on toric ideals and combinatorics of convex polytopes. We refer the reader to [2, 3] for foundations of edge polytopes. Faces of edge polytopes are studied in, e.g., [2,4, 5]. Recently, Tran and Ziegler [6] studied this extremal problem on edge polytopes. In particular, using [5, Lemma 1.4], they gave bounds for the maximum possible number ^d of edges of the edge polytope arising from finite simple graphs with d vertices. Following [1, Question 1.3], we wish to find a finite simple graph G with d vertices such that the edge polytope of G has edges and to compute Recall that a finite simple graph is a finite graph with no loops and no multiple edges. Let [d] = {1,..., d} be the vertex set and Qd the set of finite simple graphs on [d], where d > 3. Let ej denote the ith unit coordinate vector of the Euclidean space Rd. Let G G Qd and E(G) the set of edges of G. If e = {i, j} G E(G), then we set p(e) = ej + ej G Rd. The edge polytope PG of G G Qd is the convex hull of the finite set {p(e) : e G E(G)} in Rd. Let e(G) denote the number of edges, namely 1-dimensional faces, of PG. For example, consider the case of the complete graph Kd on [d]. By [5, Lemma 1.4], for edges e and f (e = f) of Kd, the convex hull of {p(e), p(f)} is an edge of the edge polytope PKd if and only if e and f have a common vertex. Hence, e (Kd) = d(d-1) = d(d - 1)(d-2)/2. On the other hand, e(Km,n) = mn(m + n - 2)/2, where Km,n is the complete bipartite graph on the vertex set [ m ] U {m + 1,..., m + n} for which m, n > 1 (see [4, Theorem 2.5]). In this paper, we are interested in = max{ e(G) : G G Qd } for d > 3. Theorem 1.1. For an integer d > 3, let Qd be the set of finite simple graphs on [d]. Given a graph G G Qd, let e(G) denote the number of edges of the edge polytope PG of G. Then, the following holds: (a) If 3 < d < 13 and G G Qd with G = Kd, then e(G) < e(Kd). (b) Let G G Q 14 with G = K14. Then e(G) < e(Ki4). Moreover, e(G) = e(Ki4) if and only if either G = K14 — K4,5 or G = K14 — K5,5. (c) If d > 15, then there exists G G Qd such that e(G) > e(Kd). We devote Section 2 to giving a proof of Theorem 1.1. At present, for d > 15, it remains unsolved to find G G Qd with = e(G) and to compute (Later, we will see that ^15 > e(K15) + 50 = 1415.) In Section 3, we study the asymptotic behavior of Recently, Tran-Ziegler [6] gave a lower bound for by a random graph: 118 1 e(G(d, 1/V3)) = — d4 + — d3--d2 + - d. ( ( , ' )) 54 + 18 27 +3 They also gave an upper bound for < (32 + o(1))d4. (However, this upper bound is not sharp. See [6, Remark].) In this paper, we succeeded in improving their lower bound by constructing a non-random graph (see Example 3.1) and a random graph whose complement is bipartite (see Theorem 3.2): 5V5 — 11 ,4 12V5 — 27 o 19V5 — 44 ,2 , e(G) = -i—--d4---d3 +---d2 + d, T. Hibi et al.: The number of edges of the edge polytope of a finite simple graph 325 where G = Kd - G(Kd/2jd/2,p) with p = 3 - %/5. These results suggest the following: Conjecture 1.2. Let G G Qd with ^d = e(G). Then, the complement of G is a bipartite graph. Note that, by Theorem 1.1, this conjecture is true for 3 < d < 14. 2 Proof of Theorem 1.1 In this section, we give a proof of Theorem 1.1. The following lemma is studied in [5, Lemma 1.4]. Lemma 2.1. Let e and f (e = f) be edges of a graph G G Qd. Then, the convex hull of {p(e),p(f)} is an edge of the edge polytope if and only if one of the following conditions is satisfied. (i) e and f have a common vertex in [d\. (ii) e = {i, j} and f = {k, l} have no common vertices, and the induced subgraph of G on the vertex set {i, j, k, l} has no cycles of length 4. The complement graph G of a graph G G Qd is the graph whose vertex set is [d] and whose edges are the non-edges of G. For a vertex i of a graph G, let degG(i) denote the degree of i in G. We translate Lemma 2.1 in terms of the complement G of G. Lemma 2.2. Let H be the complement of a graph G G Qd. Then, we have d 'd - 1 - degH(i) e(G) = £ (d - 1 -2degH(i))+ a(H)+ b(H) + c(H) 2 i=1 x 1d = (d) = e(Kd+1 — H) — e(Kd+1) — e(K — H ) + e(Kd) = t fd — degH2 (i)) — t fd — 1 — degHi + V(H) i=1 ^ 2 ' i=1 ^ 2 / + d(d — 1)(d — 2) (d +1)d(d — 1) 22 ^ + ^ ^ — degHi (i)^ — — 1 — degHi (i)^ ^ + ^(H) — 3d(d — 1) (d) + it (d — 1 — degHi (i)) + ^(H ) — ^^^ ^ ' i=1 d V>(H) — ^ degHi (i) i=1 = V(H ) — 2|E(H)|, as desired. □ Proposition 2.4. Let G e Qd and let H1, H2,..., Hm be all the nonempty connected components of G. Then, e(Kd) — e(G) = 5^j=1(e(Kd) — e(Kd — Hj)). Proof. Let H = G and let Hj = Kd — Hj for 1 < j < m. Then, it is easy to see that |E(H)| = j |E(Hj)|, £?=! degH(i) = E^ EU degH (i), «(H) = E™! a(Hj), b(H) = j b(Hj), and c(H) = E^ c(Hj). Thus, by Lemma 2.2, we are done. □ A graph G G is called bipartite if [d] admits a partition into two sets of vertices V and V2 such that, for every edge {i, j} of G, either i G V1, j G V2 or j G V1, i G V2 is satisfied. A complete bipartite graph is a bipartite graph such that every pair of vertices i, j with i g V1 and j G V2 is adjacent. Let Km,n denote the complete bipartite graph with |V1| = m and |V2| = n. Proposition 2.5. Let G = Kd — Km,n such that m + n < d and m, n > 1. Then, e(G) — e(Kd) = ^ mn(m + n — 6)d — ^ mn(3mn + 2m2 + 2n2 — 5m — 5n — 13). T. Hibi et al.: The number of edges of the edge polytope of a finite simple graph 327 Proof. Let H = Km,n. Then, (n \ (1 -0(H) — 2|E(H )| = mi j + \ 2 J — 2mn = 2 mn(m + n — 6)' Moreover, since Km+n — Km,n is the disjoint union of Km and Kn, we have m(m — 1)(m — 2) n(n — 1)(n — 2) /m\ /n ^(m + n) = -2-+-2-+V2M2 (m + n)(m + n — 1)(m + n — 2) = - mn(mn — 7m — 7n + 13) 4 by Lemma 2.1. Hence, by Proposition 2.3, e(G) — e(Kd) = — mn(m + n — 6)(d — (m + n)) + — mn(mn — 7m — 7n + 13) = — mn(m + n — 6)d — — mn(3mn + 2m2 + 2n2 — 5m — 5n — 13), as desired. □ Let k3(H) denote the number of triangles (i.e., cycles of length 3) of H. The following lemma is important. Lemma 2.6. Let H be the complement graph of G G Qd. Then, we have d2 — 16d + 29 3 e(G) < e(Kd) +-|E(H )| — -(d — 8)k3(H ). Proof. The number of pairs of edges satisfying Lemma 2.1 (i) is, by Lemma 2.2, e(Kd) — (2d — 3) |E(H)| + 1 £ti degH(i) For an edge {i, j} of H, let k3(i, j) be the number of triangles in H containing {i, j}. We define three subsets of [d] \ {i, j}: = {* G [d] \{i,j} Yij = G [d] \{i,j} Zi,j = {* G [d] \{i,j} {i,*}G E(H), {j,i}GE(H)}, {j, G E(H), {i, G E(H)}, {i, G E(H), {j, G E(H)}. It then follows that, |Xijj | + |Yi,j | + |Zijj | + ^ (i, j) = d — 2, and 1 d 1 ^E^H(i) = 2 £ (degH (i)+degH(j)) = 1 E (|Xi,j | + |Yi,j | + 2k3(i,j) + 2) {i,j}eE(H) = |E(H )| + 3k3(H ) + 2 E (|Xi,j | + |Yi,. {i,j}£E(H) 328 Ars Math. Contemp. 10 (2016) 183-192 Second, we count the number of pairs satisfying Lemma 2.1 (ii). By Lemma 2.2, this number is equal to a(H) + b(H) + c(H). Here, we count the number of the induced subgraphs H' of type (a), (b) and (c) containing an edge e = {i, j} of H. If e is an edge of H', then the other two vertices i and m of H' satisfy exactly one of the following conditions: (i) i € Xj, m € ; (ii) i € , m € Zj,j; (iii) i € Zj,j, m € Xj,j. If i, j, i, m satisfy condition (i), then one of the following holds: • H' is a path (ei, e2, e3) and e = e2 (type (a)); • H' is a cycle of length 4 and e is one of four edges (type (b)). It then follows that a(H )+4b(H )= £ |Xj,j ||Yj,j |. If i, j, i, m satisfy either condition (ii) or (iii), then one of the following holds: • H' is a path (e1, e2, e3) and e € {e1, e3} (type (a)); • H' is a path (e1, e2) with one isolated vertex and e € {e1, e2} (type (c)). It then follows that 2a(H)+2c(H)= £ (|Yj,j||Zj,j1 + |Zj,j||Xj,j|). {i,j}£E(H) Thus, we have a(H) + b(H) + c(H ) = - + £ (1 |Xj,j | + 1 ||Zj,j | + 2 |Zj,j ||Xj,j |). {i,j}£E(H) Subject to |Xj,j | + | + |Zj,j | = d - 2 - k3(i, j), we study an upper bound of a = £ [|Xj,j | + |Yj,j | + 1 |Xj,j ||Yj,j | + 2 |Yj ||Zj,j | + 2 |Zj,j ||Xj,j A . Each summand of a satisfies |Xj,j | + |Yj,j | + 1 |Xj,j ||Y,j | + 1 |Y ,j ||Zj,j | + 2 |Zj,j ||Xj,j | = 1 |Xi,j||Y,j| + 2(|Xi,j| + |Y,j|)(d - 1 - k3(i, j) - (|Xj,j| + |Y,j|)) < 1 (|Xj,j| + |Yj,j^2 + 2(|Xj,j| + |Y,j|)(d - 1 - k3(i, j) - (|Xj,j | + |Yj,j |)) = - 16(|Xj ,j | + |Y ,j |)2 + d -1 -2k3(i,j)(|Xj ,j | + |Y ,j |). T. Hibi et al.: The number of edges of the edge polytope of a finite simple graph 329 The last function has the maximum number 7(d - 1 - k3(i, j))2 when |Xjj | + | = 7(d - 1 - k3(i, j)). Hence, £ 1(d - 1 - ks(i,j))2 < £ 7(d - 1)(d - 1 - k3(i,j)) {ij}eE(H) {ij}eE(H) = 1 £ (d - 1)2 - 1 £ (d - 1)ks(i,j) {i,j}eE(H) {i,j}eE(H) 1 3 = 7(d - 1)2|E(H)|- 7(d - 1)ks(H) is an upper bound of a. Thus, 1 3 e(Kd) - (2d - 3)|E(H)| + |E(H)| + 3k3(H) + ^(d - 1)2|E(H)| - -(d - 1)k3(H) is an upper bound of e(G) as desired. □ Using Proposition 2.5 and Lemma 2.6, we prove Theorem 1.1. Proof of Theorem 1.1. (a) Let 3 < d < 13 and G G Q.d with G = Kd. If d = 3, then e(G) < e(Kd) is trivial. If d =4, then e(K4) = 12. Since |E(G)| < 6, we have e(g) < (2) =10 < e(K4). Let d > 5 and let H be the complement graph of G. By Lemma 2.6, d2 — 16d + 29 3 e(G) - e(Kd) < d ^ + 29 |E(H)| - 7(d - 8)^(H). If 8 < d < 13, then e(G) - e(Kd) < 0 since d2-16d+29 < 0, |E(H)| > 0 and k3(H) > 0. Let 5 < d < 7. Then, - f |E(H )| + fk3(H) if d =5, e(G) - e(Kd) <{ -31 |E(H)| + 6k3(H) if d =6, - f |E(H )| + 3 k3(H) if d =7. Hence, if k3(H) < 2, then e(G) - e(Kd) is negative. On the other hand, if k3(H) > 3, then |E(H)| > 5. Since ^(H) < (d), it follows that e(G) - e(Kd) is negative. (b) Let G g 014 with G = K14 and let H = G. We need to evaluate the function which appears in the proof of Lemma 2.6 more accurately by focusing on d = 14. Let |Zi,j| = 12 - k3(i, j) - |Xi,j| - |Yi,j| and f = |Xi'j | + |Yi'j | +4 |Xi,j ||Yi,j | + 2 |Yi,j ||Zi,j | + 2 |Zi,j ||Xi,j | g = - ¿(Xi'j | + |Yi'j |)2 +13 k3(i,j)(Xi'j | + |Y'j |) be functions of |Xi'j| and |Yi'j|. Recall that f < g < 7 (13 - k3(i, j))2 and g = 7(13 -k3(i,j))2 when |xi'j| + |Yi'j| = 4(13 - j)). If 1 < k3(i,j) < 12, then 1 13 11 1 13 7(13-k3(i, j))2 = 24-13k3(i, j)- 17- + 7(k3(i, j)-1)(k3(i, j)-12) < 24-13^(i, j). 330 Ars Math. Contemp. 10 (2016) 183-192 If j) = 0, then 7(13 - j))2 = 24 + 1/7. However, since 4 (|Xi'j 1 + |Yj 1 + 1 |Xi,j||Yj| + 1 ||Zi,j| + 2||Xi,j|) is an integer, the value of f is at most 24 if |Xjj | and || are non-negative integers. Thus, for k3(i, j) = 0,1,..., 12, the value of f is at most 24 - 13k3(i, j) if |Xj j| and | Yj | are non-negative integers. Thus, by the same argument in the proof of Lemma 2.6, e(G) - e(K14) is at most -24|E(H)| + 3k3(H) + 24|E(H)| - ^) - = -yMH) - ^ < 0. Therefore, e(G) < e(K14). Suppose that e(G) = e(KM). Then, -f kj(H) - > 0. Since fc^H), a(H) > 0, we have k3(H) = a(H) = 0. Moreover, |Xi'j | + |Yi'j | + 4 |Xi'j ||Yi'j | + 2 |Yi'j ||Zi'j | + 1 |Zi'j ||Xi'j | = 24 and |Xj j| + |Yj | + |Zj j| = 12 for an arbitrary edge {i, j} of H. It is easy to see that |Xi'j | + |Yj | g'{7, 8}. It then follows that, for an arbitrary {i,j} G E(H), (X^-1, |Yj |, |Zi'j |) G {(3,4,5), (4, 3,5), (4,4,4)}. In particular, the degree of each vertex is either 0, 4 or 5. Moreover, since k3(H) = 0, {j} U Xjj and {i} U Yj are independent sets. Hence, by a(H) = 0, the induced subgraph of H on {i, j} U Xjj U Yj is the complete bipartite graph K|Xi,j1 + 1'^- |+i. Suppose that an edge {i, j} of H satisfies (|Xjj1, |Yi'j |, |Zijj |) = (4,4,4). Then, the induced subgraph of H on {i, j} U Xjj U Yj is K5'5. Since the degree of any vertex of H is either, 0, 4 or 5, other four vertices are isolated. Therefore, G = K14 - K5'5. It is enough to consider the case that (|XS't|, |YS't|, |ZS't|) = (4,4,4) holds for every edge {s,t}. Suppose that (|Xj j|, |Yj|) = (3,4). Then, the induced subgraph of H on {i, j} U Xi'j U Yj is K4'5. Since (|XM|, |YM|, |ZM|) = (4,4,4) for each edge {s,t}, the degree of every vertex in {i} U Yj is 4. In this case, K4'5 is a connected component of H. Since the degree of other five vertices is at most 4, it follows that they are isolated vertices. Therefore, G = K14 - K4'5. (c) Let d > 15 and let G = - KTO'„ G By Proposition 2.5, we have e(G) - e(Kd) = ^mn(m + n - 6)d - ^mn(3mn + 2m2 + 2n2 - 5m - 5n - 13). When m = n = 5, we obtain e(G) - e(Kd) = 50(d - 14) > 0 as desired. □ 3 Asymptotic behavior of For 0 < p < 1 and an integer d > 0, let G(d,p) denote the random graph on the vertex set [d] in which the edges are chosen independently with probability p. For a graph H on the vertex set [d] and 0 < p < 1, let G(H,p) denote the random graph on the vertex set [d] in which the edges of H are chosen independently with probability p and the edges not belonging to H are not chosen. Tran-Ziegler [6] showed that, for the random graph G(d, 1/^3), 118 1 e(G(d, 1/V3)) = -d4 + -d3 - -d2 + ^d, T. Hibi et al.: The number of edges of the edge polytope of a finite simple graph 331 and hence this is a lower bound for First, for d > 0, we give an example of a (non-random) graph G on the vertex set [d] such that e(G) > e(G(d, 1/V3)). Example 3.1. Let G = Kd - Kad,ad - K(1/2-a)d,(i/2-a)d where a = ^ (7 + V2T) and d > 0. By Propositions 2.4 and 2.5, it follows that £(G) = i48d4 + Td3 - TT2d2 + d. Since 1/54 = 0.0185 and 9/448 = 0.0201, we have e(G) > e(G(d, 1/V3)) for d > 0. Second, we give a random graph G on the vertex set [d] such that e(G) >e(G(d, 1/V3)) for d > 0. Theorem 3.2. For an integer d, let G be a random graph Kd — G(Kd/2d/2,p) with p = 3 — a/5. Then, £(G)=^ - 11 d4 - ^ - 27 d3 + ^ - 44 d2 + d. v y 8 2 2 In particular, we have e (G) > e(G(d, 1/>/3)) for all d > 0. Proof. Let m = d/2 and let [d] = V U V2 be a partition of the vertex set of Km,m. The number of pairs of edges {i, j}, {i, k} satisfying Lemma 2.1 (i) is ni = m(m - 1)(m - 2) + 2m2(m - 1)(1 - p) + m2(m - 1)(1 - p)2 where each term corresponds to the case when (i) i, j, k G Vs, (ii) i, j G Vs, k G Vs and (iii) i G Vs, j, k G Vs, respectively. Next, we study the number of pairs of edges {i, j}, {k,^} satisfying Lemma 2.1 (ii). Let Gijk£ denote the induced subgraph of G on the vertex set {i, j, k, c [d]. If either "i,j,M G Vs" or "i, ^ G Vs and j, k G Vs" holds, then {i, j, k, is a cycle of Gijk£ whenever {i, j}, {k, g E(G). Hence, we consider the following two cases: Case 1. Suppose i, j G Vs and k, I G Vs. Then, Gijk£ has a cycle of length 4 if and only if either {i,k}, {j,^} G E(G) or {i,^}, {j,k} G E(G) holds. Thus, the expected number of pairs of edges is n2 = (m )2(1 - (1 - p)2)2. Case 2. Suppose that i G Vs and j, M G V hold. Then, all of {k, ¿}, {j, k} and {j, are edges of G. On the other hand, {i, j} is an edge of G with probability 1 - p. If {i, j} is an edge of G, then Gijke has a cycle of length 4 if and only if either {i, k} G E(G) or {i, G E(G) holds. Thus, the expected number of pairs of edges is ^3 = m2(m - 1)(m - 2)(1 -p)p2. Therefore, e(G) = + n2 + n3. If m = d/2 andp = 3 - a/5, then e(G)=5^ - 11 d4 - 12^ - 27d3 + 19^ - 44 d2 + d, whose leading coefficient is 5V58-ii = 0.0225425. □ 332 Ars Math. Contemp. 10 (2016) 183-192 Remark 3.3. By Theorem 3.2, the graph G in Example 3.1 does not satisfy = e(G) for d > 0. In fact, for d = 20, by Propositions 2.4 and 2.5, it follows that Let G' G n2o be the graph such that G' is the bipartite graph with E(G') = {{1,12}, {1,14}, {1,15}, {1,16}, {1,18}, {1,19}, {1, 20}, {2,11}, {2,12}, {2,13}, {2,15}, {2,17}, {2,19}, {2, 20}, {3,11}, {3,12}, {3,13}, {3,14}, {3,15}, {3,16}, {3,18}, {4,14}, {4,15}, {4,16}, {4,17}, {4,18}, {4,19}, {4, 20}, {5,11}, {5,12}, {5,13}, {5,15}, {5,17}, {5,18}, {5, 20}, {6,12}, {6,16}, {6,17}, {6,18}, {6,19}, {6, 20}, {7,11}, {7,12}, {7,13}, {7,14}, {7,16}, {7,17}, {7,19}, {8,11}, {8,12}, {8,13}, {8,14}, {8,15}, {8,18}, {8,19}, {8, 20}, {9,11}, {9,14}, {9,15}, {9,16}, {9,17}, {9,18}, {9,19}, {10,11}, {10,13}, {10,15}, Then, e(G') = 4203 > 4176. Acknowledgment. The authors are grateful to an anonymous referee for useful suggestions, and helpful comments. References [1] T. Hibi, N. Li and Y. X. Zhang, Separating hyperplanes of edge polytopes, J. Combin. Theory Ser. A 120 (2013), 218-231, doi:10.1016/j.jcta.2012.08.002. [2] H. Ohsugi and T. Hibi, Normal polytopes arising from finite graphs, J. Algebra 207 (1998), 409-426, doi:10.1006/jabr.1998.7476. [3] H. Ohsugi and T. Hibi, Toric ideals generated by quadratic binomials, J. Algebra 218 (1999), 509-527, doi:10.1006/jabr.1999.7918. [4] H. Ohsugi and T. Hibi, Compressed polytopes, initial ideals and complete multipartite graphs, Illinois J. Math. 44 (2000), 391-406. [5] H. Ohsugi and T. Hibi, Simple polytopes arising from finite graphs, in: Proceedings of the 2008 International Conference on Information Theory and Statistical Learning (ITSL), 2008 pp. 7379, (available at http://arxiv.org/abs/0 804.428 7). [6] T. Tran and G. M. Ziegler, Extremal edge polytopes, Electron. J. Combin. 21 (2014), Paper 2.57, 16. G G Q2o and each non-empty connected component of G is a complete bipartite graph {10,16}, {10,18}, {10,19}, {10, 20}}. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 333-347 Equitable coloring of corona products of cubic graphs is harder than ordinary coloring* Hanna Furmanczyk Institute of Informatics, University of Gdansk, Wita Stwosza 57, 80-952 Gdansk, Poland Marek Kubale Department of Algorithms and System Modelling, Technical University of Gdansk, Narutowicza 11/12, 80-233 Gdansk, Poland Received 21 June 2014, accepted 15 September 2015, published online 5 February 2016 A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by x=(G). In this paper the problem of determinig x= for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses x=(G) or x=(G) + 1 colors. Our algorithm is best possible, unless P=NP. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring. Keywords: Corona graph, cubic graph, equitable chromatic number, equitable graph coloring, NP-hardness, polynomial algorithm. Math. Subj. Class.: 05C15, 05C10 *This project has been partially supported by Narodowe Centrum Nauki under contract DEC-2011/02/A/ST6/00201. The authors thank Professor Staszek Radziszowski for taking great care in reading our manuscript and making useful suggestions. E-mail addresses: hanna@inf.ug.edu.pl (Hanna Furmanczyk), kubale@eti.pg.gda.pl (Marek Kubale) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract 334 Ars Math. Contemp. 10 (2016) 183-192 1 Introduction All graphs considered in this paper are connected, finite and simple, i.e. undirected, loop-less and without multiple edges, unless otherwise is stated. If the set of vertices of a graph G can be partitioned into k (possibly empty) classes Vi ,V2,... ,Vk such that each V is an independent set and the condition | |Vj| - |Vj 11 < 1 holds for every pair (i,j), then G is said to be equitably k-colorable. If |V | = l for every i = 1,2,... ,k, then G on n = kl vertices is said to be strong equitably k-colorable. The smallest integer k for which G is equitably k-colorable is known as the equitable chromatic number of G and it is denoted by x=(G) [14]. Since equitable coloring is a proper coloring with an additional constraint, we have x(G) < x=(G) for any graph G. The notion of equitable colorability was introduced by Meyer [14]. However, an earlier work of Hajnal and Szemer6di [9] showed that a graph G with maximal degree A is equitably k-colorable if k > A +1. Recently, Kierstead et al. [11] have given an O(An2)-time algorithm for obtaining a (A + 1)-coloring of a graph G on n vertices. This model of graph coloring has many practical applications. Every time when we have to divide a system with binary conflict relations into equal or almost equal conflict-free subsystems we can model this situation by means of equitable graph coloring. In particular, one motivation for equitable coloring suggested by Meyer [14] concerns scheduling problems. In this application, the vertices of a graph represent a collection of tasks to be performed and an edge connects two tasks that should not be performed at the same time. A coloring of this graph represents a partition of tasks into subsets that may be performed simultaneously. Due to load balancing considerations, it is desirable to perform equal or nearly-equal numbers of tasks in each time slot, and this balancing is exactly what equitable colorings achieve. Furmariczyk [5] mentions a specific application of this type of scheduling problem, namely, assigning university courses to time slots in a way that avoids scheduling incompatible courses at the same time and spreads the courses evenly among the available time slots. The topic of equitable coloring was widely discussed in literature. It was considered for some particular graph classes and also for several graph products: cartesian, weak or strong tensor products [13, 5] as well as for coronas [6, 10]. Graph products are interesting and useful in many situations. The complexity of many problems, also equitable coloring, that deal with very large and complicated graphs is reduced greatly if one is able to fully characterize the properties of less complicated prime factors. Moreover, corona graphs lie often close to the boundary between easy and hard problems. The corona of two graphs G and H is the graph GoH obtained by taking one copy of G, called the center graph, |V(G)| copies of H, named the outer graph, and making the i-th vertex of G adjacent to every vertex in the i-th copy of H. Such type of graph products was introduced by Frucht and Harary in 1970 [4] (for an example see Fig. 1). After that many works have been devoted to study its structure and to obtain some relationships between the corona graph and its factors [1,4, 12, 15]. In general, the problem of optimal equitable coloring, in the sense of the number of colors used, is NP-hard and remains so for corona products of graphs. In fact, Furmanczyk et al. [6] proved that the problem of deciding whether x=(G o K2) < 3 is NP-complete even if G is restricted to the line graph of a cubic graph. Let us recall some basic facts concerning cubic graphs. It is well known from Brook's theorem [2] that for any cubic graph G = K4, we have x(G) < 3. On the other hand, Chen et al. [3] proved that for any cubic graph with x(G) = 3, its equitable chromatic H. Furmanczyk and M. Kubale: Equitable coloring of corona products 335 number equals 3 as well. Moreover, since a connected cubic graph G with x(G) =2 is a bipartite graph with partition sets of equal size, we have the equivalence of the classical and equitable chromatic numbers for 2-chromatic cubic graphs. Since the only cubic graph for which the chromatic number is equal to 4 is the complete graph K4, we have 2 < x=(G) = x(G) < 4, (1.1) for any cubic graph G. In the paper we will consider the equitable coloring of coronas. We assume that in corona G o H, |V(G)| = n and |V(H)| = m. A vertex with color i is called an i-vertex. We use color 4 instead of 0, in all colorings in the paper, including cases when color label is implied by an expresion (mod4). Let • Q2 denote the class of equitably 2-chromatic cubic graphs, • Q3 denote the class of equitably 3-chromatic cubic graphs, • Q4 denote the class of equitably 4-chromatic cubic graphs. Clearly, Q4 = {K4}. Next, let Q2(t) C Q2 (Q3(t) C Q3) denote the class of bipartite (tripartite) cubic graphs with partition sets of cardinality t, and let Q3(u, v, w) C Q3 denote the class of 3-partite graphs with color classes of cardinalities u, v and w, respectively, where u > v > w > u - 1. Observe that its rrN i 4 if H e Q2, X(K4 o H)^x(H) + 1 otherwise2, (L2) In the next section we show a way to color G o H with 3 colors provided that the corona admits such a coloring. Next, in Section 3 we give a linear-time procedure for coloring corona products of cubic graphs with 5 colors. It turns out that this number of colors is sufficent for equitable coloring of any corona of cubic graphs, but in some cases less than 5 colors suffice. In Section 4 we give our main result that deciding whether G o H is equitably 4-colorable is NP-complete when H e Q3(t) and 10 divides t, in symbols 10|t. Hence, our 5-coloring algorithm of Section 3 is 1-absolute approximate and the problem of equitable coloring of cubical coronas belongs to very few NP-hard problems that have approximation algorithms of this kind. Most of our results are summarized in Table 1. H G Q2 Q3 Q4 Q2 3 or 4 [Thm. 2.3] 4 or 5* [Thms. 3.3, 4.3] 5 [Thm. 3.2] Q3 3 or 4 [Thm. 2.3] 4 or 5* [Thm. 3.4, Col. 4.4] 5 [Thm. 3.2] Q4 4 [Thm. 2.3] 4 5 [Thm. 3.2] Table 1: Possible values of x=(G o H), where G and H are cubic graphs. Asterix (*) means that deciding this case is NP-complete. To the best of our knowledge, cubical coronas are so far the only class of graphs for which equitable coloring is harder than ordinary coloring. And, since x=(G o H) < 5 and 336 Ars Math. Contemp. 10 (2016) 183-192 A(G o H) > 7, our results confirm Meyer's Equitable Coloring Conjecture [14], which claims that for any connected graph G, other than a complete graph or an odd cycle, we have x=(G) < A. 2 Equitable 3-coloring of corona of cubic graphs First, let us recall a result concerning coronas G o H, where H is a 2- or 3-partite graph. Theorem 2.1 ([6]). Let G be an equitably k-colorable graph on n > k vertices and let H be a (k — l)-partite graph. If k\n, then X=(G o H) < k. Proposition 2.2. If G and H are cubic graphs, then x=(G o H) = 3 if and only if G G Q2 U Q3, H g Q2, and G has a strong equitable 3-coloring. Proof. Since G is strong equitably 3-colorable, the cardinality of its vertex set must be divisible by 3. The thesis follows now from Theorem 2.1. Assume that x=(G o H) = 3. This implies: • H must be 2-chromatic, and due to (1.1) it must be also equitably 2-chromatic, • G must be 3-colorable (not necessarily equitably), x(G) < x=(G) < 3, which implies G G Q2 U Q3. Otherwise, we would have x(G o H) > 4 which is a contradiction. Since H G Q2 is connected, its bipartition is determined. Let H G Q2(t), t > 3. Observe that every 3-coloring of G determines a 3-partition of G o H. Let us consider any 3-coloring of G with color classes of cardinality ni,n2 and n3, respecively, where n = n1 + n2 + n3. Then the cardinalities of color classes in the implied 3-coloring of G o H form a sequence ((n2 + n3)t, (n1 + n3)t, (n1 + n2)t). Such a 3-coloring of G o H is equitable if and only if n1 = n2 = n3. This means that G must have a strong equitable 3-coloring, which, keeping in mind that x=(G o H) > 3 for all cubic graphs G and H, completes the proof. □ In the remaining cases of coronas G o H, where H g Q2, we have to use more than three colors. However, it turns out that in all such cases four colors suffice. Theorem 2.3. If G is a cubic graph, H G Q2, then Proof. Due to Proposition 2.2, we only have to define an equitable 4-coloring of G o H. The cases of G g Q2 U Q4 are easy. We start from an equitable 4-coloring of the center graph and extend it to the corona. Let us assume that G g Q3. First, we color equitably G with 3 colors and then extend this coloring to equitable 4-coloring of G o H, H = H(U, V) G Q2 (t). Since the number of vertices of cubic graph G is even, we have to consider two cases. 3 if G G Q2(s) U Q3, 3\s and G is equitably 3-colorable, 4 otherwise. H. Furmanczyk and M. Kubale: Equitable coloring of corona products 337 Case 1: n = 4k, for some k > 2. Since G is equitably 3-colorable, the color classes of equitable 3-coloring of G are of cardinalities [4k/3], [(4k - 1)/3] and [(4k - 2)/3], respectively. And, since |V(G o H)| = 4k(2t + 1), in every equitable 4-coloring of G o H each color class must be of cardinality 2kt + k. We extend our 3-coloring of G to G o H as follows (see Fig. 1a)). We color: • the vertices in one copy of H linked to a 1-vertex in G using t times color 3 (vertices in partition U), t- ([(4k-1)/3] -k) times color 2 and [(4k-1)/3] -k times color 4 (vertices in partition V), • the vertices in one copy of H linked to a 2-vertex in G using t times color 1 (vertices in partition U), t-( [(4k-2)/3] -k) times color 3 and [(4k-2)/3] -k times color 4 (vertices in partition V), • the vertices in one copy of H linked to a 3-vertex in G using t times color 2 (vertices in partition U), t - ([4k/3] - k) times color 1 and [4k/3] - k times color 4 (vertices in partition V). Figure 1: An example of coloring of W o K3,3, where W is the Wagner graph (C8 with 4 diagonals): a) partial 4-coloring; b) equitable 4-coloring. So far, colors 1, 2 and 3 have been used 2t + k times, while color 4 has been used k times. Now, we color each of uncolored copy of H with two out of three allowed colors in such a way that in this step colors 1, 2 and 3 are used (2k - 2)t times and color 4 is used 2kt times, which results in an equitable 4-coloring of the whole corona G o H (see Fig. 1b)). Case 2: n = 4k + 2, for some k > 1. Since G is equitably 3-colorable, its color classes are of cardinalities [(4k + 2)/3], [(4k +1)/3] and [4k/3], respectively, in any equitable coloring of G. Since |V(Go H) | = (4k + 2)(2t + 1) = 8kt + 4t + 4k + 2, in every equitable 4-coloring the color classes must be of cardinality 2kt +1 + k or 2kt +1 + k +1. 338 Ars Math. Contemp. 10 (2016) 183-192 We color: • the vertices in one copy of H linked to a 1-vertex of G using t times color 3 (vertices in partition U), t - ([(4k + 1)/3"| — k — 1) times color 2 and [(4k + 1)/3] — k — 1 times color 4 (vertices in partition V), • the vertices in one copy of H linked to a 2-vertex of G using t times color 1 (vertices in partition U), t — ([4k/3] — k) times color 3 and [4k/3] — k times color 4 (vertices in partition V), • the vertices in one copy of H linked to a 3-vertex of G using t times color 2 (vertices in partition U), t — ([(4k + 2)/3] — k — 1) times color 1 and [(4k + 2)/3] — k — 1 times color 4 (vertices in partition V). So far, colors 1 and 2 have been used 2t + k + 1 times, while color 3 has been used 2t + k times and color 4 has been used k times. Finally, we color still uncolored copies of H with two (out of three) allowed colors so that colors 1, 2 and 3 are used (2k — 1)t times and color4 is used 2kt times, which results in an equitable 4-colorings of the whole corona G o H. □ 3 Equitable 5-coloring of coronas of cubic graphs We start by considering cases when 5 colors are necessary for such graphs to be colored equitably. Proposition 3.1 ([6]). If G is a graph with x (G) < m + 1, then x=(G o Km) = m +1. This proposition immediately implies Corollary 3.2. If G is a cubic graph, then X=(G o K4) = 5. It turns out that 5 colors may be required also in some coronas G o H, where G G Q2 U Q3 and H g Q3. Theorem 3.3. If G G Q2(s) and H G Q3, then 4 < x=(G o H) < 5. Proof. Since H G Q3, we obviously have x=(G o H) > 4. To prove the upper bound, we consider two cases. Let H = H(U, V, W) with triparti-tionof H satisfying |U| > |V| > |W|. Case 1: s = 2k + 1, k > 1. We start with the following 4-coloring of G o H. H. Furmanczyk and M. Kubale: Equitable coloring of corona products 339 1. Color graph G with 4 colors, using each of colors 1 and 2 k times and colors 3 and 4 (k +1) times, respectively. 2. Color the vertices of each copy of H(U, V, W) linked to an ¿-vertex of G using color (i + 1) mod 4 for vertices in U, color (i + 2) mod 4 for vertices in V, and color (i + 3) mod 4 for vertices in W (we use color 4 instead of 0). Now, we have to consider three subcases, where we bound the number of vertices that have to be recolored to 5. Subcase 1.1: H G Q3(t + 1,t,t), where t = v = w. The color sequence of the 4-coloring of this corona is C4 = (ci, c5, c3, c4) = (3kt + 2k + 2t + 1, 3kt + 2k + 2t, 3kt + 2k +1 + 1, 3kt + 2k +1 + 2). In every equitable 5-coloring of the corona G o H, where G G Q5 (2k + 1) and H G Q3(t +1,t,t), every color must be used y1 = |"(12kt + 8k + 6t + 4)/5] = (2kt+t+k+|"(2kt+t+3k+4)/5]) or y5 = (2kt+t+k+|_(2kt+t+3k+4)/5_|) times. The number dj of vertices colored with i, 1 < i < 4, that have to be recolored is equal to c - Y5 or c - y5. We have d1 < c1 - Y51 < c1 - y55 = kt + t + k + 1 - |_(2kt + t + 3k + 4)/5j = = (k + 1)(t + 1) - |_(2kt + t + 3k + 4)/5j < (k + 1)(t + 1). Similarly, we have d5 < k(t +1)+1, d3 < k(t +1), and d4 < k(t +1). Subcase 1.2: H G Q3(t + 1,t + 1,t), where t = w. The color sequence of the 4-coloring of this corona is C4 = (c1, c5, c3, c4) = (3kt + 3k + 2t + 2, 3kt + 3k + 2t +1, 3kt + 3k + t + 1, 3kt + 3k + t + 2). In every equitable 5-coloring of the corona G o H, where G G Q5 (2k + 1) and H G Q3 (t + 1, t + 1, t), every color must be used y5 = |"(12kt + 12k + 6t + 6)/5] = (2kt +1 + 2k +1+ [(2kt +1 + 2k + 1)/5]) or y55 = (2kt +1 + 2k + 1 + |_(2kt +1 + 2k + 1)/5j) times. Similarly, as in Subcase 1.1, we have di < ci - y5 < ci - y5 < (k + 1)(t +1), d5 < k(t +1) + y, d3 < k(t + 1), and d4 < k(t +1). Subcase 1.3: H G Q3(t), where t = u = v = w. The color sequence of the 4-coloring of this corona is C4 = (c1, c5, c3, c4) = (3kt + k + 2t, 3kt + k + 2t, 3kt + k + t + 1, 3kt + k + t + 1). In every equitable 5-coloring of the corona G o H, where G G Q5(2k + 1) and H g Q3(t,t,t), every color must be used |"(12kt + 4k + 6t + 2)/5] = 340 Ars Math. Contemp. 10 (2016) 183-192 (2kt + t + \(2kt + t + 4k + 2)/5]) or (2kt + t + |_(2kt + t + 4k + 2)/5_|) times. Similarly, as in previous subcases, we have di < (k + 1)t, d2 < kt + t, d3 < kt, and ¿4 < kt. Consequently, in all subcases, the number of ¿-vertices that have to be recolored is bounded by: • (k + 1)u for i = 1, • ku + w for i = 2, • ku for i = 3,4. To obtain an equitable 5-coloring from the 4-coloring of G o H(U, V, W), |U| > IV| > |W|, we recolor the appropriate number of ¿-vertices in partitions U linked to (i - 1)-vertices of G for the vertices which were colored with color i. Due to the above, this is possible in the cases of colors 1, 3 and 4. In the case of 2-vertices, the number of vertices recolored in partition U in copies of H can be insufficient. In this case, we can recolor the vertices in partition W (of cardinality w) in one copy of H linked to 3-vertex of G. Case 2: s = 2k, k > 2. Again, we start with 4-coloring of G o H, as follows. 1. Color graph G with 4 colors, using each of colors 1,2, 3 and 4 k times. 2. Color the vertices of each copy of H(U, V, W) linked to an i-vertex of G using color (i + 1) mod 4 for vertices in U, color (i + 2) mod 4 for vertices in V, and color (i + 3) mod 4 for vertices in W (we use color 4 instead of 0). Notice that the resulting 4-coloring does not require recoloring: it is equitable and establishes that the lower bound is tight. □ Similar technique for obtaining an equitable coloring is used in the proof of the following theorem, by introducing the fifth color. Theorem 3.4. If G, H G Q3, then 4 < x=(G o H) < 5. Proof. Let G = G(A,B,C), where |A| > |B| > |C| > |A|-1, and let H = H(U,V,W), where |U | > |V | > |W | > |U | - 1. We start with a 4-coloring of G o H. 1. Color the vertices of graph G with 3 colors: the vertices in A with color 1, in B with 2, and in C with color 3. H. Furmanczyk and M. Kubale: Equitable coloring of corona products 341 2. Color the vertices of each copy of H linked to an ¿-vertex using color (i + 1) mod 4 for vertices in U, color (i + 2) mod 4 for vertices in V, and color (i + 3) mod 4 for vertices in W, i = 1, 2, 3 (see Fig. 2a)). Figure 2: An example of coloring of W o P, where W is the Wagner graph and P is the prism graph: a) ordinary 4-coloring; b) equitable 5-coloring. Since |V(G o H)| = (m + 1)n, the color cardinality sequence C = (ci, c2, c3, c4) of the above 4-coloring of G o H is as follows: ( [n/31 + [(n - 1)/3] [(m - 2)/3] + [(n - 2)/3] [(m - 1)/3] , [n/3] [m/31 + [(n - 1)/3l + [(n - 2)/3l |"(m - 1)/3l , [n/31 [(m - 1)/3l + [(n - 1)/3l |m/3l + [(n - 2)/3l , [n/31 [(m - 2)/3l + [(n - 1)/3l [(m - 1)/3l + [(n - 2)/3l [m/3l ), respectively. This 4-coloring is not equitable. We have to recolor some vertices colored with 1, 2, 3 and 4 into 5. The number of vertices colored with i, 1 < i < 4, that have to be recolored is equal to c - [((m + 1)n - i + 1)/5l. We have the following claims: 342 Ars Math. Contemp. 10 (2016) 183-192 C3 C4 (m + 1)n < 1 ci — 5 2 (m + 1)n — 1" < 1 5 2 (m + 1)n — 2" < 3 5 4 (m + 1)n — 3" < "1 5 2 n - 2 3 n' 3 n — 1 3 n2 m—1 1 3 = [2|C y "m" 1 3" = _2|A| m" 3" m — 2" 3 I U | , 3ibi + |v |UI, and "1 n — 1 m — 1 4 3 3 + "1 "n — 2" "m" 2 3 "3" 2 |AI |w | + 4 I B I IV | + 2 |C| I u I . (3.1) (3.2) (3.3) + (3.4) Proof of inequalities (3.1)-(3.4). Let us consider three cases, G G Q3(s), Q3(s + 1, s, s), and Q3(s +1, s +1, s), and in each case three subcases, H G Q3(t), Q3(t +1, t,t), Q3(t + 1, t + 1, t), respectively. The estimation technique for the number of vertices that have to be recolored to color 5 is similar to that used in the proof of Theorem 3.3. Case 1: G G Q3(s), where s = 2k for some k > 1. Subcase 1.1: H G Q3 (t), where t = 2/ for some / > 1. We have | V(G o H) | = (3t +1)3s = 5(7k/ + k) + k/ + k, while the color cardinality sequence C of the 4-coloring of G o H is C = (s + 2st, s + 2st, s + 2st, 3st) = (8k/ + 2k, 8k/ + 2k, 8k/ + 2k, 12k/). Since in every equitable 5-coloring of G o H each of 5 colors has to be used (7k/ + k + [(k/ + k)/5]) or (7k/ + k + |_(k/ + k)/5j) times, we have to recolor some vertices colored with 1, 2, 3 and 4 into 5. The number of vertices that have to be recolored is as follows: • the vertices colored with 1: 8k/ + 2k — 7k/ — k — [(k/ + k)/5]] < 2k/ = [ 1 |C|J |V|, • the vertices colored with 2: 8k/ + 2k — 7k/ — k — [(k/ + k — 1)/5]] < 2k/ = [ 1 |A|J |U|, • the vertices colored with 3: 8k/ + 2k — 7k/ — k — [(k/ + k — 2)/5] < 2k/ < [4|B|J |U|, • the vertices colored with 4: 12k/ — 7k/ — k — [(k/ + k — 3)/5] ^ 4k/ + [| • 2/] = = )||A|] |W| + )4|B|] |V| + ) 1 |CI] |U|. Subcase 1.2: H G Q3 (t + 1, t, t), where t = 2/ + 1 for some / > 1. We have |V(G o H)| = (3t + 2)3s = 5(7k/ + 6k) + k/, while the color cardinality sequence C of the 4-coloring of G o H is C = (s + 2st, 2s + 2st, 2s + 2st, 3st + s) = (8k/ + 6k, 8k/ + 8k, 8k/ + 8k, 12k/ + 8k). H. Furmanczyk and M. Kubale: Equitable coloring of corona products 343 Since in every equitable 5-coloring of G o H each of 5 colors has to be used (7k/ + 6k + [k//5]) or (7k/ + 6k + |k/ /5j) times, we have to recolor some vertices colored with 1, 2, 3 and 4 into 5. The number of vertices that have to be recolored is as follows: • the vertices colored with 1: k/ - [k//5] < 2k/ + k = L2|C|J |V|, • the vertices colored with 2: k(/ + 1) + k - [(k/ - l)/5] < 2k(/ + 1) = L2|A|J |U|, • the vertices colored with 3: : J k(/ +1) + k -[(k/ - 2)/5] < L4kj(2/ + 2)= L4|B|J |U|, • the vertices colored with 4: 5kl + 2k - [(k/ - 3)/5] < 4k/ + 2k + [f] (2/ + 1) = = 12|A|] |W| + 14|B|] |V| + 12|Cn |U|. Subcase 1.3: H G Q3(t + 1,t + 1,t), where t = 21 for some 1 > 1. We have |V (GoH )| = (3t+3)3s = 5(7k1+3k)+k1+3k, while the color cardinality sequence C of the 4-coloring of GoH is C = (2s+2st, 2s+2st, 3s+2st, 3st+2s) = (8k1 + 4k, 8k1 + 4k, 8k1 + 6k, 12k/ + 4k). Since in every equitable 5-coloring of G o H each of 5 colors has to be used (7k/ + 3k + [(k/ + 3k)/5]) or (7k/ + 3k + |(k/ + 3k)/5j) times, we have to recolor some vertices colored with 1, 2, 3 and 4 into 5. The number of vertices that have to be recolored is as follows: • the vertices colored with 1: L J k/ + k - [(k/ + 3k)/5] < 2k/ + k = L2|C|J |V|, • the vertices colored with 2: L J k/ + k - [(k/ + 3k - 1)/5] < 2k/ + k = L2|A|J |U|, • the vertices colored with 3: L J k/ + 3k - [(k/ + 3k - 2)/5] < L2kj (2/ + 1) = L4 |B|J |U|, • the vertices colored with 4: 5kl + k - [(k/ + 3k - 3)/5] < 4k/ k + [2] (2/ + 1) = = 12|A|] |W| + 14|B|] |V| + 11 |C|] |U|. Case 2: G G Q3(s + 1, s, s), where s = 2k + 1 for some k > 1. The proof follows by a similar argument to that in Case 1, we omit the details. Case 3: G G Q3(s + 1, s + 1, s), where s = 2k for some k > 1. The proof follows by a similar argument to that in Case 1, we omit the details. End of the proof of inequalities (3.1)-(3.4). Now, to obtain an equitable 5-coloring of G o H, we choose the vertices that have to be recolored. • Since the number of 1-vertices that have to be recolored to 5 is not greater than L2 |C|J|V|, then the vertices colored with 1 are chosen from the partitions V of L1 ¡C|J copies of H linked to the vertices from partition C of G. 344 Ars Math. Contemp. 10 (2016) 183-192 • Similarly, 2-vertices that have to be recolored are chosen from the partitions U of L1 |A|J copies of H linked to the vertices from partition A of G. • 3-vertices to be recolored are chosen from the partitions U of Lf |B|J copies of H linked to the vertices from partition B of G. • 4-vertices are chosen from: - partitions W of [ 1 |A|] copies of H linked to the vertices from the partition A of G (different copies than in recoloring of 2-vertices), - partitions V of [| |B|] copies of H linked to the vertices from the partition B of G (different copies than in recoloring of 3-vertices), - partitions U of [ 1 |C|] copies of H linked to the vertices from the partition C of G (different copies than in recoloring of 1-vertices) (see Fig. 2b)). Taking into account our claim, such recoloring is possible. □ As we have already observed, the lower bound in Theorem 3.3 is tight. Also upper bounds in Theorems 3.3 and 3.4 are tight. There are infinitely many coronas G o H, where G g Q2 U Q3 and H g Q3, that require five colors to be equitably colored. For example, in such coronas graph H g Q3 may be built of 3t (t must be even) vertices and it must contain t disjoint triangles (cycles C3) (cf. Fig. 3). Let us consider for example G = K33. In the corona K3 3 o H, where H is defined as above, the number of vertices is equal to 36k + 6, for some positive integer k. In any equitable 4-coloring of the corona, the color sequence must be (9k + 2,9k + 2, 9k + 1,9k +1). Since modifying the tripartite structure of H is impossible (it contains t = 2k disjoint triangles), such a coloring does not exist for k > 2. 4 Complexity results Although we have only two possible values, 4 and 5, for x= (G o H), where G g Q2 U Q3 and H g Q3, it is hard to decide which is correct1. All G, H are still cubic. We consider the following combinatorial decision problems: Note that the IS3(H, k) problem is NP-complete and remains so even if 10|m [8]. This is so because we can enlarge H by adding j (0 < j < 4) isolated copies of K3,3 to it so that the number of vertices in the new graph is divisible by 10. Graph H has an independent set of size at least k if and only if the new graph has an independent set of size at least k + 3j. 1 graphs considered in this section need not be connected H. Furmanczyk and M. Kubale: Equitable coloring of corona products 345 IS3 (H, k) : Given a cubic graph H on m vertices and an integer k, the question is: does H have an independent set I of size at least k? and its subproblem for m = 10q, k = 4m/10 = 4q, i.e. IS3(H, 4q). Lemma 4.1. Problem IS3(H, 4m/10) is NP-complete. Proof. Our polynomial reduction is from IS3(H, k). For an m-vertex cubic graph H, 10|m, and an integer k, let r = |4m/10 — k|. If k > 4m/10 then we construct a cubic graph G = H + rK4 + rP else we construct G = H + rK4 + 2rP + 4rK3,3, where P G Q3(2) is the prism graph. It is easy to see that the answer to problem IS3(H, k) is 'yes' if and only if the answer to problem IS3(G, 4m/10) is 'yes'. □ Lemma 4.2. Let H be a cubic graph and let k = 4/10m, where m is the number of vertices of H. The problem of deciding whether H has a coloring of type (4m/10,3m/10,3m/10) is NP-complete. Proof. We prove that H has a coloring of type (4m/10, 3m/10, 3m/10) if and only if there is an affirmative answer to IS3 (H, 4m/10). Suppose first that H has the above 3-coloring. Then the color class of size 4m/10 is an independent set that forms a solution to IS3(H, 4m/10). Now suppose that there is a solution I to IS3(H, 4m/10). Thus |I1 > 4m/10. We know from [7] that in this case there exists an independent set I' of size exactly 4m/10 such that the subgraph H — I' is equitably 2-colorable bipartite graph. This means that H can be 3-colored so that the color sequence is (4m/10,3m/10,3m/10). □ In the following we show that, given such an unequal coloring of H, we can color K3,3 o H equitably with 4 colors. (i) Color the vertices of K3,3 with 4 colors - the color sequence is (2,2,1,1). (ii) Color the vertices in copies of H = H(U, V, W), |U| = 4m/10, |V| = |W| = 3m/10, in the following way: • vertices in partitions U of H adjacent to a 1-vertex of K3 3 are colored with color 2, in partitions V - with 3, and in partitions W - with 4, • vertices in partitions U of H adjacent to a 2-vertex of K3 3 are colored with color 1, in partitions V - with 3, and in partitions W - with 4, • vertices in partition U of H adjacent to the 3-vertex of K3 3 are colored with color 1, in partition V - with 2, and in partition W - with 4, • vertices in partition U of H adjacent to the 4-vertex of K3 3 are colored with color 2, in partition V - with 1, and in partition W - with 3. Color sequence of the corona is (15m/10 + 2,15m/10 + 2,15m/10 + 1,15m/10+1). On the other hand, let us assume that the corona K3 3 o H, where H g Q3(t) and t = 10k, is equitably 4-colorable, where the color sequence for K3,3 is (2, 2,1,1). Since |V(K3,3 o H)| = 6(3t + 1) = 18t + 6 and t = 10k for some k, then each of the four 346 Ars Math. Contemp. 10 (2016) 183-192 colors in every equitable coloring is used 45k +1 or 45k + 2 times. Since color 1 (similarly color 2) can be used only in four copies of H, then in at least one copy we have to use it 12k = 12t/10 times. It follows that there must exist an independent set of cardinality 12t/10 in H. Since H has 3t vertices, the size of this set is 4m/10. The above considerations lead us to the following Theorem 4.3. The problem of deciding whether x=(K3,3 o H) = 4 is NP-complete even if H e Q3(t) and 10|t. ' □ A similar argument implies the following Corollary 4.4. The problem of deciding whether x=(P ◦ H) = 4, where P is the prism graph, isNP-complete even if H e Q3(t) and 10|t. □ In this way we have obtained the full classification of complexity for equitable coloring of cubical coronas. 5 Conclusion In this paper, we presented all the cases of corona of cubic graphs for which 3 colors suffice for equitable coloring. In the remaining cases we have proved constructively that 5 colors are enough for equitable coloring. Since there are only two possible values for x= (G o H), namely 4 or 5, our algorithm is 1-absolute approximate. Due to Theorem 4.3 and Corollary 4.4 the algorithm cannot be improved unless P=NP. Since time spend to assign a final color to each vertex is constant, the complexity of our algorithm is linear. Finally, the algorithm confirms the Equitable Coloring Conjecture [14]. Our results are summarized in Table 2. This table contains also the values of classical chromatic numbers of appropriate coronas and the complexity classification. Let us notice that all cases are polynomially solvable for ordinary coloring. H G Q2 Q3 Q4 Q2,Q3 3 3 or 4 4 4 or 5* 5 5 Q 4 4 4 4 4 5 5 Table 2: The exact values of classical chromatic number (in italics) and possible values of the equitable chromatic number (in bold) of coronas G o H. Asterix (*) means that this case is NP-complete. The other cases are solvable in linear time. References [1] S. Barik, S. Pati and B. K. Sarma, The spectrum of the corona of two graphs, SIAM J. Discrete Math. 21 (2007), 47-56 (electronic), doi:10.1137/050624029. [2] R. L. Brooks, On colouring the nodes of a network, Proc. Cambridge Philos. Soc. 37 (1941), 194-197. [3] B. L. Chen, K.-W. Lih and P.-L. Wu, Equitable coloring and the maximum degree, European J. Combin. 15 (1994), 443-447, doi:10.1006/eujc.1994.1047. H. Furmanczyk and M. Kubale: Equitable coloring of corona products 347 [4] R. Frucht and F. Harary, On the corona of two graphs, Aequationes Math. 4 (1970), 322-325. [5] H. Furmanczyk, Equitable coloring of graph products, Opuscula Math. 26 (2006), 31-44. [6] H. Furmanczyk, K. Kaliraj, M. Kubale and J. V. Vivin, Equitable coloring of corona products of graphs, Adv. Appl. Discrete Math. 11 (2013), 103-120. [7] H. Furmanczyk, M. Kubale and S. Radziszowski, On bipartization of cubic graphs by removal of an independent set, Disc. Appl. Math. (to appear), doi:10.1016/j.dam/2015.10.036. [8] M. R. Garey, D. S. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theoret. Comput. Sci. 1 (1976), 237-267. [9] A. Hajnal and E. Szemeridi, Proof of a conjecture of P. Erd6s, in: Combinatorial theory and its applications, II(Proc. Colloq., Balatonfiired, 1969), North-Holland, Amsterdam, pp. 601-623, 1970. [10] K. Kaliraj, V. J. Veninstine and V. J. Vernold, Equitable coloring on corona graph of graphs, J. Combin. Math. Combin. Comput. 81 (2012), 191-197. [11] H. A. Kierstead, A. V. Kostochka, M. Mydlarz and E. Szemeridi, A fast algorithm for equitable coloring, Combinatorica 30 (2010), 217-224, doi:10.1007/s00493-010-2483-5. [12] Y.-L. Lai and G. J. Chang, On the profile of the corona of two graphs, Inform. Process. Lett. 89 (2004), 287-292, doi:10.1016/j.ipl.2003.12.004. [13] W.-H. Lin and G. J. Chang, Equitable colorings of Cartesian products of graphs, Discrete Appl. Math. 160 (2012), 239-247, doi:10.1016/j.dam.2011.09.020. [14] W. Meyer, Equitable coloring, Amer. Math. Monthly 80 (1973), 920-922. [15] K. Williams, On the minimum sum of the corona of two graphs, in: Proceedings of the Twenty-fourth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1993), volume 94, 1993 pp. 43-49. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 349-357 Petrie polygons, Fibonacci sequences and Farey maps David Singerman , James Strudwick * Received 5 June 2015, accepted 21 September 2015, published online 5 February 2016 Abstract We consider the regular triangular maps corresponding to the principal congruence subgroups r(n) of the classical modular group. We relate the sizes of the Petrie polygons on these maps to the periods of reduced Fibonacci sequences. Keywords: Regular map, Petrie polygon, Fibonacci sequence. Math. Subj. Class.: 05C10, 11B39, 20H05 1 Introduction An interesting number theoretic problem is to determine the period of the Fibonacci sequence mod n. Here we look at the period a(n) of the Fibonacci sequence mod n up to sign. A Petrie polygon on a regular map is a zig-zag path through the map and an important invariant of a regular map is the length of a Petrie polygon. The maps we consider here are those that arise out of principal congruence subgroups r(n) of the classical modular group r. In this case It is shown that these lengths are equal to a(n). A particularly nice example is when n = 7. Here the regular map is the famous map on the Klein quartic and we find a(7) = 8 giving the title "The Eightfold Way" to the sculpture by Helaman Ferguson that represents Klein's Riemann surface of genus 3 derived from the Klein quartic. This is described in the book "The eightfold way: the beauty of Klein's quartic curve", a collection of papers related to the Klein quartic edited by Silvio Levy [5]. Let X be a compact orientable surface. By a map (or clean dessin d'enfant) on X we mean an embedding of a graph G into X such that X \ G is a union of simply-connected polygonal regions, called faces. A map thus has vertices, edges and faces. A directed edge is called a dart and a map is called regular if its automorphism group acts transitively on its darts. The platonic solids are the most well-known examples of regular maps. These are the regular maps on the Riemann sphere. We recall how we study maps using triangle groups. The universal map of type (m, n)is the tessellation of one of the three simply connected * Department of Mathematical studies, University of Southampton, Southampton SO17 1EH, United Kingdom E-mail addresses: ds@maths.soton.ac.uk (David Singerman), jes3g10@soton.ac.uk (James Strudwick) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 350 Ars Math. Contemp. 10 (2016) 183-192 Riemann surfaces, that is the Riemann sphere E, the Euclidean plane C, or the hyperbolic plane, H (depending on whether the genus of X is 0,1, or > 1) by regular m-gons with n meeting at each vertex. This map is denoted by M(m, n). The automorphism group, and also the conformal automorphism group, of M(m, n) is the triangle group r[2, m, n]. In general, a map is of type (m, n) if m is the least common multiple of the face sizes and n is the least common multiple of the vertex valencies. As shown in [3] every map of type (m, n) is a quotient of M(m, n) by a subgroup M of the triangle group r[2, m, n]. Then M is called a map subgroup of M(m, n) or sometimes a fundamental group of M(m, n), inside r[2, m, n]. A platonic surface is one that underlies a regular map. The map is regular if and only if M is a normal subgroup of r[2, m, n]. Thus a platonic surface is one of the form U/M where M is a normal subgroup of a triangle group and U is a simply connected Riemann surface. It is permissible to let m or n, or both to be to. In this paper we are particularly interested in the case where m = 3, n = to. This means that the corresponding maps are triangular though in general we are not concerned with the vertex valencies. However, if the map is regular then we must have all vertex valencies equal. For example, the icosahedron is a triangular map with all vertices of valency 5. To study triangular maps we use the triangle group [2,3, to] which is known to be the modular group r =PSL(2, Z) one of the most significant groups in mathematics. The regular maps correspond to normal subgroups of r. The most well-known normal subgroups of r are the principal congruence subgroups r(n) defined in section 5. We let M3(n) = M3(3, TO)/r(n). We call these maps principal congruence maps or PC maps. For low values of n these maps are well-known. For n = 2,3,4, 5 we get the triangle, tetrahedron, octahedron and icosahedron respectively. These are the only PC maps of genus 0. For n = 6 we get the regular map {3,6}2.2 on the torus and for n = 7 we get the Klein map on Klein's Riemann surface of genus 3. (See [2, 1]). 2 Petrie polygons A Petrie polygon in a map M is defined as a zig-zag path in the map. More precisely, we start at a vertex, then go along an edge to an adjacent vertex, the turn left and go to the next vertex and then turn right, etc., (or interchange left and right.) We have a path in which two consecutive edges belong to the same face but no three consecutive edges belong to the same face, [1, p. 54]. Eventually, in a finite regular map, we will come back to the original vertex.This path is called a Petrie path or Petrie polygon. The number of edges of this Petrie polygon is called the Petrie length of the map. We now relate the Petrie polygons to triangle groups. From the triangle group r[2, m, n], we can form the extended triangle group r(2, m, n) which is the group generated by the reflections Ri, R2, R3 in the edges of a triangle with angles n/2, n/m, n/n where we choose our ordering so that r(2, m, n) has a presentation (Ri,R2,R3|R2 = R2 = R3 = (R1R2)2 = (R2 R3)m = (R3Ri)n = 1). If we let X = R1R2, Y = R2 R3, Z = R3 R1, then we find that r[2, m, n] has a presentation (X, Y, Z |X2 = Ym = Zn = XYZ = 1). In section 5.2 of [1, p. 54], it is shown that R1R2R3 is a transformation that goes one step around a Petrie polygon. Now D. Singerman and J. Strudwick: Petrie Polygons, Fibonaci sequences and Farey maps 351 (Ri R2 R3 )2 — Ri R2 R3 Ri R2 R3 — Ri R2 R3 R2 R2 Ri R2 R3 — XY 1X 1Y showing that Petrie length is twice the order of this commutator which implies that the Petrie length is independent of the Petrie polygon chosen; it is just a property of the map. 3 The Farey map This is basically the map M(3, ro), which we abbreviate to M3. We construct it as follows. The vertices are the extended rationals Q U {ro} and two rationals | and d are joined by an edge if and only if ad — bc — ±1. This map has the following properties. (a) There is a triangle with vertices 0, i, 0 called the principal triangle. (b) The modular group r acts as a group of automorphisms of M3. (c) The general triangle has vertices |, a+d, d. Thus the Farey map (Figure 1) is a triangular map with triangular faces given by (c). In [7] it is shown that this is the universal triangular map in the sense that any other triangular map on an orientable surface is a quotient of M3 by a subgroup A of the modular group r. As M3 has vertices the extended rationals this means that every triangular map the vertices can be given coordinates which are A orbits of points in Q U {ro}. We shall denote the orbit of I by [| ]. This is illustrated in [2] where there are many examples, in particular coordinates for the triangular platonic solids are given. Also see Figure 2. Figure 1: Farey map 4 The Petrie polygons of the Farey map We consider a Petrie path in M3. By transitivity we may assume it's first edge goes from Wi — 0 to W2 — i .A left turn then takes us to W3 — i Now a right turn takes us to W4 — i. By applying a modular transformation ^ d j to the vertices ro, 0 and 1 to the 352 Ars Math. Contemp. 10 (2016) 183-192 principal triangle we find that three consecutive vertices of the Petrie polygon are a, d, a+d, that is the third vertex is the Farey median of the previous two. As the first two vertices of the Petrie polygon are 1 and\ the kth vertex of the Petrie polygon is equal to f1 where fk is the kth element of the Fibonacci sequence defined by f0 = 0, f = 1, fk+1 = fk + fk-1. for k > 1. Thus the Petrie polygon is 10 112 3 0,1,1, 2, 3, 5 ••• Lemma 4.1. The matrix P = ^^ maps each vertex of the Petrie polygon of M3 to the next one and also Pk = \fk—1 fk \ Jk Jk+1 The proof follows immediately from the definition of the Fibonacci sequence, and induction. Note that P having determinant -1 is not an element of r but T = P2 = ^ ^ is an element of r. In the following sections we will consider the Petrie polygon modulo n. As | = —a, we introduce the following concept. Definition 4.2. We call the least positive integer m with the property that fm-1 = ±1, mod n, fm = 0 mod n the semi-period ={(: d) g r:(; d)=± (;;) mod „} Now r(n) is a normal subgroup of r and so corresponds to a regular map M3 (n) which lies on the surface H*/r(n) where H* = H U Q U {to}. Another important group for us is r1(n). This is defined as r1<»>={(: d) G r:(: d)=± (0 b) m-^«} where 0 < b < n. We will not make use of this subgroup but in [2] it was shown that the left cosets of r1 (n) in r are in one-to-one correspondence with the vertices of M3(n). r(n) is a normal subgroup of r of index n3 1 ynP|n(1 - ). (1) D. Singerman and J. Strudwick: Petrie Polygons, Fibonaci sequences and Farey maps 353 6 The Petrie polygons of M3(n) Our principle object of study are the Petrie polygons of the PC-maps M3(n). We can regard M3(n) as M3(3, ro)/r(n), that is as a quotient of the Farey map. We illustrate our study with the classical regular map M3(7). This is known as the Klein map and is a map of type {3, 7}. This lies on Klein's Riemann surface of genus 3, known as the Klein quartic. Petrie polygons for this map appear on page 320 in the classic paper [4], although they were not called Petrie polygons there. In fact, Petrie polygons are named after John Flinders Petrie (1907-1972), and Klein's paper [4] was written in 1878. Three of the Petrie polygons are drawn on page 320 of "The Eightfold Way" [5]. The eight in the title comes from the fact that the size of the Petrie polygons is 8. This will be a special case of results in this paper where we determine the sizes of of the Petrie polygons in PC maps. In general we observe that the group r/r(n) has a transitive action on the Petrie polygons of M3(n). For r clearly has a transitive action on the darts of M3(to), and so r/r(n) has an induced action on the darts of M3(n). Clearly, this action will give a transitive action on the set of Petrie polygons of M3(n). The vertices of M3(n) are equivalence classes of vertices of M3(3, to). We let [a] denote the equivalence class of | in M3(n) and [|] is joined by an edge to [d] in M3(n) if and only if ad — bc = 1 mod n. The points [ 1 ], [ 0 ], • • • [ /-1 ] • • • form the vertices of a Petrie polygon which we call Pe(n). " Recall the definition of the semiperiod PSL(2, Zn) and 6(T) = P = ^ ^J, where we think of this matrix as lying in PSL(2, Zn). Now Po(n) = (fa}n)-1 fo(n) V Jo(n) Jo(n)+1/ Now fa(n) = 0 mod n and fo(n)-1 = fo(n)+1 = ±1, by the definition of a(n). Thus Po(n) = ±1 and so To(n)/2 = ±1. which is the identity in PSL (2,Zn) Thus the automorphism group of Pe(n) is generated by R and T with R2 = (RT)2 = To(n)/2 = 1 and hence (R, T} = Do(n)/2 of order a(n). □ It is interesting to see how this theorem works in practice, so let us go back to our example of n = 7 as illustrated in Figure 2. As a(7) = 8 we have an action of D4 on Pe(7) an 8-sided polygon . The element T has two cycles of length 4, namely (1,0) (1,1) (2, 3) (5,1) (1,0) (0,1) (1, 2) (3, 5) (1, 6) (0,1) and for the involution R we have (1,0) ^ (0,1), (1,1) ^ (6,1), (1,2) ^ (5,1), (2, 3) ^ (4,2). (Note that [§] = [ ] = [f] so that (3, 5) = (4,2), etc.) As r/r(n) acts transitively on the darts of M3(n) we use equation (1) in section 5 to obtain Corollary 7.3. The number of Petrie polygons on M3 (n) is equal to n3 1 np|n(1 - P2 ). 2a(n) Example. Let n = 7. Then a(7) = 8. The number of Petrie polygons of M3(7) is equal to 21. Klein drew three of them in [5]. The others can be found by rotating these through 2nk/7, for k = 1, ••• 6. 8 More about &(n) Theorem 8.1. For all positive integers m > 2, a(m) is even. Proof. Po(m) = (fo(m)-1 fo(m) \ = ±1 mod m V Jo(m) Jo{m) + 1 J Thus (detP)o(m) = 1 mod m, so (- 1)o(m) = 1 mod m and thus a(m) is even. □ Exactly the same proof shows that n(m) is even for m > 2. A much easier proof than that given in [8]. Let p = 1+2T5 (the golden ratio) and p* = . Note that pp* = -1 and p + p* = 1. Let Zn[p] = {a + bp|a, b G Z/(n)} and if a = a + bp, define a* = a + bp*. Then (aft )* = a* ft *. We define the norm N on Zn[p] by N(a) = aa*. Then N(aft) = N(a)N(ft). We call a a unit if N(a) = ±1, so that p is a unit. The units of Zn [p] form a group Z*n[p] under multiplication. Theorem 8.2. a(n) is the order of p in Z*n [p], if fo(n)-1 = 1 and is equal to half the order of p if fo(n)-1 = -1. Inallcases n(n) is equal to the order of p in Z *n[p]. 356 Ars Math. Contemp. 10 (2016) 183-192 Proof. From p2 = p +1 we can use induction to prove that pm = fmp + fm_1 Thus if m = ap. If a is a root of this polynomial then the other root is ap = 1 - a and hence so that ap+1 = a - a2 = -1. Thus, by Theorem 8.2 and Lemma 9.1, n(p) = 2a(p). □ Theorem 9.3. Let p = 11,19 mod20. Then n(p) = a(p). Proof. We have p = ±1 mod 5 and so 5 has a square root in Fp the finite field with p elements and hence p e Fp. Its multiplicative group has order p - 1. Now pn(p) = fn(p)p + fn(p)_ 1 = 1mod p Therefore n(p) is a divisor of p - 1. Now p = 3 mod 4 so that p = 4k + 3 for some integer k. This p - 1 = 4k + 2. If n(p) = 2a(p), then a(p) is a divisor of 2k + 1 and thus a(p) is odd contradicting Theorem 8.1. Therefore a(p) = n(p). □ We would like to thank Tom Harris for helping us with the results in section 9 and the referee for his careful reading of the manuscript. D. Singerman and J. Strudwick: Petrie Polygons, Fibonaci sequences and Farey maps 357 References [1] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Springer-Verlag, Berlin-New York, 4th edition, 1980. [2] I. Ivrissimtzis and D. Singerman, Regular maps and principal congruence subgroups of Hecke groups, European J. Combin. 26 (2005), 437-456, doi:10.1016/j.ejc.2004.01.010. [3] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), 273-307. [4] F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14 (1878), 428-471, doi:10.1007/BF01677143. [5] S. Levy, The eightfold way: the beauty of Klein's quartic curve, volume 35, Cambridge University Press, 1999. [6] E. Schulte and J. M. Wills, A polyhedral realization of Felix Klein's map {3, 7}8 on a Riemann surface of genus 3, J. London Math. Soc. (2) 32 (1985), 539-547, doi:10.1112/jlms/s2-32.3.539. [7] D. Singerman, Universal tessellations, Rev. Mat. Univ. Complut. Madrid 1 (1988), 111-123. [8] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 359-370 Odd edge-colorability of subcubic graphs* Risto Atanasov Department of Mathematics and Computer Science, Western Carolina University, 28723 Cullowhee, NC, USA Mirko Petrusevski Department of Mathematics and Informatics, Faculty of Mechanical Engineering, Sts. Cyril and Methodius University, 1000 Skopje, Republic of Macedonia Riste Skrekovski Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Faculty of Information Studies, 8000 Novo Mesto, Slovenia University of Primorska, FAMNIT, 6000 Koper, Slovenia Received 16 October 2015, accepted 21 January 2016, published online 1 March 2016 An edge-coloring of a graph G is said to be odd if for each vertex v of G and each color c, the vertex v either uses the color c an odd number of times or does not use it at all. The minimum number of colors needed for an odd edge-coloring of G is the odd chromatic index xO(G). These notions were introduced by Pyber in [7], who showed that 4 colors suffice for an odd edge-coloring of any simple graph. In this paper, we consider loopless subcubic graphs, and give a complete characterization in terms of the value of their odd chromatic index. Keywords: Subcubic graph, odd edge-coloring, odd chromatic index, odd edge-covering, T-join. Math. Subj. Class.: 05C15 1 Introduction 1.1 Terminology and notation Throughout the article we mainly follow the terminology and notation used in [1, 11]. A graph G = (V(G), E(G)) is always regarded as being finite, i.e. having a finite nonempty *This work is partially supported by ARRS Program P1-0383. E-mail addresses: ratanasov@email.wcu.edu (Risto Atanasov), mirko.petrushevski@gmail.com (Mirko Petrusevski), skrekovski@gmail.com (Riste Skrekovski) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 360 Ars Math. Contemp. 10 (2016) 183-192 set of vertices V(G) and a finite (possibly empty) set of edges E(G). An edge with identical ends is called a loop, and an edge with distinct ends a link. Two or more links with the same pair of ends are said to be parallel edges. The parameters n(G) = |V(G)| and m(G) = |E(G)| are called order and size of G, respectively. A graph of order 1 is said to be trivial, whereas a graph of size 0 empty. For every v e V(G), EG(v) denotes the set of edges incident to v, and the size of EG (v) (every loop being counted twice) is the degree of v in G, with notation dG (■v). The maximum (resp. minimum) vertex degree in G is denoted by A(G) (resp. ¿(G)). We speak of G as a subcubic graph whenever A(G) < 3. Each vertex v having an even (resp. odd) degree dG(v) is an even (resp. odd) vertex. In particular, if dG(v) equals 0 (resp. 1), we say that v is an isolated (resp. pendant) vertex of G. Any vertex of degree d is also called a d-vertex. A graph is even (resp. odd) whenever all its vertices are even (resp. odd). The set of neighboring vertices of v e V (G) is denoted by NG(v). For every u e NG(v), the edge set EG(u) n EG(v) is called the uv-bouquet in G, with notation Buv. The maximum size of a bouquet in G is its multiplicity. We say that G is a simple graph whenever it is loopless and of multiplicity at most 1. Whenever the underlying graph G is clear from the context, the edge-complement of a subgraph H is denoted by H, i.e. H = G - E(H). A co-forest in G is a subgraph whose edge-complement is a forest. Every maximal path whose interior consists entirely of 2-vertices (of G) is called an open thread; similarly, every cycle all of whose vertices except one are 2-vertices of G is a closed thread. For every connected graph G that is not a cycle, each of its 2-vertices belongs to a unique thread, either open or closed. 1.2 Odd edge-colorings and odd chromatic index Any mapping y : E(G) ^ S is referred to as an edge-coloring of G, and then S is called the color set of y. We say that y is a k-edge-coloring when |S| < k. Since the nature of the colors is irrelevant, it is conventional to use S = [k] := {1,2,..., k} whenever the color set is of size k. For each color c e S, Ec(G, y) denotes the color class of c, being the set of edges colored by c (i.e. Ec(G, y) = y-1(c)); whenever G and y are clear from the context, we denote the color class of c simply by Ec. Given an edge-coloring y and a vertex v of G, we say the color c appears at v if Ec n EG(v) = 0. Any decomposition {H1;..., Hk} of G can be interpreted as its k-edge-coloring for which the color classes are E(Hi),...,E (Hfc). An odd edge-coloring of a given graph G is an edge-coloring such that each nonempty color class induces an odd subgraph. In other words, at each vertex v, for any appearing color c the degree dG[Ec](v) is odd. Equivalently, an odd edge-coloring can be seen as a decomposition of G into (edge disjoint) odd subgraphs. Such decompositions represent a counterpart to decompositions into even subgraphs, which were mainly used while proving various flow problems (see e.g. [6, 9]). Historically speaking, as a topic in graph theory, decomposing into subgraphs of a particular kind started with the paper of Erdos et al. [2]. An odd edge-coloring of G using at most k colors is referred to as an odd k-edge-coloring, and then we say that G is odd k-edge-colorable. Whenever G is odd edge-colorable, the odd chromatic index xO(G) is defined to be the minimum integer k for which G is odd k-edge-colorable. It is obvious that a necessary and sufficient condition for odd edge-colorability of G is the absence of vertices incident only to loops. Apart from this, the presence of loops does not influence the existence nor changes the value of the index xO(G). Therefore, while studying these matters it is enough to confine to loopless graphs. R. Atanasov, M. Petrusevski and R. Skrekovski: Odd edge-colorability of subcubic graphs 361 Figure 1: A simple graph with odd chromatic index equal to 4. As a notion, odd edge-coloring was introduced by Pyber in his survey on graph coverings [7]. The mentioned work considers simple graphs and (among others) contains a proof of the following result. Theorem 1.1 (Pyber, 1991). For every simple graph G, it holds that xO(G) < 4. Pyber remarked that the upper bound is realized by the wheel on four spokes W4 (see Fig. 1). This upper bound of four colors does not apply to the class of all looplees graphs G. For instance, Fig. 2 depicts four graphs with the following characteristic property: each of their odd subgraphs is of order 2 and size 1, i.e. a copy of K2. Consequently, for each of them the odd chromatic index equals the size of the graph. Figure 2: Four Shannon triangles (the smallest one of each type). As defined in [4], a Shannon triangle is a loopless graph on three pairwise adjacent vertices. Observe that for any Shannon triangle, as a direct consequence of the handshake lemma, the edge set of every odd subgraph is fully contained in a single bouquet. Let p, q, r be the parities of the sizes of the bouquets of a Shannon triangle G in non-increasing order, with 2 (resp. 1) denoting that a bouquet consists of an even (resp. odd) number of parallel edges. Then G is a Shannon triangle of type (p, q, r), and it holds that xO(G) = p + q + r. The following result was proven in [4]. Theorem 1.2. For every connected loopless graph G, it holds that xO(G) < 6. Equality is achieved if and only if G is a Shannon triangle of type (2,2,2). In this paper we study the odd chromatic index for the class of loopless subcubic graphs G. We shall prove that over that class of graphs holds maxG xO(G) = 4. Moreover, we will give a complete characterization of the loopless subcubic graphs in terms of the value of their odd chromatic index. In doing so, we will use methods such as eliminating characteristic subtrees and unicyclic subgraphs, or odd co-forests, developed in [3, 10, 12]. The rest of the article is divided into three sections. In the next one, as a preliminary, are collected several 'easy' results (most of them previously known). Section 3 is devoted to a derivation of our main result - a characterization of the loopless subcubic graphs G in 362 Ars Math. Contemp. 10 (2016) 183-192 terms of the value of x0(G). The final section briefly conveys some ideas on odd edge-coverability of loopless subcubic graphs. 2 Preliminary results We begin by recalling the definition of a T-join. For a graph G, let T be an even-sized subset of V(G). Following [1], a spanning subgraph H of G is said to be a T-join if dH(v) is odd for all v e T and even for all v e V(G) \ T. For example, if P is an x-y path in G, the spanning subgraph of G with edge set E(P) is an {x, y}-join. Observe that the symmetric difference of a T-join and an S-join is a T A S-join. With the use of this simple fact and the mentioned example, it can be readily deduced (see [8]) that for any connected graph G and any even-sized subset T of V(G), there exists a T-join of G. Note also that by taking S = 0 we infer that the symmetric difference of a T-join and a spanning even subgraph is again a T-join. In particular, removal (resp. addition) of the edges of an edge-disjoint cycle from (resp. to) a T-join, furnishes a T-join. Thus, whenever a T-join of G exists, there also exists such a forest (resp. co-forest). The above discussion yields the following conclusion. Lemma 2.1. Given a connected graph G of even order, there exists an odd co-forest in G. The next lemma originally appears in [7]. For a proof we refer the reader to [4]. Lemma 2.2. If F is a forest, then x0(F) < 2. With the use of Lemmas 2.1 and 2.2, it can be easily shown that every connected graph of even order is odd 3-edge-colorable. Proposition 2.3. For every connected graph G of even order, it holds that x0 (G) < 3. Proof. There exists an odd co-forest H in G. Take an odd edge-coloring of H with the color set {1,2} and extend it to E(G) by coloring E(H) with 3. Note that we have thus constructed an odd 3-edge-coloring of G. □ Corollary 2.4. Let v be a 2-vertex in a connected graph G of odd order. Then G admits a 3-edge-coloring that is nearly odd with the only exception being that EG(v) is monochromatic. Proof. Suppress the vertex v, i.e. remove it and then add an edge e with ends in NG(v) (the edge e is either a link or a loop depending on whether NG (v) is of size 1 or 2). Denote the obtained graph by H. Since H is connected and of even order, the previous proposition assures its odd 3-edge-colorability. Apply such an edge-coloring to H, and then 'reinstate' the vertex v on the edge e. We thus regain the graph G with a required edge-coloring. □ 3 Odd edge-colorability As already mentioned, throughout this section we consider loopless subcubic graphs. We begin by showing that four colors suffice for an odd edge-coloring of any such graph. Proposition 3.1. If G is a loopless subcubic graph, then x0(G) < 4. R. Atanasov, M. Petrusevski and R. Skrekovski: Odd edge-colorability of subcubic graphs 363 Proof. We may assume that G is connected and non-trivial. Moreover, by Proposition 2.3 we may assume that n(G) is odd. In case ¿(G) = 1 it is easily shown that xO(G) < 3. Indeed, say v is one of its pendant vertices. Since the graph G - v is connected and of even order, by Lemma 2.1 there exists an odd co-forest K in G - v. Let us denote its edge-complement in G by F, i.e. F = G - E(K). Then {K, F} is a decomposition of G into an odd subgraph K and a forest F. By coloring E(K) with 1, and applying to F an odd 2-edge-coloring with the color set {2,3}, we furnish an odd 3-edge-coloring of G. Henceforth we assume that ¿(G) = 2. Let v be one of its non-cut vertices. Either dG(v) = 2 or dG(v) = 3. We study first the case when dG(v) = 2 (see Fig. 3). G-v G-v Figure 3: The two possibilities when dG (v) = 2. v v f e Let EG(v) = {e, f}. By Lemma 2.1, consider a decomposition {K, F} of G - v consisting of an odd subgraph K and a forest F. Then the graph F + e is also a forest. Color E(K) with 1, the edge f by 2, and combine with an odd edge-coloring of F + e with the color set {3,4}. This confirms that G is odd 4-edge-colorable. G - v Figure 4: The only possibility when dG (v) = 3. Now we study the case when dG (v) = 3 (see Fig. 4). Denote by u the neighbor of v for which the uv-bouquet is of size 2. Clearly, u is a pendant vertex of G - v. Select an odd co-forest K in G - v. Observe that in its edge-complement K (taken in G - v), the vertex u is isolated. Color E(K) with 1; apply to the forest K an odd edge-coloring with the color set {2, 3}; color the bouquet Buv with 2 and 3; finally, color the remaining non-colored edge (incident to v) by 4. This gives an odd 4-edge-coloring of G. □ The established upper bound (of four colors) for the odd chromatic index of any loop-less subcubic graph is sharp. For example, consider the smallest Shannon triangle G of type (2,1,1) (the second of the graphs depicted in Fig. 2). As already observed in the introduction, xO(G) = 4. Note that this particular G can be obtained from a cubic bipartite v u 364 Ars Math. Contemp. 10 (2016) 183-192 graph (of order 2) by a single edge subdivision. As it turns out, every subcubic graph obtainable in this manner requires four colors for an odd edge-coloring. On the other hand, for any other connected loopless graph three colors suffice. In order to prove this assertion we will use the following lemma. Lemma 3.2. Let G be a connected graph having at least two 2-vertices. Then there exists a tree T in G that satisfies the following two conditions: (i) every 2-vertex of G belongs to V(T), (ii) every pendant vertex of T is a 2-vertex of G. Proof. We argue by induction on the number k of 2-vertices in G. In case k = 2, we merely take T to be a path in G connecting the only two 2-vertices. Assume that k > 2 and let the statement be true whenever the number of 2-vertices is less than k. Suppress a 2-vertex v of G, i.e. remove v and add a new edge e between its neighbors; denote the obtained graph by G'. The inductive hypothesis provides us with a tree T' satisfying the conditions (i) and (ii) for G'. If e G E(T'), then by reversing the suppression, i.e. by subdividing e, we arrive at the desired tree. Otherwise, T' is a subtree of G - v. If that is the case, then let P be a v-V(T') path in G and set T = P U T'. Note that T is a tree in G for which both (i) and (ii) hold. □ Proposition 3.3. For any connected loopless subcubic graph G, the following two statements are equivalent: (i) xO(G) = 4; (ii) G is obtainable from a cubic bipartite graph by a single edge subdivision. Proof. (i) ^ (ii): Let G be a connected loopless subcubic graph that cannot be obtained from a cubic bipartite graph by a single edge subdivision. We shall prove that xO(G) < 3. As in the proof of Proposition 3.1, we may assume that n(G) is odd and ¿(G) = 2. There are two cases to be considered. Case 1: G has at least two 2-vertices. Let T be a tree in G as in Lemma 3.2. Note that for each non-isolated vertex u of its edge-complement T the degree df (u) is odd. Therefore, the combination of an odd 2-edge-coloring of T with the color set {1,2} and a monochromatic edge-coloring of T with the color 3 constitutes an odd 3-edge-coloring of G. Case 2: G has a unique 2-vertex. Denote this particular vertex by v. Assume first the existence of an odd cycle (i.e. a cycle of odd length) Co in G that does not pass through v. Since G is connected, there exists a nontrivial v-V(Co) path P. Let w be the other endpoint of P (besides v) and consider the subgraph G' = P U Co. Note that w is the only isolated vertex of G'; moreover, every other vertex of G' has an odd degree. Color the set E(G') with 3; use the color 1 for EG(w); color the remaining non-colored edges of P and Co alternately by 1 and 2 such that the obtained edge-coloring of G' fails to be proper only at w (such a 2-edge-coloring of G' is possible because Co is an odd cycle). This completes an odd 3-edge-coloring of G. Assume now that such an odd cycle does not exist in G, meaning that every cycle avoiding v is even. We claim that there exists an even cycle Ce passing through v. To prove this, we argue as follows. Suppress the vertex v, and let e be the new edge. The obtained graph G* is cubic (since v is the only 2-vertex of G and ¿(G) = 2), which further R. Atanasov, M. Petrusevski and R. Skrekovski: Odd edge-colorability of subcubic graphs 365 implies that G* is not bipartite (otherwise, G would be obtainable from the cubic bipartite graph G* by a single edge subdivision). Consider an odd cycle C* of G*. By our current assumption, C* is not a cycle in G, which implies that e G E(C*). Therefore, v U V(C*) constitutes the vertex set of an even cycle Ce passing through v. Once the existence of Ce is established, we can construct an odd 3-edge-coloring of G as follows: take a proper 2-edge-coloring of Ce with the color set {1, 2}; then color the edge set of Ce with 3. (ii) ^ (i): Let G be obtainable from a cubic bipartite graph by a single edge subdivision. We shall show that xO(G) = 4. Denote by v the unique 2-vertex of G, and let G' be the graph obtained from G by suppressing v. Since G' is bipartite, there exists a partition X,Y of V(G') such that E(G') = E(X,Y). By Proposition 3.1, xO(G) < 4. Suppose this inequality is strict, i.e. suppose there exists an odd 3-edge-coloring of G with the color set {1,2,3}. Without loss of generality, we may assume that the v-X edge is colored by 1, whereas the v-Y edge is colored by 2. Let xi; x2, x3, xi23 be respectively the number of vertices u from X such that EG(u) is colored entirely with 1, entirely with 2, entirely with 3, or by all the three colors 1, 2, 3. Analogously, we employ notation yi; y2, y3, yi23 for the sizes of the respective subsets of Y. By double counting the color class Ei, we derive the equality 3xi + X123 = 1 + 3yi + yi23 . (3.1) Reasoning similarly for the class E2, we deduce 1 + 3x2 + xi23 = 3y2 + yi23 . (3.2) Let us now consider the difference xi23 - yi23. From (3.1) it follows that xi23 - yi23 = 1(mod3). On the other hand, (3.2) yields xi23 - yi23 = — 1(mod3). This is the desired contradiction. □ It is a trivial task to characterize the connected loopless subcubic graphs G that are odd 1-edge-colorable. Namely, xO(G) =0 if and only if G is Ki, whereas xO(G) = 1 precisely when G is odd. We proceed to characterize odd 2-edge-colorability. Proposition 3.4. If G is a connected loopless subcubic graph, then the following two statements are equivalent: (i) xO(G) < 2; (ii) for every cycle C of G, the set {v G V(C) : dG(v) = 2} is even-sized. Proof. (i) ^ (ii): Assume (i) and apply to G an odd 2-edge-coloring. Consider an arbitrary cycle C of G. Note that for every v G V(C) the edge set EC(v) is either monochromatic (when dG(v) = 3) or dichromatic (when dG(v) = 2). This clearly implies that the set {v G V(C) : dG(v) = 2} is even-sized. (ii) ^ (i): Assume that (ii) holds. In case G is a cycle, it is readily seen that (i) follows. Henceforth, we prove that (i) holds when G is not a cycle. For each pair x, y of non-even vertices of G consider an arbitrary x-y walk W, and count the number of traversed 2-vertices, i.e. count the 2-vertices of G appearing (possibly with repetition) in the interior of W. We claim that the parity of this number is an invariant of the unordered pair x, y, i.e. does not dependent on the choice of W. Indeed, if we suppose the existence of an x-y walk W' which presents a counterexample combined with W, then the symmetric 366 Ars Math. Contemp. 10 (2016) 359-370 difference E(W) © E(W') must contain the edge set of a cycle C of G for which the set {v e V(C) : dG (v) = 2} is odd-sized. Let us employ notation x ~ y (resp. x « y) whenever the parity of the considered number is odd (resp. even). Seen as binary relations on the set on non-even vertices, both ~ and « are symmetric. Moreover, by concatenating suitable walks, one readily deduces that « is an equivalence relation, whereas ~ is non-transitive (i.e. x ~ y & y ~ z ^ x « z). This means that there are at most two equivalence classes of In other words, the set of non-even vertices of G can be written as a disjoint union of two (possibly empty) subsets A, B such that x ~ y holds if and only if x and y belong to distinct subsets. Note that there is no A-B edge in G. For each u e A color EG(u) with 1; similarly, for each u e B color Eg (u) with 2. This gives a partial edge-coloring of G such that any non-colored edge is incident to a 2-vertex. Apply the following procedure: as long as there exists a 2-vertex, say v, with EG(v) not fully colored, consider the unique thread H that contains v. Two edges of H are already colored, and this pre-coloring extends to an edge-coloring of H with the color set {1,2} that is proper at each 2-vertex belonging to V(H). (In case the two pre-colored edges received the same color then the length of H is odd; on the other hand, if they are of different colors, then the length is even.) This eventually completes an odd 2-edge-coloring of G. □ Since all the threads of a given connected loopless subcubic graph G can be detected in linear time, it is linearly decidable whether the set on non-even vertices of G admits a partition into two (possibly empty) subsets A and B as in the proof of the implication (ii) ^ (i). Thus, it can be checked in linear time whether x'o(G) < 2. Moreover, the proof of (ii) ^ (i) suggests the following constructive characterization of odd 2-edge-colorability. Corollary 3.5. Every connected loopless subcubic graph G satisfying xO(G) < 2 is either an even cycle or can be obtained from a connected odd subcubic graph Go (loops allowed) in the following manner: 1. split V(Go) arbitrarily into two (possibly empty) subsets A and B; 2. subdivide an odd number of times each edge from E(A, B); 3. subdivide an even non-zero number of times each loop from E(Go); 4. subdivide an even (possibly zero) number of times each link whose endvertices belong to the same set from the pair A, B. To summarize this section, we state the promised characterization of all connected loop-less subcubic graphs in terms of their odd chromatic index. Theorem 3.6. Let G be a connected loopless subcubic graph. Then if G is empty; if G is odd; if G has 2-vertices, with an even number of them on each cycle; if G is obtained from a cubic bipartite graph by a single edge subdivision ; otherwise. x'o(G) 3 R. Atanasov, M. Petrusevski and R. Skrekovski: Odd edge-colorability of subcubic graphs 367 The above comments on the algorithmic aspects of odd 2-edge-colorability, combined with the well-known fact that the decision problem whether a given graph is bipartite can be solved in polynomial time (by using Breadth-First Search), assure that our characterization is good. Corollary 3.7. For any loopless subcubic graph G, the odd chromatic index xO(G) can be determined in polynomial time of n(G). 4 Odd edge-coverability In this section we present an application of Theorem 3.6 while briefly studying the odd edge-coverability of subcubic graphs, a related concept to odd edge-colorability. An edge-covering of a graph G is a family {Hi,..., Hk} of subgraphs such that Ui=1 E(H) = E(G). Any edge-covering of G can be interpreted as a 'generalized edge-coloring', i.e. a mapping f * : E(G) ^ P*([k]) assigning to each edge of G a nonempty subset of the set of colors {1,..., k}. In other words, we pass from edge-colorings to edge-coverings by allowing more than one color per edge. In the context of an edge-covering f *, the color class Ec of any color c G [k] consists of the edges e G E(G) for which c G f * (e). If each non-empty color class induces an odd subgraph, then we speak of an odd edge-covering of G. More verbosely, we say that G is odd k-edge-coverable whenever it admits an odd edge-covering with at most k colors. The minimum size (i.e. minimum number of colors) of an odd edge-covering of G is denoted by covo(G). Similar to odd edge-colorability, a given graph G is odd edge-coverable if and only if there are no vertices incident only to loops, and apart from this, the presence of loops does not influence the existence nor changes the value of covo(G). Therefore, any study of odd edge-coverability should be restricted to loopless graphs. Since every odd edge-coloring of G is also an odd edge-covering, it holds that cov0(G) < xO(G). (4.1) As a notion, odd edge-covering was introduced in [5]. The scope of the mentioned work are all simple graphs, and the following result is proven. Theorem 4.1 (Mitrai, 2006). For every simple graph G, it holds that covo(G) < 3. In this section we consider the possible values of the index covo(G) taken over all connected loopless subcubic graphs G. When G is the smallest Shannon triangle of type (2,1,1) (the second graph in Fig. 2), the handshake lemma readily implies that covo(G) = 4; indeed, for every graph G of order n(G) = 3 the equality covo(G) = xO(G) holds. We shall prove that this is the only exception to odd 3-edge-coverability of connected loopless subcubic graphs. For this we should note that, according to (4.1) and Theorem 3.6, any exception must be obtainable from a cubic bipartite graph by a single edge subdivision. Thus, it is enough to consider the odd 3-edge-coverability of that particular class of graphs. Proposition 4.2. Apart from the smallest Shannon triangle of type (2,1,1), every other connected loopless subcubic graph is odd 3-edge-coverable. Proof. Suppose the opposite, i.e. let G present a counterexample. Hence, G can be obtained from a cubic bipartite graph H by a single edge subdivision. Say the subdivided edge e G E(H) has endpoints x and y, and let v be the introduced 2-vertex. Denote by ex and ey the respective 'parts' of e in G (see Fig. 5). 368 Ars Math. Contemp. 10 (2016) 359-370 v —^^ y H G Figure 5: The graphs H and G. (The possibility of another xy-edge in H, i.e. an xy-edge in G, is not excluded.) Let Bxy be the xy-bouquet of H. Since H is a cubic graph and G is not the smallest Shannon triangle of type (2,1,1), the size of Bxy is either 1 or 2. We claim the latter. To confirm this, we argue by contradiction. Suppose the opposite, i.e. let x and y be non-adjacent in G. First we show that the graph G - v is connected. Otherwise, it must consist of two components Hx and Hy, containing x and y, respectively. Moreover, since the only even vertex of the graph Hx (resp. Hy) is the 2-vertex x (resp. y), the handshake lemma implies that both n(Hx) and n(Hy) are odd. By Corollary 2.4, there exists an edge-coloring ^>x (resp. ) of Hx (resp. Hy) with the color set {1,2,3} that is nearly odd, the only exception being that the edge set EHx (x) is colored with 1 (resp. the edge set EHy (y) is colored with 2). Apply ^>x U ^>y, and then color ex by 1 and ey by 2. We thus obtain an odd 3-edge-coloring of G, a contradiction. This confirms that G - v is indeed connected. Denote by P ashortest x-y path in G-v,andsay x, u1;... ,uk_i,y are the consecutive vertices met on a traversal of P. Since H is bipartite with x and y belonging to different partite sets, the length k is an odd integer greater than 2. Suppose that NG_v(x) = {u1}, i.e. let the bouquet Bxui be of size 2. We can then apply to G the following edge-covering with the color set {1, 2, 3}: color ex with 1; for ey use both 2 and 3; color Bxui with 1; for the u1M2-edge of P use both 1 and 3; color the rest of E(G) with 3. This clearly implies covo(G) < 3, a contradiction. Therefore, it must be that, besides u1, there exists another neighbor of x in G - v; let us denote this particular vertex by u. The choice of P assures u 1, has been studied for: G = K3 by D. Jungnickel, R. C. Mullin, S. A. Vanstone [13], Y. Zhang and B. Du [25]; G = K4 by M. J. Vasiga, S. Furino and A.C.H. Ling [22]; G = C4 by M.X. Wen and T.Z. Hong [17]. In this paper we investigate the existence of an a-resolvable A-fold (K4 - e)-design (where K4 - e is the complete graph K4 with one edge removed). In what follows, by (a, b, c; d) we will denote the graph K4 - e having {a, b, c, d} as vertex-set and {{a, 6}, {a, c}, {b, c}, {a, d}, {b, d}} as edge-set. Basing on the definitions given above, we can derive the following necessary conditions: (1) Av(v - 1) = 0 (mod 10); (2) av = 0 (mod 4); (3) 2A(v - 1) = 0 (mod 5a). Note that, since the number of a-parallel classes of an a-resolvable A-fold (K4 - e)-design of order v is 2A(5vQ-i) and every vertex appears exactly a times in each of them, we have the following theorem. Theorem 1.1. Any a-resolvable A-fold (K4 - e)-design is balanced. From Conditions (1) - (3) we can desume minimum values for a and A, say a0 and A0, respectively. Similarly to Lemmas 2.1, 2.2 in [22], we have the following lemmas. Lemma 1.2. If an a-resolvable A-fold (K4 - e)-design of order v exists, then a0| a and Ao| A. Lemma 1.3. If an a-resolvable A-fold (K4 - e)-design of order v exists, then a ta-resolvable nA-fold (K4 - e)-design of order v exists for any positive integers n and t with 11 ^oi). 1 5a The above two lemmas imply the following theorem (for the proof see Theorem 2.3 in [22]). M. Giongriddo, et al.: The spectrum of a-resolvable X-fold (K4 — e)-designs 373 Theorem 1.4. If an a0-resolvable Xo-fold (K4 — e)-design of order v exists and a and A satisfy Conditions (1) — (3), then an a-resolvable A-fold (K4 — e)-design of order v exists. Therefore, in order to show that the necessary conditions for a-resolvable designs are also sufficient, we simply need to prove the existence of an a0-resolvable A0-fold (K4 — e)-design of order v, for any given v. 2 Auxiliary definitions A (AKnijn2i...jnt, G)-design is known as a A-fold group divisible design, G-GDD in short, of type {ni, n2,..., nt} (the parts are called the groups of the design). We usually use an "exponential" notation to describe group-types: the group-type 1®2j3f... denotes i occurrences of 1, j occurrences of 2, etc. When G = Kn we will call it an n-GDD. If the blocks of a A-fold G-GDD can be partitioned into partial a-parallel classes, each of which contains all vertices except those of one group, we refer to the decomposition as a A-fold (a, G)-frame; when a = 1, we simply speak of A-fold G-frame (n-frame if additionally G = Kn). In a A-fold (a, G)-frame the number of partial a-parallel classes missing a specified group of size g is 2a|£(G)|. An incomplete a-resolvable A-fold G-design of order v + h, h > 1, with a hole of size h is a (A(Kv+h \ Kh), G)-design in which there are two types of classes, A(h—1g(G)G)I partial classes which cover every vertex a times except those in the hole and 2a|E(G)| full classes which cover every vertex of Kv+h a times. 3 v = 0 (mod 4) In [4, 5, 23] it was showed that there exists a resolvable (K4 - e)-design of order v = 16 (mod 20); while, for every v = 0,4, 8,12 (mod 20) Gionfriddo et al. ([7]) proved that there exists a resolvable 5-fold (K4 - e)-design of order v. Hence the necessary conditions are also sufficient. 4 v = 1 (mod 2) 4.1 v = 1 (mod 10) If v = 1 (mod 10), then A0 = 1 and a0 = 4 and so a solution is given by a cyclic (K4 - e)-design ([2]), where every base block generates a 4-parallel class. If v = 10k + 1, k > 4, the desired design can be obtained by developing in Z10f+1 the base blocks listed below: (1 + 2i, 4k + 1 + i, 1; 2k + 2), i = 3,4,..., f J; (2k + 3 - 2i, 5k + 2 - i, 1; 2k + 2), i = 1, 2,..., [f ]; (1,4k + 1, 3; 6k); (1, 2k + 2, 5; 6k + 1); where [x\ (or [x]) denote the greatest (or lower) integer that does not exceed (or that exceed) x. If v = 11, 21, 31, the base blocks are: v = 11: (1,10,2; 5) developed in Zn; v = 21: (1,11,3; 15), (1,7, 2; 10) developed in Z21; v = 31: (2,13,1; 5), (1, 27,10; 11), (1,7, 3; 14) developed in Z31. 374 Ars Math. Contemp. 10 (2016) 183-192 4.2 v = 3, 5, 7, 9 (mod 10) If v = 3, 5, 7,9 (mod 10), then A0 = 5 and a0 = 4 and so a solution is given by a cyclic 5-fold (K4 - e)-design, where every base block generates a 4-parallel class. The required design is obtained by developing in Zv the following blocks: (1 + i,v - 1 - i, 0; 1), i = 1, 2,..., v—■; (0,1, 2; v - 1). 5 v = 2 (mod 4) 5.1 v = 6 (mod 20) If v = 6 (mod 20), then A0 = 1 and a0 = 2. In order to prove the existence of a 2-resolvable (K4 - e)-design of order v for every v = 6 (mod 20), preliminarly we need to construct one of order 6. Lemma 5.1. There exists a 2-resolvable (K4 - e)-design of order 6. Proof. Let V = {0,1, 2, 3,4,5} be the vertex-set and {(0,1,2; 3), (2, 3,4; 5), (4, 5,0; 1)} be the class. □ For constructing a 2-resolvable (K4 - e)-design of any order v = 6 (mod 20) and for later use, note that starting from a (K4 - e)-frame of type hn also a A-fold (2, K4 - e)-frame of type hn can be obtained for any A > 0, since necessarily h = 0 (mod 5) and so the number of partial parallel classes missing any group is even. Lemma 5.2. For every v = 6 (mod 20), there exists a 2-resolvable (K4 - e)-design of order v. Proof. Let v = 20k + 6. The case k = 0 follows by Lemma 5.1. For k > 0, consider a (2, K4 - e)-frame of type 54fc+1 ([5]) with groups G^ i = 1,2,..., 4k + 1 and a new vertex to. For each i = 1, 2,..., 4k + 1, let Pi the unique partial 2-parallel class which misses the group Gi. Place on Gi U {to} a copy of a 2-resolvable (K4 - e)-design of order 6, which exists by Lemma 5.1, and combine its full class with the partial class Pi so to obtain the desired design. □ 5.2 v = 2,10,14,18 (mod 20) To prove the existence of an a-resolvable A-fold (K4 - e)-design of order v = 2,10,14,18 (mod 20), with minimum values A0 = 5 and a0 = 2, we will construct some small examples most of which will be used as ingredients in the constructions given by the following theorems. Theorem 5.3. Let v, g, u, and h be positive integers such that v = gu + h. If there exists i) a 5-fold (2, K4 - e)-frame of type gu; ii) a 2-resolvable 5-fold (K4 - e)-design of order g; iii) an incomplete 2-resolvable 5-fold (K4 - e)-design of order g + h with a hole of size h; then there exists a 2-resolvable 5-fold (K4 - e)-design of order v = gu + h. M. Giongriddo, et al.: The spectrum of a-resolvable X-fold (K4 — e)-designs 375 Proof. Take a 5-fold (2, K4 - e)-frame of type gu with groups Gj, i = 1,2,..., u and a set H of size h such taht H n (UU=1Gj) = 0. For j = 1,2,..., g, let Pj,j be the j-th 2-partial class which misses the group Gj. Place on HUG1 a copy D1 of a 2-resolvable 5-fold (K4 -e)-design of order g + h having g + h - 1 classes R1,1, R1,2,...,, R1,g, H1,1, H1,2,..., H1,h-1. For i = 2,3,..., u, place on H U Gj a copy Dj of an incomplete 2-resolvable 5-fold (K4 - e)-design of order g + h with H as hole and having h - 1 partial classes Hj,1, Hj,2,..., Hj,h-1 and g full classes Rj,1, Rj,2,...,, Rj,g. Combine the g partial classes P1,j with the full classes R1,1, R1,2,...,, R1,g of D1 and for i = 2,3,..., u the g partial classes Pj,j of Dj with the full classes Rj,1, Rj,2,..., Rj,g so to obtain gu 2-parallel classes on H U (uu=1Gj). Combine the classes H1,1, H1,2,..., H1,h-1 with the partial classes Hj,1, Hj,2,..., Hj,h-1 so to obtain h - 1 2-parallel classes. The result is a 2-resolvable 5-fold (K4 - e)-design of order gu + h with gu + h -1 2-parallel classes. □ The following lemma gives an input design in the construction of Theorem5.5. Lemma 5.4. There exists a 2-resolvable 5-fold (K4 - e)-GDD of type 23. Proof. Let {0, 3}, {1,4} and {2,5} be the groups and consider the following classes: P1 = {(0, 2,1; 4), (1, 5,0; 3), (3,4, 2; 5)}, P2 = {(3, 5,1; 4), (1, 2, 0; 3), (0, 4, 2; 5)}, P3 = {(0, 5,1; 4), (2,4,0; 3), (1, 3, 2; 5)}, P4 = {(2, 3,1; 4), (4, 5, 0; 3), (0,1, 2; 5)}. □ Theorem 5.5. Let v, g, m, h and u be positive integers such that v = 2gu + 2m + h. If there exists i) a 3-frame of type m1gu; ii) a 2-resolvable 5-fold (K4 - e)-design of order 2m + h; iii) an incomplete 2-resolvable 5-fold (K4 - e)-design of order 2g + h with a hole of size h; then there exists a 2-resolvable 5-fold (K4 - e)-design of order 2gu + 2m + h. Proof. Let F be a 3-frame with one group G of cardinality m and u groups Gj, i = 1,2,..., u of cardinality g; such a frame has y partial classes which miss G, each containing gu triples, and, for i = 1,2,..., u, 2 partial classes which miss Gj, each containing g("-31)+m triples. Expand each vertex 2 times and add a set H of h new vertices. Place on H U (G x{1, 2}) a copy D of a 2-resolvable 5-fold (K4 - e) -design of order 2m + h having 2m + h - 1 classes R1, R2,..., R2m, H1, H2,..., Hh-1. For each i = 1,2,..., u place on H U (Gj x {1,2}) a copy Dj of an incomplete 2-resolvable 5-fold (K4 - e)-design of order 2g + h with H as hole and having h - 1 partial classes Hj,j with j = 1, 2,..., h - 1 and 2g full classes Rj,i, t = 1, 2,..., 2g. For each block b = {x, y, z} of a given class of F place on b x {1, 2} a copy of a 2-resolvable 5-fold (K4 - e)-GDD of type 23 from Lemma 5.4, having {x1, x2}, {y1, y2} and {z1, z2} as groups. This gives 2m partial classes (whose blocks are copies of K4 - e) which miss G x {1,2} and 2g partial classes which miss Gj x {1,2}, i =1,2,..., u. Combine the 2m partial classes which miss the group G x {1,2} with the classes R1, R2,..., R2m so to obtain 2m classes. For i = 1,2,..., u combine the 2g partial classes which miss the group Gj x {1, 2} with the full classes of Dj so to obtain 2gu classes. Finally, combine the h - 1 classes H1, H2,..., Hh-1 of D with the partial classes of Dj so to obtain h - 1 classes. This gives a 2-resolvable 5-fold (K4 - e)-design of order v and v - 1 2-parallel classes. □ 376 Ars Math. Contemp. 10 (2016) 183-192 Theorem 5.6. Let v, k and h be non-negative integers. If there exists i) an incomplete a-resolvable X-fold (K4 — e)-design of order v + k + h with a hole of size k + h; ii) an incomplete a-resolvable X-fold (K4 — e) -design of order k + h with a hole of size h; then there exists an incomplete a-resolvable X-fold (K4 — e)-design of order v + k + h with a hole of size h. Lemma 5.7. There exists a resolvable (K4 — e )-GDD of type 52101. Proof. Let Z10 U {to0, to^ ..., to9} be the vertex-set and 2Z10, 2Z10 + 1, {to0, to^ ..., to9} be the groups. The desired design is obtained by adding 2 (mod 10) to the following base blocks, including the subscripts of to: (0,1, to0; to1), (2, 5, to0; to1), (4,9, to0; to1), (6,3, to0; to1), (8,7, to0; to1). The parallel classes are generate by every base block. □ Lemma 5.8. There exists a 2-resolvable 5-fold (K4 — e)-GDD of type 103. Proof. Start with the 2-resolvable 5-fold (K4 — e)-GDD G of type 23 of Lemma 5.4 with groups Gj, i = 1, 2,3. For each block b = (x, y, z; t) of a given 2-parallel class of G consider a copy of a resolvable (K4 — e)-GDD of type 52101 where {x} x Z5, {y} x Z5, {z, t} x Z5 are the groups. □ Lemma 5.9. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 6 with a hole of size 2. Proof. On V = Z4 U H, where H = {to1, to2} is the hole, consider the partial class {(1,3,0; 2), (0,2,1; 3)} and the four full classes obtained by developing {(0,2, to1; to2), (to1, 1, 0; 3), (to2, 2, 3; 1)} in Z4, where to, + 1 = to» for i = 1, 2. □ Lemma 5.10. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 10 with a hole of size 2. Proof. On V = Z8 U H, where H = {to1, to2} is the hole, consider the partial class {(0,4,2; 6), (1,5,3; 7), (2,6,4; 0), (3,7, 5; 1)} and the eight full classes obtained by developing {(0,1, to1; 3), (2, 3, to2; 7), (to1, 5, 6; 2), (to2, 6,4; 5), (4, 7,1; 0)} in Z8, where TOj + 1 = TOj for i = 1,2. □ Lemma 5.11. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 14 with a hole of size 4. Proof. Let V = Z10 U H be the vertex-set, where H = {to1, to2, to3, to4} is the hole. The partial classes are obtained by adding 2 (mod 10) to the base blocks (2,6,9; 5), (5,9, 2; 8), (8, 7,6; 9), each block generating a partial class; while, the full classes are obtained by adding 2 (mod 10) to the following base blocks partitioned into two full classes, each class generating five full classes: {(0,8, to1; to2), (1,5, to3; to4), (to1, 4, 0; 9), (to2, 6, 2; 3), (to3, 3, 7; 8), (TO4, 9,1;4), (2, 7, 6; 5)}, {(1, 5, TO1; TO2), (0, 8, to3; to4), (to 1, 3, 9; 4), (to2, 9, 7;0), (to3, 2, 6; 1), (to4, 6, 8; 3), (4, 7, 2; 5)}, where to, + 1 = oofor i = 1,2, 3,4. □ M. Giongriddo, et al.: The spectrum of a-resolvable X-fold (K4 — e)-designs 377 Lemma 5.12. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 14 with a hole of size 2. Proof. On V = Z12 U H, where H = {to^ to2} is the hole, consider the partial class {(0, 6,3; 9), (1,7,4; 10), (2, 8,5; 11), (3, 9,6; 0), (4,10, 7; 1), (5,11, 8; 2)} and the twelve full classes obtained by developing {(0,1, to^ 11), (2,4, to2; 10), (to1, 10,6; 5), (to2, 9, 2; 0), (3, 7,8; 1), (5, 8,7; 9), (6,11, 3; 4)} in Z12, where to, + 1 = to, for i = 1,2. □ Lemma 5.13. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 6. Proof. Let V = Z16 U H be the vertex-set, where H = {to1, to2, ..., to6} is the hole. In Zi6 develop the full 2-parallel base class {(0,3, toi; 12), (1,5, TO2;2), (8,13, to3;4), (14, 15, to4; 11), (6,11, to5; to6 ), (to1, 2,1;3), (to2, 4,13; 8), (to3, 7, 0; 14), (to4, 9, 6; 10), (to5, 10, 5; 15), (to6, 12,7; 9)}. Additionally, include the partial 2-parallel class {(0,8, 2; 10), (1, 9, 3; 11), (2,10,4; 12), (3,11, 5; 13), (4,12, 6; 14), (5,13, 7; 15), (6,14, 8; 0), (7, 15, 9;1)} repeated five times. □ As consequence of Lemmas 5.9 and 5.13, by Theorem 5.6 the following lemma follows. Lemma 5.14. There exists a 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 2. Lemma 5.15. There exists a 2-resolvable 5-fold (K4 — e)-design of order 10. Proof. Let V = Z9 U {to} be the vertex-set. The required design is obtained by developing the base class {(to, 0,6; 5), (1,5,4; 3), (7,8,1; to), (2, 6, 7; 8), (3,4, 2; 0)} in Z9. □ Lemma 5.16. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 30 with a hole of size 10. Proof. Start from a 2-resolvable 5-fold (K4 — e)-GDD of type 103 (which exists by Lemma 5.8) having G,, i = 1, 2, 3, as groups. Fill in the groups G2 and G3 with a copy of a 2-resolvable 5-fold (K4 — e)-design of order 10, which exists by Lemma 5.15. This gives an incomplete 2-resolvable 5-fold (K4 — e)-design of order 30 with G1 as hole. □ Lemma 5.17. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 38 with a hole of size 12. Proof. Let V = Z26 U H be the vertex-set, where H = {to1, to2, ..., to12} is the hole. The partial classes are: {(i, 13 + i, 2 + i; 15 + i) : i = 0,1,..., 12}, repaeated five times; {(2i, 10 + 2i, 3 + 2i; 7 + 2i) : i = 0,1,..., 12} and {(1 + 2i, 11 + 2i, 4 + 2i; 8 + 2i) : i = 0,1,..., 12}, repeated twice; {(2i, 10+2i, 1 + 2i; 9+2i) : i = 0,1,..., 12}; {(1 + 2i, 11 + 2i, 2 + 2i; 10 + 2i) : i = 0,1,..., 12}. The full classes are obtained by developing in V = Z26 the full base class {(toi, 2,1;7), (to2, 12,3; 24), (to3, 16,4; 11), (to4, 13,5; 25), (to5, 15, 9; 22), (to6, 17,11; 23), (to7, 19,18; 20), (to8, 14,10; 18), (to9, 4, 0; 8), (to10, 9,17; 19), (to 11, 7, 2; 12), (to12, 15, 3; 24), (1, 5, to1; to2), (10, 20, to3; to4), (6, 23, to5; to6), (16, 21, to7; to8), (22, 25, to9; to10), (13, 21, to11; to12), (0,14, 6; 8)}. □ As consequence of the existence of a 2-resolvable 5-fold (K4 — e)-design of order v = 4,12 (see Section 3 and Theorem 1.4) and Lemmas 5.1, 5.11, 5.13, 5.16, 5.17, 5.15, by Theorem 5.6 the following lemma follows. 378 Ars Math. Contemp. 10 (2016) 183-192 Lemma 5.18. There exists a 2-resolvable 5-fold (K4-e)-design of order v = 14,22,30,38. Lemma 5.19. There exists a 2-resolvable 5-fold (K4 — e)-design of order v = 42,234. Proof. Start with a resolvable 3-GDD of type 36 ([20]). Expand each vertex 2 times and for each triple b of a given parallel class place on b x {1, 2} a copy of a 2-resolvable 5-fold (K4 — e)-GDD of type 23, which exists by Lemma 5.4. Finally, fill each group of size 6 with a copy of a 2-resolvable 5-fold (K4 — e)-design of order 6, which exists by Lemma 5.1. □ Lemma 5.20. There exists a 2-resolvable 5-fold (K4 — e)-design of order v = 50,62. v-2 Proof. Start from a 3-frame of type 6([3]) and apply Contraction 5.5 with m = g = 6, h = 2 and u = to obtain a 2-resolvable 5-fold (K4 — e)-design of order v = 50,62 (the input designs are: a 2-resolvable 5-fold (K4 — e)-design of order 14, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 — e)-GDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 14 with a hole of size 2, which exists by Lemma 5.12). □ Lemma 5.21. There exists a 2-resolvable 5-fold (K4 — e)-design of order v = 34,274. v-2 Proof. Start from a 3-frame of type 4([3]) and apply Theorem 5.5 with m = g = 4, h = 2 and u = ^^g10 to obtain a 2-resolvable 5-fold (K4 — e)-design of order v = 34, 274 (the input designs are: a 2-resolvable 5-fold (K4 — e)-design of order 10, which exists by Lemma 5.15; a 2-resolvable 5-fold (K4 — e)-GDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 10 with a hole of size 2, which exists by Lemma 5.10). □ Lemma 5.22. There exists a 2-resolvable 5-fold (K4 — e)-design of order 70. Proof. Start from a 3-frame of type 84 ([3]) and apply Theorem 5.5 with m = g = 8, h = 6 and u = 3 to obtain a 2-resolvable 5-fold (K4 — e)-design of order 70 (the input designs are; a 2-resolvable 5-fold (K4 — e)-design of order 22, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 — e)-RGDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 6, which exists by Lemma 5.13). □ Lemma 5.23. For every v = 2 (mod 20), there exists a 2-resolvable 5-fold (K4 — e)-design of order v. Proof. Let v=20k + 2. The case v = 22,42,62 are covered by Lemmas 5.18, 5.19 and 5.20. For k > 4, start from a 5-fold (2, K4 — e)-frame of type 20k ([5]) and apply Theorem 5.3 with h = 2 to obtain a 2-resolvable 5-fold (K4 — e)-design of order v (the input designs are a 2-resolvable 5-fold (K4 — e)-design of order 22, which exists by Lemma 5.18, and an incomplete 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 2, which exists by Lemma 5.14). □ Lemma 5.24. For every v = 10 (mod 20), there exists a 2-resolvable 5-fold (K4 — e)-design of order v. M. Giongriddo, et al.: The spectrum of a-resolvable X-fold (K4 — e)-designs 379 Proof. Let v=20k + 10. The case v = 10, 30, 50, 70 are covered by Lemmas 5.15, 5.18, 5.20 and 5.22. For k > 4, start from a 5-fold (2, K4 - e)-fTame of type 20k ([5]) and apply Theorem 5.3 with g = 20 and h =10 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are a 2-resolvable 5-fold (K4 - e)-design of order 10, which exists by Lemma 5.15, and an incomplete 2-resolvable 5-fold (K4 - e)-design of order 30 with a hole of size 10, which exists by Lemma 5.16). □ Lemma 5.25. For every v = 14 (mod 20), there exists a 2-resolvable 5-fold (K4 - e)-design of order v. Proof. Let v=20k + 14. The case v = 14, 34, 234,274 are covered by Lemmas 5.18, 5.19 and 5.21. For k > 2, k £ {11,13}, start from a 5-fold (2, K4 - e)-frame of type 102fc+1 ([5]), apply Theorem 5.3 with h = 4 and proceed as in Lemma 5.24. □ Lemma 5.26. For every v = 18 (mod 60), there exists a 2-resolvable 5-fold (K4 - e)-design of order v. Proof. Let v=60k + 18. Take a resolvable 3-GDD of type 310k+3 ([6]). Expand each vertex 2 times and for each block b of a parallel class place on b x {1, 2} a copy of a 2-resolvable 5-fold (K4 - e)-GDD of type 23 which exists by Lemma 5.4, so to obtain a 2-resolvable 5-fold (K4 - e)-GDD of type 610k+3. Finally, fill in each group of size 6 with a copy of a 2-resolvable 5-fold (K4 - e)-design, which exists by Lemma 5.1. □ Lemma 5.27. For every v = 38 (mod 60), there exists a 2-resolvable 5-fold (K4 - e)-design of order v. Proof. Let v = 60k + 38. The case v = 38 follows by Lemmas 5.18. For k > 1, start from a 3-fTame of type 65k+3 ([6]) and apply Theorem 5.5 with m = g = 6, h = 2 and u = 5k + 2 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are: a 2-resolvable 5-fold (K4 - e)-design of order 14, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 - e)-GDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 - e)-design of order 14 with a hole of size 2, which exists by Lemma 5.11) □ Lemma 5.28. For every v = 58 (mod 120), there exists a 2-resolvable 5-fold (K4 - e)-design of order v. Proof. Let v = 120k + 58. Start from a 3-frame of type 415k+7 ([6]) and apply Theorem 5.5 with m = g = 4, h = 2 and u = 15k + 6 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are: a 2-resolvable (K4 - e)-design of order 10, which exists by Lemma 5.15; a 2-resolvable 5-fold (K4 - e)-RGDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 - e)-design of order 10 with a hole of size 2, which exists by Lemma 5.10). □ Lemma 5.29. For every v = 118 (mod 120), there exists a 2-resolvable 5-fold (K4 - e)-design of order v. Proof. Let v = 120k + 118. Start from a 3-frame of type 101415k+12, k > 0, ([6]) and apply Theorem 5.5 with h = 2 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are: a 2-resolvable 5-fold (K4 - e)-design of order 22, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 - e)-RGDD of type 23, which exists by Lemma 380 Ars Math. Contemp. 10 (2016) 183-192 5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 10 with a hole of size 2, which exists by Lemma 5.10). □ 6 Main result The results obtained in the previous sections can be summarized into the following theorem. Theorem 6.1. The necessary conditions (1) — (3) for the existence of a-resolvable X-fold (K4 — e)-designs are also sufficient. References [1] J.-C. Bermond, K. Heinrich and M.-L. Yu, Existence of resolvable path designs, European J. Combin. 11 (1990), 205-211, doi:10.1016/S0195-6698(13)80120-5, http://dx.doi. org/10.1016/S0195-6698(13)80120-5. [2] J.-C. Bermond and J. 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Wang, Completing the spectrum of resolvable (K4 — e)-designs, Ars Combin. 105 (2012), 289-291. [24] L. Wang and R. Su, On the existence of maximum resolvable (K4 — e)-packings, Discrete Math. 310 (2010), 887-896, doi:10.1016/j.disc.2009.10.007, http://dx.doi.org/10. 1016/j.disc.2009.10.007. [25] Y. Zhang and B. Du, a-resolvable group divisible designs with block size three, J. Combin. Des. 13 (2005), 139-151, doi:10.1002/jcd.20028, http://dx.doi.org/10.10 02/ jcd.20028. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 383-392 The endomorphisms of Grassmann graphs* Li-Ping Huang School of Mathematics, Changsha University of Science and Technology, Changsha, 410004, China Benjian Lv Kaishun Wang Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China Received 17 December 2014, accepted 17 December 2015, published online 15 March 2016 A graph G is a core if every endomorphism of G is an automorphism. A graph is called a pseudo-core if every its endomorphism is either an automorphism or a colouring. Suppose that Jq(n, m) is a Grassmann graph over a finite field with q elements. We show that every Grassmann graph is a pseudo-core. Moreover, J2(4,2) is not a core and Jq (2k + 1, 2) (k > 2) is a core. Keywords: Grassmann graph, core, pseudo-core, endomorphism, maximal clique. Math. Subj. Class.: 05C60, 05C69 1 Introduction Throughout this paper, all graphs are finite undirected graphs without loops or multiple edges. For a graph G, we let V(G) denote the vertex set of G. If xy is an edge of G, then x and y are said to be adjacent, and denoted by x ~ y. Let G and H be two graphs. A homomorphism p from G to H is a mapping p : V(G) ^ V(H) such that p(x) ~ p(y) whenever x ~ y. If H is the complete graph Kr, then p is a r-colouring of G (colouring for short). An isomorphism from G to H is a bijection p : V(G) ^ V(H) such that x ~ y ^ p(x) ~ p(y). Graphs G and H are called isomorphic if there is an isomorphism from * Projects 11371072, 11301270, 11271047, 11371204, 11501036 supported by NSFC. Supported by the Fundamental Research Funds for the Central University of China, Youth Scholar Program of Beijing Normal University (2014NT31) and China Postdoctoral Science Foundation (2015M570958). t Corresponding author E-mail address: lipingmath@163.com (Li-Ping Huang), bjlv@bnu.edu.cn (Benjian Lv), wangks@bnu.edu.cn (Kaishun Wang) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 384 Ars Math. Contemp. 10 (2016) 183-192 G to H, and denoted by G = H .A homomorphism (resp. isomorphism) from G to itself is called an endomorphism (resp. automorphism) of G. Recall that a graph G is a core if every endomorphism of G is an automorphism. A subgraph H of G is a core of G if it is a core and there exists a homomorphism from G to H. Every graph has a core, which is an induced subgraph and is unique up to isomorphism [5]. A graph is called core-complete if it is a core or its core is complete. A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. Every core is a pseudo-core. Any pseudo-core is core-complete but not vice versa. For more information, see [2, 6, 9]. For a graph G, an important and difficult problem is to distinguish whether G is a core [2, 5, 6, 7, 11, 15]. If G is not a core or we don't know whether it is a core, then we need to judge whether it is a pseudo-core because the concept of pseudo-core is the most close to the core. Recently, Godsil and Royle [6] discussed some properties of pseudo-cores. Cameron and Kazanidis [2] discussed the core-complete graph and the cores of symmetric graphs. The literature [10] showed that every bilinear forms graph is a pseudo-core which is not a core. One of the latest result is from [9], where it was proved that every alternating forms graph is a pseudo-core. Moreover, Orel [13, 12] proved that each symmetric bilinear forms graph (whose diameter is greater than 2) is a core and each Hermitian forms graph is a core. Suppose that Fq is the finite field with q elements, where q is a power of a prime. Let V be an n-dimensional row vector space over Fq and let be the set of all m-dimensional subspaces of V. The Grassmann graph Jq (n, m) has the vertex set , and two vertices are adjacent if their intersection is of dimension m - 1. If m = 1, we have a complete graph and hence it is a core. Since Jq(n, m) = Jq(n, n - m), we always assume that 4 < 2m < n in our discussion unless specified otherwise. The number of vertices of Jq (n, m) is the Gaussian binomial coefficient: nq 1 -1. (1.1) qi-1 For Jq(n, m), the distance of two vertices X and Y is d(X, Y) := m — dim(X n Y). Any Grassmann graph is distance-transitive [1, Theorem 9.3.3] and connected. By [6, Corollary 4.2], every distance-transitive graph is core-complete, thus every Grassmann graph is core-complete. The Grassmann graph plays an important role in geometry, graph theory, association schemes and coding theory. Recall that an independent set of a graph G is a set of vertices that induces an edgeless graph. The size of the largest independent set is called the independence number of G, denoted by a(G). The chromatic number x(G) of G is the least value of k for which G can be k-colouring. A clique of a graph G is a complete subgraph of G. A clique C is maximal if there is no clique of G which properly contains C as a subset. A maximum clique of G is a clique with the maximum size. The clique number of G is the number of vertices in a maximum clique, denoted by w(G). By [6, p.273], if G is a distance-transitive graph and x(G) > w(G), then G is a core. Unluckily, applying the eigenvalues or the known results of graph theory for Grassmann graph, to prove the inequality x(G) > w(G) is difficult. Thus, it is a difficult problem to verify a Grassmann graph being a core. However, there are some Grassmann graphs which are not cores (see Section 4). Therefore, we need to judge whether a Grassmann graph is a pseudo-core. So far, this is an open problem. We solve this problem as follows: L.-P. Huang, B. Lv, K. Wang: The endomorphisms of Grassmann graphs 385 Theorem 1.1. Every Grassmann graph Jq (n, m) is a pseudo-core. The paper is organized as follows. In Section 2, we give some properties of the maximal cliques of Grassmann graphs. In Section 3, we shall prove Theorem 1.1. In Section 4, we discuss cores on Grassmann graphs. We shall show that J2(4,2) is not a core, Jq (2k +1, 2) (k > 2) is a core. 2 Maximal cliques of Grassmann graph In this section we shall discuss some properties of the maximal cliques of Grassmann graphs. We will denote by |X| the cardinal number of a set X. Suppose that V is an n-dimensional row vector space over Fq. For two vector subspaces S and T of V, the join S V T is the minimal dimensional vector subspace containing S and T. We have the dimensional formula (cf. [8, Lemma 2.1] or [16]): dim(S V T) = dim(S) + dim(T) - dim(S n T). (2.1) Throughout this section, suppose that 4 < 2m < n. For every (m - 1)-dimensional subspace P of V, let [P}m denote the set of all m-dimensional subspaces containing P, which is called a star. For every (m + 1)-dimensional subspace Q of V, let (Q]m denote the set of all m-dimensional subspaces of Q, which is called a top. By [4], every maximal clique of Jq (n, m) is a star or a top. For more information, see [14]. By [16, Corollary 1.9], qn-m+1 _ 1 qm+1 _ 1 |[P}m| = --:-, |(Q]m| = --r-. (2.2) q — 1 q — 1 If n > 2m, then every maximum clique of Jq(n, m) is a star. If n = 2m, then every maximal clique of Jq (n, m) is a maximum clique. By (2.2) we have w(Jq(n,m))= [n"m+1] ifn > 2m. (2.3) Since n > 2m, we have |[P}m| > |(Q]m|, and |[P}m| > |(Q]m| if n > 2m. (2.4) Lemma 2.1. If [P}m n (Q]m = 0, then the size of [P}m n (Q]m is q + 1. Proof. Since [P}m n (Q]m = 0, one gets P C Q. It follows that [P}m n (Q]m consists of all m-dimensional subspaces containing P in Q. By [16, Corollary 1.9], the desired result follows. □ Lemma 2.2. ([8, Corollary 4.4]) Let M1 and M2 be two distinct stars (tops). Then |Mi n M2| < 1. Lemma2.3. Suppose [A}m = [B}m. Then [A}mn [B}m = 0 ifandonlyif dim(AnB) = m — 2. In this case, [A}m n [B}m = {A V B}. Proof. Since dim(A) = dim(B) = m — 1 and A = B, one gets dim(A V B) > m. If [A}m n [B}m = 0, then by Lemma 2.2, there exists a vertex C of Jq (n, m) such that {C} = [A}mn[B}m. Itfollows from (2.1) and A, B c C that C = AVB anddim(AnB) = m—2. Conversely, if dim(A n B) = m — 2, then Lemma 2.2 and (2.1) imply that C := A V B is a vertex of Jq(n, m) and hence {C} = [A}m n [B}m. □ 386 Ars Math. Contemp. 10 (2016) 183-192 Lemma 2.4. Suppose (P ]m = (Q]m. Then (P ]m n(Q]m = 0 if and only if dim(P n Q) = m. In this case, (P]m n (Q]m = {P n Q}. Proof. By dim(P) = dim(Q) = m +1 and P = Q, we have dim(P n Q) < m. If (P]m n (Q]m = 0, then Lemma 2.2 implies that there exists a vertex C of Jq(n, m) such that {c} = (P]m n(Q]m. Since C c Pn Q, we get that C = Pn Q and dim(Pn Q) = m. Conversely, if dim(P n Q) = m, then by P n Q G (P]m n (Q]m and Lemma 2.2, we have {P n Q} = (P]m n(Q]m. □ In the following, let f be an endomorphism of Jq(n, m) and Im(f) be the image of f. Lemma 2.5. If M is a maximal clique, then there exists a unique maximal clique containing f (M). Proof. Suppose there exist two distinct maximal cliques M' and M" containing f (M). Then f (M) CM' n M''. By Lemmas 2.1 and 2.2, |M' n M''| < q +1. Since |M| = |f (M) |, by (2.2) we have |f (M) | > q + 1, a contradiction. □ Lemma 2.6. Let M be a star and N be a top such that |f (M) n f (N)| > q +1. Then f(N) C f (M). Proof. LetN' be the maximal clique containing f (N). Then |f (M) nN'| >q +1. One gets f (M) = N' by Lemmas 2.1 and 2.2. □ Lemma 2.7. Suppose there exist two distinct stars [A)m and [B)m such that [A)m n [B)m = {X}, f([A)m) = f([B)m). If f ([A)m) is a star, then f is a colouring of Jq(n, m). Proof. Write M := f ([A)m). Then f ([B)m) = M and f (X) G M. Assume that M is a star. If Im(f) = M, then f is a colouring of Jq(n, m). Now we prove Im(f) = M as follows. Suppose that Y is any vertex with Y ~ X. Since G := Jq(n, m) is connected, it suffices to show that there exist two distinct stars [C)m and [D)m such that {Y} = [C)m n [D)m and f([C)m) = f([D)m) = M. In fact, if we can prove this point, then we can imply that f (Z) G M for all Z G V(G). We prove it as follows. Since X G (X V Y]m n [A)m n [B)m, using Lemma 2.2 we get |(X V Y]m n [A)m n [B)m| = 1. By Lemma 2.1 we obtain |(X V Y]m n [A)m| = |(X V Y]m n [B)m | = q +1. It follows that |(X V Y]m n ([A)m U [B)m)| = 2q +1. Observe that f((XvY]mn([A)mu [B)m)) c f((XvY]m)nf([A)mu[B)m) c f((XvY]m)nM. Since the restriction of f on a clique is injective, one gets |f ((X V Y]m) n M| > 2q + 1 > q + 1. L.-P. Huang, B. Lv, K. Wang: The endomorphisms of Grassmann graphs 387 Thus, Lemma 2.6 implies that ^((X V Y]m) CM. (2.5) So ) € M. Write C := X n Y. Since every vertex of [C)m \ {X} is adjacent to X, by our claim we have y([C)m) = M. Pick a vertex Z such that Z — Y and the distance from X is 2. Write D = Yn Z. Since Y € [D)m n (X V Y]m, by Lemma 2.1 we have |[D)m n (X V Y]m| = q + 1. It follows from (2.5) that |y([D)m) n M| > q +1. Thus Lemma 2.2 implies that y([D)m) = M. Since {Y} = [C)m n [D)m, [C)m and [D)m are the desired stars. □ 3 Proof of Theorem 1.1 For the proof of Theorem 1.1, we only need to consider the case 4 < 2m < n. We divide the proof of Theorem 1.1 into two cases: n > 2m and n = 2m. Lemma 3.1. If n > 2m, then every Grassmann graph Jq (n, m) is a pseudo-core. Proof. Suppose that n > 2m > 4. Then by (2.4), every maximum clique of Jq (n, m) is a star. Let ^ be an endomorphism of Jq(n, m). Then the restriction of ^ on any clique is injective, so ^ transfers stars to stars. Suppose ^ is not a colouring. It suffices to show that ^ is an automorphism. Write Gr := Jq (n, r), where 1 < r < m - 1. By Lemma 2.7, the images under ^ of any two distinct and intersecting stars are distinct. Hence by Lemma 2.3, ^ induces an endomorphism ym-i of Gm-1 such that y([A)m) = [ym-i(A))m. Let X be any vertex of Jq (n, m). Then there exist two vertices X' and X" of Gm-1 such that X = X 'VX ".Then [X ')m n[X ")m = {X } and y(X) € m-1 is not a colouring of Gm-i for m > 3. For any two vertices A1 and A3 of Gm-i at distance 2, we claim that ^m-i(Ai) = ^m-i(A3). There exists an A2 € V(Gm-i) such that Ai - - A3. Write Yi := Ai V A2 and Y2 := A2 V A3. Then Yi - Y>, so ^(Yi) = ^(Y>). By (3.1), ^(Yi) = ^m-i(Ai) V ^m-i(A2), ^(Y2) = ^m-i(A2) V ^m-1 (A3). Thus our claim is valid. Otherwise, one has ^(Y1) = y(Y2), a contradiction. Pick a star N of Gm-1. Since the diameter of Gm-1 is at least two, there exists a vertex A4 € V (Gm-1) \ N that is adjacent to some vertex in N .If B € N such that A4 is not adjacent to B, then d(A4, B) = 2. By our claim, ^>m-1(A4) = y(B) and hence ^m-1(A4) € ^m-1(N). Therefore, y>m-1 is not a colouring. By induction, we may obtain induced endomorphism ^>r of Gr for each r. Furthermore, y(X) = ^ (Xfcl) V Vk2 (Xfc2) V • • • V Vks (Xks), (3.2) 388 Ars Math. Contemp. 10 (2016) 183-192 where X = Xkl V Xkl V ■ ■ ■ V Xks G V(Gm) and 1 < dim(Xki) = k, < m - 1. In order to show that ^ is an automorphism, it suffices to show that ^ is injective. Assume that X and Y are any two distinct vertices in Gm with d(X, Y) = s. Thus dim(X n Y) = m — s. If s = 1, then y(X) = ^(Y). Now suppose s > 2. There are 1-dimensional row vectors X,, Y,, i = 1,... .s, such that X, Y can be written as X = (Xn Y)VXiV---VXs, Y = (XnY)VYiV---VYs.LetZ = (XnY)VXiV---VXs_iVYs G V(Gm). By X - Z, dim(^(X) V y>(Z)) = m + 1. Applying (3.2), one has that y(X) = ^m_s(X n Y) V ^i(Xi) v ■ ■ ■ v ^i(Xs), y>(Y) = ^m_s(X n Y) v ^i(Yi) v ■ ■ ■ v ^i(Ys) and ^(Z) = ^m_s(X n Y) V ^(Xi) V ■ ■ ■ V y>i(Xs_i) V ^(Ys). Therefore, we get y(X) V ^(Z) C y>(X) V ^(Y). It follows that y(X) = ^(Y). Otherwise, one has y(X) V ^(Z) C ^>(X), a contradiction to dim(^(X) V <^(Z)) = m + 1. Hence, ^ is an automorphism, as desired. By above discussion, Jq(n, m) is a pseudo-core when n > 2m. □ Lemma 3.2. If n = 2m, then every Grassmann graph Jq (n, m) is a pseudo-core. Proof. Suppose that n = 2m > 4. For a subspace W of V, the dual subspace WL of W in V is defined by W^ = {v G V | wvt = 0, V w G W}, where vt is the transpose of v. For an endomorphism ^ of Jq (2m, m), define the map ^ : V(Jq(2m, m)) —> V(Jq(2m, m)), A i—> ^(A)^. Then ^ is an endomorphism of Jq (2m, m). Note that ^ is an automorphism (resp. colouring) whenever ^ is an automorphism (resp. colouring). For any maximal clique M of Jq(2m, m), <^(M) and <^(M) are of different types. Next we shall show that Jq(2m, m) is a pseudo-core. Case 1. There exist [A)m and (X]m such that [A)m n (X]m = 0 and y([A)m), ¥>((X]m) are of the same type. By Lemma 2.1, the size of [A)m n (X]m is q +1. Then |y([A)m) n y((X]m)| > q +1. Since y([A)m) and y((X]m) are of the same type, by Lemma 2.2 one gets ^([A)m) = ^((X ]m). (3.3) Note that A C X. Pick any Y G [J+J satisfying A C Y and dim(X n Y) = m. Then (Y]m n [A)m = 0. By Lemma 2.1 we have |y((Y]m) n y([A)m)| > q +1. By Lemma 2.2 and (3.3) we obtain either y>((Y]m) = ^((X]m) or y>((Y]m) and ^((X]m) are of different types. Case 1.1. There exists a Y g [mV J such that <^((Y]m) and ^((X]m) are of different types. For any B G [m_T], we have that B C Y and B C X. Since |[B)m) n (X]m| = | [B)m) n (Y]m| = q + 1, we have similarly b([B)m) n ^((X]m)| > q +1 and |^([B)m) n ^((Y]m)| > q +1. Since y((Y]m) and ^((X]m) are of different types, Lemma 2.2 implies that y([B)m) = ^((X]m) or ^([B)m) = ^((Y]m) for any B G [^^i]. Since the size of [m^Yl is at least 3, by above discussion, there exist two subspaces Bi, B2 g [m^5!] such that m (because (A v C) c w n X). Since T € (W]m, y(T) € y((W]m). By the condition, y((W]m) = y((X]m). Then y((W]m) = y([A)m) by (3.3). It follows that y(T) € y([A)m) for all T € [C)m, and so y([C)m) C y([A)m). Hence, y([C)m) = y([A)m). Since [C)m n [A)m = 0, similar to the proof of Case 1.1, y is a colouring. Case 2. For any two maximal cliques of different types containing common vertices, their images under y are of different types. In this case, y maps the maximal cliques of the same type to the maximal cliques of the same type. Case 2.1. y maps stars to stars. In this case y maps tops to tops by Lemmas 2.1 and 2.2. If there exist two distinct stars M and M' such that MnM' = 0 and y(M) = y(M'), then y is a colouring by Lemma 2.7. Now suppose y(M) = y(M') for any two distinct stars M and M' with M n M' = 0. By Lemma 2.3, y induces an endomorphism ym-1 of Jq(2m, m - 1) such that y([A)m) = [ym-1(A))m. By Lemma 3.1, Jq(2m, m - 1) is a pseudo-core. Thus, ym-1 is an automorphism or a colouring. We claim that ym-1 is an automorphism of Jq(2m, m - 1). For any C € [V] and B € L-J, since C € [B)m and y([B)m) = [ym_1(B))m, we have y(C) € [ym_1(B))m. Then ym-1(B) C y(C), which implies that ym-1((C]m-1) is a top of Jq(2m, m - 1). If m = 2, our claim is valid. Now suppose m > 3 and ym-1 is a colouring. Then Im(ym-1) is a star of Jq(2m, m - 1). Note that ym-1((C]m-1) C Im(ym-1) and |ym-1 ((C]m-1)| > q +1, contradicting to Lemma 2.1. Hence, our claim is valid. Therefore, y maps distinct stars onto distinct stars, and y is an automorphism. Case 2.2. y maps stars to tops. In this case y maps tops to stars by Lemmas 2.1 and 2.2. Note that y^ maps stars to stars. By Case 2.1, y^ is an automorphism. Hence, y is an automorphism. By above discussion, we have proved that every Grassmann graph Jq (2m, m) is a pseudo-core. □ By Lemmas 3.1 and 3.2, we have proved Theorem 1.1. 4 Cores on Grassmann graphs In this section, we shall show that J2 (4,2) is not a core and Jq(2k + 1, 2) (k > 2) is a core. It is well-known (cf. [3, Theorem 6.10 and Corollary 6.2]) that the chromatic number of G satisfies the following inequality: X(G) > max{w(G), |V(G)|/a(G)} 390 Ars Math. Contemp. 10 (2016) 383-392 By [15, Lemma 2.7.2], if G is a vertex-transitive graph, then x(G) »^f > .(G). a(G) (4.1) Lemma 4.1. Let G be a Grassmann graph. Then G is a core if and only if x(G) > w(G). In particular, if ^(^j1 is not an integer, then G is a core. Proof. By [6, Corollary 4.2], every distance-transitive graph is core-complete, thus G is core-complete. Then, x(G) > w(G) implies that G is a core. Conversely, if G is a core, then we must have x(G) > w(G). Otherwise, there exists an endomorphism f of G such that f (G) is a maximum clique of G, a contradiction to G being a core. Thus, G is a core if and only if x(G) > w(G). By [2, p.148, Remark], if the core of G is complete, then |V(G)| = w(G)a(G). Assume that is not an integer. Then |V(G)| = w(G)a(G). Therefore, the core of G is not complete and hence G is a core. □ Denote by F^-*" the set of m x n matrices over F0 and F" «udFq = F^*". Let G = Jq(n,m) where n > m. If X is a vertex of G, then X = [ai,..., am] is an m-dimensional subspace of the vector space F", where {ai,..., am} is a basis of X. Thus, X has a matrix representation € F^*" (cf. [8, 16]). For simpleness, the matrix representation of X € V(G) is also denoted by X. For matrix representations X, Y of two vertices X and Y, X ~ Y if and only if rank ^ X j = m +1. Note that if X is a matrix representation then X = PX (as matrix representation) for any m x m invertible matrix P over Fq. Then, V (G) has a matrix representation V(G) = {X : X € F^*", rank(X) = m} . Now, we give an example of Grassmann graph which is not a core as follows. Example 4.2. Let G = J2(4,2). Then G is not a core. Moreover, x(G) = w (G) = 7 and a(G) = 5. Proof. Applying the matrix representation of V(G), G = J2(4, 2) has 35 vertices as fol- lows: Ai = 1 0 0 1 0 0 0 0 ^ A 2 = 1 0 0 1 1 0 0 0 A3 A5 = 1 0 0 0 ), A6 = 1 0 0 0 , A 7 0 1 1 0 0 1 0 1 Ag = 1 0 0 1 0 1 0 1 ), A10 = 1 0 0 1 1 0 0 1 , A11 A 13 = 1 0 0 1 1 0 11 ^ A14 = 1 0 0 1 11 0 1 , A15 A 7 = 0 0 0 0 1 0 0 1 ), A18 = 1 0 0 0 1 0 0 1 , A19 A 21 = 0 0 0 1 1 0 0 1 ^ A22 = 11 0 0 1 0 0 1 , a23 A 25 = 0 0 1 1 1 0 0 1 ), A26 = 11 11 1 0 0 1 , A27 A 29 = 0 0 1 0 0 0 11 ), A30 = 1 0 0 0 0 0 0 1 , A31 A 33 = 1 0 1 0 0 0 0 1 ^ A 34 = 11 0 0 0 0 11 , A35 A4 = As = A12 : A16 = A20 = A24 = A28 = a32 = ai a 0 0 0 0 0 L.-P. Huang, B. Lv, K. Wang: The endomorphisms of Grassmann graphs 391 Suppose that £1 = {A1, A10, A12, A15, A17}, £2 = {A2, A6, A20, A19, A34}, £3 = {A3, Ag, A21, A22, A35}, £4 = {A5, A9, A18, A24, A29}, £5 = {A7, A14, A23, A27, A33}, £5 = {A4, A13, A25, A28, A30}, and £7 = {An, A16, A26, A31, A32}. It is easy to see that V(G) = £1 U£2 U • • • U £7 and £1,..., £7 are independent sets. Thus x(G) < 7. On the other hand, (4.1) implies that x(G) > w(G) = 7. Therefore, x(G) = w(G) = 7. It follows from Lemma 4.1 that G is not a core. By (4.1) again, we have a(G) = 5. □ We believe that Jq(2k, 2) (k > 2) is not a core for all q (which is a power of a prime). But this a difficult problem. Next, we give some examples of Grassmann graphs which are cores. Example 4.3. If k > 2, then Jq (2k + 1,2) is core. Proof. When k > 2, let G = Jq(2k + 1, 2). Applying (1.1) and (2.3) we have |V(G)| _ q2fc+1 - 1_ q2fc+1 - q 1 w(G) q2 - 1 q2 - 1 q + 1' Thus is not an integer for any q (which is a power of a prime). By Lemma 4.1, G is a core. □ Acknowledgement We are grateful to the referees for useful comments and suggestions. Projects 11371072, 11301270, 11271047, 11371204, 11501036 supported by NSFC. This research also supported by the Fundamental Research Funds for the Central University of China, Youth Scholar Program of Beijing Normal University (2014NT31) and China Postdoctoral Science Foundation (2015M570958). References [1] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1989, doi:10.1007/978-3-642-74341-2, http://dx.doi. org/10.1007/978-3-642-74341-2. [2] P. J. Cameron and P. A. Kazanidis, Cores of symmetric graphs, J. Aust. Math. Soc. 85 (2008), 145-154, doi:10.1017/S1446788708000815, http://dx.doi.org/10.1017/ S1446788708000815. [3] G. Chartrand and P. Zhang, Chromatic graph theory, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2009. [4] W.-L. Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32-67. [5] C. Godsil and G. 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ARS MATHEMATICA CONTEMPORANEA 10 (2016) 393-410 On convergence of binomial means, and an application to finite Markov chains* David Gajser IMFM, Jadranska 19, 1000 Ljubljana, Slovenia, and Department of Mathematics, FMF, University of Ljubljana, Slovenia Received 15 July 2014, accepted 21 December 2015, published online 13 April 2016 Abstract For a sequence {an}n>0 of real numbers, we define the sequence of its arithmetic means {an}n>0 as the sequence of averages of the first n elements of {an}n>0. For a parameter 0 < p < 1, we define the sequence of p-binomial means {an}n>0 of the sequence {an}n>0 as the sequence of p-binomially weighted averages of the first n elements of {an }n>0. We compare the convergence of sequences {an}n>0, {an}n>0 and {an}n>0 for various 0 < p < 1, i.e., we analyze when the convergence of one sequence implies the convergence of the other. While the sequence {an}n>0, known also as the sequence of Cesaro means of a sequence, is well studied in the literature, the results about {an}n>0 are hard to find. Our main result shows that, if {an}n>0 is a sequence of non-negative real numbers such that K}„>0 converges to a G R U {to} for some 0 < p < 1, then {an}n>0 also converges to a. We give an application of this result to finite Markov chains. Keywords: Sequence, convergence, Cesaro mean, binomial mean, finite Markov chain. Math. Subj. Class.: 00A05 1 Introduction For a sequence {an}n>0 of real numbers and for a parameter 0 < p < 1, define the sequence of its arithmetic means {a^ }n>0 and the sequence of its p-binomial means {an }n>0 as and an = ^ i "" jpiqn-iai, i=0 ^1 ' ai n + 1 ^ i=0 1 a *This work is partially funded by the Slovenian Research Agency, Research Program P1-0297. E-mail address: david.gajser@fmf.uni-lj.si (David Gajser) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 394 Ars Math. Contemp. 10 (2016) 183-192 where q = 1 - p. We see that a« is a uniformly weighted average of the numbers ao, —i,.. ., an and an is a binomially weighted average of the numbers ao, ai,. .., an. In this article, we will analyse the relationship between the convergence of sequences {an}n>0, {a«}n>0 and {a«}n>0. Our results are presented in the following table. {an}n>0 K1 }n>0 K2 }n>0 {an}n>0 {an}n>0 - - - K1 }n>0 /s ? an>0 an>0 - - K2 }n>0 /s an>0 —^ - - {an}n>0 /s /s /s —^ —^ —^ Table 1: The table shows whether the convergence of a sequence in the leftmost column implies the convergence of a sequence in the first row, for 0 < p < p2 < 1. The symbol means that the implication holds, and the symbol means that there is a counterexample with an G {0,1}, for all n G N. If there is a condition above , then the implication does not hold in general, but it holds if the condition is true. If there is a ? before the condition, we do not know whether the condition is the right one (an open problem), but the implication does not hold in general. The sequence {a«}n>0 is also known as the sequence of Cesaro means and is well studied in the literature [1, 4]. On the other hand, information about the convergence of p-binomial means is hard to find. Also, the notion of p-binomial means is coined especially for the purpose of this article. However, there are a few definitions that are close to ours [1, 4, 5]. First, we have to mention the Hausdorff means [1, 4]: the p-binomial means as well as the arithmetic mean are its special cases. Unfortunately, the Hausdorff means are a bit too general for our purposes in the sense that the known results that are useful for this paper can be quite easily proven in our special cases. One of the closest notions to the k-binomial mean is the one of k-binomial transform [5]: = E (n)k"ai, which coincides with {an }n>0 for k = p = 0.5, but is different for other p and k. Another similar definition is given with Euler means [4, pages 70, 71]: - f (n +1 an =2«+^ U +1 i=0 v Some results, like the first row and the first column of Table 1, are not hard to prove (Section 3). Other results (Sections 4 and 5) require more careful ideas. This is true especially for the main result of this paper, Theorem 5.1, which proves, using the notation from Table 1, that {a«}n>0 ==>■ {a«}n>0. In Section 6 we give an application of this theorem to finite Markov chains. D. Gajser: On the convergence of binomial means . 395 2 Preliminaries Let N, R+ and R+ be the sets of non-negative integers, positive real numbers and nonnegative real numbers, respectively. For a e R, let |_aj be the greatest integer not greater than a and let [a] be the smallest integer not smaller than a. We will allow a limit of a sequence to be infinite and we will write a < to (which means exactly a e R) to emphasize that a is finite. For functions f, g : N ^ R+ we say that • f (n) = O(g(n)) if there is some C > 0 such that f (n) < Cg(n) for all sufficiently large n, • f (n) = ©(g(n)) if there are some Ci, C2 > 0 such that Cig(n) < f (n) < C2g(n) for all sufficiently large n, f(n) f (n) = o(g(n)) if g(n) is non-zero for all large enough n and lim —— n^TO g(n) 0. The following lemma will be useful later. Lemma 2.1. Let u : N ^ R\{0} and k : N ^ R be functions such that lim u(n)k(n) = lim u(n) = 0. Then n^w lim n (1 + u(n)) k(n)/u(n) ek(n) 1. Proof. Because ex = ^ ^y- and ex > 1 + x, there is an analytic function g : R ^ R+ such that ex = 1 + x + g(x)x2 and g(0) = 1. Hence, if we omit writing the argument of functions u and k, lim n (1 + u)k/u lim n eu — g(u)u2 2 \ k/u ukg(u) lim n 1 g(u)u2 \ S(")"' g( u) u2 Because lim -= 0 and because lim(1 — x)1/x = e-1, we have n^w eu x^Q lim 1 g(u)u2\ s(«)u2 _1 From the result follows. lim ukg(u) n^w eu □ Some properties of probability mass function of binomial distribution Let X be a random variable having a binomial distribution with parameters p e (0,1) and n e N. For q =1 — p and i e Z, we have by definition Pr[X = i] = Bn (P) = f (nyqn-i if 0 < i < else. 2 k u u e e e u e 0 n 396 Ars Math. Contemp. 10 (2016) 183-192 In this subsection, we state and mathematically ground some properties that can be seen from a graph of binomial distribution (see Fig. 1). The results will be nice, some of them folklore, but the proofs will be technical. 0.10 0.08 0.06 0.04 0.02 0.00 0 50 100 150 200 250 300 Figure 1: Binomial distribution with n = 300 andp = 0.2 (red), p = 0.5 (green), p = 0.7 (blue). The graphs show Bln(p) with respect to i. It is well known (see some basic probability book) that the expected value of X is E(X) = pn. First, we will prove that also the "peak" of the probability mass function is roughly at pn. Lemma 2.2. For p e (0,1), n e N and for 0 < i < n, Bn (p) > Bn-1(p) ^ i < (n +1)p. Proof. The expression Bn (p) = (n - i + 1)p Bn-1(p) i(1 - p) is at least 1 iff i < p(n + 1). □ Next, we state a Chernoff bound proven in [3, inequalities (6) and (7)], which explains why the probability mass function for binomial distribution "disappears" (see Fig. 1), when i is far enough from pn. Theorem 2.3. Let X be a binomially distributed random variable with parameters p e (0,1) and n e N. Then for each 6 e (0,1), Pr [|X - np| > np6] < 2e-,52np/3. We will only use the following corollary of the theorem. It is not difficult to prove and the proof is omitted. Corollary 2.4. For p e (0,1), let a : N ^ R+ be some function such that a(n) < p^n for all n. Then, for all n e N, it holds E Bn(p) < 2e-a2(n)/(3P). i: \i-np\">^fna.(n) B3 oo(0.2) B3 oo(0.5) B3 oo(0.7) D. Gajser: On the convergence of binomial means . 397 This corollary also tells us that, for large n, roughly everything is gathered in an O( -,/n) neighborhood of np. What is more, the next lemma implies that in o(v/n) neighborhood of np, Bn (p) does not change a lot. Lemma 2.5. Let p G (0,1) be a parameter and let ß(n) : N ^ R be a function such that |ß(n)| = O(y/n) and lim |ß(n)| = to. Then, for all large enough n, it holds n^w BjinpJ(p) 1 LßWJ2 Proof. For all large enough n for which 0(n) > 0, we have Snnpj-Lß(n)j(p) Wj-W)j) (1 - p)Lß(n)J = LßTT 1 (n - Lnp_l + Lß(n)J - i)p (2 1) ¿=o (LnpJ - i)(1 -p) . Lß(n)J-1 / i I M M < TT i+ 1 • IM V p(1 - p) n In the last inequality we used the fact that (n - |npj + (n)j - i)p ^ 1 + 1 _ L0(n)j ([npj — i)(1 — p) _ p(1 — p) n holds for large enough n, which is true because it is equivalent to Lnpj (np - InpJ) + (Iß(n)J - i)p + ¿(1 - p) +--< -Iß(n)J, pn np where • np — |_npj < 1, • (|_0(n)j — i)p + ¿(1 — p) < L0(n)j • max{p, 1 — p}, since i < |_0(n)j and • ^^ = O(1), since 0(n) = O(^). Using the fact that (1 + x) < ex for all x G R, we see that Bn PJ (p) < ^ i 1 + 1 L0(n)j Bnnpj-Lß(n)j(pr ,-=o v p(1 - p) n Lß(n)J-1 n1 Lß(n)J e p(1-p) n i=o 1 Lß(n) J 2 = e p(1—p) n . 398 Ars Math. Contemp. 10 (2016) 183-192 For all large enough n for which fi(n) < 0, we write b(n) = ||_fi(n)J| and we have g,i"pJ(p) _ (Lnpj)(i—p)b(n} B^^^fr) (L„pJ+6(„})pb(n} b(n}-1 , , _ TT (LnPJ + b(n) - i)(l - p) (n — |_npJ — i)p b(n} —1 , , < TT (nP + b(n) — i)(1 — p) < i=o (n(1 — p) — i)p < 1 (n — Ln(1 — p)j +b(n) — i)(1 — p) i=0 (|n(1 — p)J — i)p which is the same as (2.1) in the case fi (n) > 0, only that p and (1 — p) are interchanged. □ Now we know that the values of Bn (p) around the peaks in Fig. 1 are close to the value of the peak. The next lemma will tell us that the peak of Bn (p) is asymptotically %/2nP(1-p}n Lemma 2.6. For 0 < p < 1, it holds lim v/2np(1 — p)nBnnpJ(p) _ 1. n—^^o Proof. Using Stirling's approximation n ! lim n—TO V2nn (n)n we see that lim v/2np(1 — p)nBnnpJ(p) V2np(1 — p)n ■ n)"pLnpJ(1 — p)n-Lnp lim e n^oo lim LnpJ ^-;-rr- y2ninpJ (inp1) np ■ V2n(n — LnpJ) ( nnpLnpJ (1 — p)n-L"pJ n—TO |_npJ L"pj ■ (n — LnpJ)n-L"PJ np I / \ n — np np n — np lim | npJ n — | np . np — |_npJ \ LnpJ / np — |_npJ ^ n Lnp n—TO \ [npJ y y n — |_npJ _ lim e"P-L"PJ . e-(nP-L"pJ} _ 1 n—TO where the last line follows by Lemma 2.1. □ 1 D. Gajser: On the convergence of binomial means . 399 3 Comparing convergence of {an}n>0 with convergence of {aPn}n>o and R}n>o In this section we show that the convergence of {an}n>0 implies the convergence of {<}„>0 and {<}„>o to the same limit. It is well known [4] that if {a„}„>0 converges to a € R U {to}, then so does {an}n>0. The next theorem tells us that in this case, {an}n>0 also converges to the same limit. Theorem 3.1. If {an}n>0 converges to a € R U {to}, then {an}n>0 and {an}n>0 converge to a for all 0 < p < 1. Proof. The case a = to is straightforward to handle, so suppose a < to. Take any e > 0 and such N that |an — a| < e for all n > N. Then, for n > N, k- a| 1 < n + 1 1 Ek - a) ¿=0 n E - al n +1 ¿=0 1 A, . 1 < —T E |ai - a| + ~TT • e(n - N). n + 1 f—n +1 ¿=0 The last line converges to e when n goes to infinity, which implies that {an}n>0 converges to a. To prove the convergence of binomial means, denote q = 1 — p. For n > N, we get K - al E (n)piqn-i(«i - «) ¿=0 ^1 ' < E (nV«n~v - ai+e è ¿=0 ^ ' ¿=N+ N 0 also converges to a. □ One does not need to go searching for strange examples to see that convergence of {a"}n>0 or {a"}n>0 does not imply the convergence of {an}n>0. We state this as a proposition. Proposition 3.2. There exists a sequence {an}n>0 of zeros and ones that does not converge, whereas {a"}n>0 and {a"}n>0 converge for all 0 < p < 1. Proof. Define 0 if n is odd, an = 1 if n is even. 400 Ars Math. Contemp. 10 (2016) 183-192 Then {an}n>0 does not converge and {an}n>0 converges to 1, as can easily be verified. Next, we will prove that {an}n>0 converges to 1. First, we see that, for 0 < p < 1 and q = 1 - p, the value of q - p is strictly between -1 and 1, thus (q - p)n converges to 0 when n goes to infinity. Hence, E inVqn-i - E inVq"-i = (q - p)n converges to 0. Because i is odd E Op' we have that {an }n>0 converges to 2. i is odd + E I )p*qn-i = □ 4 Comparing convergence of binomial means In this section we compare convergence of sequences {an}n>0 for different parameters p € (0,1). We will see that if 0 < pi < p2 < 1, then the convergence of {a^2 }n>0 implies the convergence of {an }n>0 to the same limit, while the convergence of {a^1 }n>0 does not imply the convergence of {an }n>0 in general. We leave as an open problem whether for an > 0 it does. First, let us prove the main lemma in this section, which tells us that the sequence of p2-binomial means of the sequence of p1-binomial means of some sequence is the sequence of (pip2)-binomial means of the starting sequence. Lemma 4.1. For 0 0, let {bn}n>0 be the sequence ofpi-binomialmeansof {an}n>0, i.e., bn = an1 for all n. Then bn = aniP2 for all n. Proof Denote qi = 1 - pi and q2 = 1 - p2. We change the order of summation, consider (j)(nt = enten-it for i < j and replace j by k = j - i: bn2 = E' ,p i j n-j p2 q2 j ai 7 = 0 ¿=0 E< i=0 n PiP2 i j -i j n-j Piqi p2?2 n En j=i j - « j—i j—i n—j qi p2 q2 j i=0 Ea^" )PiP2 Er-1 ) (qiP2 )k qn-i-k k=0 ^ = E M i^)pip2(qip2 + q2)n ¿=0 = E aM J (pip2)i(1 - pip2)n ¿=0 V«/ The last line equals aniP2. □ n j j n j n — i D. Gajser: On the convergence of binomial means . 401 The next theorem will now be trivial to prove. Theorem 4.2. For 0 < p1 < p2 < 1 and for a sequence {an}n>0, if {an,2 }n>o converges to a G R U {to}, then {an1 }n>0 also converges to a. Proof. From Lemma 4.1 we know that {an }n>0 is the sequence of ^-binomial means of the sequence {an }n>0. By Theorem 3.1, it converges to a. □ The next proposition tells us that the condition 0 < p1 < p2 < 1 in the above theorem cannot be left out in general. Proposition 4.3. For 0 < p1 < p2 < 1, there exists a sequence {an}n>0, such that {a^1 }n>0 converges to 0, but {an,2 }n>0 does not converge. Proof. Denote q1 = 1 - p1 and define {an}n>0 as an = an for some parameter a G R. If a > -1, {an}n>0 converges (possibly to to), so let us examine the case when a < -1. In this case we have an1 = £ f^aVlqr' = (api + qi)n = (pi(a - 1) + 1)n, i=0 ^% ' which converges iff p1 < j^a. So we can choose such an a that p1 < t^ < P2, i.e., 1 - < a < 1 - .It follows that {an,1 }n>0 converges to 0, but {an }n>0 does not converge. □ The sequence {an}n>0 in the above proof is growing very rapidly in absolute value and the sign of its elements alternates. We think that this is not a coincidence and we state the following open problem. Open problem 4.4. Let {an}n>0 be a sequence of non-negative real numbers. Is it true that, for all 0 < p1?p2 < 1, the sequence {an1 }n>0 converges to a iff {an }n>0 converges to a? If the answer is no, is there a counterexample where an G {0,1}? Note that the condition an > 0 is also required for the main result of the paper, Theorem 5.1. If the answer on 4.4 were yes, then we would only have to prove Theorem 5.1 in a special case, e.g. for p = 1. The (possibly negative) answer would also make this paper more complete (see Table 1). In the rest of this section we will try to give some insight into this problem and we will present some reasons for why we think it is hard. Suppose we have 0 < p1 < p2 < 1 and a sequence {an}n>0 of non-negative real numbers such that {an }n>0 converges to a G R (the case when {an }n>0 converges is covered by Theorem 4.2). The next lemma implies that {an}n>0 has a relatively low upper bound on how fast its elements can increase, ruling out too large local extremes. Lemma 4.5. Let {an}n>0 be a sequence of non-negative real numbers and let 0 < p < 1. If {an }n>0 converges to a < to, then an = O^^/n). Proof. We know that an > a i npi (p), where(p) « , 1 „ by Lemma 2.6 L j i/2np(1-p)n and an « a for large n. Hence, a|npj = O(^n). □ 402 Ars Math. Contemp. 10 (2016) 183-192 To see whether {af2 }„>0 converges, it makes sense to compare apf/pi j with apf/p2j, since the peaks of the "weights" B[f/pi j (pi) and B*n/P2j (p2) (roughly) coincide at n (see Fig 2). Now the troublesome thing is that, for large n, the peaks are not of the same height, but rather they differ by a factor 1 - P2 1 - pi by Lemma 2.6. Because the weights B[f/pi j(p1) and B*n/P2j(p2) are (really) influential only in the O(v/n) neighborhood of n (Corollary 2.4 and Lemma 2.5), where the p1-weights are only a bit "downtrodden" p2-weights, it seems that the convergence of {af1 }n>0 could imply the convergence of {af2 }„>0. Figure 2: The graphs show B[f/pij (p1) and j (p2) with respect to i in the neighborhood of n for n = 300, p1 = 0.4 (red) andp2 = 0.7 (green). On the other hand, one could take an = 0 for all except for some n where there would be outliers of heights ©( Jn). Those outliers would be so far away from each other that the weights Bf(p1) could "notice" two consecutive outliers, while the weights Bf(p2), which are slimmer, could not (in Fig. 2, the two outliers could be at 280 and 320). Then {af1 }n>0 could converge because there would be a small difference between [when the weights Bf(p1) amplify one outlier] and [when they "notice" two outliers] (these two events seem to be the most opposite). On the other hand, {af2 }n>0 would not converge. From Chernoff bound (Corollary 2.4) and from Lemma 2.5 it follows that the (horizontal) distance between outliers should be roughly CJn for some C. What C would be the most appropriate? 5 Comparing convergence of {apn}n>o with convergence of {a*n}n>o This section contains the main result of this paper, which is formulated in the next theorem. The proof will be given later. Theorem 5.1. Let {a„}„>0 be a sequence of non-negative real numbers such that {af }f>0 converges to a G R U {to} for some 0 < p < 1. Then {af}„>0 converges to a. An example of how this theorem can be used is given in Section 6.1. Here we give an example where {aj^^o converges to a G R U {to} for all 0 < p < 1, but {af}„>0 does not converge. D. Gajser: On the convergence of binomial means . 403 Proposition 5.2. For the sequence {an}n>0 given by an = ( — 1)nn, {an }n>0 converges to 0for all 0 < p < 1 and {an}n>0 does not converge. Proof. Take 0 < p < 1 and denote q =1 — p. It holds = £ (—1)ii(n )pV-i = —np £ ( —1)i-1( i — J) pi-1qn-1-(i-1) = —np(—p + q)n 1. Because q — p is strictly between —1 and 1, {an }n>0 converges to 0. However, the induction shows that a|n+1 = — 1 and a|n = ^n+r, which implies that {an}n>0 does not converge. □ Next, we show that we cannot interchange {an}n>0 and {an}n>0 in Theorem 5.1. Proposition 5.3. There exists a sequence {an}n>0 of zeros and ones such that {an}n>0 converges to 0 and {an }n>0 diverges for all 0 < p < 1. Proof. Define 1 if there is some k G N such that In — 22k | < 2kk an 0 else. So {on|„>o has islets of ones in the sea of zeros. The size of an islet at position N is ©(%/N log(N)) and the distance between two islets near position N is ©(N). It is easy to see that the sequence a*n converges to zero. Now let 0 < p < 1. By Chernoff bound (Corollary 2.4) we see that Bln (p) is concentrated around i = |_-pj and that, for |i - np| > V—log(n), we have roughly nothing left. It is easy (but tedious) to show formally that |ap22fc/pj j ^ converges to 1 and that |ap22fc-i/p^ > converges to 0, which implies that {an}n>0 diverges. □ Now we go for the proof of Theorem 5.1. First, for a sequence {an}n>0 and 0

0 as a sequence of arithmetic means of the sequence {an}n>0. We get 1n an = £ ap n+1 j=oj -i ^ j / • > 1 - ^ \ .-iqj-i n j n +1 EE ¿)pq' j=0i=0 v 1 n n nrr E aE i=0 j=i v ' where q = 1 — p. p an 404 Ars Math. Contemp. 10 (2016) 183-192 It makes sense to define weights wln(p) = J2 j=i (j)piqj-i, so that it holds 1 n < = nrr^ wn (p)ai. i=0 Figure 3: The graph shows w\00(0.3) with respect to i. We see a steep slope at i = 90 plunging from height approximately 03 to 0. We can see in Fig. 3 that the weights wn (p) have a very specific shape. They are very close to P for i < np — e(n) and very close to 0 for i > np + e(n), for some small e(n). Such a shape can be well described using the next lemma (and its corollary), which gives another way to compute wni ( p) . 50 100 150 200 250 300 Lemma 5.4. For 0

2 1 else. Now the following corollary holds. Corollary 5.5. For 0 1 — n- 0(l°g(n)), p wLnp+^(n)J (p) < n- 0(l°g(n)). Proof. Use the Chernoff bound (Corollary 2.4) on the expression for wn (p) from Lemma 5.4. □ For 0

0, define sequences {an(p)}n>o, ian(p)} }n>0 j x=q p j 406 Ars Math. Contemp. 10 (2016) 183-192 and {<(p)}n>o as Lpn-e(n)J (p) = wn (pH i=0 Lpn+e(n)J-1 «n(p) = wn (p)ai i= [pn-e(n)J + 1 < < (p) = wn (pK = Lpn+e(n)J Hence, we have n + 1 («n(p)+«n(p) + anw). From Corollary 5.5 we see that the weights in an(p) are very close to ?, which suggests that n+i«n (p) can be very close to a*npj (see Lemma 5.8 below). From the same corollary we see that n+r«n(p) can be very close to 0 (see Lemma 5.7 below). And because we have a sum of only ©(e(n)) elements in an(p), n+r«n(p) could also be very close to 0 (see Lemma 5.6 below). We have just described the main idea for the proof of the main theorem, which we give next. It will use three lemmas just mentioned (one about «n(p), one about «n (p) and one about «n(p)), that will be proven later. Proof of Theorem 5.1. Suppose that an > 0 for all n and suppose that {an}n>0 converges to a € R U {to} for some 0 < p < 1. We know that this implies the convergence of {<}n>0 to a (Theorem 3.1). First, we deal with the case a = to. We can use the fact that wn(p) < J for all i (see Lemma 5.4), which gives an 1n wn (p)a n + 1 • 0 i=0 1 ^ 1 < - y —aj n + 1 f—' p i=0 p Hence, {an}n>0 converges to a = to. In the case a < to, we can use Lemma 5.6 and Lemma 5.7 to see that and | n+ran(p)} ^^ converge to 0. Hence, j n+ran(p) j ^ converges to a. Lemma 5.8 tells us that in this case {an}n>0 also converges to a. □ Now we state and prove the remaining lemmas. Lemma 5.6. Let 0 < p < 1 and let {an}n>0 be a sequence of non-negative real numbers such that {an}n>0 converges to a < to. Then j n+raK(p)} > converges to 0. 1 a?* = n a 1 y a n+rn^' n0 D. Gajser: On the convergence of binomial means . 407 Proof. Fix e > 0, define 6(n) = |_log2(n)J and let k : N ^ N be such that pn — e(n) < k(n) < pn + e(n) — 6(n) holds for all n. We claim that fc(n) + 5(n) E ai = 0(v«), i=k(n) where the constant behind the O is independent of k. To prove this, define N = N(n) = fc(n) P . It follows that N = n ± ©(e(n)). Note that, for large enough n, fc(n)+5(n) N E «¿BNr(p) < E «¿BNr(p) < a + ?, i=fc(n) i=0 because {an}n>0 converges to a. From Lemma 2.5 which bounds the coefficients BN (p) around i = pN it follows that, for all k(n) < i < k(n) + J(n), BN(p) > e-o(1)BNNpJ(p). Using N = n ± ©(e(n)) and the bound bNNpJ(p) 1 ©(%/N ) from Lemma 2.6, we get fc(n) + 5(n) E ai < (a + ?)eo(1) ©("n) = O("n). i=k(n) Next, we can see that Lpn+e(n)J-1 , E ai = O^ n \ log n ) i= [pn-e(n)J + 1 Just partition the sum on the left-hand side into ^(y sums of at most 6(n) elements. Then we have Lpn+e(n)J-1 / e(n) ^ \ „ I n ai = " ' i I <5(n) I log n y Lpn-e(n)j + 1 V V ' 7 V 6 7 Now using wn (p) < p from Lemma 5.4, we get p Lpn+e(n)J-1 -, -, LPn+c(n)J 1 -, / an(p) < \ V ai = \ O ( — n +1 n(p) " (n +1)p ^ i (n + 1)p ^logn i= [pn-e(n)J + 1 which implies the convergence of j n+1 an (p) } > to 0. □ 408 Ars Math. Contemp. 10 (2016) 183-192 Lemma 5.7. Let 0 < p < 1 and let {an}n>0 be a sequence of non-negative real numbers such that {<}„>0 converges to a < <. Then j n+rja^(p) j > converges to 0. Proof. From Lemma 5.4 we see that the weights wln(p) decrease with i, so *(P) " (n +1) ai. n +1 " (n +1) , Lpn+e(n)J Corollary 5.5 gives us w„ (p) — n 0(l°s(n)), while Lemma 4.5 implies a = O(%/I). Hence, j n+r «n(p) f converges to 0. □ ^ + J n>0 Lemma 5.8. Let 0 < p < 1 and let {an}n>0 be a sequence of non-negative real numbers such that \ n+r«n(p) f converges to a < to. Then {an}n>0 converges to a. ^ + J n>0 Proof. Because the weights wln(p) are bounded from above by p (Lemma 5.4), we have «n(p) (n+i)p < * n +1 ^ [pn - e(n)J + 1 - Lpn-e(n)J' where the left side converges to a. Because the weights wn (p) decrease with i (Lemma 5.4) and because wnnp-e(n)^ (p) > ' - n- 0(l°g(n)) (Corollary 5.5), we have p «X (p) n +1 ^ nv _ 1 Lpn—e(n)J - n + 1 ^ (P - n- 0(l°g(n))) • ([pn - e(n)J + 1)' where the right side converges to a. Hence, a*pn_e(n)j is sandwiched between two sequences that converge to a. It follows that {an}n>0 converges to a. □ 6 Application of Theorem 5.1: a limit theorem for finite Markov chains For a stochastic matrix1 P, define the sequence {Pn}n>0 as Pn = Pn. As in the one-dimensional case, we define the sequence {P,t}n>0 as P^ = n+rj J2"=0 Pn. We say that {Pn}n>0 converges to A if, for all possible pairs (i, j), the sequence of (i, j)-th elements of Pn converges to (i, j)-th element of A. In this section, we will prove the following theorem. Theorem 6.1. For any finite stochastic matrix P, the sequence {P^ }n>0 converges to some stochastic matrix A, such that AP = PA = A. This theorem is nothing new in the theory of Markov chains. Actually, it also holds for (countably) infinite transition matrices P. Although we did not find it formulated this way in literature, it can be easily deduced from the known results. The hardest thing to show 1A stochastic matrix is a (possibly infinite) square matrix that has non-negative real entries and for which all rows sum to 1. Each stochastic matrix represents transition probabilities of some discrete Markov chain. No prior knowledge of Markov chains is needed for this paper. D. Gajser: On the convergence of binomial means . 409 is the convergence of {Pn}n>0 [2, page 32]. After we have it, we can continue as in the proof of Theorem 6.1 below. We will give a short proof of Theorem 6.1, using only linear algebra and Theorem 5.1. First, we prove a result from linear algebra. Lemma 6.2. Let P be a finite stochastic matrix and let P = 1 ^P + /J. Then a) for all eigenvalues A of P, it holds |A| < 1, b) for all eigenvalues A of P for which |A| = 1, it holds A = 1, c) the algebraic and geometric multiplicity of eigenvalue 1 of P are the same. Proof. Since the product and convex combination of stochastic matrices is a stochastic matrix, P" and P" are stochastic matrices for each n G N. First, we will prove by contradiction that, for all eigenvalues A for P, it holds |A| < 1. Suppose that there is some eigenvalue A for P such that |A| > 1. Let w be the corresponding eigenvector and let its ¿-th component be non-zero. Then |(P"w)j | = | An | • | w^ |, where the right side converges to œ and the left side is bounded by maxj |wj | (since P" is a stochastic matrix). This gives a contradiction. Hence, for all eigenvalues A for P, it holds |A| < 1. Because P is also stochastic, the same holds for P. We see that we can get all eigenvalues of P by adding 1 and dividing by 2 the eigenvalues of P. Because P has all eigenvalues in the unit disc around 0, P has all eigenvalues in a disc centered in 1 of radius 2. Hence, for all eigenvalues A of P, for which |A| = 1, it holds A = 1. 2 2 For the last claim of the lemma, suppose that the algebraic and geometric multiplicity of eigenvalue 1 of P are not the same. Then, by Jordan decomposition, there is an eigenvector v for eigenvalue 1 and a vector w, such that Pw = v + w. Then, for each n G N, we have P"w = nv + w. Because v has at least one non-zero component and because all components of P"w are bounded in absolute value by maxj |wj |, we have come to contradiction. Hence, the algebraic and geometric multiplicity of eigenvalue 1 of P are the same. □ Proof of Theorem 6.1. For the matrix P = 1 ^P + /), let P = XJX-1 be its Jordan decomposition. From Lemma 6.2 a) and b) it follows that the diagonal of J consists only of ones and entries of absolute value strictly less than one. From Lemma 6.2 c) it follows that the Jordan blocks for eigenvalue 1 are all 1 x 1. It follows that J" converges to some matrix J0 with only zero entries and some ones on the diagonal. Hence, P" converges to A = X J0X-1. Since P" is a stochastic matrix for all n, the same is true for A. Using P" = P", we see that {P"}">0 is just a sequence of 0.5-binomial means of the sequence {P"}">0, hence by Theorem 5.1 {P"}">0 also converges to A. Thus, we have AP = ( lim —1—— V PM P \ "^œ n +1 / V i=o / = lim 212 (-L- vP• -_L-/) "^œ n +1\ n + 2^ n + 2 I = A. The same argument shows also PA = A. □ 410 Ars Math. Contemp. 10 (2016) 183-192 An application of Theorem 6.1 in formal language theory. The following application was suggested by an anonymous reviewer. To each formal language L C E* where E is a finite alphabet, we can assign the sequence _ |En n Ll Jn (L) _ |E„| of relative frequencies of words of length n in L. If this sequence is convergent, then its limit can be taken as a measure for the size of L, which provides interesting information about L. Unfortunately, the sequence fn(L) can be divergent even if L is a regular language, such as, for example, the language E of all words of even length. But using Theorem 6.1 we can show that f* (L) converges for every regular L as follows. If L is regular, it is recognised by some deterministic finite automaton (Q, q0, F, E, S) where Q _ {qo, qi,..., qm-i} is the set of states, q0 e Q is the starting state, F C Q is the set of final states, and S : Q x E ^ Q is the transition function. Define the matrix T e Qmxm with elements _ |{a e E; S(qra) = qj }|, _o, 1 m - 1. |e| ' >■> ' ' ' Then T is stochastic and Jn(L) _ J2qjeF(Tn)0 ,j, so by Theorem 6.1, f*(L) is convergent and we can define ^(L) _ f*(L) to be the (finitely additive) measure of L. For example, returning to the language E of words of even length, we find that ^(E) _ 0.5. Acknowledgements. The author wishes to thank the anonymous reviewer for a very constructive review, and Sergio Cabello, MatjaZ Konvalinka and Marko Petkovsek for valuable comments and suggestions. All figures were done with Mathematica 8.0.1.0. References [1] J. Boos and P. Cass, Classical and Modern Methods in Summability, Oxford University Press, 2000. [2] K. L. Chung, Markov Chains with Stationary Transition Probabilities, Springer, 1960. [3] T. Hagerup and C. Rub, A guided tour of Chernoff bounds, Inf. Process. Lett. 33 (1990), 305-308, doi:10.1016/0020-0190(90)90214-I, http://dblp.uni-trier.de/ db/journals/ipl/ipl33.html\#HagerupR90. [4] G. H. Hardy, Divergent Series, Clarendon Press, 1st edition, 1956. [5] M. Z. Spivey and L. L. Steil, The fc-binomial transforms and the Hankel transform, J. Integer Seq. 9 (2006). ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 411-425 An infinite class of movable 5-configurations Leah Wrenn Berman * Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, Alaska, USA Elliott Jacksch Seattle, Washington, USA Lander Ver Hoef NOAA Corps of Commissioned Officers, Ketchikan, Alaska, USA Received 11 September 2015, accepted 3 March 2016, published online 13 April 2016 A geometric 5-configuration is a collection of points and straight lines, typically in the Euclidean plane, in which every point has 5 lines passing through it and every line has 5 points lying on it; that is, it is an (n5) configuration for some number n of points and lines. Using reduced Levi graphs and two elementary geometric lemmas, we develop a construction that produces infinitely many new 5-configurations which are movable; in particular, we produce infinitely many 5-configurations with one continuous degree of freedom, and we produce 5-configurations with k - 2 continuous degrees of freedom for all odd k > 2. Keywords: Configurations, incidence geometry. Math. Subj. Class.: 51A20, 51A45, 51E30, 05B30 A geometric k-configuration is a collection of points and straight lines, typically in the Euclidean plane, where every point lies on k lines and every line passes through k points. Geometric 3-configurations have been studied since the mid-1800s, and geometric 4-con-figurations since the late 1900s, with the first intelligible drawing of a 4-configuration appearing in a 1990 paper by Griinbaum and Rigby [15]. However, the situation for more highly incident configurations, that is, for (pq, nk ) configurations with at least one of q, k > 4, is poorly understood in general. Two constructions that produce infinite families of 5-configurations with a reasonably small number of points and lines are known [7, 9]. The (485) configuration shown in Figure 1a * Research supported by a grant from the Simons Foundation (#209161 to L. Berman) E-mail addresses: lwberman@alaska.edu (Leah Wrenn Berman), ecjacksch@alaska.edu (Elliott Jacksch), lander.verhoef@gmail.com (Lander Ver Hoef) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 412 ArsMath. Contemp. 10(2016)411-425 is the smallest known geometric 5-configuration and is the smallest example of the construction in [9]; a reasonably small example of the construction discussed in [7] is shown in Figure 1b (the smallest example is not intelligible at small scale). In his monograph on configurations [14, Section 4.1], Griinbaum spends only 5 pages (mostly pictures) discussing the little that is known about 5-configurations. (a) A (485) configuration with 4 symmetry classes of (b) A (645) configuration with 8 symmetry classes of points and lines points and lines Figure 1: Examples of known small 5-configurations In this paper, we present a new construction that produces infinitely many new geometric 5-configurations which are movable: that is, there is at least one continuous degree of freedom in the construction while fixing 4 points in general position. This construction significantly generalizes the construction presented in [9] and removes the need to complete the construction via a continuity argument, instead providing an entirely ruler-and-compass construction for those configurations, given an initial m-gon. The new construction technique uses two elementary geometric lemmas, the Circumcircle Construction Lemma and the Crossing Spans Lemma, which previously have been used separately in other configuration construction techniques. 1 Definitions; Levi and reduced Levi graphs Given any (pq, nk) configuration, whether geometrically realizable or not, it is possible to construct a corresponding bipartite graph, called a Levi graph, which has one white vertex for each point of the configuration and one black vertex for each line of the configuration, with two vertices in the graph incident if and only if the corresponding point and line are incident in the configuration. More details on Levi graphs and configurations may be found in Grunbaum [14, Section 1.4] and Coxeter [12]. We say that a geometric k-configuration is symmetric if there exist non-trivial isometries L. W. Berman, E. Jacksch and L. Ver Hoef: An infinite class of movable 5-configurations 413 of the Euclidean plane that map the configuration to itself. Note that in other places in the literature, the word 'symmetric' has been used to mean (pq, nk) configurations where q = k (and thus p = n), i.e., k-configurations. Since we are interested in emphasizing the geometric nature of the configuration, we—following Griinbaum [14, p. 16]—refer to k-configurations as balanced, and reserve the word 'symmetric' to refer to the geometric structure. The symmetry class of an element (point or line) is the orbit of the element under the symmetry group of the configuration. If a geometric configuration has the property that every symmetry class under some fixed cyclic subgroup of the geometric symmetry group contains the same number of elements, then the configuration is called polycyclic; polycyclic configurations were first described by Boben and Pisanski [11]. Given a polycyclic geometric configuration with cyclic symmetry group Zm, it is possible to construct an edge-labelled bipartite graph, called the reduced Levi graph, by associating one vertex of the graph to each symmetry class of points and of lines in the configuration, and connecting two vertices of the graph with an edge precisely when the corresponding elements of the configuration are incident. Suppose the elements of each symmetry class of elements are labelled cyclically counterclockwise, beginning from some chosen 0th element in each class; for example, line class L is labelled (L)0,..., (L)m-1 and vertex class v is labelled (v)0,..., (v)m-1. If for each i, line Lj and vertex vj+a are incident (with indices computed modulo m), the corresponding directed edge in the reduced Levi graph from vertex L to vertex v is labelled a; in the case where Lj and vertex vj are incident (that is, where a = 0), then we use an undirected thick edge. When vertices vj and vj+a both lie on line Lj, or from an alternate point of view, when lines Lj and Lj-a intersect at point vj, then the reduced Levi graph contains a double arc If p and q are any two points, we denote the line L passing through p and q as p V q. Similarly, if L and M are any two lines, we denote their point of intersection as L A M (possibly at infinity if L||M). Given points v0,..., vm-1 that form the vertices of a regular m-gon centered at O, we say that a line is span b if it passes through vj and vj+b for some i, with all indices computed modulo m; span b lines correspond to double arcs in the reduced Levi graph. A circle C is a circumcircle of span b if it passes through vj; O, and vj-b for some i; to specify which i, we say that C is a circumcircle of span b through vd. (Note that span b lines are constructed by moving counterclockwise from the initial point, and span b circumcircles by moving clockwise!) 2 Two construction lemmas In 2006, one of the authors (LWB) discovered the Crossing Spans Lemma [3] (somewhat restated here): Lemma 2.1 (Crossing Spans Lemma (CSL)). Given a regular m-gon with vertices cyclically labelled as u0, u1,..., um-1 and lines Lj = uj V uj+a of span a and Mj = uj V uj+b of span b, where 1 < a = b < m, suppose that v0 is an arbitrary point on M0 (different from u0, ub to avoid degeneracies), and construct other points vj to be the rotations of v0 through . Let Nj = vj V vj+a and let wj = Nj A Nj-b. Then wj also lies on Lj. Although easy to state and prove, the Crossing Spans Lemma has been used to produce a 414 Ars Math. Contemp. 10(2016)411-425 -3--^ /--N --2- N Z7 (a) Illustrating the Crossing Spans Lemma; m = (b) The reduced Levi graph corresponding to Figure 2; 7, a = 2,b = 3. Only point w0 in class w has been the dashed edge corresponds to the forced incidence. shown, to better illustrate that the three lines Li, Ni-b, and Ni really do intersect three at a time (that is, no almost-incidences are covered by points). Figure 2: Illustrating the Crossing Spans Lemma 2 3 number of novel constructions for configurations [3, 5, 8, 9]. The Crossing Spans Lemma and its associated reduced Levi graph "gadget" are shown in Figure 2. In fact, it is straightforward to show (by relabelling symmetry classes and applying duality arguments) that given either of the labelled subgraphs in a reduced Levi graph that are shown in Figure 3, the incidence given by the dashed line is induced, where white nodes correspond to point classes and gray nodes to line classes. These subgraphs, with various choices of labels, are used extensively in the proof of Theorem 4.1. b + y y c x— a + xx- c Figure 3: In either of these subgraphs in a reduced Levi graph (over Zm), the dashed line corresponds to a forced incidence via the CSL; ,c,x,y are integers between 0 and m - 1, and 1 < a = b < m. Gray vertices correspond to line classes and white vertices to point classes. In the construction in Section 4, we typically take c = 0, x = 0, and y = 0 or S. In [14, p. 116-118], Branko Grunbaum described a geometric technique to constructing a certain class of 3-configurations. This technique was extended in [7] to the Circumcir-cle Construction Lemma. Although the lemma can be stated as a more general incidence theorem [8], we state it as follows in order to facilitate the main construction in Section 4. L. W. Berman, E. Jacksch and L. Ver Hoef: An infinite class of movable 5-configurations 415 Lemma 2.2 (Circumcircle Construction Lemma (CCL)). Let v0,v1,..., vm-1 and w0, w1,..., wm-1 form the vertices, labelled cyclically counterclockwise, of two regular convex m-gons centered at O. The point w0 lies on the circle passing through vd, vd-b, O if and only if the points w0, wb, vd are collinear. That is, if w0 lies on the circumcircle of span b through vd, then the line L0 of span b through w0 passes through vd, and conversely. By symmetry, the line L-d will also pass through the point w0, and in general, if w0 is defined to also lie on some other line M0, then each rotated image wj will lie on the three lines Li, Li-d and Mi. The Circumcircle Construction Lemma, along with its reduced Levi graph structure, is illustrated in Figure 4. © 3 3'' ^^ g Z7 (a) Illustrating the Circumcircle Construction Lemma; m = 7, b = 2, d = 3. The green line is L0, and the dashed gray line is a possible other line M0 passing through w0 (i.e., w0 could be defined as the intersection of Mo and C); other elements of line classes L and M have been suppressed for clarity. (b) The "gadget" in a reduced Levi graph corresponding to Figure 4a. (The connection between w and M is optional, depending on whether there happens to be a line M0 passing through w0; this is the typical situation in applications of the CCL.) Figure 4: The Circumcircle Construction Lemma. 3 Celestial 4-configurations The building blocks for the new construction of 5-configurations presented in Section 4 are the celestial 4-configurations, which are configurations that have the property that every point has two lines from each of two symmetry classes of lines passing through it, and every line has two points from each of two symmetry classes of points lying on it. An example of such a configuration is shown in Figure 5, along with a general reduced Levi graph. Celestial 4-configurations were first described in detail (aside from a handful of examples, e.g., [15, 16]) in Boben and Pisanski's article Polycyclic Configurations [11], as the main class of 4-configurations analyzed in that paper. Their description was expanded in Griinbaum's monograph Configurations of Points and Lines [14, Sections 3.5-3.8], although in that chapter, he unfortunately called them k-astral configurations (even though as he defined previously [14, p. 34], a k-astral configuration is simply a configuration with k symmetry classes of points and of lines, and there exist k-astral 4-configurations that are not k-celestial [13]). 416 Ars Math. Contemp. 10 (2016) 183-192 Every k-celestial 4-configuration can be described by a celestial symbol m#(si,ti;... ; sk,tfc) that satisfies four axioms: Axiom 1: (order condition) sj = tj = sj+1 (with indices taken modulo m) k Axiom 2: (even condition) (sj — tj) = for some integer ^ j=1 Axiom 3: (cosine condition) IT cos ( ^^ ) = IT cos j=i m _-i j=i Axiom 4: (substring condition) no substring sj, tj;... ; sj, tj or tj; sj+1,..., tj; sj+1 sat- A symbol satisfying the 4 axioms is said to be valid. Although celestial 4-configurations are probably the most well-understood class of 4-configuration, they are still poorly understood in general. The collection of 2-celestial configurations is completely classified ([2], with a clearer proof in [14, p. 210-211]), but general k-celestial configurations are not completely classified, and the problem appears to be non-tractable (since it depends on being able to solve certain trigonometric diophantine equations). However, some known families of valid k-celestial configurations, primarily for k = 3,4, were presented in [1]. Given a valid symbol, there is a corresponding cohort m#S; T, where S = {s^ ...,sk} and T = {t1,... ,tk} (as sets), which corresponds to a collection of valid symbols; in particular, the sets in a cohort must satisfy the even and cosine conditions, and it must be possible to find an ordering of the sj and tj that satisfies the order condition. To construct a k-celestial 4-configuration m#(s1, t1;...; sk,tk) with k point classes v1,..., vk and k line classes L1,..., Lk, do the following: Algorithm 1 (Constructing a celestial 4-configuration). Input: A valid celestial symbol m#(s1, t1;...; sk, tk). 1. Construct the vertices of a regular m-gon centered at O, labelled (v1)0,... (v1)m-1. 2. Let L1 be the collection of lines of span s1 with respect to point class v1: that is, let (L1)j = (v1)j V (v1)i+Sl. 3. Construct point class v2 to be the set of t1-st intersection points of the lines L1: that is, (v2)j = (L1)j A (L1 )j_ti. 4. Continue in this fashion; line class L2 is the set of lines of span s2 with respect to point class v2, point class v3 is the set of t2-nd intersection points of the lines L2, etc., stopping after the construction of line class Lk. Because the symbol m#(s1, t1;...; sk, tk) is valid, the point class vk+1 corresponds, as a set, to point class v1, and in particular, (vk+1)0 = (v1)^, where = ^ik=1(sj —tj). The general reduced Levi graph for the configuration m#(s1, t1;...; sk, tk) is shown in Figure 5b; the "twist" [11], is guaranteed to be an integer by the even condition. In isfies the previous axioms. L. W. Berman, E. Jacksch and L. Ver Hoef: An infinite class of movable 5-configurations 417 general, the underlying graph for every reduced Levi graph of a celestial 4-configuration is a double cycle of even length; that is, an even cycle in which every edge is replaced by a pair of parallel edges. (a) The celestial 4-configuration 9#(4, 3; 2, 3; 1, 3). (b) The reduced Levi graph, a double cycle, for The 0th element of each symmetry class is shown larger a general celestial 4-configuration, where & = (points) or thicker (lines), and elements in different 2 (s; - i;). symmetry classes are distinguished by color (class 1 is red, class 2 is blue, and class 3 is green). 4 Constructing movable 5-configurations The general idea of the construction is to produce a 5-configuration whose reduced Levi graph consists of concentric double cycles, each of which corresponds to a particular celestial 4-configuration, where the double cycles are successively linked by single edges by applying the CSL, and finally, the innermost cycle is linked to the outermost cycle using the CCL; if k > 2 the construction will produce a movable 5-configuration. The reduced Levi graph is shown in Figure 6. More specifically, the reduced Levi graph contains k concentric double cycles, each of which corresponds to a k-celestial 4-configuration with cohort m#S; T where Sn T = 0. If the outermost cycle corresponds to the configuration with symbol then each successive cycle has the sj's permuted cyclically one step while the Vs remain fixed: that is, the second cycle has symbol Figure 5: Celestial 4-configurations m#(si,ti; S2,t2;... ; sfc_i,ifc_i; sk,tk), m#(s2,ti; S3,t2;... ; sk,tk_i; s i,tk), the third has symbol m#(s3,ti; sA,t2;... ; si,tk_i; S2,tk), 418 Ars Math. Contemp. 10(2016)411-425 and so on, so that the innermost cycle has symbol TO#(sfc «1,^2;...; «fc-2,ifc-i; «fc-i,ifc). The point classes of the celestial configuration corresponding to cycle j are labelled vj,... vjj and the line classes Lj,... L^; that is, the superscript indicates the cycle, and the subscript the symmetry class in the celestial configuration. In Figure 6, the first point class of each celestial configuration is highlighted. Given a valid configuration symbol m#(s1, t1;...; sk, tk) with cohort m#S; T with the property that SnT = 0, the geometric construction algorithm to produce a 5-configuration with k—2 continuous degrees of freedom is given in Algorithm 2. If k = 2 the configuration is static and has been described previously in [9]; however, the construction algorithm given here, which uses the CCL to complete the construction, eliminates the need for completing the configuration via a continuity argument as described in that paper. Algorithm 2 (Constructing a 5-configuration). Input: A valid celestial symbol m#(s1, t1;...; sk, tk) with the property that S n T = 0. 1. Construct the first k-celestial 4-configuration with symbol m#(s1,t1;...; sk,tk), with point classes v1,..., v£ and line classes L1,..., L\. 2. If k > 2, for j = 2,... ,k — 1: (a) Place a new point (vj )o arbitrarily on line (L1-1) o, and construct the rest of the points (vj)j in point class v2 by rotating (v1 )0 by for i = 0,... m — 1. (b) Using the point class vj as the starting m-gon, construct the configuration m#(sj,t1; sj+1,t2;...; Sj-2,tk-1; sj-1,tfc) (where the sequence s1, s2,..., sk-1, sk has been cyclically permuted j steps but the sequence t1 , . . . , tk remains fixed). 3. To construct the k-th celestial configuration: (a) Construct a circumcircle C of span sk through (v1)c, choosing c (and varying continuous parameters if possible/necessary) so that C intersects line (L^-1)0. (b) Let ( v\)o be the intersection of C with line (L^ 1)o, and let (vk)i be the rotation of (v£)0 through about O. (c) Construct configuration m#(sk,t1; S1,t2;...; Sfc-2,tfc-1; sfc-1,tfc) using the points (vk )j as the initial set of points. Theorem 4.1. Algorithm 2, beginning with m#(s1, t1;...; sk, tk), creates a valid 5-configuration with mk2 points, mk2 lines and k — 2 continuous degrees of freedom. L. W. Berman, E. Jacksch and L. Ver Hoef: An infinite class of movable 5-configurations 419 Figure 6: The reduced Levi graph, over Zm, for a movable 5-configuration with k2 point classes and k2 line classes. It consists of k concentric double cycles, each corresponding to a particular celestial 4-configuration, with the double cycles linked by arcs. The arcs shown red and dashed are induced by the Crossing Spans Lemma, with example CSL gadgets inducing the dashed edges highlighted in yellow and green, while the structure shown in blue is constructed via the Circumcircle Construction Lemma. 420 Ars Math. Contemp. 10(2016)411-425 Proof. First, note that Algorithm 2 constructs k celestial configurations; each celestial con-fguration contains k symmetry classes of points and of lines, and each symmetry class contains m elements, for a total of mk2 points and mk2 lines. Second, for j = 2,..., k - 1, the point (vj)o is placed arbitrarily on line (L1- 1) o , for (k - 1) - 2+1 = k - 2 continuous degrees of freedom. Thus, the nontrivial part of the proof is to show that every point lies on 5 lines, and every line passes through 5 points. Recall that the symbol for celestial configuration j is m#(sj,t i; sj+i,t2;...; sj+^i*;...; sj- 1 ). By construction, for each j = 1,..., k - 1, each line (Lj)j passes through the point (vj+1)i (that is, the first symmetry class of points in celestial configuration j + 1 lies on the first symmetry class of lines in celestial configuration j), as well as through points (vj)j, (vj )j+sj, (vj )j, and (vj )j+tl from celestial configuration j. By careful choice of labels and the Crossing Spans Lemma, it follows that for all I = 2,..., k - 1 (with I indexing the symmetry classes in the celestial configuration j), each line (Lj )j passes through point (vj+^^as well as through points (vj )j, (vj )i+Sj+£, (v|+1)j and (vj+1)i+ti from celestial configuration j. A CSL gadget showing that points v| are incident with lines L1, (dashed red line) beginning with the input that points v12 are constructed incident with lines L12 (solid black line) is highlighted in Figure 6 in yellow. Finally, again by the CSL, line (L{)j passes through point (vj+1)i, as well as through points (vj)j, (vj,)j+Sj_1, ( vj )j+ 4 and 0 < p, r < q is a valid family of celestial 4-configuration cohorts. Suppose that r = p, r = §, p = § and p + r = q. Under these conditions, the sets S and T will always be disjoint. To see this, first note that q — p = q — r, because p = r; q — p = r, because p + r = q; and q — p = q — 2p because p = 0. Next, p = q — r because p + r = q; p = r by hypothesis; and p = q — 2p since p = q/3. Finally, q — 2r = q — r because r = 0; q — 2r = r since r = q/3; and q — 2r = q — 2p because r = p. Thus, the sets are disjoint. Hence the cohort is valid as input for Algorithm 2. In particular, p =1 and r = 2 produces the valid input cohort 2q#{q — 1,1, q — 4}; {q — 2,2, q — 2} for any q > 4. □ 2k + 1 Lemma 5.3. The cohort 3q#{1,2,..., 2k-1}; {q, q,..., q} for q = -, k odd and --^-' 3 k k > 2 is a valid celestial cohort. Proof. Note that the cohort 9#{1,2,4}; {3,3,3} can be viewed as the case k = 3 of this cohort. 2k + 1 To show the cohort is valid, we need to show that q = —-— is an integer and that the cohort satisfies the cosine and even conditions. 422 Ars Math. Contemp. 10(2016)411-425 Figure 7: The smallest movable 5-configuration produced by Algorithm 2, an (815 ) configuration, with initial celestial configuration 9#(4, 3; 2,3; 1, 3) shown in red, second celestial configuration 9#(2, 3; 1,3; 4, 3) shown in blue, and final celestial configuration 9#(1, 3; 4,3; 2,3) shown in green. The point (v\)0 is highlighted in red, the line (L\)0 is the thickest red line, the point (v2)0 is highlighted in blue, and the line (L\)0 is the thickest blue line. The point (v3)0, which was constructed via the intersection of (L \)0 with the black circumcircle of span 1 through (v\)0, is highlighted in green, and (L\)0 is the thickest green line. Other 0th elements of symmetry classes are shown at medium weights. Already we have reached the limits of intelligibility of a small-scale diagram. L. W. Berman, E. Jacksch and L. Ver Hoef: An infinite class of movable 5-configurations 423 If k = 2j + 1 for some integer j, it is straightforward to show that 2 j 2k + 1 = 22j+1 + 1 = (2 + 1^(-1)i2i, ¿=0 so 22j + 1 is clearly divisible by 3, and q = J22= 0(-1)i2i, which is odd. >i=0V oi — 1 _ ok Moreover, if s, = 2i—1, then £k=1 2i—1 = 2k + 1. Thus, if t, = q for i = 1,..., k, then ¿(Si - ti) = (2k + 1) - (2j + 1)q i=i is even, since both terms are odd. It remains to show the cosine condition is fulfilled: that is, we need to show that for q 3 ' n cos (^)=n cos (). (5,) The right-hand side of equation (5.1) clearly evaluates to 2k .To see the left-hand side also evaluates to , we use the following trigonometric identity, which can be proved using the identity sin(2$) = 2 sin(0) cos(0) and induction (see [10]): , , . ) sin (2ja) 2k TT cos (2j a) = / . (5.2) sin( a) j=0 v 7 Applying this identity to the left-hand side of (5.1), we see that ,'2i-1n) A ( 2i-1n ) 1 /sin (2^) HcosH^ =nc°s i=i V V2k + U 2k Uni \2k + 1 1 ( n \ ( n = —T sin n--;- csc 2k V 2k + 1J \2k + 1 = ¿k (sin(n) cos ( ) - cos(n) sin ( ^^ ) ) csc ^ 2k V \2k + 1J v 7 V 2k + 1// V 2k + 1 1 (0 - (-1)si^)cs/ ^ 2k 2k + 1 2k + 1 _ 1 = , so the cosine condition is satisfied. □ Theorem 5.4. There exists at least one 5-configuration with s continuous degrees of freedom, for infinitely many values of s. 2k + 1 424 Ars Math. Contemp. 10(2016)411-425 Proof. Use the cohort 3q#{1, 2,..., 2k-1}; {q,q,..., q} for q = , k odd and k > s-V-' 3 k 2 from Lemma 5.3; clearly, the sets S and T are disjoint. This produces a movable 5-configuration with k - 2 degrees of freedom for all odd k > 3. □ 6 Open Questions Question 1. In [8], the Crossing Spans Lemma is generalized to allow larger and differently labelled subgraphs, as the Extended Crossing Spans Lemma. Are there interesting movable configurations that can be constructed from this generalization? Question 2. This construction depends on two very simple geometric lemmas, which are straightforward to prove using basic Euclidean geometry. Are there other such useful lemmas? What techniques can be used, and which incidence theorems, to construct new configurations from known configurations while retaining useful symmetry properties? Question 3. Finding movable 3-configurations is easy [6], and there are a number of known classes of movable 4-configurations [3, 4, 8, 14]. This paper presents a class of movable 5-configurations. Are there movable k-configurations for any k > 5? For all k > 5? In particular, are there movable 6-configurations? References [1] A. Berardinelli and L. W. Berman, Systematic celestial 4-configurations, Ars Math. Contemp. 7 (2014), 361-377. [2] L. W. Berman, A characterization of astral (n4) configurations, Discrete Comput. Geom. 26 (2001), 603-612. [3] L. W. Berman, Movable (n4 ) configurations, Electron. J. Combin. 13 (2006), Research Paper 104, 30, http://www.combinatorics.org/Volume_13/Abstracts/ v13i1r104.html. [4] L. W. Berman, A new class of movable (n4) configurations, Ars Math. Contemp. 1 (2008), 44-50. [5] L. W. Berman, Constructing highly incident configurations, Discrete Comput. Geom. 46 (2011), 447-470, doi:10.1007/s00454-010-9279-7, http://dx.doi.org/10.10 07/ s00454-010-9279-7. [6] L. W. Berman, Geometric constructions for 3-configurations with non-trivial geometric symmetry, Electron. J. Combin. 20 (2013), Paper 9, 29. [7] L. W. Berman and J. R. Faudree, Highly incident configurations with chiral symmetry, Discrete Comput. Geom. 49 (2013), 671-694, doi:10.1007/s00454-013-9494-0, http://dx.doi. org/10.1007/s00454-013-9494-0. [8] L. W. Berman, J. R. Faudree and T. Pisanski, Polycyclic movable 4-configurations are plentiful, Discrete Comput. Geom. (Published Online 16 November 2015), doi:10.1007/ s00454-015-9749-z. [9] L. W. Berman and L. Ng, Constructing 5-configurations with chiral symmetry, Electron. J. Combin. 17 (2010), Research Paper 2, 14, http://www.combinatorics.org/ Volume_17/Abstracts/v17i1r2.html. L. W. Berman, E. Jacksch and L. Ver Hoef: An infinite class of movable 5-configurations 425 [10] W. A. Beyer, J. D. Louck and D. Zeilberger, Math Bite: A Generalization of a Curiosity that Feynman Remembered All His Life, Math. Mag. 69 (1996), 43-44, http://www.jstor. org/stable/2 691393?origin=pubexport. [11] M. Boben and T. Pisanski, Polycyclic configurations, European J. Combin. 24 (2003), 431-457, doi:10.1016/S0195-6698(03)00031-3, http://dx.doi.org/10. 1016/S0195-6698(03)00031-3. [12] H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413-455. [13] B. Grunbaum, Musings on an example of Danzer's, European J. Combin. 29 (2008), 19101918, doi:10.1016/j.ejc.2008.01.004, http://dx.doi.org/10.1016/j.ejc.2008. 01.004. [14] B. Grimbaum, Configurations of points and lines, volume 103 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2009. [15] B. Grunbaum and J. F. Rigby, The real configuration (214), J. London Math. Soc. (2) 41 (1990), 336-346, doi:10.1112/jlms/s2-41.2.336, http://dx.doi.org/10.1112/ jlms/s2-41.2.336 . [16] D. Marusic and T. Pisanski, Weakly flag-transitive configurations and half-arc-transitive graphs, European J. Combin. 20 (1999), 559-570, doi:10.1006/eujc.1999.0302, http://dx.doi. org/10.1006/eujc.1999.0302. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 427-437 Odd automorphisms in vertex-transitive graphs Ademir Hujdurovic *, Klavdija Kutnar t University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia University of Primorska, UP FAMNIT, Glagoljaska 8, 6000 Koper, Slovenia Dragan Marušič * University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia University of Primorska, UP FAMNIT, Glagoljaska 8, 6000 Koper, Slovenia IMFM, Jadranska 19, 1000 Ljubljana, Slovenia Received 24 February 2016, accepted 10 July 2016, published online 25 July 2016 An automorphism of a graph is said to be even/odd if it acts on the set of vertices as an even/odd permutation. In this article we pose the problem of determining which vertex-transitive graphs admit odd automorphisms. Partial results for certain classes of vertex-transitive graphs, in particular for Cayley graphs, are given. As a consequence, a characterization of arc-transitive circulants without odd automorphisms is obtained. Keywords: Graph, vertex-transitive, automorphism group, even permutation, odd permutation. Math. Subj. Class.: 20B25, 05C25 1 Introduction Apart from being a rich source of interesting mathematical objects in their own right, vertex-transitive graphs provide a perfect platform for investigating structural properties of transitive permutation groups from a purely combinatorial viewpoint. The recent outburst *This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0032, N1-0038, and J1-7051). tThis work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0032, N1-0038, J1-6720, J1-6743, and J1-7051), in part by WoodWisdom-Net+, W3B, and in part by NSFC project 11561021. * This work is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0032, N1-0038, J1-5433, J1-6720, and J1-7051), and in part by H2020 Teaming InnoRenew CoE. E-mail address: ademir.hujdurovic@upr.si (Ademir Hujdurovic), klavdija.kutnar@upr.si (Klavdija Kutnar)dragan.marusic@upr.si (Dragan Marusic) Abstract ©® This work is licensed under http://creativecommons.Org/licenses/by/3.0/ 428 Ars Math. Contemp. 10 (2016) 427-437 of research papers on this topic should therefore come as no surprise. Most of these papers have arisen as direct attempts - by developing consistent theories and strategies - to solve open problems in vertex-transitive graphs; the hamiltonicity problem [17], for example, being perhaps the most popular among them. In this context knowing the full (or as near as possible) automorphism group of a vertex-transitive graph is important because it provides the most complete description of its structure. While some automorphisms are obvious, often part of the defining properties, there are others, not so obvious and hence more difficult to find. Consider for example bicirculants, more precisely, n-bicirculants, that is, graphs admitting an automorphism p with two orbits of size n > 2 and no other orbits. There are three essentially different possibilities for such a graph to be vertex-transitive depending on whether its automorphism group contains a swap and/or a mixer, where a swap is an automorphism interchanging the two orbits of p, and a mixer is an automorphism which neither fixes nor interchanges the two orbits of p. For example, the Petersen graph has swaps and mixers, prisms (except for the cube) have only swaps, while the dodecahedron has only mixers. Clearly, swaps are the "obvious" automorphisms and mixers are "not so obvious" ones (see Figure 1). Figure 1: The Petersen graph, the 5-prism and the dodecahedron - the first two admit a swap, while the third one does not. In this paper we propose to approach the sometimes elusive separation line between the obvious and not so obvious automorphisms via the even/odd permutations dichotomy. Let us call an automorphism of a graph even/odd if it acts on the vertex set as an even/odd permutation. Further, a graph is said to be even-closed if all of its automorphisms are even. The Petersen graph and odd prisms have odd automorphisms, the swaps being such automorphisms. On the other hand, the dodecahedron has only even automorphisms [13]. Furthermore, consider the two cubic 2k-bicirculants, k > 1, shown in Figure 2 for k = 4. Both have swaps which are even automorphisms. More precisely, all of the automorphisms of the 2k-prism on the left-hand side are even. As for the graph on the right-hand side - the Cayley graph Cay(Z4k, {±1,2k}) on the cyclic group Z4k = (1) - any generator of the left regular representation of Z4k is an odd automorphism (note that the bicirculant structure of this graph arises from the action of the square of any generator of the left regular representation of Z4k). This brings us to the following natural question: Given a transitive group of even automorphisms H of a graph X, is there a group G < Aut(X) containing odd automorphisms A. Hujdurovic et al.: Odd automorphisms in vertex-transitive graphs 429 of X and H as a subgroup? In particular, we would like to focus on the following problem. Problem 1.1. Which vertex-transitive graphs admit odd automorphisms? Of course, in some cases, the answer to the above problem will be purely arithmetic. Such is for example the case with cycles. Clearly, all cycles of even length admit odd automorphisms, while cycles of odd length 2k + 1 admit odd automorphisms if and only if k is odd. The answer for some of the well studied classes of graphs, however, suggest that the above even/odd question goes beyond simple arithmetic conditions and is likely to uncover certain more complex structural properties. For example, while the general distinguishing feature for cubic symmetric graphs (with respect to the above question) is their order 2n, n even/odd, there are exceptions on both sides. Namely, there exist cubic symmetric graphs without odd automorphisms for n odd, and with odd automorphisms for n even, see [13]. In this paper a special emphasis is given to certain classes of Cayley graphs (see Section 3), such as circulants for example. Theorem 3.15 gives a necessary and sufficient condition for a normal circulant to be even-closed. This result combined together with certain other results of this section then leads to a characterization of even-closed arc-transitive circulants, see Theorem 3.16. In Section 4 the even/odd question is discussed in the more general context of vertex-transitive graphs. 2 Preliminaries Here we bring together definitions, notation and some results that will be needed in the remaining sections. For a finite simple graph X let V(X), E(X), A(X) and Aut(X) be its vertex set, its edge set, its arc set and its automorphism group, respectively. A graph is said to be vertex-transitive, edge-transitive and/or arc-transitive (also symmetric) if its automorphism group acts transitively on the set of vertices, the set of edges, and/or the set of arcs of the graph, respectively. A non-identity automorphism is semiregular, in particular (m,n)-semiregular if it has m cycles of equal length n in its cycle decomposition, in other words m orbits of equal length n. An n-circulant (circulant, in short) is a graph admitting a (1, n)-semiregular automorphism, and an n-bicirculant (bicirculant, in short) is a graph admitting a (2, n)-semiregular automorphism. 430 Ars Math. Contemp. 10 (2016) 427-437 Given a group G and a symmetric subset S = S-1 of G \ {1}, the Cayley graph X = Cay(G, S) has vertex set G and edges of the form {g, gs} for all g G G and s G S. Every Cayley graph is vertex-transitive but there exist vertex-transitive graphs that are not Cayley, the Petersen graph being the smallest such graph. Cayley graphs are characterized in the following way. A graph is a Cayley graph of a group G if and only if its automorphism group contains a regular subgroup GL, referred to as the left regular representation of G, isomorphic to G, see [19]. Using the terminology and notation of Cayley graphs, note that an n-circulant is a Cayley graph Cay(G, S) on a cyclic group G of order n relative to some symmetric subset S of G \ {id}, usually denoted by Circ(n, S). The first of the two group-theoretic observations below reduces the question of existence of odd automorphisms to Sylow 2-subgroups of the automorphism group. Proposition 2.1. A permutation group G contains an odd permutation if and only if its Sylow 2-subgroups contain an odd permutation. Proof. Since any odd permutation a is of even order, we can conclude that ak, where k is the largest odd number dividing the order of a, is a non-trivial odd permutation belonging to a Sylow 2-subgroup of G. □ Proposition 2.2. A permutation group G acting semiregularly with an odd number of orbits admits odd permutations if and only if its Sylow 2-subgroups are cyclic and non-trivial. Proof. Note that any Sylow 2-subgroup of G must also have an odd number of orbits. Thus if a Sylow 2-subgroup is cyclic and non-trivial, the corresponding generators are odd permutations. On the other hand, if a Sylow 2-subgroup J is not cyclic (or is trivial) then the semiregularity of G implies that all of the elements of J must be even permutations. By Proposition 2.1 G itself consists solely of even permutations. □ As a consequence of Proposition 2.2, for some classes of graphs the existence of odd automorphisms is easy to establish. For instance, in Cayley graphs the corresponding regular subgroup contains odd automorphisms if and only if its Sylow 2-subgroup is cyclic and non-trivial. When a Sylow 2-subgroup is not cyclic, however, the search for odd automorphisms has to be done outside this regular subgroup, raising the complexity of the problem. 3 Cayley graphs In this section we give some general results about the existence of odd automorphisms in Cayley graphs and discuss the problem in detail for circulants. The first proposition, a corollary of Proposition 2.2, gathers straightforward facts about the existence of odd automorphisms in Cayley graphs. (A graph is said to be a graphical regular representation, or a GRR, for a group G if its automorphism group is isomorphic to G and acts regularly on the vertex set of the graph.) Proposition 3.1. A Cayley graph on a group G admits an odd automorphism in GL if and only if G has cyclic Sylow 2-subgroups. In particular, • a Cayley graph of order 2 (mod 4) admits odd automorphisms, • a GRR admits an odd automorphism if and only if the Sylow 2-subgroups of G are cyclic. A. Hujdurovic et al.: Odd automorphisms in vertex-transitive graphs 431 By Proposition 3.1, Cayley graphs of order twice an odd number admit odd automorphisms (they exist in a regular subgroup of the automorphism group). As for Cayley graphs whose order is odd or divisible by 4 the answer is not so simple. The next proposition answers the question of existence of odd automorphisms in particular subgroups of automorphisms of Cayley graphs on abelian groups. Proposition 3.2. Let X = Cay(G, S) be a Cayley graph on an abelian group G and let t € Aut(G) be such that t(i) = -i. Then (Gl,t} < Aut(X), and there exists an odd automorphism in (GL, t } if and only if one of the following holds: (i) |G| = 3 (mod 4) (in which case t is an odd automorphism), (ii) |G| = 2 (mod 4), (iii) |G| = 0 (mod 4) and a Sylow 2-subgroup of G is cyclic. Proof. First recall that the mapping t : G ^ G defined by t(i) = -i is an automorphism of the group G if and only if G is abelian. Moreover, since S = -S it is easy to see that t € Aut(X). Clearly, when |G| = 1 (mod 4) there are no odd automorphisms in (GL, t}. Suppose now that |G| = 1 (mod 4). If |G| = 3 (mod 4) then the involution t has 2k + 1 cycles of length 2 and one fixed vertex in its cyclic decomposition, and so it is an odd automorphism. If |G| = 2 (mod 4) then there exist odd automorphisms in GL < (Gl,t} by Proposition 3.1. We are therefore left with the case |G| = 0 (mod 4). Hence suppose that G is of such order. If a Sylow 2-subgroup J of GL is cyclic then a generator of J is a product of an odd number |G|/| J| of cycles of length | J|, and is thus an odd automorphism. On the other hand, if J is not cyclic then every element of J has an even number of cycles in its cyclic decomposition. As for t, an element of G is fixed by t if and only if it is an involution. In other words, it fixes the largest elementary abelian 2-group T inside the Sylow 2-subgroup J, say of order 2k. Consequently, the number of transpositions in the cyclic decomposition of t equals |G|/2 - 2k, which is an even number if and only if k > 1. Consequently, t is an odd automorphism if and only if T = Z2 and hence J is cyclic. □ Corollary 3.3. Let X = Circ(n, S), where S is a symmetric subset of Zn, and either n is even or n = 3 (mod 4). Then X admits odd automorphisms. When n = 1 (mod 4) the situation is more complex. For example, cycles C4k+1 = Circ(4k + 1, {±1}) admit only even automorphisms. On the other hand, the circulant Circ(13, {±1, ±5}) is an example of a (4k + 1)-circulant admitting odd automorphisms. Namely, one can easily check that the permutation (0)(1,5,12,8)(2,10,11,3)(4, 7, 9, 6) arising from the action of 5 € is one of its odd automorphisms. (For a positive integer n we use Z*n to denote the multiplicative group of units of Zn.) We therefore propose the following problem. Problem 3.4. Classify even-closed circulants of order n = 1 (mod 4). A partial answer to this problem is given at the end of this section, see Corollary 3.11 and Theorem 3.16. We start with the class of connected arc-transitive circulants. The classification of such circulants, obtained independently by Kovacs [12] and Li [16], is essential to this end. In order to state the classification let us recall the concept of normal Cayley graphs and certain graph products. 432 Ars Math. Contemp. 10 (2016) 427-437 Let X and Y be graphs. The wreath (lexicographic) product X[Y] of X by Y is the graph with vertex set V(X) x V(Y) and edge set {{(xi,yi), (x2,y2)}: {xi,x2} G E(X), or xi = x2 and {y1,y2} G E(Y)}. The deleted wreath (deleted lexicographic) product X ld Y of X by Y is the graph with vertex set V (X) x V(Y) and edge set {{(xi,yi), (x2,y2)}: {xi,X2} G E(X) and yi = y2, or xi = X2 and {yi,y2} G E(Y)}. If Y = Kb = bKi then the deleted lexicographic product X ld Y is denoted by X [Kb] - bX. Let X = Cay(G, S) be a Cayley graph on a group G. Denote by Aut(G, S) the set of all automorphisms of G which fix S setwise, that is, Aut(G, S) = {a G Aut(G)|SCT = S}. It is easy to check that Aut(G, S) is a subgroup of Aut(X) and that it is contained in the stabilizer of the identity element id G G. Following Xu [25], X = Cay(G, S) is called a normal Cayley graph if GL is normal in Aut(X), that is, if Aut(G, S) coincides with the vertex stabilizer id G G. Moreover, if X is a normal Cayley graph, then Aut(X) = Gl x Aut(G, S) (see [11]). Proposition 3.5. [12, 16] Let X be a connected arc-transitive circulant of order n. Then one of the following holds: (i) X = K„; (ii) X = Y [Kd], where n = md, m, d > 1 and Y is a connected arc-transitive circulant of order m; (iii) X = Y[Kd] — dY, where n = md, d > 3, gcd(d, m) = 1 and Y is a connected arc-transitive circulant of order m; (iv) X is a normal circulant. The proof of the next proposition is straightforward. Proposition 3.6. Complete graphs Kn and their complements Kn, n > 2, admit odd automorphisms. Propositions 3.7, 3.8, 3.9, and 3.10 deal with the existence of odd automorphisms in the framework of (deleted) lexicographic products of graphs. Proposition 3.7. Let Z be a graph admitting an odd automorphism. Then a lexicographic product Y [Z] of the graph Z by a graph Y admits odd automorphisms. In particular, Y[Kd], d > 1, admits odd automorphisms. Proof. An odd automorphism is constructed by taking a map that acts trivially on all blocks (that is, copies of the graph Z) but one, where it acts as an odd automorphism of the graph Z. By Proposition 3.6, Kd admits an odd automorphism, so such a map exists when Z = Kd. □ Proposition 3.8. Let X be the deleted lexicographic product X = Y id Z of a graph Y by a graph Z, where Z has odd automorphisms and Y is of odd order. Then X admits odd automorphisms. Proof. An odd automorphism is constructed by taking a map that acts as the same odd automorphism on each of the odd number of copies of the graph Z. □ A. Hujdurovic et al.: Odd automorphisms in vertex-transitive graphs 433 Proposition 3.9. Let X be the deleted lexicographic product X = Y id Z of a graph Y by a graph Z, where Z is of odd order and Y has odd automorphisms. Then X admits odd automorphisms. Proof. Let a' be an odd automorphism of Y. Let a : V (X) ^ V(X ) be defined with a((y, z)) = (a'(y), z). It is easy to see that a e Aut(X), and the fact that |V(Z)| is odd implies that a is an odd automorphism of X. □ Propositions 3.8 and 3.9 combined together imply existence of odd automorphisms in arc-transitive circulants belonging to the family given in Proposition 3.5(iii). Proposition 3.10. Let X be an arc-transitive circulant isomorphic to the deleted lexicographic product Y[dKi] — dY, where Y is an arc-transitive circulant of order coprime with d > 1. Then X has an odd automorphism. Proof. Suppose first that Y is of odd order. Then, since, by Proposition 3.6, d,K1 admits odd automorphisms, the existence of odd automorphisms in Aut(X ) follows from Proposition 3.8. Suppose now that Y is of even order. Then any generator of a regular cyclic subgroup of Aut(Y) is an odd automorphism. Since in this case d is odd the existence of odd automorphisms in Aut(X) follows from Proposition 3.9. □ Corollary 3.3 and Propositions 3.6, 3.7 and 3.10 combined together imply that even-closed arc-transitive circulants may only exist amongst normal arc-transitive circulants of order 1 (mod 4). In all other cases an arc-transitive circulant admits an odd automorphism. Corollary 3.11. An even-closed arc-transitive circulant is normal and has order 1 (mod 4). For the rest of this section we may, in our search for odd automorphisms, therefore restrict ourselves to normal circulants. Let X = Circ(n, S) be a normal arc-transitive circulant of order order 1 (mod 4) and let s e S. Then for any s' e S there must be an automorphism a of G such that a(s) = s', and so s and s' are of the same order. Thus if s is not of order n then Circ(n, S) is not connected. Hence it has at least three components (since n is not even), and has an automorphism that fixes all but one component while rotating that component, but this automorphism does not normalize the regular cyclic subgroup of Aut(X). This shows that we may assume that 1 e S (note that additive notation is used for Zn). This fact is used throughout this section. The following lemma about the action of the multiplicative group of units is needed in this respect. For a positive integer n we use np to denote the highest power of p dividing n. Lemma 3.12. Let p be an odd prime, and let k > 1 be a positive integer. Then Zp k, in its natural action on Zpk, admits an odd permutation if and only if k is odd. Proof. By Proposition 2.1 it suffices to consider the Sylow 2-subgroup J of Z*pk. Since Z*k is a cyclic group, J is cyclic too. Let a be a generator of J. We claim that (a) acts semiregularly on Zpk \ {0}. Suppose on the contrary that this is not the case. Then there exist m e N such that am = 1 and am(x) = x for some x e Zpk \ {0}. This is equivalent to (am — 1)x = 0 (mod pk). 434 Ars Math. Contemp. 10 (2016) 427-437 The above equation admits a non-trivial solution if and only if am - 1 is divisible by p. Suppose that j 1 the number (2i ) (Apj )i is divisible by pj+1. Consequently, 2sApj is divisible by pj+1, and so we conclude that p divides 2s A, a contradiction. As claimed above, this shows that a acts semiregularly on Zpk \ {0} with p - 1 pk - 1=(1+p + ... +pk"l)- p - 1 (pk — pk 1)2 p — 1 (pk — pk 1)2 cycles of even length (pk — pk-1)2 = (p — 1)2 in its cycle decomposition (since a is a generator of J). Since the parity of 1 + p + ... + pk-1 depends on whether k is even or odd, it follows that a is an odd permutation if and only if k is odd. The result follows. □ Corollary 3.13. Let p be an odd prime, and let k > 1 be a positive integer such that pk = 1 (mod 4). Then a normal arc-transitive circulant X = Cay(Zpk, S) admits an odd automorphism if and only if k is odd and S contains the Sylow 2-subgroup of Z*pk. Proof. Recall that Aut(X) = Zpk x S, and thus X admits odd automorphisms if and only if S contains an element giving rise to an odd permutation on Zpk (generators of Sylow 2-subgroups of Zpk are odd permutations on Zpk). The result is thus obtained by combining together Proposition 2.1 and Lemma 3.12. □ Lemma 3.14. Let n = p2kl+1 • • • paka+1q2il ... q2'6 be a prime decomposition of an odd integer n, andletZn = P1 ©• • •©Pa©Q1 ©• • •©Qb, where Pj = Z^+i, i € {1,..., a}, and Qi = Zq2, i € {1,..., b}. Further, let aj and Pi, respectively, be generators of the Sylow 2-subgroup of P* and the Sylow 2-subgroup of Q*. Then, for each i, we have that ai is an odd permutation on Zn, and is an even permutation on Zn. Proof. Observe that each cycle in the cycle decomposition of ai € Pi (considered as a permutation of Zp2^+i) is lifted to n/p2ki+1 cycles of the same length in the cycle decomposition of ai (when considered as a permutation of Zn). By Lemma 3.12, ai is an odd permutation on Zp2^+i for each i. Similarly, is an even permutation on Zq2^ for each i. Since n is odd, the result follows. □ We introduce the fo a positive integer n, let We introduce the following notation. Let n = p^ • • • pj;a be a prime decomposition of Zn = © P,-, where P ^ Z --11 i, where P - = Zpki and let J(p,) be the Sylow 2-subgroup of p,:*. In the next theorem a necessary and sufficient condition for a normal circulant to be even-closed is given. One of the immediate consequences is, for example, that a normal circulant of order n2, n odd, is even-closed. A. Hujdurovic et al.: Odd automorphisms in vertex-transitive graphs 435 Theorem 3.15. Let n = pk1 ■ ■ ■ p'ka be a prime decomposition of a positive integer n, and let X = Circ(n, S) be a normal arc-transitive circulant on Zn = ©a=1Pj. Then X is even-closed if and only if n = 1 (mod 4) and for every a = ©a=1 aj G S we have ± 6i(a) = 0 (mod 2), where 6+a) = { J Jp^ <*> and ki is odd . i= l ^ ' Proof. By Corollary 3.3 for n = 1 (mod 4) the graph X admits odd automorphisms. We may therefore assume that n = 1 (mod 4). By Lemma 3.14, the existence of odd automorphisms in X depends solely on the parity of the exponents ki and the containment of the generators of the corresponding Sylow 2-subgroups in S, and the result follows. (Recall, that we are using the assumption that 1 G S and the fact that in a normal arc-transitive circulant, every element of S is conjugate, so every a G S is odd if and only if X has an odd automorphism.) □ Combined together with Corollary 3.11 and Theorem 3.15 we have the following characterization of even-closed arc-transitive circulants. Theorem 3.16. Let X be an even-closed arc-transitive circulant of order n and let n = Pi1 • • • Paa be a prime decomposition of n. Then X is a normal circulant X = Circ(n, S) on Zn = ®'a=1Pi, n = 1 (mod 4) and for every a = ®a=1ai G S we have ]T 0i(a) = 0 (mod 2), where 6(a) = { 1 ^¿^ and k is odd . i= 1 ^ ' 4 Vertex-transitive graphs It is known that every finite transitive permutation group contains a fixed-point-free element of prime power order (see [9, Theorem 1]), but not necessarily a fixed-point-free element of prime order and, hence, a semiregular element (see for instance [3, 9]). In 1981 the third author asked if every vertex-transitive digraph with at least two vertices admits a semiregular automorphism (see [20, Problem 2.4]). Despite considerable efforts by various mathematicians the problem remains open, with the class of vertex-transitive graphs having a solvable automorphism group being the main obstacle. The most recent result on the subject is due to Verret [24] who proved that every arc-transitive graph of valency 8 has a semiregular automorphism, which was the smallest open valency for arc-transitive graphs (see [7, 10, 23] and [15] for an overview of the status of this problem). While the existence of such automorphisms in certain vertex-transitive graphs has proved to be an important building block in obtaining at least partial solutions in many open problems in algebraic graph theory, such as for example the hamiltonicity problem (see [14, 2, 17]), the connection to the even/odd problem is straightforward. Proposition 4.1. An even-closed vertex-transitive graph does not have even order semiregular automorphisms with an odd number of orbits. This suggest that in a search for odd automorphisms a special attention should be given to semiregular automoprhisms of even order. Furthermore, for those classes of vertex-transitive graphs for which a complete classification (together with the corresponding automorphism groups) exists, the answer to 436 Ars Math. Contemp. 10(2016)411-425 Problem 1.1 is, at least implicitly, available right there - in the classification. Such is, for example, the case of vertex-transitive graphs of order a product of two primes, see [6, 8, 21, 22, 18], and the case of vertex-transitive graphs which are graph truncations, see [1]. The hard work needed to complete these classifications suggest that the even/odd question is by no means an easy one. Let us consider, for example, the class of all vertex-transitive graphs of order 2p, p a prime. In the completion of the classification of these graphs, the classification of finite simple groups is an essential ingredient in handling the case of primitive automorphism groups. We know, by this classification, that the Petersen graph and its complement are the only such graphs with a primitive automorphism group. Of course, they both admit odd automorphisms. As for imprimitive automorphism groups, it all depends on the arithmetic of p. When p = 3 (mod 4), the graphs are necessarily Cayley graphs (of dihedral groups) and hence must admit odd automorphisms. (Namely, reflections interchanging the two orbits of the rotation in the dihedral group are odd automorphisms.) When p = 1 (mod 4), then it follows by the classification of these graphs [20] that there is an automorphism of order 2k, k > 1, interchanging the two blocks of imprimitivity of size p, having one orbit of size 2 and 2(p — 1)/2k orbits of size 2k, thus an odd number of orbits in total (since 2k divides p — 1). We have thus shown that every vertex-transitive graph of order twice a prime number admits an odd automorphism. However, no "classification of finite simple groups free" proof of the above fact is known to us. In conclusion we would like to make the following comment with regards to cubic vertex-transitive graphs. Recall that the class of cubic vertex-transitive graphs decomposes into three subclasses depending on the number of orbits of the vertex-stabilizer on the set of neighbors of a vertex. These subclasses are arc-transitive graphs (one orbit), graphs with vertex-stabilizers being 2-groups (two orbits) andGRR graphs (three orbits), see [4]. (Note that there are two types of cubic GRR graphs, those with connecting set consisting of three involutions and those with connecting set consisting of an involution, a non-involution and its inverse, see [5].) For the first and third subclass the answer to Problem 1.1 is given in [13] and Proposition 3.1, respectively, while the problem is still open for the second subclass. Examples given in Section 1 (see also Figure 2) show, however, that this second subclass contains infinitely many even-closed graphs as well as infinitely many graphs admitting odd automorphisms. Problem 4.2. Classify cubic vertex-transitive graphs with vertex-stabilizers being 2-groups that admit odd automorphisms. References [1] B. Alspach and E. Dobson, On automorphism groups of graph truncations, Ars Math. Contemp. 8 (2015), 215-223. [2] K. K. and, and p. sparl, hamilton paths and cycles in vertex-transitive graphs of order 6p, Discrete Math 309 (2009), 5444-5460. [3] P. J. Cameron, M. Giudici, W. M. Kantor, G. A. Jones, M. H. Klin, D. M. sic and L. A. Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc 66 (2002), 325-333. [4] P. P. cnik, P. Spiga and G. Verret, Cubic vertex-transitive graphs on up to 1280 vertices, J. Symbolic Computation 50 (2013), 465-477. A. Hujdurovic et al.: Odd automorphisms in vertex-transitive graphs 437 [5] M. D. E. Conder and T. Pisanski, and a. Zitnik, Vertex-transitive graphs and their arc-types, arXiv:, http://arxiv.org/abs/150 5.02 02 9. [6] E. Dobson, Automorphism groups of metacirculant graphs of order a product of two distinct primes, Combin. Probab. Comput 15 (2006), 105-130. [7] E. Dobson, A. M. c, D. Marusic and L. A. Nowitz, Semiregular automorphisms of vertex-transitive graphs of certain valencies, J. Combin. Theory, Ser. B 97 (2007), 371-380. [8] E. Dobson and D. Witte, Transitive permutation groups of prime-squared degree, J. Algebraic Combin 16 (2002), 43-69. [9] B. Fein, W. M. Kantor and M. Schacher, Relative brauer groups ii, J. Reine Angew. Mat 328 (1981), 39-57. [10] M. Giudici and G. Verret, Semiregular automorphisms in arc-transitive graphs of valency 2p, in preparation. [11] C. D. Godsil, On the full automorphism group of a graph, Combin 1 (1981), 243-256. [12] I. Kovacs, Classifying arc-transitive circulants, J. Algebr. Combin 20 (2004), 353-358. [13] K. Kutnar and D. M. sics, Cubic symmetric graphs via odd automorphisms, manuscript. [14] K. Kutnar and D. M. sic, Hamiltonicity of vertex-transitive graphs of order 4p, European J. Combin 29 (2008), 423-438. [15] K. Kutnar and D. M. sic, Recent trends and future directions in vertex-transitive graphs, Ars Math. Contemp 1 (2008). [16] C. H. Li, Permutation groups with a cyclic regular subgroup and arc-transitive circulants, J. Algebr. Combin 21 (2005), 131-136. [17] L. Lovasz, Combinatorial structures and their applications, (proc, Calgary Internat. Conf., Calgary Alberta 1969 (1970), 243-246. [18] C. E. Praeger and M. Y. Xu, Vertex-primitive graphs of order a product of two distinct primes, J. Combin. Theory, Ser. B 59 (1993), 245-266. [19] G. Sabidussi, On a class of fixed-point-free graphs, in: Proc. Amer. Math. Soc, 9, 1958 pp. 800-804. [20] D. M. sic, On vertex symmetric digraphs, Discrete Math 36 (1981), 69-81. [21] D. M. sic and R. Scapellato, Characterizing vertex-transitive pq-graphs with imprimitive automorphism subgroups, J. Graph Theory 16 (1992), 375-387. [22] D. M. sic and R. Scapellato, Classifying vertex-transitive graphs whose order is a product of two primes, Combinatorics 14 (1994), 187-201. [23] D. M. sic and R. Scapellato, Permutation groups, vertex-transitive digraphs and semiregular automorphisms, European J. Combin 19 (1998), 707-712. [24] G. Verret, Arc-transitive graphs of valency 8 have a semiregular automorphism, Ars Math. Contemp 8 (2015), 29-34. [25] M. Y. Xu, Automorphism groups and isomorphisms of cayley digraphs, Discrete Math 182 (1998), 309-320. ARS MATHEMATICA CONTEMPORANEA Author Guidelines Before submission Papers should be written in English, prepared in ETEX, and must be submitted as a PDF file. The title page of the submissions must contain: • Title. The title must be concise and informative. • Author names and affiliations. 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