ARS MATHEMATICA CONTEMPORANEA Volume 8, Number 1, Spring/Summer 2015, Pages 1-234 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. Operation is partially funded by the European Union through the European Social Fund and the Ministry of Education, Science and Sport. The operation is performed under the Operational Programme Human Resources Development 2007-2013, the development priority 3: Human resource development and lifelong learning policy; priorities 3.3.: Quality, competitiveness and responsiveness of higher education. ARS MATHEMATICA CONTEMPORANEA Editoral The present issue of Ars Mathematica Contemporanea contains a selection of articles related to the topics presented at the International Conference on Graph Theory and Combinatorics, held form 1 to 3 May 2013, in Koper. The conference was organised in honour of Professor Dragan Marušic on the occasion of his 60th birthday. It brought together many of the researchers with whom Dragan's discussions on mathematics have been (and still are) the most enjoyable and fruitful. There were 14 invited talks at the conference, covering some of Dragan's favourite topics related to symmetries of graphs and other combinatorial structures, such as the hamil-tonicity problem in vertex-transitive graphs and the polycirculant conjecture. The 15 articles in this special issue of Ars Mathematica Contemporanea have been selected for publication after the same thorough refereeing process that papers go through for regular issues of this journal. We are convinced that these high quality articles will have a great impact, and will positively influence future research on the topics concerned. Klavdija Kutnar, Štefko Miklavic, Tomaž Pisanski and Primož Šparl ARS MATHEMATICA CONTEMPORANEA DM=60 - A Surprise Conference On 1 May 2013, a prominent Slovenian mathematician Dragan Marušic turned 60. His former students and colleagues decided to mark this event in a way that scientists know best: by organising a special scientific conference and publishing a Festschrift, in honour of Dragan. Following the tradition of similar scientific festivities, and in order to reduce costs, the conference was organised by invitation only. A list of speakers was very carefully hand-picked from eminent mathematicians, Dragan's long-time collaborators and his students. Not all were able to attend, but there was a twist in the organisation that makes DM=60 unique. The conference was a surprise conference. In 2013 the University of Primorska's FAMNIT organised 14 international conferences, and Dragan was led to believe that his 60th birthday would be commemorated later that year at the annual summer school at Rogla. Figure 1: Professor Dragan Marušic with his favorite CFSG-free proof that S5 and A5 are the only simple primitive groups of degree 2p I will never forget the expression on Dragan's face when he entered the big lecture room that morning, on his birthday. The auditorium that was supposed to be empty (due to a national holiday) was full of mathematicians. The conference started promptly and ran for three days. What a joy for Mathematics! Students were able to meet first class mathematicians, and colleagues were able to work with each other on various unsolved problems, and discuss future projects together. All the lectures at the conference touched upon the work that Dragan has done in the past, and the influence he has had on his colleagues and students for years. Reprints of Dragan's published papers were bound into a volume and Figure 2: The DM-60 conference photo, with signatures of the speakers presented to him, together with the signatures of all invited speakers. (See Figure 2.) This volume comprises a remarkable 1729 pages of original mathematics - about a page for every week of Dragan's life after his PhD! (See Figure 3.) Dragan is Rector of the University of Primorska, which is based in the city of Koper, on the Adriatic coast. Currently among its students of mathematics about one third are from abroad. And although the University of Primorska is ten times smaller than the University of Ljubljana, it attracted four times as many new mathematics PhD students in 2014. These things highlight Dragan's successful leadership of this young University. It has also been very interesting to watch the unfolding of the career of Professor Dragan Marušic. His talents were clearly visible already in high school, but it took a long time before his work was recognised as mainstream. He submitted his first paper as un undergraduate, and completed a very good PhD at the University of Reading, UK, under the guidance of the legendary Crispin Nash-Williams. But this was not sufficient to get him position in the main Mathematics department in Slovenia. As a free thinker, a follower of Grateful Dead, an anti-nuclear activist, and long-haired vegetarian, Dragan did not fit he traditional mould of Slovenian society. Even after returning home from the USA, where he began his postdoctoral academic career, the doors to the Mathematics department in Ljubljana remained closed for Dragan. What a waste for Slovenian mathematics! Dragan had started dreaming about democracy and the University of Primorska a long time earlier, and at the first free elections in Slovenia, he ran as a candidate for parliament, for the opposition alliance DEMOS. Creating the University of Primorska was his the main subject on his agenda. (See Figure 4.) Although he won in his district, the votes ARS MATHEMATICA CONTEMPORANEA Figure 3: Collection of the 1729 pages of original mathematics produced by Dragan Marušic over a 30 year period he gained helped a politician who was higher in the party list to enter the parliament. But his campaign to create a new university was taken up and realised by others who were less passionate about it but in a better position politically to succeed. Dragan is the third rector of the University, but the first critical intellectual to hold this position. (The previous two rectors were government ministers before wearing their University insignia.) His primary goal is to promote and ensure the quality and visibility of his University. He will not compromise his integrity, and has no fear of exposing corruption among officials. Sadly, a lot of people in Slovenia are envious of Dragan's scientific achievements and the flourishing of mathematics at the University of Primorska under his leadership. A few months after the conference took place, an anonymous pamphlet was distributed to the media, ministries, and prosecutors, claiming that the DM=60 conference never took place, and that Dragan spent public money on his own birthday party. I am glad that this special volume gives proof of the quality of the DM=60 conference, and that its funding was well-spent. Tomaž Pisanski ARS MATHEMATICA CONTEMPORANEA Figure 4: Poster of Dragan Marušič running for the Slovenian Parliament in 1990 - including a slogan about creating the University of Primorska ARS MATHEMATICA CONTEMPORANEA Dragan Marušic, Bibliography: 1980-2013 [1] B. Alspach, M. D. E. Conder, D. Marušic and M.-Y. Xu, A classification of 2-arc-transitive circulants, J. Algebraic Combin. 5 (1996), 83-86. [2] B. Alspach, D. Marušic and L. Nowitz, Constructing graphs which are 1/2-transitive, J. Austral. Math. Soc. Ser. A 56 (1994), 391-402. [3] A. Araluze, I. Koväcs, K. Kutnar, L. Martinez and D. Marušic, Partial sum quadruples and bi-Abelian digraphs, J. Combin. Theory Ser. A 119 (2012), 1811-1831. [4] A. Araluze, K. Kutnar, L. Martinez and D. Marušic, Edge connectivity in difference graphs and some new constructions of partial sum families, European J. Combin. 32 (2011), 352-360. [5] A. Blokhuis, G. Kiss, I. Koväcs, A. Malnic, D. Marušic and J. Ruff, Semiovals contained in the union of three concurrent lines, J. Combin. Des. 15 (2007), 491501. [6] P. J. Cameron, M. Giudici, G. A. Jones, W. M. Kantor, M. H. Klin, D. Marušic and L. A. Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc. (2) 66 (2002), 325-333. [7] V. Carli, N. Jovanovic, A. Podlesek, A. Roy, Z. Rihmer, S. Maggi, D. Marušic, C. Cesaro, A. Marušic and M. Sarchiapone, The role of impulsivity in self-mutilators, suicide ideators and suicide attempters: a study of 1265 male incarcerated individuals, J. Affective Disord. 123 (2010), 116-122. [8] M. D. E. Conder, A. Malnic, D. Marušic, T. Pisanski and P. Potocnik, The edge-transitive but not vertex-transitive cubic graph on 112 vertices, J. Graph Theory 50 (2005), 25-42. [9] M. D. E. Conder, A. Malnic, D. Marušic and P. Potocnik, A census of semisymmetric cubic graphs on up to 768 vertices, J. Algebraic Combin. 23 (2006), 255-294. [10] M. D. E. Conder and D. Marušic, A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer, J. Combin. Theory Ser. B 88 (2003), 67-76. [11] E. Dobson, A. Malnic, D. Marušic and L. A. Nowitz, Minimal normal subgroups of transitive permutation groups of square-free degree, Discrete Math. 307 (2007), 373-385. [12] E. Dobson, A. Malnic, D. Marušic and L. A. Nowitz, Semiregular automorphisms of vertex-transitive graphs of certain valencies, J. Combin. Theory Ser. B 97 (2007), 371-380. [13] E. Dobson and D. Marušic, An unusual decomposition of a complete 7-partite graph of order 28, Discrete Math. 308 (2008), 4595-4598. [14] E. Dobson and D. Marušic, On semiregular elements of solvable groups, Comm. Algebra 39 (2011), 1413-1426. [15] S. Du, A. Malnič and D. Marušic, Classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008), 1349-1372. [16] S. Du and D. Marušič, Biprimitive graphs of smallest order, J. Algebraic Combin. 9 (1999), 151-156. [17] S. Du and D. Marušic, An infinite family of biprimitive semisymmetric graphs, J. Graph Theory 32 (1999), 217-228. [18] S. Du, D. Marušic and A. O. Waller, On 2-arc-transitive covers of complete graphs, J. Combin. Theory Ser. B 74 (1998), 276-290. [19] T. Durkee, D. Marušic and V. Poštuvan, Prevalence of pathological internet use among adolescents in europe: demographic and social factors, Addiction 107 (2012), 2210-2222. [20] Y.-Q. Feng, K. Kutnar, A. Malnic and D. Marušic, On 2-fold covers of graphs, J. Combin. Theory Ser. B 98 (2008), 324-341. [21] H. H. Glover, K. Kutnar, A. Malnic and D. Marušic, Hamilton cycles in (2, odd, 3)-Cayley graphs, Proc. Lond. Math. Soc. (3) 104 (2012), 1171-1197. [22] H. H. Glover, K. Kutnar and D. Marušic, Hamiltonian cycles in cubic Cayley graphs: the (2,4k, 3) case, J. Algebraic Combin. 30 (2009), 447-475. [23] H. H. Glover and D. Marušic, Hamiltonicity of cubic Cayley graphs, J. Eur. Math. Soc. (JEMS) 9 (2007), 775-787. [24] M. Hladnik, D. Marušic and T. Pisanski, Cyclic Haar graphs, Discrete Math. 244 (2002), 137-152, algebraic and topological methods in graph theory (Lake Bled, 1999). [25] R. Jajcay, A. Malnic and D. Marušic, On the number of closed walks in vertex-transitive graphs, Discrete Math. 307 (2007), 484-493. [26] G. Kiss, A. Malnic and D. Marušic, A new approach to arcs, Acta Math. Hungar. 84 (1999), 181-188. [27] S. Klavžar, D. Marušic, B. Mohar and T. Pisanski, Preface [Algebraic and topological methods in graph theory], Discrete Math. 307 (2007), 299. [28] S. Klavžar, D. Marušic, B. Mohar and T. Pisanski, Preface: Algebraic and topological graph theory (Bled'07), Discrete Math. 310 (2010), 1651-1652, held at Lake Bled, June 2007. [29] I. Koväcs, K. Kutnar and D. Marušic, Classification of edge-transitive rose window graphs, J. Graph Theory 65 (2010), 216-231. [30] I. Koväcs, K. Kutnar, D. Marušic and S. Wilson, Classification of cubic symmetric tricirculants, Electron. J. Combin. 19 (2012), Paper 24, 14. [31] I. Koväcs, A. Malnic, D. Marušic and Š. Miklavic, One-matching bi-Cayley graphs over abelian groups, European J. Combin. 30 (2009), 602-616. [32] I. Koväcs, D. Marušic and M. E. Muzychuk, Primitive bicirculant association schemes and a generalization of Wielandt's theorem, Trans. Amer. Math. Soc. 362 (2010), 3203-3221. [33] I. Koväcs, D. Marušic and M. E. Muzychuk, On dihedrants admitting arc-regular group actions, J. Algebraic Combin. 33 (2011), 409-426. [34] I. Koväcs, D. Marušic and M. E. Muzychuk, On G-arc-regular dihedrants and regular dihedral maps, J. Algebraic Combin. 38 (2013), 437-455. [35] K. Kutnar, U. Borštnik, D. Marušic and D. Janežic, Interconnection networks for parallel molecular dynamics simulation based on hamiltonian cubic symmetric topology, J. Math. Chem. 45 (2009), 372-385. [36] K. Kutnar, A. Malnic and D. Marušic, Chirality of toroidal molecular graphs, J. Chem. Inf. Mod. 45 (2005), 1527-1535. [37] K. Kutnar, A. Malnic, D. Marušic and Š. Miklavic, Distance-balanced graphs: symmetry conditions, Discrete Math. 306 (2006), 1881-1894. [38] K. Kutnar, A. Malnic, D. Marušic and Š. Miklavic, The strongly distance-balanced property of the generalized Petersen graphs, Ars Math. Contemp. 2 (2009), 41-47. [39] K. Kutnar and D. Marušic, Hamiltonicity of vertex-transitive graphs of order 4p, European J. Combin. 29 (2008), 423-438. [40] K. Kutnar and D. Marušic, On cyclic edge-connectivity of fullerenes, Discrete Appl. Math. 156 (2008), 1661-1669. [41] K. Kutnar and D. Marušic, Recent trends and future directions in vertex-transitive graphs, Ars Math. Contemp. 1 (2008), 112-125. [42] K. Kutnar and D. Marušic, A complete classification of cubic symmetric graphs of girth 6, J. Combin. Theory Ser. B 99 (2009), 162-184. [43] K. Kutnar and D. Marušic, Hamilton cycles and paths in vertex-transitive graphs— current directions, Discrete Math. 309 (2009), 5491-5500. [44] K. Kutnar and D. Marušic, On certain graph theory application, Lecture notes in economics and mathematical systems 613 (2009), 283-291. [45] K. Kutnar and D. Marušic, Some topics in graph theory, Lecture notes in economics and mathematical systems 613 (2009), 3-22. [46] K. Kutnar, D. Marušic and D. Janežic, Fullerenes via their automorphism groups, MATCH Commun. Math. Comput. Chem. 63 (2010), 267-282. [47] K. Kutnar, D. Marušic, Š. Miklavic and P. Šparl, Strongly regular tri-Cayley graphs, European J. Combin. 30 (2009), 822-832. [48] K. Kutnar, D. Marušic, D. W. Morris, J. Morris and P. Šparl, Hamiltonian cycles in Cayley graphs whose order has few prime factors, Ars Math. Contemp. 5 (2012), 27-71. [49] K. Kutnar, D. Marušic and P. Šparl, An infinite family of half-arc-transitive graphs with universal reachability relation, European J. Combin. 31 (2010), 1725-1734. [50] K. Kutnar, D. Marušič and D. Vukičević, On decompositions of leapfrog fullerenes, J. Math. Chem. 45 (2009), 406-416. [51] K. Kutnar, D. Marušic and C. Zhang, Hamilton paths in vertex-transitive graphs of order 10p, European J. Combin. 33 (2012), 1043-1077. [52] K. Kutnar, D. Marušic and C. Zhang, On cubic non-Cayley vertex-transitive graphs, J. Graph Theory 69 (2012), 77-95. [53] C. Li, D. Marušic and J. Morris, Classifying arc-transitive circulants of square-free order, J. Algebraic Combin. 14 (2001), 145-151. [54] C. H. Li, Z. P. Lu and D. Marušic, On primitive permutation groups with small suborbits and their orbital graphs, J. Algebra 119 (2004), 749-770. [55] M. Lovrecic Saražin and D. Marušic, Vertex-transitive expansions of (1, 3)-trees, Discrete Math. 310 (2010), 1772-1782. [56] A. Malnic and D. Marušic, Imprimitive groups and graph coverings, in: Coding theory, design theory, group theory (Burlington, VT, 1990), Wiley, New York, Wiley-Intersci. Publ., pp. 227-235, 1993. [57] A. Malnic and D. Marušic, Constructing 4-valent 1 -transitive graphs with a non-solvable automorphism group, J. Combin. Theory Ser. B 15 (1999), 46-55. [58] A. Malnic and D. Marušic, Constructing 1 -arc-transitive graphs of valency 4 and vertex stabilizer Z2 x Z2, Discrete Math. 245 (2002), 203-216. [59] A. Malnic, D. Marušic, Š. Miklavic and P. Potocnik, Semisymmetric elementary abelian covers of the Möbius-Kantor graph, Discrete Math. 301 (2007), 2156-2175. [60] A. Malnic, D. Marušic, R. G. Möller, N. Seifter, V. Trofimov and B. Zgrablic, Highly arc transitive digraphs: reachability, topological groups, European J. Combin. 26 (2005), 19-28. [61] A. Malnic, D. Marušic and P. Potocnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004), 71-97. [62] A. Malnic, D. Marušic and P. Potocnik, On cubic graphs admitting an edge-transitive solvable group, J. Algebraic Combin. 20 (2004), 99-113. [63] A. Malnic, D. Marušic, P. Potocnik and C. Wang, An infinite family of cubic edge-but not vertex-transitive graphs, Discrete Math. 280 (2004), 133-148. [64] A. Malnic, D. Marušic and N. Seifter, Constructing infinite one-regular graphs, European J. Combin. 20 (1999), 845-853. [65] A. Malnic, D. Marušic, N. Seifter, P. Šparl and B. Zgrablic, Reachability relations in digraphs, European J. Combin. 29 (2008), 1566-1581. [66] A. Malnic, D. Marušic, N. Seifter and B. Zgrablic, Highly arc-transitive digraphs with no homomorphism onto Z, Combinatorica 22 (2002), 435-443. [67] A. Malnic, D. Marušic and P. Šparl, On strongly regular bicirculants, European J. Combin. 28 (2007), 891-900. [68] A. Malnic, D. Marušic, P. Šparl and B. Frelih, Symmetry structure of bicirculants, Discrete Math. 307 (2007), 409-414. [69] A. Malnic, D. Marušic and C. Wang, Cubic edge-transitive graphs of order 2p3, Discrete Math. 274 (2004), 187-198. [70] A. Marušic, K. Belšak and D. Marušic, Sensitization of emotions as a risk factor for ischemic heart disease, Psychiatria Danub. 20 (2008), 30-35. [71] A. Marušic and D. Marušic, Suicidology - expanding its research tools, Crisis 23 (2002), 95-97. [72] D. Marušic, Vertex minimal planar cyclic graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 1981 (1980), 161-169 (1981). [73] D. Marušic, On vertex symmetric digraphs, Discrete Math. 36 (1981), 69-81. [74] D. Marušic, Cayley properties of vertex symmetric graphs, Ars Combin. 16 (1983), 297-302. [75] D. Marušic, Hamiltonian circuits in Cayley graphs, Discrete Math. 46 (1983), 4954. [76] D. Marušic, Some problems in vertex symmetric graphs, in: Finite and infinite sets, Vol. I, II (Eger, 1981), North-Holland, Amsterdam, volume 37 of Colloq. Math. Soc. Jànos Bolyai, pp. 581-589, 1984. [77] D. Marušic, Vertex transitive graphs and digraphs of order pk, in: Cycles in graphs (Burnaby, B.C., 1982), North-Holland, Amsterdam, volume 115 of North-Holland Math. Stud., pp. 115-128, 1985. [78] D. Marušic, Hamiltonian cycles in vertex symmetric graphs of order 2p2, Discrete Math. 66 (1987), 169-174. [79] D. Marušic, On vertex-transitive graphs of order qp, J. Combin. Math. Combin. Com-put. 4 (1988), 97-114. [80] D. Marušic, Research problems: Problem 92, problem 93, Discrete Math. 70 (1988), 215. [81] D. Marušič, Strongly regular bicirculants and tricirculants, Ars Combin. 25 (1988), 11-15, eleventh British Combinatorial Conference (London, 1987). [82] D. Marušic, Strong regularity and circulant graphs, Discrete Math. 78 (1989), 119125. [83] D. Marušic, Hamiltonicity of tree-like graphs, Discrete Math. 80 (1990), 167-173. [84] D. Marušic, On tree-like cubic vertex-transitive graphs, Ars Combin. 29 (1990), 9196, twelfth British Combinatorial Conference (Norwich, 1989). [85] D. Marušic, The seven "weirdos", Obzornik Mat. Fiz. 37 (1990), 105-108. [86] D. Marušic, Hamiltonicity of vertex-transitive pq-graphs, in: Fourth Czechoslo-vakian Symposium on Combinatorics, Graphs and Complexity (Prachatice, 1990), North-Holland, Amsterdam, volume 51 of Ann. Discrete Math., pp. 209-212, 1992. [87] D. Marušic, Research problems: Problems 211-212, Discrete Math. 131 (1994), 397-398. [88] D. Marušic, A family of one-regular graphs of valency 4, European J. Combin. 18 (1997), 59-64. [89] D. Marušic, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory Ser. B 73 (1998), 41-76. [90] D. Marušic, Recent developments in half-transitive graphs, Discrete Math. 182 (1998), 219-231, graph theory (Lake Bled, 1995). [91] D. Marušic, Constructing cubic edge- but not vertex-transitive graphs, J. Graph Theory 35 (2000), 152-160. [92] D. Marušic, In memoriam: some personal impressions of Crispin Nash-Williams, Discrete Math. 244 (2002), 2-4, algebraic and topological methods in graph theory (Lake Bled, 1999). [93] D. Marušic, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003), 162-196, dedicated to Crispin St. J. A. Nash-Williams. [94] D. Marušic, Quartic half-arc-transitive graphs with large vertex stabilizers, Discrete Math. 299 (2005), 180-193. [95] D. Marušic, Corrigendum to: "On 2-arc-transitivity of Cayley graphs" [J. Combin. Theory Ser. B 87 (2003), no. 1, 162-196],(2006), 761-764. [96] D. Marušic, Hamilton cycles and paths in fullerenes, J. Chem. Inf. Mod. 47 (2007), 732-736. [97] D. Marušic, Preface: Special issue of discrete mathematics on Hamiltonicity problem for vertex-transitive (Cayley) graphs, Discrete Math. 309 (2009), 5425. [98] D. Marušic and J. Morris, Normal circulant graphs with noncyclic regular subgroups, J. Graph Theory 50 (2005), 13-24. [99] D. Marušic and R. Nedela, Maps and half-transitive graphs of valency 4, European J. Combin. 19 (1998), 345-354. [100] D. Marušic and R. Nedela, On the point stabilizers of transitive groups with non-self-paired suborbits of length 2, J. Group Theory 4 (2001), 19-43. [101] D. Marušic and R. Nedela, Partial line graph operator and half-arc-transitive group actions, Math. Slovaca 51 (2001), 241-257. [102] D. Marušic and R. Nedela, Finite graphs of valency 4 and girth 4 admitting halftransitive group actions, J. Aust. Math. Soc. 73 (2002), 155-170. [103] D. Marušic and T. D. Parsons, Hamiltonian paths in vertex-symmetric graphs of order 5p, Discrete Math. 42 (1982), 227-242. [104] D. Marušic and T. D. Parsons, Hamiltonian paths in vertex-symmetric graphs of order 4p, Discrete Math. 43 (1983), 91-96. [105] D. Marušic and T. Pisanski, Weakly flag-transitive configurations and half-arc-transitive graphs, European J. Combin. 20 (1999), 559-570. [106] D. Marušic and T. Pisanski, The Gray graph revisited, J. Graph Theory 35 (2000), 1-7. [107] D. Marušic and T. Pisanski, The remarkable generalized Petersen graph G(8, 3), Math. Slovaca 50 (2000), 117-121. [108] D. Marušic and T. Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croat. Chem. Acta 73 (2000), 969-981. [109] D. Marušic, T. Pisanski and S. Wilson, The genus of the GRAY graph is 7, European J. Combin. 26 (2005), 377-385. [110] D. Marušic and P. Potocnik, Semisymmetry of generalized Folkman graphs, European J. Combin. 22 (2001), 333-349. [111] D. Marušic and P. Potocnik, Bridging semisymmetric and half-arc-transitive actions on graphs, European J. Combin. 23 (2002), 719-732. [112] D. Marušic and P. Potocnik, Classifying 2-arc-transitive graphs of order a product of two primes, Discrete Math. 244 (2002), 331-338, algebraic and topological methods in graph theory (Lake Bled, 1999). [113] D. Marušic and C. E. Praeger, Tetravalent graphs admitting half-transitive group actions: alternating cycles, J. Combin. Theory Ser. B 75 (1999), 188-205. [114] D. Marušic and R. Scapellato, Characterizing vertex-transitive pq-graphs with an imprimitive automorphism subgroup, J. Graph Theory 16 (1992), 375-387. [115] D. Marušic and R. Scapellato, A class of non-Cayley vertex-transitive graphs associated with PSL(2,p), Discrete Math. 109 (1992), 161-170, algebraic graph theory (Leibnitz, 1989). ARS MATHEMATICA CONTEMPORANEA [116] D. Marušic and R. Scapellato, Imprimitive representations of SL(2,2k), J. Combin. Theory Ser. B 58 (1993), 46-57. [117] D. Marušic and R. Scapellato, A class of graphs arising from the action of PSL(2, q2) on cosets of PGL(2, q), Discrete Math. 134 (1994), 99-110, algebraic and topological methods in graph theory (Lake Bled, 1991). [118] D. Marušic and R. Scapellato, Classification of vertex-transitive pq-digraphs, Istit. Lombardo Accad. Sci. Lett. Rend. A 128 (1994), 31-36 (1995). [119] D. Marušic and R. Scapellato, Classifying vertex-transitive graphs whose order is a product of two primes, Combinatorica 14 (1994), 187-201. [120] D. Marušic and R. Scapellato, Permutation groups with conjugacy complete stabilizers, Discrete Math. 134 (1994), 93-98, algebraic and topological methods in graph theory (Lake Bled, 1991). [121] D. Marušic and R. Scapellato, Permutation groups, vertex-transitive digraphs and semiregular automorphisms, European J. Combin. 19 (1998), 707-712. [122] D. Marušic, R. Scapellato and N. Zagaglia Salvi, A characterization of particular symmetric (0,1) matrices, Linear Algebra Appl. 119 (1989), 153-162. [123] D. Marušic, R. Scapellato and N. Zagaglia Salvi, Generalized Cayley graphs, Discrete Math. 102 (1992), 279-285. [124] D. Marušic, R. Scapellato and B. Zgrablic, On quasiprimitive pqr-graphs, Algebra Colloq. 2 (1995), 295-314. [125] D. Marušic and P. Šparl, On quartic half-arc-transitive metacirculants, J. Algebraic Combin. 28 (2008), 365-395. [126] D. Marušic and A. O. Waller, Half-transitive graphs of valency 4 with prescribed attachment numbers, J. Graph Theory 34 (2000), 89-99. [127] D. Marušic and M.-Y. Xu, A 1 -transitive graph of valency 4 with a nonsolvable group of automorphisms, J. Graph Theory 25 (1997), 133-138. [128] T. Pisanski, M. Boben, D. Marušic, A. Orbanic and A. Graovac, The 10-cages and derived configurations, Discrete Math. 275 (2004), 265-276. [129] R. Šcepanovic, G. Ringel, D. Marušic, G. L. Chia and B. 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ARS MATHEMATICA CONTEMPORANEA Contents Odd-order Cayley graphs with commutator subgroup of order pq are hamil- tonian Dave Witte Morris .............................. 1 Arc-transitive graphs of valency 8 have a semiregular automorphism Gabriel Verret.................................29 Hamilton paths in Cayley graphs on Coxeter groups: I Brian Alspach................................. 35 Polarity graphs revisited Martin Bachraty, Jozef Sirän......................... 55 Tight orientably-regular polytopes Marston Conder, Gabe Cunningham.....................69 Reachability relations, transitive digraphs and groups Aleksander Malnic, Primož Potočnik, Norbert Seifter, Primož Šparl .... 83 The clone cover Aleksander Malnic, Tomaž Pisanski, Arjana Žitnik.............95 Unstable graphs: A fresh outlook via TF-automorphisms Josef Lauri, Russell Mizzi, Raffaele Scapellato ...............115 A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two Primož Potocnik, Pablo Spiga, Gabriel Verret................133 Some recent discoveries about half-arc-transitive graphs Marston D. E. Conder, Primož Potocnik...................149 A note on m-factorizations of complete multigraphs arising from designs György Kiss, Christian Rubio-Montiel....................163 Regular embeddings of cycles with multiple edges revisited Kan Hu, Roman Nedela, Martin Škoviera..................177 Strongly regular m-Cayley circulant graphs and digraphs Luis Martinez.................................195 On automorphism groups of graph truncations Brian Alspach, Edward Dobson .......................215 The equalization scheme of the residual voluntary health insurance in Slovenia Boris Zgrablic.................................225 Volume 8, Number 1, Spring/Summer 2015, Pages 1-234 ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 1-28 Odd-order Cayley graphs with commutator subgroup of order pq are hamiltonian* Dave Witte Morris Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K3M4, Canada Received 1 May 2012, accepted 8 August 2013, published online 11 April 2014 We show that if G is a nontrivial, finite group of odd order, whose commutator subgroup [G, G] is cyclic of order pMqv, where p and q areprime, then every connected Cayley graph on G has a hamiltonian cycle. Keywords: Cayley graph, hamiltonian cycle, commutator subgroup. Math. Subj. Class.: 05C25, 05C45 1 Introduction It has been conjectured that there is a hamiltonian cycle in every connected Cayley graph on any finite group, but all known results on this problem have very restrictive hypotheses (see [2, 13, 15] for surveys). One approach is to assume that the group is close to being abelian, in the sense that its commutator subgroup is small. This is illustrated by the following theorem that was proved in a series of papers by Marušic [12], Durnberger [3, 4], and Keating-Witte [10]: Theorem 1.1 (D.Marušic, E.Durnberger, K.Keating, and D.Witte, 1985). If G is a nontrivial, finite group, whose commutator subgroup [G, G] is cyclic of order pM, where p prime and ^ G N, then every connected Cayley graph on G has a hamiltonian cycle. Under the additional assumption that G has odd order, we extend this theorem, by allowing the order of [G, G] to be the product of two prime-powers: Theorem 1.2. If G is a nontrivial, finite group of odd order, whose commutator subgroup [G, G] is cyclic of order pMqv, where p and q are prime, and g N, then every connected Cayley graph on G has a hamiltonian cycle. *To Dragan Marušic on his 60th birthday. E-mail address: Dave.Morris@uleth.ca, http://people.uleth.ca/~dave.morris/ (Dave Witte Morris) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Remark 1.3. Of course, we would like to prove the conclusion of Theorem 1.2 without the assumption that |G| is odd, or with a weaker assumption on the order of [G, G]. If M, v < 1, then there is no need to assume that [G, G] is cyclic: Corollary 1.4. If G is a nontrivial, finite group of odd order, whose commutator subgroup [G, G] has order pq, where p and q are distinct primes, then every connected Cayley graph on G has a hamiltonian cycle. This yields the following contribution to the ongoing search [11] for hamiltonian cycles in Cayley graphs on groups whose order has few prime factors: Corollary 1.5. If p and q are distinct primes, then every connected Cayley graph of order 9pq has a hamiltonian cycle. Here is an outline of the paper. Miscellaneous definitions and preliminary results are collected in Section 2. (Also, Corollaries 1.4 and 1.5 are derived from Theorem 1.2 in §2E.) The paper's main tool is a technique known as "Marusic's Method." A straightforward application of this method is given in Section 3, and some other consequences are in Section 4. The proof of Theorem 1.2 is in Section 5, except that one troublesome case is postponed to Section 6. Acknowledgments. I thank D. Marusic for suggesting this research problem. I also thank him, K. Kutnar, and other members of the Faculty of Mathematics, Natural Sciences, and Information Technologies of the University of Primorska (Koper, Slovenia), for their excellent hospitality that supported the early stages of this work. Furthermore, I am grateful to an anonymous referee for helpful comments on an earlier version of this manuscript. 2 Preliminaries 2A Assumptions, definitions, and notation Assumption 2.1. (1) G is always a finite group. (2) S is a generating set for G. Definition 2.2. The Cayley graph Cay (G; S ) is the graph whose vertex set is G, with an edge from g to gs and an edge from g to gs-1, for every g g G and s G S. Notation 2.3. • We let G' = [G, G] and G = G/G'. Also, for g g G, we let g = gG' be the image of g in G. • For g, h G G, we let gh = h-1gh and [g, h] = g-1h-1 gh. • If H is an abelian subgroup of G and k g Z, we let Hk = { hk | h g H }. This is a subgroup of H (because H is abelian). Notation 2.4. For g G G and S1;..., sn e S U S 1, we use [g](si,..., sn) to denote the walk in Cay(G; S) that visits (in order), the vertices g, gsi, gSiS2, gSlS2S3, .. . , gSlS2 • • • Sn. We often write (s1;..., sn) for [e](s1,..., sn). Definition 2.5. Suppose • N is a normal subgroup of G, and • С = (sj)n=1 is a hamiltonian cycle in Cay(G/N; S). The voltage of C is ПП=1 Sj. This is an element of N, and it may be denoted ПС. Remark 2.6. If С = [g](sb ..., Sn), and N is abelian, then ПП=1 Sj = (nC)g. Proof. There is some i with (s1s2 • • • s^)g e N. Then С = (s£+1, S£+2, .. ., Sn, S1, S2, .. ., S£), so (ПС)g = g-1(S£+1S£+2 • • • Sn S1S2 • • • S£)g = [(S1S2 • • • S£)g] 1 (S1S2 • • • S£)(S£+1S£+2 • • • Sn) [(S1S2 • • • S£)g] = (S1S2 ••• Sn)(sis2 = S1S2 • • • Sn. □ 2B Factor Group Lemma and Marušič's Method Lemma 2.7 ("Factor Group Lemma" [15, §2.2]). Suppose • N is a cyclic, normal subgroup of G, • (Si)m=1 is a hamiltonian cycle in Cay(G/N; S), and • the product s1s2 • • • Sm generates N. Then (s1; S2,..., Sm)|N 1 is a hamiltonian cycle in Cay(G; S). The following simple observation allows us to assume |N| is square-free whenever we apply the Factor Group Lemma (2.7). Lemma 2.8 ([10, Lem. 3.2]). Suppose • N is a cyclic, normal subgroup of G, • N = N/Ф is the maximal quotient of N that has square-free order, • G = G/Ф, • (s1 , S2,..., Sm) is a hamiltonian cycle in Cay(G/N; S), and • the product s^ s2 • • • Srn. generates N. Then S1S2 • • • Sm generates N, so (s1; s2, ..., Sm)|N 1 is a hamiltonian cycle in Cay(G; S). Remark 2.9 (cf. [7, Thm. 5.1.1]). When applying Lemma 2.8, it is sometimes helpful to know that if • N, N = N/Ф, and G = G/Ф are as in Lemma 2.8, and • S is a minimal generating set of G. Then S is a minimal generating set of G. Lemma 2.10 ("MarusiC's Method" [12], cf. [10, Lem. 3.1]). Suppose • So C S, • (S0) contains G', • there are hamiltonian cycles C1;..., Cr in Cay( (S0)/G'; S0) that all have an oriented edge in common, and • for every 7 g G', there is some i, such that (7 • nCj) = G'. Then there is a hamiltonian cycle in Cay (G/G'; S ) whose voltage generates G'. Hence, the Factor Group Lemma (2.7) provides a hamiltonian cycle in Cay(G; S). Corollary 2.11. Assume G' = Zp x Zq, where p and q are distinct primes. Then, in the situation of Marusic's Method (2.10), the final condition (*) can be replaced with either of the following: (1) r = 3, and ((nCi)-1(nCj)) = G' whenever 1 < i < j < 3. (2) r = 4, and • ( (nCi)-1 (nC2 )) contains Zp, and • ((nCi)-1(nC3 )) = ((nC2)-1(nC4)) = Zq. Proof. Let 7 g G'. (1) Consider the three elements 7 •nC1,7 ^C2, and 7 nC3 of Zp xZq. By assumption, no two have the same projection to Zp, so only one of them can have trivial projection. Similarly for the projection to Zq. Therefore, there is some i, such that 7 • nCj projects nontrivially to both Zp and Zq. Therefore (7 • nCj) = G'. (2) There is some i g {1, 2}, such that 7 • nCj projects nontrivially to Zp. We may assume the projection of 7 • nCj to Zq is trivial (otherwise, we have (7 • nCj) = G', as desired). Then 7 • nCj+2 has the same (nontrivial) projection to Zp, but has a different (hence, nontrivial) projection to Zq. So (7 • nCj+2) = G'. □ 2C Some known results We recall a few results that provide hamiltonian cycles in Cay(G; S) under certain assumptions. Theorem 2.12 (Witte [14]). If |G| = pM, where p is prime and ^ > 0, then every connected Cayley digraph on G has a directed hamiltonian cycle. Theorem 2.13 (Ghaderpour-Morris [6]). If G is a nontrivial, nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle. Theorem 2.14 (Ghaderpour-Morris [5]). If |G| = 27p, where p is prime, then every connected Cayley graph on G has a hamiltonian cycle. The proof of the preceding theorem has the following consequence. Corollary 2.15. If G is a finite group, such that |G/G'| = 9 and G' is cyclic of order pM • 3V, where p > 5 is prime, then every connected Cayley graph on G has a hamiltonian cycle. Proof. Let G = G/(G')3p. Then |G| = 27p and |G'| = 3p, so the proof of [5, Props. 3.4 and 3.6] provides a hamiltonian cycle in Cay (G/G'; S) whose voltage generates G'. Then Lemma 2.8 provides a hamiltonian cycle in Cay(G; S). □ Theorem 2.16 (Alspach [1, Thm. 3.7]). Suppose • s G S, • (s) 9. We may assume S is minimal, so #S = 2; write S = {a, b}. Then we have the following two hamiltonian cycles in Cay(G; S): C1 = (a2, b)3 and C2 = (a2,b-1)3. Since Lemma 2.21(2) tells us (xy)3 = x3 y3 for all x,y G G, and we have x3 G G' = Z (G) for all x g G, we see that (nCi)-1(nC2) = ((a2b)3)-1(a2b-1)3 = ((a2)3b3)-1 ((a2)3(b-1)3) = b-6 = e, since |b| > 9. □ We will use the following version of this result in Subcase ii of Case 5.12. Proposition 4.2. Suppose • | G| is odd, • G' = Zp has prime order, • Z is a subgroup of Z (G), • S n G'Z = 0, and • G is not nilpotent. Then there exist hamiltonian cycles C1 and C2 in Cay(G/(G'Z); S) that have an oriented edge in common, such that ((ПС1)-1(ПС2)) = G'. Proof. Choose a, b G S with [a, b] = e. Since G is not nilpotent, we may assume a does not centralize G'. Furthermore, since we are using Marusic's Method (2.10), there is no harm in assuming S = {a, b}. If b G (a, G', Z), then the proof of [10, Case 5.3] provides two hamiltonian cycles C = (si)n=1 and C = (ti)n=1 in Cay(G/(G'Z); a, b), such that ПС1 = ПС2 (and the two cycles have an oriented edge in common). From the construction, it is clear that (si)n=1 is a permutation of (ti)n=1, so (ПС0-1(ПС2) G G'. We may now assume b G (a, G', Z). Then, letting n = |G : (a, G', Z)|, there is some i, such that bi g aiG'Z and 0 < i < n. Therefore, we have the following two hamiltonian cycles in Cay(G/(G'Z); S) that both contain the oriented edge (b): C1 = (b, a-(i-1),b,an-i-1), C2 = (b, an-i-1,b,a-(i-1)) = [a-1]C1. The sequence of edges in C2 is a permutation of the sequence of edges in C1, therefore (ПС1)-1(ПС2) g G'. Also, since a does not centralize G', it is not difficult to see that (ПС1)-1(ПС2) is nontrivial, and therefore generates G'. □ Lemma 4.3. Assume • G = Zp^ x Zqv, where p and q are prime, • S n G' = 0, • there exist a, b G S U S-1, with a = b, such that aG' = bG', • the generating set S is minimal, and • |G| is odd. Then there is a hamiltonian cycle in Cay(G; S). Proof. Write b = a7, with 7 g G'. Case 1. Assume (7) = G'. We apply Marusic's Method (2.10), so Lemma 2.8 allows us to assume G' = Zp x Zq. Since |a| > 3, it is easy to find an oriented hamiltonian cycle C0 in Cay(G; S) that has (at least) 2 oriented edges a1 and a2 that are labeled a. We construct two more hamiltonian cycles C1 and C2 by replacing one or both of a1 and a2 with a b-edge. (Replace one a-edge to obtain C1; replace both to obtain C2.) Then there are conjugates 71 and 72 of 7, such that (nC0)-1(nC1) = 71, (nC1)-1(nC2) = 72, (nC0)-1(nC2) = 7172. By the assumption of this case, we know that 71 and 72 generate G'. Also, since |G| is odd, we know that no element of G inverts any nontrivial element of G', so 7172 also generates G'. Therefore, Marusic's Method 2.11(1) applies. Case 2. Assume (7) = G'. Since S is minimal, we know (7) contains either Zp^ or Zqv. By the assumption of this case, we know it does not contain both. So let us assume (7) = N x Zqv, where N is a proper subgroup of Zp^. Assume, for the moment, that G/(G')p is not the nonabelian group of order 27 and exponent 3. We use Marusic's Method (2.10), so Lemma 2.8 allows us to assume G' = Zp x Zq. Applying Theorem 4.1 to G/Zq provides us with hamiltonian cycles C1 and C2 in Cay(G/G'; S \ {b}), such that ((nC1)-1(nC2)) contains Zp. (Furthermore, the two cycles have an oriented edge in common.) Since S is a minimal generating set, we know that Cj contains an edge labelled a±1. (In fact, more than one, so we can take one that is not the edge in common with the other cycle.) Assume, without loss of generality, that it is labelled a. Replacing this edge with b results in a hamiltonian cycle Cj, such that ((nCj)-1(nCj)) = (7) = Zq. Then Marusic's Method 2.11(2) applies. We may now assume that G/(G')p is the nonabelian group of order 27 and exponent 3. Then G/ (7) is a 3-group, so Theorem 2.12 )ells us there is a directed hamiltonian cycle C0 in the Cayley digraph Cay(G/ (7); S \ {b}). Since S \ {b} is a minimal generating set of G/(7), there must be at least two edges a1 and a2 that are labeled a in C. Now the proof of Case 1 applies (but with (7) in the place of G'). □ 5 Proof of Theorem 1.2 Assumption 5.1. We always assume: (1) The generating set S is minimal. (2) S n G' = 0 (see Corollary 2.17). (3) p and q are distinct (see Theorem 1.1). (4) G is not nilpotent (see Theorem 2.13). This implies G/(G')pq is not nilpotent [9, Satz 3.5, p. 270]. (5) There do not exist a, b g S U S-1 with a = b and aG' = bG' (see Lemma 4.3). (6) There does not exist s g S, such that G' C (s) (see Theorem 2.16). Remark 5.2. We consider several cases that are exhaustive up to permutations of the variables a, b, and c, and interchanging p and q. Here is an outline of the cases: • There exist a, b G S, such that ([a, b]) = G'. (5.3) b e (a). (5.4) b e (a) and |a| > 5. (5.5) |a| = |b| = 3 and (a) = (b). • There exist a, b, c e S, such that Zp^ С ([a, b]) and Zqv С ([a, c]). (5.7) b,c e (a). (5.8) (a) C (a,b) C (a,b,c). (5.9) a centralizes G'/(G')pq. (5.10) b,c e (a). (5.11) c e (a) and b e (a). • There do not exist a, b, c e S, such that ([a, b], [a, c]) = G'. (5.12) Case 5.3. Assume there exist a, b e S, such that ([a, b]) = G' and b e (a). Proof. We use Marusic's Method (2.11), so there is no harm in assuming S = {a, b}. Then (a) = (a, b) = G. Furthermore, Lemma 2.8 allows us to assume G' — Z pq. Let n = |a| = |G|, fix k with b = ak, and choose 7 e G', such that b = ak7. Note that • an = e (since Corollary 2.20 implies that a cannot centralize a nontrivial subgroup of G'), and • (7) = G' (since (a) к (7) = (a, b) = G). We may assume 1 < k < n/2, by replacing b with its inverse if necessary. We may also assume n > 5 (otherwise, we must have k =1, contrary to Assumption 5.1(5)). Therefore n - k - 2 > 0. We have the following three hamiltonian cycles in Cay(G; a, b): C1 = (an), C2 = (an-k-1,b,a-(k-1),b), C3 = (an-k-2, b, a-(k-1), b, a). Their voltages are nC1 = an = e, nC2 = an-k-1ba-(k-1)b = an-k-1(ak7)a-(k-1)(ak7) = an • a-17a7 = 7a7, nC3 = an-k-2ba-(k-1)ba = a-1(an-k-1ba-(k-1) b)a = (nC2)a. Since |G| is odd, we know that a does not invert Zp or Zq. Therefore nC2 generates G'. Hence, the conjugate nC3 must also generate G'. Furthermore, as was mentioned above, we know that a does not centralize any nontrivial element of G', so (nC2)(nC3)-1 also generates G'. (Also note that all three hamiltonian cycles contain the oriented edge (a).) Hence, Marusic's Method 2.11(1) applies. □ Case 5.4. Assume there exist a, b e S, such that ([a, b]) = G' and be (a). Also assume |a| > 5. Proof (cf. proof of [10, Case 4.3]). We use Marusic's Method (2.11), so there is no harm in assuming S — {a, b}. Furthermore, Lemma 2.8 allows us to assume G' — Zpq. Let d — |G/(a)|, so there is some r with b ar — e and 0 < r < |a|. We may assume r < |a| — 2, by replacing b with its inverse if necessary. Applying Corollary 3.3 to the hamiltonian cycle (b-d) yields hamiltonian cycles C0, Gi, and C2 (since 2 — 5 — 3 < |a| — 3). Note that all of these contain the oriented edge b(b-1). Furthermore, the voltage of Ck is nCk — n[a-k,b] [a-k,b]a-, where n — ПС0 is independent of k. Since [a-1, b] generates G', and a does not invert any nontrivial element of G' (recall that |G| is odd), it is easy to see that G' is generated by the difference of any two of e, [a-1, b], and [a-2, b] — [a-1, b][a-1, b]a-. Using again the fact that a does not invert any element of G', this implies that G' is generated by the difference of any two of the three voltages, so Marusic's Method 2.11(1) applies. □ Case 5.5. Assume there exist a, b G S, such that ([a, b]) — G', |a| — |b| — 3 and (a) — (6). Proof. This proof is rather lengthy. It can be found in Section 6. □ Assumption 5.6. Henceforth, we assume there do not exist a, b G S U S-1, such that ([a, b]) — G'. Case 5.7. Assume Zp^ C ([a, b]), Zqv С ([a, c]), and (b,c) С (a). Proof. We use Marusic's Method (2.11), so there is no harm in assuming S — {a, b, c}. (Furthermore, Lemma 2.8 allows us to assume G' — Zpq, so ([a, b]) — Zp and ([a, c]) — Zq.) Then, since b, c G (a), we must have (a) — G. Therefore, Corollary 2.20 tells us that a does not centralize any nonidentity element of G'. Fix k and i with b — and c — We may write b — ak y1 and c — a£Y2, for some y1 g Zp and y2 G Yq. Since 1, k, and i are distinct (see Assumption 5.1(5)), we may assume 1 < k < i < n/2, by interchanging b and c and/or replacing b and/or c with its inverse if necessary. Therefore i > 3 and k + i < n — 2, so we have the following three hamiltonian cycles in Cay(G; a, b, c): C1 — (a-n) C2 — (a-(£-1),c, b, a-(k-1), b, an-fc-£-2, c) Сз — (a-(£-2U b, a-(k-1), b, an-fc-£-2, c,a-1). Note that each of these contains the oriented edge (a-1). Since a does not centralize any nonidentity element of G', we know nC1 — e. A straightforward calculation shows nC2 — (Y1Y?-1 r" (Ya-1 Y2), which generates G'. Therefore, nC3 — (nC2)° 1 and (nC2)-1(nC3) also generate G'. (For the latter, note that a-1 does not centralize any nonidentity element of G'.) Therefore Marusic's Method 2.11(1) applies. □ Figure 3: A hamiltonian cycle X. e Case 5.8. Assume Zp^ С ([a, b]), Zqv С ([a, c]), and there exists s G {a, b}, such that (a) C (a, s) C (a, b, C). Proof. We use Marusic's Method (2.11), so there is no harm in assuming S = {a, b, c}. Furthermore, Lemma 2.8 allows us to assume G' = Zpq, so ([a, b]) = Zp and ([a, c]) = Zq. Choose A, B, C > 3, such that aA = e, and every element of G can be written uniquely in the form 0 < x < A, cz with 0 < y < B, 0 < z < C. More precisely, we may let JA = |a|, B = |(a,b) : (a)|, C = |G : (a, b)| if s = b, \ A = |a|, C = |(a, c) : (a)|, B = |G : (a,c)| if s = c. Then we have the following hamiltonian cycle X in Cay(G; a, b, c) (see Figure 3): X = [a,(aA-2, (b,a-(A-1),b,aA-1) (a-(A-1), b-1, aA-1, b-1)(B-1)/2, a-(A-2), c) b, a-1, bB-2, a, (aA-2, b-1, a-(A-2), b-1)(B-3)/2, aA-2,b-1 ,a-(A-3),b-1,aA-2,c-(C-1) ,a - - b,a" 0(B-1)/2,c, ,(C-1)/2 a ^ b We obtain a new hamiltonian cycle Xp by replacing a subpath of the form [g] (aA 1, b, a-(A-1)) with [g] (a-(A-1) ,b,aA-1). Then (nX )-1(nXp) is a conjugate of (aA-1ba-(A-1))-1 (a-(A-1)baA-1 ) = [b, aA-1]a[b, aA-1]. Similarly, replacing a subpath of the form [g](aA-1, c, a-(A-1)) with [g^a-(A-1),c, aA-1) results in a hamiltonian cycle Xq, such that (nX)-1(nXq) is a conjugate of [c, aA-1]a[c, aA-1]. Furthermore, doing both replacements results in a hamiltonian cycle Xp, such that (nXp)-1 (nXp) is also a conjugate of [c, aA-1]a[c, aA-1]. Note that all four of these hamiltonian cycles contain the oriented edge c(c-1). Since G' С (a) (see Assumption 5.1(6)), we may assume aA G Zp (by interchanging p and q if necessary). Since [c, a] G Zq, this implies that c centralizes aA, so [c, aA-1] = Figure 4: A hamiltonian cycle Y1. e (C-1)/2 [c, a-1] generates Zq. Since a does not invert any nontrivial element of Z (recall that G has odd order), this implies that [c, aA-1]°[c, aA-1] generates Zq. Assume, for the moment, that [b, aA-1] generates Zp. Since a does not invert any nontrivial element of Zp, this implies that [b, aA-1]°[b, aA-1] generates Zp. Therefore, Marusic's Method 2.11(2) applies. We may now assume [b, aA-1] does not generate Zp. This means [b, aA-1] = e. Since [b, a-1] = e, we conclude that [b, aA] = e, so b does not centralize Zp. We have the following hamiltonian cycle Y1 in Cay(G; a, b, c) (see Figure 4): Y1 = (b, (bB-3, (a, b-(B-2), a, bB-2)(A-1)/2, b, a-(A-1), c, aA-1, b-1, (b-(B-2), a-1, bB-2, a-1)(A-1)/2, b-(B-3), c) bB-2, a, (aA-2, b-1, a-(A-2), b-1)(B-1)/2, aA-1, c-(C-1^ . We create a new hamiltonian cycle Y2 by replacing a subpath of the form [g] (a-(A-1), c, aA-1) with [g^aA-1, c, a-(A-1)). This is the same as the construction of Xq from X, but with a and a-1 interchanged, so the same calculation shows (ПУ1)-1(ПУ2) is a conjugate of [c, a-(A-1)f-1 [c, a-(A-1)], which generates Zq. Furthermore, since Y1 and Y2 both contain the oriented path [bB-3](b, a, b-1), and either the oriented edge [bB-2](a) or the oriented edge [bB-2a](a-1), Remark 3.2 provides hamiltonian cycles У/ and Y2', such that (ПУ4)-1 (ПУ/) generates Zp. Since all four hamiltonian cycles contain the oriented edge [c](c-1 ), Marusic's Method 2.11(2) applies. □ Case 5.9. Assume Zp^ C ([a, b]}, Zqv С ([a, c]}, and a centralizes G'/(G')pq. Proof. We use Marusic's Method (2.11), so there is no harm in assuming S = {a, b, c}. Furthermore, Lemma 2.8 allows us to assume G' = Zpq, so ([a, b]} = Zp and ([a, c]} = Zq. Note that [a, b-1 , c] G Zp, [c, a 1,b] G Zq, and [b, c 1 ,a] — e (because a centralizes G'). Since Zp n Zq = {e}, and the Three-Subgroup Lemma [7, Thm. 2.3, p. 19] tells us [a, b-1, cf[b, c-1, a]c[c, a-1, b]a = e, we conclude that [a, b-1, c] = [c, a-1, b] = e, so c centralizes Zp and b centralizes Zq. We know G' C Z (G), because G is not nilpotent (see Assumption 5.1(4)). Since a centralizes G', this implies we may assume c does not centralize G' (by interchanging b and c if necessary). So c does not centralize Zq. Since a, b, and G' all centralize Zq, this implies c G (a, b, G'). In other words, c G (a, b). Furthermore, applying Corollary 2.20 to the group (a, b) tells us that (a) = (a, b). Therefore (a) C (a, b) C (a, b, c), so Case 5.8 applies. □ Case 5.10. Assume Zp^ C ([a, b]), Zqv C ([a, c]), andb, cG (a). Proof. We use Marusic's Method (2.11), so there is no harm in assuming S = {a, b, c}. Furthermore, Lemma 2.8 allows us to assume G' = Zpq, so ([a, b]) = Zp and ([a, c]) = Zq. We may assume (a, b) = (a, c) = G, for otherwise Case 5.8 applies. Let us begin by showing that a does not centralize any nontrivial element of G'. Suppose not. Then we may assume that a centralizes Zp. Let G = G/Zq = G/([a, c]). Since (a, c, G') = G, we know that (a, c, Zp) = G , so a is in the center of G. This contradicts the fact that ([a, b]) = Zp is nontrivial. Since G is abelian (and because b,c G (a)), it is easy to choose a hamiltonian cycle (sj)d=1 in Cay(G/(a); S) that contains both an edge labeled b (or b-1) and an edge labeled c (or c-1). Note that C = ((sj)dt1, a 1 a| -1, (s--2j+1, a-( 1 a| -2), s--M, a 1 a 1 -2)£-1)/2, a) is a hamiltonian cycle in Cay(G; S). Subcase i. Assume |a| > 3. We may assume s1 = b-1 and s2 = c-1. Then C0 contains the four subpaths (b-1), [b-1a2](a-1, b, a), [b-1](c-1), [b-1 c-1a-2](a, c, a-1). Therefore, we may let g be either b-1 or b-1c-1 in Lemma 3.1, so Remark 3.2(2) tells us we have hamiltonian cycles Cb and Cc, such that (nC0)-1 (nCb) is a generator of Zp, and (nC0)-1(nCc) is a generator of Zq. Since |a| > 3, we see that Cb, like C0, contains [b-1]( c 1) and [b 1c 1a 2](a, c, a 1), so Remark 3.2(2) provides a hamiltonian cycle Cb, such that (nCb)-1 (nCb) is a generator of Zq. Therefore, Marusic's Method 2.11(2) applies (since each of these four hamiltonian cycles contains the oriented edge [a-1](a)). Subcase ii. Assume d > 3. We may assume s1 = b-1 and s3 = c-1. Then C0 contains the four subpaths (b-1), [b-1a2](a-1, b, a), [s1s2](c-1), [s1s2c-1a2](a-1, c, a). Therefore, we may let g be either b-1 or s1s2c-1 in Lemma 3.1, so Remark 3.2(2) tells us we have hamiltonian cycles Cb and Cc, such that (nC0)-1(nCb) is a generator of Zp, and (nC0)-1(nCc) is a generator of Zq. It is clear that Cb, like C0, contains [s1s2](c-1) and [s1s2c-1a2](a-1, c, a), so Remark 3.2(2) provides a hamiltonian cycle Cb, such that (ПСЬ)-1(ПС*) is a generator of Zq. Therefore, Marusic's Method 2.11(2) applies (since each of these four hamiltonian cycles contains the oriented edge [a-1](a)). Subcase iii. Assume |a| = 3 and d =3. Since d = 3, we may assume b = С (mod(a)) (by replacing c with its inverse if necessary). Let Co = (b-1, c-1, a2, c, a-1, b, a2), so C0 is a hamiltonian cycle in Cay(G; S). Then C0 contains the four subpaths (b-1), [b-1a2](a-1, b, a), [b-1](c-1), [b-1c-1a-2](a, c, a-1). Therefore, we may let g be either b-1 or b-1c-1 in Lemma 3.1, so Remark 3.2(2) tells us we have hamiltonian cycles Cb = (a, b-1, a-1, c-1, a2, c, b, a) and Cc = (b-1, a-1, c-1, a2, c, b, a2), such that (ПC0)-1 (П^) is a generator of Zp, and (ПC0)-1(ПCc) is a generator of Zq. Furthermore, Cc contains the oriented paths [ab-1](b) and [a-1](a, b-1, a-1), so, by letting g = a in Lemma 3.1 (and replacing b with b-1), Remark 3.2(2) tells us we have a hamiltonian cycle Cbc = (a2, b-1 ,c-1, a2, c, a-1, b), suchthat (ПCc)-1(ПCьc) is a generator of Zp. Therefore Marusic's Method 2.11(2) applies (since all four of these hamiltonian cycles contain the oriented edge [b-1 c-1](a)). □ Case 5.11. Assume Zp^ C ([a, b]), Zqv C ([a, c]), c G (a), andbG (a). Proof. We use Marusic's Method (2.10), so there is no harm in assuming S = {a, b, c}. Furthermore, Lemma 2.8 allows us to assume G' = Zpq, so ([a, b]) = Zp and ([a, c]) = Zq. Also note that, from Assumption 5.1(5), we know c G {a±1}, so we must have |a| > 3. Let d = |G/(a)|. Since c G (a), we have (a, b) = G, so (bd) is a hamiltonian cycle in Cay(G/(a); S). Choose r such that arbd G G' and 0 < r < |a| - 1. Assume r < |a|/2 (so r < |a| - 3), by replacing b with its inverse if necessary. Then letting k = |a| - 3 in Corollary 3.3 provides us with a hamiltonian cycle C0 = C|a|-3. Choose I with c = and write c = a£7, where Zq C (7). We may assume 0 < I < |a|/2 (by replacing c with its inverse, if necessary). Then I < |a| - 3, so we see from Figure 2 that C|a|-3 contains the path [a£b](a-(£+1)). Replacing this with the path [a£b](c-1, a£-1, c-1) results in a hamiltonian cycle C1, such that ^C0)-1(ПC1) is a conjugate of c-V-1c-1 • a£+1 = (a^7)-1a£-1(a£7)-1 • a£+1 = 7-1(7-1)a. Since |G| is odd, we know that a does not invert any nontrivial element of G', so this is a generator of (7), which contains ([a, c]) = Zq. Furthermore, from Figure 2, we see that C|a|-3 contains both the oriented edge [b-1a-1](b) and the oriented path [b-1a](a-1, b, a). Then, by construction, C1 also contains these paths. Therefore, we may apply Lemma 3.1 with g = b-1a-1, so Remark 3.2(1) tells us we have hamiltonian cycles Go and G1, such that (nCi)-1(nĆ7i) is a generator of Zp. Therefore Marusic's Method 2.11(2) applies (since there are many oriented edges, such as [a- 1](a-1 ), that are in all four hamiltonian cycles). □ Case 5.12. Assume there do not exist a, b, c G S, such that ([a, b], [a, c]) = G'. Proof. Let G = G/(G')pq, so G' = Zpq. The assumption of this case implies that we may partition S into two nonempty sets Sp and Sq, such that • Sp centralizes Sq in G, and • for r G {p, q}, and a, b G Sr, we have [a, b] G Z^. Let Gp = (Sp ), Gq = (Sq ), and Z = Gp n Gq C Z (G). Since G is notnilpotent (see Assumption 5.1(4)), we know that G' C Z (G). Therefore, we may assume Zq C Z (G) (by interchanging p and q if necessary). Since Gp n Gq C Z (G), this implies Zq C Gp. Subcase i. Assume there exist ap, bp, aq, bq G S, such that ([ap, bp]) = Zp, ([aq, bq]) = Zq, and {bp, bq } is a minimal generating set of (ap,bp,aq,bq )/(ap,aq). We use Marusic's Method (2.10) with So = {ap, bp, aq, bq}. Assume, for simplicity, that S = So. Lemma 2.8 allows us to assume G' = Zpq, so G = G. After perhaps replacing some generators with their inverses, it is easy to find: • a hamiltonian cycle (si)m=1 in Cay((ap,aq); ap,aq), such that sm-2 = ap and sm-1 = aq, and • a hamiltonian cycle (tj)n=1 in Cay(G/(ap, aq); bp, bq), such that t1 = bp and = bq. We have the following hamiltonian cycle Co in Cay(G; S): П ((t < -1 Ш-Ь )(m-1)/2 / Nn-W.-1 ™-1 ^ Co ^((si)i=1 , t2j-1, (sn-1-i)i-1 ,t2^j = 1 , (si)i=1 , (tm-j)j = 1 , snJ . Much as in the proof of Lemma 3.1, we construct a hamiltonian cycle C1 by • replacing the oriented edge [sm1bp](b-1) with the path [s^1 bp](a-1, b-1, aq), and • the oriented path [s^1 a-1a-1](ap, bp, a-1) with [s^a-1 a-1](bp). Then there exist g, h G G, such that (nCo)-1(nC1) = [b-1,aq]g [a-1 ,bp]h = eg • [a-1,bp]h = [a-1,bp]h, which generates Zp. Similarly, we may construct hamiltonian cycles Co and C1 from Co and C1 by • replacing the oriented edge [sm1t1t2bq](b-1) with the path [sm1t1 t2bq](a-1,b-1, ) , and the oriented path [s^,1 a-1a-1t1t2](ap, bq, a-1) with [sm1a-1a-1t1t2](bq). a Then, for k g {0,1}, essentially the same calculation shows there exist g', h' g G, such that (nCk)-1(nCk) = [b-1,aq]g' [a-1, bq]h' = [b-1,aq]g' • eh' = [b-1,aq]g', which generates Zq. All four hamiltonian cycles contain the oriented edge ( s 1 ), so Marusic's Method 2.11(2) applies. Subcase ii. Assume Gp is not the nonabelian group of order 27 and exponent 3. We will apply Marusic's Method (2.11), so Lemma 2.8 allows us to assume G' = Zpq, which means G = G. Claim. We may assume Sq П (G'Z) = 0. Suppose aq g Sq П (G'Z). By the minimality of S, we know aq G Gp. Since Z and Zp are contained in Gp, this implies G' C (Gp, aq}. Therefore, the minimality of S implies that Sq \ {aq} is a minimal generating set of G/(Gp, aq}. So Subcase i applies. This completes the proof of the claim. Now, applying Proposition 4.2 to Gq tells us there exist hamiltonian cycles Cq and Cq in Cay(Gq/Z; Sq), such that Cq and Cq have an oriented edge in common, and ((nCq)-1 (ncq )} = Zq. _ Also, Theorem 4.1 provides hamiltonian cycles Cp and Cp in Cay(Gp; Sp), such that Cp and Cp have an oriented edge in common, and ((ncp)-1(ncp)} = Zp. For r g {p, q}, write Cr = (srji)n= 1 and C = (tr,i)n=r1. Since Cr and C have an edge in common, we may assume sr,nr = tr,nr. Let C = ((sp,i)n=pi , (sq,i1 , (S-,np-2i+1, (sg,n,-jj i , S-,np-2i, (sq,j2 )(=1 , Sq,n^ • (5.12A) Then C is a hamiltonian cycle in Cay(G; S). For r g {p, q}, a path of the form [g](srji )n= -1 appears near the start of C. We obtain a new hamiltonian cycle Cr in Cay(G; ^ by replacing this with [g](trii)™r-1. We can also construct a hamiltonian cycle Cp q by making both replacements. Then ((nC )-1(nCr )} = ((nCr )-1(nCr )} = Zr, and ((nCq )-1 (nCp,q )} = ((nCp)-1(nCp )} = Zp, so Marusic's Method 2.11(2) applies (since all four hamiltonian cycles contain the oriented edge [s-n,](sq,n,)). Subcase iii. Assume Gp is the nonabelian group of order 27 and exponent 3. We have p = 3, and Lemma 2.21(rTtells us ^ =1; i.e., G' = Z3 x Zqv. Therefore G = G/(G')q. Let Cp = (sp,i)?:=1 be a hamiltonian cycle in Cay(Gp; S^. Also, for r = q, Theorem 4.1 provides hamiltonian cycles Cq = (sqji)n= 1 and Cq = (tqji)n= 1 in Cay(Gq; Sq), such that sq,nq = tq,nq and (nCq)-1 (nCq) generates Zqv. Define the hamiltonian cycle C as in (5.12A) (with np = 27). We obtain a new hamiltonian cycle Cq in Cay(G; S) by e si s 2 S3 s 3 s1 Figure 5: A hamiltonian cycle C0. replacing an occurrence of (sqji)™!11 with the path (tq,i)i= 1 \ Much as in Subcase ii, we have ((nC )-1(nCq )) = {(U€1)-1(UCL)) = Zq, so nC and nCq cannot both be trivial. Therefore, applying the Factor Group Lemma (2.7) with N = Zq provides a hamiltonian cycle in Cay(G; S), and then Lemma 2.8 tells us there is a hamiltonian cycle in Cay(G; S). □ 6 Proof of Case 5.5 In this section, we prove Case 5.5. Therefore, the following assumption is always in effect: Assumption 6.1. Assume there exist a, b G S, such that ([a, b]) = G', |a| = |b| = 3, and (a) = (b). The proof will consider two cases. Case I. Assume #S > 2. Proof. Let c be a third element of S, and let i = |G : (a,b) |. (Since S is a minimal generating set, and G' = ([a, b]) C (a, b), we must have i > 1.) We use MarusiCs Method (2.10) with S0 = {a, b, c}; assume, for simplicity, that S = S0. Lemma 2.8 allows us to assume G' = Zpq. Let (si)3= 1 = ((b, c, b-1, c)('-1)/2, b2, c-('-1), b), so (si)3= 1 is a hamiltonian cycle in Cay(G/(a); b, c). Note that s1 = s5 = b. \n„-1 From the definition of (si)?^ 1, it is easy to see that П3= 1 si = b° = e, so we have the following hamiltonian cycle C0 in Cay(G; a, b, c) (see Figure 5): 1 Co = ((sj)j=1 , a , s3'-2, s3'-1, a , s3', / -1 \3('-1)/2 -1 ) (a, s2j-1,a , s2j)j=i , s3'-2, a , s3'-1,s3^. Since s1 = b, we see that C0 contains the oriented edge (b), and it also contains the oriented path [a-2](a, b, a-1), so Lemma 3.1 provides a hamiltonian cycle C1, such that (nC0)-1(nC1) is a conjugate of [a, b-1][a, b-1]a. Similarly, since s5 = b and s^s^ = c2, we see that C contains both the oriented edge [c2](b) and the oriented path [c2a-2](a, b, a-1), soLemma3.1 provides a hamiltonian cycle C2, such that (nCi)-1(nC2) is also a conjugate of [a, b-1][a, b-1 ]a. Since no element of G inverts any nontrivial element of G' (recall that |G| is odd), this implies that (nCj)-1(nCj) generates G' whenever i = j. So Marusic's Method 2.11(1) applies (since all three hamiltonian cycles contain the oriented edge [s1](s2). □ Case II. Assume #S = 2. Proof. We have S = {a, b}, so |G| = 9pMqv. We may assume p, q > 3, for otherwise Corollary 2.15 applies (perhaps after interchanging p and q). One very special case with a lengthy proof will be covered separately: Assumption 6.2. Assume Proposition 6.4 below does not provide a hamiltonian cycle in Cay(G; S). Under this assumption, we will always use the Factor Group Lemma (2.7) with N •pq. G', so Lemma 2.8 allows us to assume G' = Z Let C = (a-2, b-1, a, b-1, a-2,b2), so C is a hamiltonian cycle in Cay(G; a, b). We have nC = a-2b-1ab-1a-2b2 = [a, b]a[a, b][a, b]b(a-3)b'. (6.2A) Let G = G/Zp, so G' = Zq. Since p,q > 3, we know gcd(|G|, |G'|) = 1, so G = G к G' [7, Thm. 6.2.1(i)]. Therefore G' П Z (G) is trivial, so we may assume that a does not centralize Zq (perhaps after interchanging a with b). Therefore a acts on Zq via a nontrivial cube root of unity. Since the nontrivial cube roots of unity are the roots of the polynomial x2 + x + 1, this implies that [a, b]a [a, b]a [a, b] = e, so [a, b]a[a, b] = ([a,b]a2)-1 = ([a,b]a-1 )-1 (since |a| = 3). Furthermore, a-3 = e (since a has trivial centralizer in Zq). Hence, nC = [a, b]a [a, b] [a, b]b(a-3)b2 = ([a, b]°-1 )-1 [a, b]b e ([a, b]°- )-1 [a, b]b. Therefore nC = e unless yb = ya for all y G Zq. (6.2B) Hence, we may assume (nC} contains Zq (by replacing b with its inverse if necessary). Subcase i. Assume a centralizes Zp. Since G' П Z (G) is trivial, we know that b does not centralize Zp. Also, we may assume (nC) — G', for otherwise the Factor Group Lemma (2.7) applies. Therefore nC must project trivially to Zp. Fixing r, k G Z with [a, b]b — [a, b]r and a-3 — [a,b]k (and using the fact that r2 + r + 1 = 0 (modp)), we see from (6.2A) that this means 0 = 1 + 1 + r + kr2 = 1 — r2 + kr2 = r2(r — 1 + k) (modp), so k = 1 — r (mod p). Therefore k = 0 (modp) (since r is a primitive cube root of unity). Also, since a centralizes Zp, we have [a-1, b-1]-kr = ([a,b-1]-1)-kr — ([a,b]b-1 )-kr — [a, b]-k — a3 — (a-1)-3 (modZq). Therefore, replacing a and b with their inverses replaces k with —kr (modulo p), and it obviously replaces r with r2. Hence, we may assume that we also have —kr = 1 — r2 = r3 — r2 — —(1 — r) r2 = —kr2 (mod p) , so r = 1 (modp). This contradicts the fact that b does not centralize Zp. Subcase ii. Assume a does not centralize Zp. We may assume that the preceding subcase does not apply when a and b are interchanged (and perhaps p and q are also interchanged). Therefore, we may assume that either • b centralizes both Zp and Zq, in which case, interchanging p and q in (6.2B) tells us that nC projects nontrivially to both Zp and Zq, so the Factor Group Lemma (2.7) applies, or • b has trivial centralizer in G'. Henceforth, we assume a and b both have trivial centralizer in G'. We may assume уЪ — ya for у g Zq, by replacing b with its inverse if necessary. We may also assume (nC) — G' (for otherwise the Factor Group Lemma (2.7) applies). Since (nC) contains Zq, this means that (nC) does not contain Zp. By interchanging p and q in (6.2B), we conclude that хЪ — xa for x g Zp. We are now in the situation where a hamiltonian cycle in Cay(G; a, b) is provided by Proposition 6.4 below. □ The remainder of this section proves Proposition 6.4, by applying the Factor Group Lemma (2.7) with N — Zqv. To this end, the following lemma provides a hamiltonian cycle in Cay (G/Zqv ; S). Lemma 6.3. Assume • G — Zpm X (Z3 X Z3) — (x) X ((a) X (b0)), with p > 3, • b — xbo, • xb — xa 1 — xr, where r is a primitive cube root of unity in Zp^, • k G Z, such that o k = 1 (mod 3), o k = r (modpM), and o 0 < k < 3pM, • i is the multiplicative inverse of k, modulo 3pM (and 0 < i < 3pM), • C = ( a, b-2, (a-1, b2)k-1, a-2, b2, (a, b-2)£-fc-1, a-2, (b-2, a)^-^-1 ), and • C is the walk obtained from C by interchanging a and b, and also interchanging k and i. Then either C or C is a hamiltonian cycle in Cay(G; a, b). Proof. Define v2i+e = (ba)V for e G {0,1}, Wj = (ba)j b-1, and let V = {vj} and W = {wj}. Note that, since xab = x, we have |ab| = 3pM, so #V = 6pM and #W = 3pM, so G is the disjoint union of V and W. With this in mind, it is easy to see that C1 = (b-2, a)3pM is a hamiltonian cycle in Cay(G; a, b). Removing the edges of the subpaths (b-2) and [(ba)k](b-2, a, b-2) from C1 results in two paths: • path P1 from b-2 = b to (ba)k, and • path P2 from (ba)fc+1b to e (since (ba)k(b-2ab-2) = (ba)k(bab) = (ba)fc+1b). The union of P1 and P2 covers all the vertices of G except the interior vertices of the removed subpaths, namely, all vertices except b-1, (ba)kb-1, (ba)kb, (ba)k+1, and (ba)k+1b-1. By ignoring y in calculation (6.4A) below, we see that b-1a-1 = (a-1b-1)k, which means ab = (ba)k. Since b-2 = b, this implies ab-2 = (ba)k. Also, since a-1 = a2, we have ba-1b2 = ba2b2 = (ba)(ab)b = (ba)((ba)k)b = (ba)k+1b. Therefore Q1 = (a, b-2) is a path from the end of P2 to the end of P1, and Q2 = [b](a-1,b2) is a path from the start of P1 to the start of P2. So, letting —P1 be the reverse of the walk P1, we see that C2 = Q1 U —P1 U Q2 U P2 is a closed walk. Note that the interior vertices of Q1 are a = (ab)b-1 = (ba)kb-1 and ab-1 = (ab)b = (ba)k b, and the interior vertices of Q2 are ba-1 = ba2 = (ba)(ab)b-1 = (ba)(ba)kb-1 = (ba)k+1b-1 and ( ) ba-1b = ((ba)k+1b-1)b = (ba)k+1. These are all but one of the vertices that are not in the union of P1 and P2, so C2 is a cycle that covers every vertex except b-1. Notice that the only a-edge removed from C1 is [(ba)kb-2](a) = [(ba)kb](a). Since k2 = (r2)2 = r4 = r = 1 (modp^), and I is the multiplicative inverse of k, modulo 3pM, we know k = so this removed edge is not equal to [(ba)£b](a). Therefore [(ba)£b](a) is an edge of C2. Now, we create a walk C * by removing this edge from C2, and replacing it with the path [(ba)£b](a-2). Since ( ) (ab)£ = ((ba)k ) = (ba)fc£ = ba, we see that the interior vertex of this path is [(ba)£b]a-1 = [b(ab)£]a-1 = [b(ba)]a-1 = b2 = b-1. Therefore C* covers every vertex, so it is a hamiltonian cycle. Since ab = (ba)k and ba = (ab)£, it is obvious that interchanging a and b will also interchange k and £ Therefore, we may assume k < Д by interchanging a and b if necessary. Then the edge [(ba)£b](a) is in P2, rather than being in P1. If we let P2' be the path obtained by removing this edge from P2, and replacing it with [(ba)£b](a-2), then we have C = ( (a, b-2), (a-1, b2)k-1, a-1, (a-1,b2), (a, b-2)£-fc-1, a-2, (b-2, a)3p"-£-1 ) = Q1 U -P1 U Q2 U P2 = C* is a hamiltonian cycle in Cay(G; a, b). □ Proposition 6.4. Assume • G = Z3 X Z3, • G' = Zpm X Zqv, with p = q and p, q > 3, • S = {a, b} has only two elements, • a and b have trivial centralizer in G', and ab centralizes Zp^ and ab 1 centralizes Zqv. Then Cay(G; a, b) has a hamiltonian cycle. Proof. Since gcd(|G|, |G'|) — 1, we have G = G' X G — (Zpm X Zqv) X (Z3 X Z3). Write Zpm — (x) and Zqv — (y). Since a does not centralize any nontrivial element of G', we may assume a G Z3 x Z3 (after replacing it by a conjugate). Write b — y60, with Y G G' and b0 G Z3 x Z3. Since (a, b) — G, we must have (y) — G', so we may assume Y — xy; therefore b — xyb0. Choose r G Z with xa 1 — xr. Since |a| — 3 and a does not centralize any nontrivial element of Zp^, we know that r is a primitive cube root of unity, modulo pM. Also, since ab centralizes Zp^, we have xb — xr. Define k and i as in Lemma 6.3. Then, letting G — G/Zqv (and perhaps interchanging a with b), Lemma 6.3 tells us that C — (a, b-2, (a-1, b2)k-1, a-2, b2, (a, b-2)£-fc-1, a-2, (b-2, a)3PM-£-1) is a hamiltonian cycle in Cay (G; a, b). To calculate the voltage of C, choose s G Z with ya — ys, and let y1 — ysM1^-^-1) — ys2-1 (since 1 + s + s2 = 0 (mod q) and k = 1 (mod 3)), and note that (a-1b-1)k — (a-1(xyb0)-1) (6.4A) ( -U-1 -1 -1)fc — a b0 y x — x-k (a-1b-1y-1)fc (x commutes with a-1b-1 and y) x-r (a-1b0-1)ky-(1+s+s2+ -+sk-1) ( -1kfJ= r (maobdpM) and "j V 0 ) y Vya Ъо — ya Ъо — ys — ysj — x r b° a y s y1 a and b0 commute, k = 1 у (mod 3), and definition of y1 b°1x-1y-1a-1y1 (xr — xЪо and ys' — ya 2 — ya 1 ) b 1a 1ys2 1 (b — xybo and y1 — ys2 1). Therefore ПС = ab-2(a-1b2)k-1a-2b2(ab-2)£-k-1a-2(b-2a)3^-£-1 = ab(a-1b-1)k-1ab(b(ab)£-k-1a) (ba)3^-£-1 = ab(a-1b-1)k (a-1b-1)-1ab(ba)£-k (ba)-£-1 = ab(a-1b-1)k (ba)ab(ba)-k(ba)-1 = ab(b-1a-1 ys2-1)ba2b(b-1a-1ys2-1)(a-1b-1) = ys~-1bays -1a-1b-1 = У yv y (s2-1)(1+s) (|a| = |b| =3) (|ba| = (ba)-k = (a-1b-1)k = b-1a-1ys2-1 ,a-1b- 22 . .a b ys = ys Since s is a primitive cube root of unity modulo qv, we know s ф ±1 (mod q). Therefore, the exponent of y is not divisible by q, which means ПС ^ (yq}, so ПС generates Zqv. Hence, the Factor Group Lemma (2.7) provides the desired hamiltonian cycle in Cay(G; a,b). □ References [1] B.Alspach, Lifting Hamilton cycles of quotient graphs, Discrete Math. 78 (1989), 25-36. MR 1020643 [2] S.J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey, Discrete Math. 156 (1996), 1-18. MR 1405010 [3] E. Durnberger, Connected Cayley graphs of semidirect products of cyclic groups of prime order by abelian groups are Hamiltonian, Discrete Math. 46 (1983), 55-68. MR 0708162 [4] E. Durnberger, Every connected Cayley graph of a group with prime order commutator group has a Hamilton cycle, in: B.Alspach, C. Godsil (eds.), Cycles in Graphs (Burnaby, B.C., 1982), North-Holland, Amsterdam, 1985, 75-80. MR 0821506 [5] E.Ghaderpour and D.W.Morris, Cayley graphs of order 27p are hamiltonian, Internat. J. Comb. 2011 (2011), Article ID 206930, 16 pages. MR 2822405 [6] E. Ghaderpour and D. W. Morris, Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian, Ars Contemp. Math. 7 (2014), 55-72. MR 3029452 [7] D. Gorenstein, Finite Groups, Chelsea, New York, 1980. MR 0569209 [8] M. J. Hall, The Theory of Groups, Macmillan, New York, 1959. MR 0103215 [9] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967. MR 0224703 [10] K.Keating and D.Witte, On Hamilton cycles in Cayley graphs with cyclic commutator subgroup, in: B.R. Alspach, C.D.Godsil (eds.), Cycles in Graphs (Burnaby, B.C., 1982), North-Holland, Amsterdam, 1985, 89-102. MR 0821508 [11] K.Kutnar, D.Marušic, J.Morris, D.W.Morris, and P.Sparl, Hamiltonian cycles in Cayley graphs whose order has few prime factors, Ars Math. Contemp. 5 (2012), 27-71. MR 2853700 [12] D. Marušic, Hamiltonian circuits in Cayley graphs, Discrete Math. 46 (1983), 49-54. MR 0708161 [13] I. Pak and R. Radoicic, Hamiltonian paths in Cayley graphs, Discrete Math. 309 (2009), 55015508. MR 2548568 [14] D.Witte, Cayley digraphs of prime-power order are Hamiltonian, J. Combin. Theory Ser. B 40 (1986), 107-112. MR 0830597 [15] D. Witte and J. A. Gallian, A survey: Hamiltonian cycles in Cayley graphs, Discrete Math. 51 (1984), 293-304. MR 0762322 ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 29-34 Arc-transitive graphs of valency 8 have a semiregular automorphism* Gabriel Verretf Centre for Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia. Received 23 May 2013, accepted 16 October 2013, published online 14 April 2014 One version of the polycirculant conjecture states that every vertex-transitive graph has a non-identity semiregular automorphism that is, a non-identity automorphism whose cycles all have the same length. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open valency. Keywords: Arc-transitive graphs, polycirculant conjecture, semiregular automorphism. Math. Subj. Class.: 20B25, 05E18 1 Introduction All graphs considered in this paper are finite and simple. An automorphism of a graph Г is a permutation of the vertex-set V(r) which preserves the adjacency relation. The set of automorphisms of Г forms a group, denoted Aut(r). A graph Г is said to be G-vertex-transitive if G is a subgroup of Aut(r) acting transitively on V(r). Similarly, Г is said to be G-arc-transitive if G acts transitively on the arcs of Г. (An arc is an ordered pair of adjacent vertices). When G = Aut^), the prefix G in the above notation is sometimes omitted. A group G acting on a set Q is semiregular on Q if the stabiliser Gu is trivial for every w G Q; note that this implies that the action is faithful. An element g G G is called semiregular provided that it generates a semiregular group. Equivalently, an element g G G is semiregular if (g) acts faithfully on Q and that all of the cycles of the permutation induced by g have the same length. * Dedicated to Dragan Marusic on the occasion of his 60th birthday. t The author is supported by UWA as part of the Australian Research Council grant DE130101001. E-mail address: gabriel.verret@uwa.edu.au (Gabriel Verret) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract In 1981, Marušic conjectured [7] that every vertex-transitive graph with at least two vertices has a non-identity semiregular automorphism. This is sometimes called the poly-circulant conjecture. While there has been a lot of work on this conjecture and some of its variants, it is still wide open. There has been progress in some directions (see [6] for a survey). For example, the conjecture has been settled for graphs of certain orders [2, 7, 8] and for graphs of valency at most four [1, 8]. In the arc-transitive case, slightly more can be said, but we first need a few definitions. If Г is a graph and G < Aut(T) then denotes the permutation group induced by the action of the vertex-stabiliser Gv on the neighbourhood r(v) of the vertex v. A permutation group is called quasiprimitive if each of its nontrivial normal subgroup is transitive, while a vertex-transitive graph Г is called locally-quasiprimitive if Aut(r)^(v) is quasiprimitive. Using this terminology, we have the following theorem due to Giudici and Xu. Theorem 1.1. [5, Theorem 1.1] A locally-quasiprimitive vertex-transitive graph has a non-identity semiregular automorphism. A transitive group of prime degree is clearly quasiprimitive (in fact primitive) and hence it follows from Theorem 1.1 that arc-transitive graphs of prime valency have a non-identity semiregular automorphism. In an upcoming paper [4], the author and Giudici deal with the case of arc-transitive graphs of valency twice a prime. The main result of this paper is to prove the polycirculant conjecture for arc-transitive graphs of valency 8, the smallest open valency. Theorem 1.2. An arc-transitive graph of valency 8 has a non-identity semiregular automorphism. The prime 2 plays a special role in our proof. In some sense, we use the fact that any proper factorisation of 8 must include 2 as one of the factors. In particular, if Г is a G-arc-transitive graph of valency 8 and N is a normal subgroup of G that is not semiregular and that has at least three orbits then the quotient graph Г/N is an arc-transitive graph of valency either 2 or 4. In the former case, we can view Г/N as an asymmetric arc-transitive digraph of out-valency 4, which turns out to be very useful (see Theorem 2.7). In the latter case, the valency of Г is exactly twice the valency of Г/N which gives us very precious information about the kernel of the action of G on N-orbits (see the proof of Theorem 1.2). This special role of the prime 2 is also evident in the proof of the case of valency twice a prime [4]. In the more general case of valency a product of two primes, Xu [13] has made significant progress and it seems the bottleneck is the case when G is solvable. (See [3] for some results in this case.) 2 Preliminaries 2.1 Permutation groups We start with a few very basic lemmas about permutation groups. Since the proofs are short, we include them for the sake of completeness. Lemma 2.1. Let p be a prime and let G be a transitive permutation group of degree a power of p. Then G contains a non-identity semiregular element. Proof. Let S be a Sylow p-subgroup of G. By [12, Theorem 3.4'], S is transitive. Let z be a non-identity element of the centre of S. If z fixes a point then it must fix all of them, which is a contradiction. It follows that z is semiregular. □ Lemma 2.2. If G is a transitive permutation group with a non-trivial abelian normal subgroup N that has at most two orbits, then N contains a non-identity semiregular element. Proof. If N is semiregular then we are done. We may thus assume that there exists a non-identity element n G N fixing some point v. Since N is abelian, n fixes the N-orbit vN pointwise. It follows that N has exactly two orbits. Let u be a point not in vN and let g G G such that vg = u. Then n acts semiregularly on uN, while ng fixes uN pointwise and acts semiregularly on vN. It follows that nng is a non-identity semiregular element of N. □ Lemma 2.3. Let G bea permutation group and let K bea normal subgroup of G such that G/K acts faithfully on the K-orbits. If G/K contains a semiregular element gK of order r coprime to |K | then G contains a semiregular element of order r. Proof. There exists k G K such that gr = k. Let h = g|k|. Since |k| and r are coprime, it follows that |h| = r and hK is a non-identity power of gK hence it is semiregular. We now show that h is semiregular. Suppose h fixes a point v for some integer i. Then h1 K = (hK У fixes vK and, since hK is semiregular, hlK = K. This implies that h1 G K. Since h has order coprime to |K|, it follows that h = 1. □ Lemma 2.4. A transitive permutation group of degree 8 is either primitive or it is a {2, 3}-group. Proof. Let G be a transitive permutation group of degree 8 that is not primitive. Then G has a non-trivial system of imprimitivity, with blocks of size either 2 or 4. It follows that G is isomorphic to a subgroup of Sym(4) I Sym(2) or Sym(2) I Sym(4) and hence it is a {2,3}-group. □ We also need the following folklore lemma Lemma 2.5. Let p be a prime, let Г be a connected graph and let G < Aut(r). If, for every v G V(r), |Gr(v) | is coprime to p, then so is |Gu | for every u G V(r). Proof. We prove this by contradiction. Let p be a prime dividing |Gu | for some u g V(r) and let g G Gu have order p. Since g is non-trivial, it must move some vertex. Let w be a vertex of Г moved by g at minimal distance from u. By the connectivity of Г, there is a path u, ui,..., ut, w such that g fixes each щ. Then g G Gut and g acts nontrivially on r(ut). Thus p divides |Gr(ut) | and the result follows. □ 2.2 Quotient graphs We will need some basic facts about quotient graphs, which we now collect. (For a reference, see [11] for example.) Let Г be a graph and let N < Aut(r). The quotient graph Г/N is the graph whose vertices are the N-orbits with two such N-orbits vN and uN adjacent whenever there is a pair of vertices v' g vN and u' G uN that are adjacent in Г. Clearly, if Г is connected then so is Г/N. Let Г be a G-arc-transitive graph, let N < G and let K be the kernel of the action of G on N-orbits. Then K = NKv < G, G/K < Aut^/N) and Г/N is G/K-arc-transitive. If N has at least 3 orbits then the valency of Г/N is at least 2 and divides the valency of Г. If Г/N has the same valency as Г then N = K and K is semiregular. Some of these facts are used in the proof of the following lemma. Lemma 2.6. Let p be an odd prime and let Г be a connected 4-valent G-arc-transitive graph such that p divides |V(T)|. If G is solvable then G contains a semiregular element of order p. Proof. The proof goes by induction on |V(T)|. Let v e V(r). If G^(v) is a 2-group then it follows from Lemma 2.5 that Gv is a 2-group and every element of G of order p is semiregular. We thus assume that G^(v) is not a 2-group and, since it is a transitive group of degree 4, it is 2-transitive. Since G is solvable, it contains a non-trivial normal elementary abelian q-group N. If N has at most two orbits then p = q and the result follows from Lemma 2.2. From now on, we assume that N has at least three orbits. Since G^(v) is 2-transitive, this implies that Г/N is 4-valent and hence N is semiregular and G/N acts faithfully on Г/N .If p = q then N contains a semiregular element of order p and the conclusion holds. Suppose now that p = q. Then Г/N is a connected, 4-valent G/N-arc-transitive graph. Since p is coprirne to |N |, it divides |V^/N )|. By the induction hypothesis, G/N contains a semiregular element of order p. The result then follows from Lemma 2.3. □ 2.3 Digraphs Finally, we will need a few notions about digraphs. We follow the terminology of [10] closely. A digraph Г consists of a finite non-empty set of vertices V(f) and a set of arcs А(Г) С V x V, which is an arbitrary binary relation on V. A digraph Г is called asymmetric provided that the relation А(Г) is asymmetric. If (u, v) is an arc of Г then we say that v is an out-neighbour of u and that u is an in-neighbour of v. The symbols f+(v) and T-(v) will denote the set of out-neighbors of v and the set of in-neighbors of v, respectively. We also say that u is the tail and v the head of (u, v), respectively. The digraph Г is said to be of out-valence k if |Г+ (v) | = k for every v e V(f). An automorphism of Г is a permutation of V(f) which preserves the relation А(Г). The set of automorphisms of Г forms a group, denoted А^(Г). We say that Г is arc-transitive provided that Aut(f) acts transitively on А(Г). We say that two arcs a and b of Г are related if they have a common tail or a common head. Let R denote the transitive closure of this relation. The alternet of Г (with respect to a) is the subdigraph of Г induced by the R-equivalence class R(a) of the arc a. (i.e. the digraph with vertex-set consisting of all heads and tails of arcs in R(a) and whose arc-set is R(a)). If the alternet with respect to (u, v) contains an arc of the form (v, w) then this alternet is called degenerate. If the alternet of the arc (u, v) is non-degenerate then it is a connected bipartite digraph where the first bipartition set consists only of sources while the second bipartition set contains only sinks. An important case occurs when this alternet is in fact a complete bipartite digraph in which case we will simply say that the alternet is complete bipartite. We say that Г is loosely attached if Г has no degenerate alternets and the intersection of the set of sinks of one alternet intersects the set of sources of another alternet in at most one vertex. We define the digraph of alternets А1(Г) of Г as the digraph the vertices of which are the alternets of Г and with two alternets A and B forming an arc (A, B) of А1(Г) whenever the intersection of the set of sinks of A with the set of sources of B is non-empty. We are now ready to prove the following theorem. Theorem 2.7. Let Г be a connected asymmetric arc-transitive digraph of out-valence 4. Then Г has a non-identity semiregular automorphism. Proof. The proof goes by induction on |V(T)|. Since Г is connected and arc-transitive (and finite), it follows that it is vertex-transitive and strongly connected (see for example [9, Lemma 2]). Let v g V(T) and let G = Aut(T). Without loss of generality, we may assume that every prime that divides |G| also divides |Gv |. If Gv is a 2-group then the conclusion follows from Lemma 2.1. We may thus assume that Gv is not a 2-group and hence neither r+(v) r+(v) is Gv . Since Gv is a transitive permutation group of degree 4, it is 2-transitive. Since Г has out-valence 4, Gr (v) is a {2,3}-group, hence so are Gv and G and therefore G is solvable by Burnside's Theorem. It follows that G has an abelian minimal normal subgroup N. If N is semiregular then the conclusion holds. We may thus assume that N is not semiregular and hence N^ (v) is a non-trivial normal subgroup of Gr (v) and therefore is transitive. The same argument yields that Nj (v) is also transitive. Let (u, v) be an arc of Г. We have just seen that Nur+(") and Nr (v) are both transitive. This implies that Г+(и) C vN and T-(v) C uN. On the other hand, N is abelian and hence N„ fixes uN pointwise and Nv fixes vN pointwise. It follows that the alternet of Г with respect to (u, v) is not degenerate and is complete bipartite. If Г is not loosely attached then it follows that, for every vertex x, there exists at least one other vertex y such that Г- (x) = Г- (y) and Г+ (x) = Г+ (y). It follows easily that Г has a non-identity semiregular automorphism in this case. We may thus assume that Г is loosely attached. Let Г' = А1(Г) be the digraph of alternet of Г. It follows from [10, Lemmas 3.1-3.3] that Г' is a connected asymmetric digraph of out-valence 4 and that G = Aut(F ). It also follows easily that an automorphism which is semiregular on Г' is also semiregular on Г. Note that |V(F )| = |V^)|/4 and hence we may apply the induction hypothesis to Г' to conclude that it has a non-identity semiregular automorphism g. Together with the observation in the previous sentence, this concludes the proof. □ 3 Proof of Theorem 1.2 Let Г be an arc-transitive graph of valency 8. We must show that Г has a non-identity semiregular automorphism. Clearly, we may assume that Г is connected. Let G = Aut(F). If Г is locally-quasiprimitive then the result follows from Theorem 1.1. We therefore assume that GIv(v) is not quasiprimitive. By Lemma 2.4, GIv(v) is a {2,3}-group and hence, by Lemma 2.5, so is Gv. We may also assume that G itself is a {2,3}-group and hence it is solvable by Burnside's Theorem. It follows that G has an elementary abelian minimal normal subgroup N. If N has at most two orbits then the result follows from Lemma 2.2. We may thus assume that N has at least three orbits and, in particular, nV(v) is intransitive. If N is semiregular then the conclusion follows. We may thus assume that Nv = 1 and hence nV(v) = 1. In particular, Г/N has valency strictly less than 8 and nV(v) is anon-trivial, intransitive normal subgroup of cl{v). Since the orbits of Nr(v) are blocks of it follows that Nr(v) is a 2-group and hence N is an elementary abelian 2-group. Let K be the kernel of the action of G on N-orbits. If r/N is 4-valent then v is adjacent to at most two vertices from any N-orbit. It follows that Kr(v) is a 2-group and hence so are Kv and K. If |V(T/N)| is a power of 2 then so is |V(r)| and the result follows from Lemma 2.1. We may thus assume that |V(T/N)| is not a power of 2 and, by Lemma 2.6, G/K contains a semiregular element of odd order. It then follows from Lemma 2.3 that G contains a semiregular element of odd order. It remains to deal with the case when Г/N is 2-valent. In this case, there is a natural orientation of Г as a connected asymmetric 4-valent digraph Г and Aut(r) is a subgroup of index 2 in G. By Theorem 2.7, Aut(r) has a semiregular element. This concludes the proof. Acknowledgements. We thank the anonymous referees for their valuable advice. References [1] E. Dobson, A. Malnic, D. Marušic and L. A. Nowitz, Semiregular automorphisms of vertex-transitive graphs of certain valencies, J. Combin. Theory, Ser. B 97 (2007), 371-380. [2] E. Dobson, A. Malnic, D. Marušic and L. A. Nowitz, Minimal normal subgroups of transitive permutation groups of square-free degree, Discrete Math. 307 (2007), 373-385. [3] E. Dobson and D. Marušic, On semiregular elements of solvable groups, Comm. Algebra 39 (2011), 1413-1426. [4] M. Giudici and G. Verret, Semiregular automorphisms in arc-transitive graphs of valency 2p, in preparation. [5] M. Giudici and J. Xu, All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism, J. Algebraic Combin. 25 (2007), 217-232. [6] K. Kutnar and D. Marušic, Recent Trends and Future Directions in Vertex-Transitive Graphs, Ars. Math. Contemp. 1 (2008), 112-125. [7] D. Marušic, On vertex symmetric digraphs, Discrete Math. 36 (1981), 69-81. [8] D. Marušic and R. Scapellato, Permutation groups, vertex-transitive digraphs and semiregular automorphisms, European J. Combin. 19 (1998), 707-712. [9] P. M. Neumann, Finite Permutation Groups, edge-coloured graphs and matrices, in: M. P. J. Curran (ed.), Topics in groups theory and computations, Academic Press, London (1977). [10] P. Potocnik and G. Verret, On the vertex-stabiliser in arc-transitive digraphs, J. Combin. Theory, Ser. B 100 (2010), 497-509. [11] C. E. Praeger, Imprimitive symmetric graphs, Ars Combin. 19 (1985), 149-163. [12] H. Wielandt, Finite permutation groups, translated from German by R. Bercov, Academic Press, New York (1964). [13] J. Xu, Semiregular automorphisms of arc-transitive graphs with valency pq, European J. Com-bin. 29 (2008), 622-629. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 35-53 ARS MATHEMATICA CONTEMPORANEA Hamilton paths in Cayley graphs on Coxeter groups: I* Brian Alspach School of Mathematical and Physical Sciences, University of Newcastle Callaghan, NSW 2308, Australia Received 11 July 2013, accepted 7 January 2014, published online 18 April 2014 Abstract We consider several families of Cayley graphs on the finite Coxeter groups An, Bn, and Dn with regard to the problem of whether they are Hamilton-laceable or Hamilton-connected. It is known that every connected bipartite Cayley graph on An, n > 2, whose connection set contains only transpositions and has valency at least three is Hamilton-laceable. We obtain analogous results for connected bipartite Cayley graphs on Bn, and for connected Cayley graphs on Dn. Non-bipartite examples arise for the latter family. Keywords: Hamilton path, Cayley graph, Coxeter group, Hamilton-connected, Hamilton-laceable. Math. Subj. Class.: 05C25, 05C70 1 Introduction The motivational stream for this paper is a confluence of many rivulets varying in age and intrigue. We now explore this history and do so in spite of postponing definitions until completing the brief excursion. The oldest is Lovasz's 1969 question [15] asking whether every connected vertex-transitive graph has a Hamilton path. A closely related thread arose more or less simultaneously, namely, the question of whether every connected Cayley graph has a Hamilton cycle. The latter question has attracted considerable attention for more than forty years. There have been three survey papers of which I am aware [2, 9, 17] and many, many individual papers dealing with the question. References [10, 11, 14] are examples of some recent papers on the topic. Altshuler [6] studied Hamilton cycles in certain embeddings of trivalent graphs on the torus where all faces are hexagons. He was unable to completely settle the problem of * Dedicated to Dragan Marušič on the occasion of his sixtieth birthday. E-mail address: brian.alspach@newcastle.edu.au (Brian Alspach) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ whether all such graphs are hamiltonian. Many of these graphs (but not all) are Cayley graphs on dihedral groups. This was the first specific instance of looking for Hamilton cycles in Cayley graphs on dihedral groups. The author and Zhang [5] proved that all connected trivalent Cayley graphs on dihedral groups are hamiltonian. Unfortunately, we were unable to extend the result to any larger valency. So the general problem of determining whether all connected Cayley graphs on dihedral groups are hamiltonian remains unresolved. There has been some progress. It has been known for almost thirty years that the problem has an affirmative answer if it could be proved that the answer is yes when all the elements in the connection set are reflections. As a corollary of a general result in a recent paper [4], we now know that all connected Cayley graphs on dihedral groups are hamiltonian whenever the order of the group is a multiple of 4. Two apparently unrelated threads come from computer science. The older of the two is related to algorithms for generating all the permutations of an n-set [13, 16]. Several of the algorithms correspond to a Hamilton path in a Cayley graph on the symmetric group Sn with a connection set composed of transpositions. Recently, there has been considerable interest in other Cayley graphs on symmetric groups, where the connection set contains only transpositions. The most celebrated graph of this type is the star graph of dimension n [1] (as it frequently is called). The penultimate thread is fairly new and arises in computational biology. Analysis of genomes evolving by inversions leads to a graph theoretic interpretation that involves signed permutations [12]. This involves the Cayley graphs we study in Sections 3 and 4. An older paper [8] actually ties these threads together (although it may not yet be apparent). In that paper the following theorem appears. Theorem 1.1. Let G be a finite group generated by reflections R1,..., Rn. Then there is a hamiltonian circuit in the Cayley diagram for G corresponding to these generators. Upon reading Theorem 1.1, we might think this settles the problem for connected Cayley graphs on dihedral groups because, as mentioned above, it suffices to settle that problem when all the members of the connection set are reflections. However, a careful reading of [8] leads to the discovery that they prove there is a presentation for every finite Coxeter group so that the corresponding Cayley graph has a Hamilton cycle. It is highly likely that Theorem 1.1 is true, but it is yet to be proven. The final thread arises out of a strong generalization of Theorem 1.1 for symmetric groups (see Theorem 2.3 in the next section). The purpose of this paper is to start extending Theorem 2.3 to Cayley graphs on other Coxeter groups. The two main results are Theorems 4.3 and 5.2. We are ready to start useful background information. The Cayley graph Cay(G; S) on the group G with connection set S is the graph whose vertex set is the set of elements of G, with an edge joining g and h if and only if h = g s for some s G S. There is a restriction on the connection set S, namely, 1 G S and S is inverse-closed, that is, s G S if and only if s-1 G S. We shall use two notations for permutations because each of them is convenient for certain contexts. One common notation is cyclic notation in which a permutation is written as a product of disjoint cycles. The cycles are enclosed in parentheses and upon observing a cycle ( • • • i j • • • ), this is to be interpreted as meaning the permutation maps i to j. There are no commas in this notation, instead, the elements in the cycles are separated by extra space. We also adopt the convention that fixed points, that is, 1-cycles, are not written down. Thus, the transposition interchanging i and j is written (i j). The other notation for permutations we use is range notation. That is, if we write a permutation as a1a2... an, we mean the permutation that maps i to ai for i = 1, 2,... ,n. Thus, the identity permutation in the symmetric group S5 is written as 12345. We shall employ the Orbit-Stabilizer Theorem and state it now for convenience. Theorem 1.2. If G is a permutation group acting on a finite set A, then the order of G is given by |G| = №)||Gx|, where x e A, O(x) denotes the orbit of G containing x, and Gx is the stabilizer of x. A graph X is Hamilton-connected whenever one can find a Hamilton path joining any two arbitrarily chosen vertices. Similarly, a bipartite graph X, for which both parts have the same cardinality, is Hamilton-laceable whenever one can find a Hamilton path joining any two arbitrarily chosen vertices lying in different parts. 2 The symmetric groups A Coxeter group is a group generated by reflections R1,R2,... ,Rn such that the only other relations are of the form (RiRj)k = 1. Given a Coxeter group G, we associate a graph with it, called a Coxeter diagram, where there is a vertex associated with each of the generating reflections. It is easy to see that Ri and Rj commute if and only if (RiRj)2 = 1. So we do not place an edge in the Coxeter diagram if and only if (RiRj )2 = 1. If (RiRj )3 = 1, then we join Ri and Rj by an edge and do not label the edge. Finally, if (RiRj)k = 1 and k > 3, then we join Ri and Rj by an edge and label the edge with k. The symmetric group Sn is a Coxeter group corresponding to the Coxeter diagram given in Figure 1 with the last edge removed. (Thus, we see that Sn is the Coxeter group An-1.) The generator Ri, 1 < i < n — 1, is the reflection of En, n-dimensional euclidean space, through the orthogonal complement of the vector with -1 in coordinate i, 1 in coordinate i + 1 and zeros in all other coordinates. We present the known results for the symmetric groups for completeness and because it takes little space. Definition 2.1. Let S be a collection of transpositions in Sn. We define an auxiliary graph aux(S ) by letting the vertices be labelled 1, 2,... ,n and joining i and j with an edge if and only if (i j) e S. The following proposition is easily proved by induction using Theorem 1.2. Proposition 2.2. If X = Cay(Sn; S), where S consists of transpositions only, then X is connected if and only if aux(S ) is connected. The proof given in [8] for Theorem 1.1 applies to the connection set consisting of the transpositions (1 2), (2 3),..., (n — 1 n). The next theorem is a strong generalization in that it tells us that all of the connected Cayley graphs on the symmetric group, whose connection sets contain only transpositions, are Hamilton-laceable. This vastly extends the connection sets involved, and strongly extends the conclusion as well. Of course, Hamilton-laceable is the best we can hope for because the graphs are bipartite. Theorem 2.3. (Araki [7]) If X = Cay(Sn; S) is connected, S consists of transpositions only, and n > 4, then X is bipartite and Hamilton-laceable. 3 Path extension and Johnson graphs The proofs of the main results to follow are variations on a single theme. Namely, choose a single vertex u and a target vertex v with the object of finding a Hamilton path joining u to v. We proceed by building longer and longer paths from u until we have a path from u that spans all the vertices and terminates at v. We call this technique path extension. The following lemma is the path extension lemma and is used many times. We employ the notion of a quotient graph arising from a partition of the vertex set. It is defined as follows. Given a graph X and a partition A1, A2,..., At of its vertex set, we define the quotient graph with respect to the partition to be the graph of order t whose vertices correspond to the parts, where two vertices are adjacent if and only if there was at least one edge in X between the corresponding parts. We remind the reader that a k-matching is a set of k vertex-disjoint edges. Lemma 3.1. Let X be a graph whose vertex set is partitioned into parts Ai, A2,..., At, let Yi denote the subgraph induced on Ai, i = 1,2,... ,t, and let X/A denote the quotient graph with respect to the partition. We are interested in two scenarios. (i) If each Yi is Hamilton-connected, then we assume that whenever two parts are joined by an edge, there is in fact a 3-matching between the parts. In this case, if there is a Hamilton path in X/A from Ai to A j, then there is a Hamilton path in X joining any vertex in Ai to any vertex in A j. (ii) If each Yi is Hamilton-laceable, let Bi, Ci denote the parts of the bipartition of Yi. We now assume that whenever two parts Ai and Aj are joined by an edge, then there is a 2-matching between Ai and Aj so that the four end vertices of the two edges intersect each of the sets Bi, Bj ,Ci, C j. In this case, if there is a Hamilton path in X/A from Ai to A j, then there is a Hamilton path joining any vertex of Ci to any vertex of one of Bj or C j, and any vertex of Bi to any vertex of the other one of Bj or C j. Proof. In scenario (i), choose an arbitrary vertex u in Ai. Let v be the target vertex in A j. There is a Hamilton path P' joining Ai and Aj in X/A. Let Ak be the second vertex of P '. There must be a vertex u' g Ai distinct from u such that u' has a neighbor w g Ak because there is a 3-matching between Ai and Ak. Thus, take a path from u to u' that spans the vertices of Ai. Then add on the edge from u' to w. There now must be a vertex w' g Ak, distinct from w, with a neighbor in the next part. We then extend the path by adding on a path from w to w' that spans the vertices of Ak. We continue in the obvious way noting that we may enter A j, the last part in P ', at a vertex distinct from v because there is a 3-matching between Aj and the preceding part on P '. We then complete the path to a Hamilton path from u to v by adding a path spanning Aj that terminates at v. The proof when each Yi is bipartite is essentially the same outside of respecting the bipartition of the subgraphs. □ Recall that the Johnson graph J(n, r) has all the r-subsets of an n-set as its vertices, where two vertices are adjacent if and only if their corresponding subsets have exactly r - 1 elements in common. We need to define another graph. Let C = {a1, a2,..., am} be a non-empty subset of {0,1,2,... ,n} such that the elements are listed in the order a1 < a2 < • • • < am. We define the graph QJ(n, C) in the following way. For each ai G C, we include a copy of the Johnson graph J(n, a^. Thus far the Johnson graphs are vertex-disjoint with no edges between them. We then insert edges between J(n, ai) and J(n, ai+1 ), for each i, using set inclusion, that is, we join an ai-subset S1 and an ai+1-subset S2 if S1 is contained in S2. The graph QJ(n, C) can be pictured as having levels made up of Johnson graphs with edges between successive levels based on set inclusion. The following theorem is proved in [3]. Theorem 3.2. The graph QJ (n, C ) is Hamilton-connected for every non-empty C. 4 Wreath products Because we shall be working with signed permutations throughout the rest of the paper, we adopt a convention that simplifies notation. Instead of writing —k for a positive integer k, we write k. We extend this in the obvious way in that k = k, and use x for —x. Consider the Coxeter diagram shown in Figure 1. The generator Ri, 1 < i < n — 1, is the reflection of En through the orthogonal complement of the vector with 1 in coordinate i, 1 in coordinate i +1 and zeros in all other coordinates. The generator Rn is the reflection of En through the orthogonal complement of the vector with 1 in coordinate n and zeros in all other coordinates. This is the Coxeter group Bn and it is easy to see that R1 R2 R3 Rn 1 Rn Figure 1 4 it is isomorphic to the wreath product Sn l S2. This group may be visualized as the set of all permutations acting on the set {1,1,2,2,... ,n, n} such that if f (i) = y, then f (i) = y. This then gives us a compact notation for the elements of Sn l S2, namely, we write a1 a2 . . . an to be the permutation mapping i to ai and i to a for i = 1,2,..., n. That is, the elements are all the signed permutations of 1,2,... ,n. Note that Sn l S2 is imprimitive with the complete block system composed of the blocks {i, i} for i = 1,2,..., n. Thus, there is a natural homomorphism ^ : Sn i S2 ^ Sn, with kernel isomorphic to S2, representing the action on the block system. If y>(f ) = (i j) is a transposition in Sn, then either f = (i j)(i j)g or f = (i j)(i j)g, where g is in the kernel. When g = 1, we call such an element of Sn l S2 a double transposition. If f G Sn l S2 is a transposition, it is easy to see that f G ker(y), that is, f = (i i) for some i in cyclic notation. Let X = Cay(Sn l S2; S), where S contains only transpositions and double transpositions. We define an auxiliary graph aux(S) in this case similar to what we did for Cayley graphs on Sn. The vertices are again the integers 1, 2,3,... ,n. We join i and j by an edge if and only if there is a double transposition f e S for which the homomorphic image 4>(f) = (ij). The next result provides some useful structural information. Lemma 4.1. If S is a collection of n — 1 double transpositions in Sn l S2 such that aux(S ) is a tree, then the subgroup (S) generated by S is isomorphic to Sn, has two orbits such that i and i belong to different orbits for 1 < i < n, has index 2n in Sn l S2, and the graphs induced on the left cosets of (S) are precisely the components of Cay(Sn l S2 ; S). Proof. We induct on n. When n = 2, S contains a single double transposition. We can check easily that the conclusions follow as there are only two possibilities for the double transposition. Let S be a collection of double transpositions satisfying the hypotheses for some n > 2, and assume the result holds for n — 1. We may assume the elements on which Sn l S2 is acting are labelled so that n is a leaf of the tree aux(S). Remove from S the double transposition т involving the element n and let S' denote the set of n — 1 transpositions left over. The subgroup (S') fixes n and n, and by induction satisfies the conclusions of the theorem when restricted to {1,1,2,2,...,n — 1,n—1}. Thus, (S)n, the stabilizer of n, has order (n — 1)! and is isomorphic to Sn-1. If т = (n y)(n y), then let the orbit of (S') containing y be 01. Because т maps n to y and the other n — 1 double transpositions map elements of O1 to elements of O1, the orbit of (S) containing n is {n} U 01. So the stabilizer of n has order (n — 1)! and the orbit containing n has cardinality n. We then know that |(S)| = n! by Theorem 1.2. Hence, (S) is isomorphic to Sn. Clearly, the orbit containing n is {n} U 02, where 02 is the other orbit of the restriction of Sn-1 to {1,1,2, 2,...,n — 1, n — 1}. Hence, (S ) has two orbits such that {i, i} intersects both orbits for 1 < i < n. Because |Sn l S2| = 2nn! and |(S)| = n!, it certainly is the case that (S) has index 2n in the Sn l s2. So that property holds. Examining the Cayley graph Cay(Sn l S2; S), we know that we have a component consisting of the vertices corresponding to the elements of (S). Because left-multiplication is an automorphism of a Cayley graph, the components of this Cayley graph are induced on the left cosets of (S). □ Lemma 4.2. If X = Cay(Sn l S2; S), where S contains only transpositions and double transpositions, then X is connected if and only if S contains at least one transposition and aux(S) is connected. Proof. If aux(S) is not connected, then it is clear that X is not connected. Thus, if X is connected, then aux(S) is connected. Multiplying on the right by a double transposition either switches two positions, or switches two positions and negates both entries. Thus, if S contains only double transpositions, the signed permutation 123 .. .n is not in the same component as a signed permutation with a single negative entry. Hence, if X is connected, then S must contain a transposition. This completes the proof of one direction. We now assume that aux(S) is connected and S contains a transposition (i i). Let T be a spanning tree of aux(S). Let Y = Cay(Sn i S2; T). We know from Lemma 4.1 that Y has 2n components so let Y' be the component containing 123 ... n. Let I (A) denote the involution consisting of the product of the transpositions (k k) as k runs through elements of A, where A is a subset of {1,2,..., n}. Left multiplication by I (A) is an automorphism of Y mapping Y ' to the component containing I (A). Consider the component containing I(k), where we are writing k rather than {k}. If we choose any element a1a2... an of Y' with a G {k, k}, then the corresponding element of the component containing I(k) has all entries the same except that in coordinate i it has ai. Thus, there is an edge joining these two vertices via (i i). In a similar way, there is an edge from Y ' to any component containing I (A), where A is a singleton. In a similar manner, we can find an edge from the component containing I(k) to any component containing I (k, I), where I = k and we do not include the set brackets around k and I. It now is obvious that we can use edges generated by (i i) to connect all the components of Y into a single component of X. □ Theorem 4.3. If X = Cay(Sn i S2; S) is connected, has valency at least three, and S contains only double transpositions and transpositions, then X is bipartite and Hamilton-laceable. Proof. First we show that X is bipartite. Let A consist of the signed permutations f = a1a2 • • • an such that y(f ) is an even permutation and f has an even number of negative terms, or y>(f ) is an odd permutation and f has an odd number of negative terms. Let B be the remaining elements of Sn i S2. It is easy to see that if we multiply any element of A on the right by an element of S, we obtain an element of B and vice versa. We conclude that X is bipartite. Small values of n produce some anomalous situations and we investigate them separately. When n = 2, all of the possibilities giving valency 3 are isomorphic to the cartesian product of a 4-cycle and K2. This is known to be Hamilton-laceable. When the valency is 4, the graph is isomorphic to K4,4 which is Hamilton-laceable. Hence, the result is true for n = 2. We cannot apply induction for the n = 3 case because the valency may be 3 and upon deleting an element from the connection set, we obtain a subgraph whose components are even length cycles. Even cycles are not Hamilton-laceable so that we must do this case separately. Let X satisfy the hypotheses and n = 3. Because X is connected, aux(S) is connected and contains a spanning tree. The spanning tree must be a path of length 2 because n = 3. By relabelling the elements on which the group Sn i S2 acts, if necessary, we may assume the spanning tree is 123. Note that the spanning tree does not uniquely determine the connection set for X. For example, the edge 12 arises from at least one of the double transpositions (1 2)(T 2) and (1 2)(1 2) belonging to S. (Of course, both of these double transpositions could belong to S.) „,„ 231 321 _ ,„„ 213___312 ,„„ 123 m---— rn rn------- 132 Figure 2 If the double transpositions (1 2)(T 2) and (2 3)(2 3), and the transposition (3 3) belong to S, then they generate a spanning subgraph of X isomorphic to the graph shown in Figure 2. In fact, no matter which double transpositions are chosen corresponding to the spanning tree together with either of the transposition (1 1) or (3 3), we obtain a spanning subgraph of X isomorphic to the graph in Figure 2. If we have the transposition (2 2), we obtain edges between the eight 6-cycles such that the two edges joining two fixed 6-cycles are incident with diametrically opposed vertices on the 6-cycles instead of neighboring vertices as in the graph shown in Figure 2. The essential point is that we have two trivalent bipartite graphs of order 48 that need to be directly checked whether they are Hamilton-laceable. This may seem to be a daunting task, but as a matter of fact it is fairly straightforward. Consider the graph in Figure 2. It suffices to find a Hamilton path from the vertex 123 to any vertex in the other part of the bipartition because X is vertex-transitive. For example, suppose you want a Hamilton path terminating at the vertex 312 in the figure. Construct a path starting with 123 that spans the 6-cycle containing 123 and terminates at 132. Continue by taking the edge to 132 followed by using all the vertices of this 6-cycle and terminating at the vertex 312. We now have a path starting at 123 and terminating at 312. It is easy to see how to transform this starting path into a Hamilton path terminating at 312. Remove the edge joining 231 and 321 from the path and take the two edges down to the vertices 231 and 321 in another 6-cycle. Join 321 and 231 using all the vertices of that 6-cycle. It is easy to see that we can continue to delete an edge from the expanding path and move to another 6-cycle and pick up all of its vertices until reaching a Hamilton path. The preceding technique together with Posa exchanges establishes that the graph of Figure 2 is Hamilton-laceable. The other possible isomorph is a little harder to work with, but it is still fairly easy to establish that it is Hamilton-laceable. Hence, the theorem is true for n = 3. We continue the proof by induction on n. Let n > 4 and assume the theorem holds for n - 1. Because aux(S) is connected, it has a spanning tree T. Moreover, because T has at least two leaves, T has a leaf j such that the connection set S contains a transposition (i i) for which i = j. Relabel the elements 1,2,..., n, if necessary, so that a double transposition g satisfying p(g) = (n - 1 n) belongs to S, where j is relabelled as n. Now let S' denote all the elements of S that fix n. The group (S'} generated by S' is isomorphic to Sn-1i S2 by Lemma 4.2, and the Cayley graph Y = Cay(Sn-1i S2; S') is connected and bipartite. We know that Y is Hamilton-laceable by induction. The Cayley graph X' = Cay(Sn i S2; S') is disconnected with components isomorphic to Y. This follows because (S'} generates all signed permutations of 1,2,33,... ,n with n fixed and this is the component of X' containing the identity. If we left multiply (S'} by any element of h g Sn i S2, we get all the signed permutations for which n is mapped to h(n), that is, the last coordinate is h(n). Left multiplication is an automorphism of both X and X' so that all the left cosets of (S'} induce isomorphs of Y. This sets the stage for the induction proof via Lemma 3.1. We use the components of X ' to give us the partition of V (X ). Let C (z) denote the component consisting of the signed permutations ending with z. If there is an edge from one part of the component C(x) to one part of the component C(y), x = y, then left multiplication by a double transposition from S' gives an edge joining the other two parts of the same two components. Thus, a crucial hypothesis of Lemma 3.1 is satisfied. It suffices to show that there are Hamilton paths in X from 123 • • • n to every vertex of B because X is vertex-transitive. The double transposition g satisfying ip(g) = (n - 1 n) is either (n — 1 n)(n — 1 n) or (n — 1 n)(n — 1 n). We first consider the case that g = (n — 1 n)(n — 1 n). Let x = y such that x = y. There is a signed permutation in C(x) ending yx. Right multiplication by g gives an edge from C(x) to C(y). Let v be an arbitrary vertex in B in a component C(x) such that x = n. By Lemma 3.1 it suffices to find a Hamilton path in the quotient graph X/A from C(n) to C(x). We claim there is a sequence y1,y2,..., y2n composed of the elements 1,1,..., n, n such that y1 = n, y2n = x and j, j are never consecutive. Letting i < n, use the sequence n, n — 1,..., i +1,1, 2,..., n, 1, 2,..., i when x = i. When x = i, we negate every term in this sequence other than the first. When x = n, use n, n — 1,..., 1, 2,1, 3,..., n. From the above remark, there are edges joining consecutive components corresponding to the sequence and the desired Hamilton path exists. We have to modify the approach somewhat when x = n, that is, the target vertex v = a1a2 • • • an-1n also lies in C(n). We start with a path P from 123 • • • n to v that spans the vertices of C(n). We examine an-1. As we traverse P backwards, find the first vertex w for which the n — 1 entry is different from an-1. In other words, the subpath P [123 • • • n, w] terminates in a vertex w whose n—1 entry is not an-1, but every vertex of the subpath P(w, a1a2 • • • n] has an-1 in coordinate n — 1. Let w' be the successor of w on P. Remove the edge ww' from P. The vertices wg and w'g lie in different components because they differ in coordinate n. It is easy to see how to slightly modify the preceding argument so that we obtain a path from w to w' spanning all the vertices of the remaining components. This yields a Hamilton path joining 123 • • • n and v. The other case is g = (n n — 1)(n - 1 n). It is different because multiplying on the right by g not only switches the elements in coordinates n - 1 and n, it also changes the signs of both. However, it takes only a small modification of the above procedure to handle this case. Use the same sequences y1,y2,..., y2n, but when terminating the path spanning a left coset, stop at a vertex whose last two coordinates are y+yi. Multiplying on the right by g then takes you to a vertex in the correct left coset. This completes the proof. □ 5 An Index 2 Subgroup Of Sn I S2 The groups we considered in the preceding section were the signed permutations of length n. A natural subgroup for each of these groups is the collection of signed permutations with an even number of negative terms. This group is the Coxeter group Dn. It is easy to see that Dn has index 2 in Sn i S2. The Coxeter diagram for the group Dn is shown in Figure 3. The generator Ri, 1 < i < n — 1, is the reflection through the orthogonal complement of the vector with 1 in coordinate i, 1 in coordinate i + 1, and zeros elsewhere. The generator Rn is the reflection through the orthogonal complement of the vector with 1 in the last two coordinates and zeros elsewhere. The first step is to decide which connection sets we are going to allow for the Cayley graphs on Dn. We shall use double transpositions as we have done for Sn i S2, but now we require a product of two transpositions that negates two coordinates, that is, a permutation f G Sn i S2 such that f (i) = i, f (j) = j, and f fixes all other elements k, where i = j and k G {i, j}. We call such a permutation a double negator. In order to set the stage for what follows, we examine n = 2, 3 ahead of time. The case of n = 2 is particularly simple, and not particularly edifying, because |D2| = 4. So the only Cayley graph on Dn of valency 3 is K4. It is not bipartite and certainly is Hamilton-connected. In general, we let X = Cay(Dn; S) be a Cayley graph on Dn such that S contains only double transpositions and double negators. We again define aux(S) by letting the vertices be 1, 2,... ,n, and joining i and j with an edge if and only if there is a double transposition f G S such that f ) = (i j). We return to our consideration of the two smallest values of n. There is considerably more complexity when n = 3. Note that |D3| = 24. If aux(S) is not connected, then it must have a singleton component. Without loss of generality we may assume vertex 3 is a singleton. This means that every double transposition in the connection set S fixes both 3 and 3, and double negators fix all the blocks. Thus, the block {3,3} is fixed by the group (S} generated by S. Hence, (S} is a proper subgroup of D3 which implies that X is not connected. Therefore, we see that if X is connected, then aux(S) is connected. We are assuming that X is connected so that aux(S) contains a spanning tree. The spanning tree must be a path of length 2 because aux(S) has order 3. Without loss of generality we may assume the set on which the group D3 is acting is labelled so that the path forming the spanning tree consists of the edges 12 and 23. We now consider possible special subgraphs of X. First, suppose the double transpositions generating the edges 12 and 23 are (1 2)(1 2) and (2 3)(2 3). These two double transpositions generate the subgraph shown in Figure 4. 123 213 231 321 312 132 »--•-•-•-* ............• 123 213 231 321 312 132 123 213 231 321 312 132 »............ »-•-•-* ............»............ »-•-•-»____• 123 213 231 321 312 132 »............ »-•-•-* ............ Figure 4 In order for X to be connected, we need either a double negator or a negative double transposition in S (where a double transposition is negative when it has the form (i j)(i j)). If we have the double negator (1 1)(2 2) in S, we obtain the trivalent spanning subgraph Y1 shown in Figure 5. The graph Y1 is not bipartite and it can be verified directly that it is Hamilton-connected. Figure 5: The subgraph Yi If we use the double negator (2 2)(3 3), we obtain a graph that is isomorphic to Y1. The same conclusions then follow. If instead we use the double negator (1 1)(3 3), we obtain the graph Y2 shown in Figure 6. The graph Y2 also is not bipartite and it can be verified directly that Y2 is Hamilton-connected. Figure 6: The subgraph Y2 If we partition the vertices of X so that part A contains the permutations f for which y(f ) is an even permutation in S3, and part B contains the permutations f such that y(f ) is an odd permutation, then the edges generated by any double transposition have one end in A and one end in B. Hence, if S contains no double negator, then X is bipartite. Moreover, if S contains no double negator, then there must be a double transposition in S not contained in the group generated by (1 2)(1 2) and (2 3)(2 3). If we use the negative double transposition (1 2)(1 2) to connect the components of the spanning graph shown in Figure 4, we obtain the graph Y3 shown in Figure 7. This graph is bipartite and it is easy to verify that it is Hamilton-laceable. Figure 7: The subgraph Y3 If we use the negative double transposition (2 3)(2 3), we obtain a graph isomorphic to Y3. This leaves the negative double transposition (1 3)(1 3). In this case we obtain the graph Y4 shown in Figure 8. It is bipartite and easily shown to be Hamilton-laceable. Figure 8: The subgraph Y4 It is obvious that two Cayley graphs on the same group G, with respective connection sets S' and g S'g-1 for g g G, are isomorphic via conjugation by g. Therefore, if S contains exactly one positive double transposition, or S contains no positive double transpositions, then each Cayley graph we obtain is isomorphic via conjugation to one of those we have considered above. This completes the analysis of the case that n — 3. What we have seen is that a spanning tree of aux(S) always produces a 2-factor composed of four 6-cycles. If any double negator lies in the connection set S, then the resulting spanning trivalent subgraph is connected, not bipartite and Hamilton-connected. This implies, of course, that X is Hamilton-connected. Continuing in this vein, if S contains no double negators, then X is bipartite and Hamilton-laceable when it is connected. The subgraph generated by any two double transpositions т1,т2, corresponding to a spanning tree, is a 2-factor composed of four 6-cycles. Any double transposition т3 g (т1, т2) results in a disconnected spanning trivalent subgraph. So for X to be connected, we need a double transposition that is not an element of the group (ть т2). Lemma 5.1. If X — Cay(Dn; S) is a Cayley graph on Dn such that S contains only double transpositions and double negators, then X is connected if and only if aux(S) is connected and one of the following two conditions holds: (a) S contains a double negator, or (b) S contains no double negators, but if т1,т2,..., тп-1 are elements of S corresponding to a spanning tree of aux(S), then there is a т g S such that т does not belong to the group (т1, т2,..., тп-1) generated by т1, т2,..., тп-1. Moreover, X is not bipartite when n > 2 and (a) holds, whereas, X is bipartite when (b) holds. Proof. Observe that if aux(S) is not connected, then it is obvious that X is not connected. Thus, if X is connected, then aux(S) also is connected. If X is connected, let т1 ,т2,... ,тп-1 be the double transpositions for some spanning tree of aux( S). The group H — (т1,т2,... ,тп-1) is a proper subgroup of Dn by Theorem 4.1. So if S does not contain a double negator, then in order for X to be connected, there must be a double transposition т such that т G H. Hence, if X is connected, then aux(S) is connected and at least one of the two conditions holds. Now let aux(S) be connected and let condition (a) or (b) hold. Let т1,т2,... ,тп-1 be as in the preceding paragraph, let X' be the subgraph of X generated by these n — 1 double transpositions, and let Y be the component of X' containing 123 • • • n. First suppose that condition (a) holds and that the double negator is (i i)(j j) The proof that X is connected closely mirrors the corresponding proof of Lemma 4.2. First, recall that I (A), where A is a subset of {1,2,..., n}, denotes the permutation which is a product of the transpositions (i i) as i runs through A and fixes all other elements. The components of X ' are then the subgraphs containing the permutations I (A) as A runs over all subsets of even cardinality of {1,2,... ,n}. We then show that X is connected using an argument that is the same as that used for Lemma 4.2 except that we now use two coordinates at a time instead of one. This takes care of connectivity. We cannot claim that X is not bipartite for n = 2 because the graph may be a cycle of length 4. To conclude the proof for condition (a), we must show that X is not bipartite when S contains a double negator and n > 3 (n = 3 was done above). Let т denote the double negator (i i)(j j). We can always find a permutation f = a1a2 • • • an, where ai and a j are fixed, with f e A by inverting two elements in two coordinates different from i and j if necessary. Let f1 denote a permutation in Y such that ai e {i, i}, a j e {r, r}, r = j, and f1 e A. There is then an edge joining f1 and f1т in the left coset I (i, r)H. In a similar manner, there is a permutation f2 in Y such that ai e {j, j}, a j e {r, r}, and f2 e A. There is then an edge joining f2 and f2т in the left coset I(j,r)H. We then find a permutation f3 in the left coset I(i, r)H belonging to A with ai e {i, i} and aj e {j, j. This is then adjacent to f3т in the left coset I (j, r)H and f3т e A. The paths joining f1 and f2 in Y, f3 and f33т in I (i, r)H, and f2т and f33т in I (j, r)H all have even lengths because the vertices all belong to A. Thus, we have an odd length cycle in X so that it is not bipartite. When condition (b) holds, then S contains no double negators but it does contain a double transposition т not contained in the group H. The double transposition т satisfies ^(т) = (i j) for some i = j. We know there is an element т' e H such that у(т') = (i j) because H is isomorphic to Sn by Lemma 4.1. Moreover, Lemma 4.1 informs us that k and k are in different orbits for all k so that т' fixes all elements not in {i, i,j,j}. This implies that т'т = (i i)(j j) which is a double negator. Because multiplying on the right by т' keeps one in the same left coset of H, we see that т joins the same left cosets as the preceding double negator. Therefore, X is connected by the argument for condition (a). Note that if we let A be all the permutations f e Dn for which y>(f ) is an even permutation and let B be all those for which y>(f ) is an odd permutation, then any edge generated by a double transposition has one end vertex in A and one end vertex in B. Thus, if S contains no double negators, then X is bipartite. □ Theorem 5.2. If X = Cay(Dn; S) is a connected Cayley graph of valency at least 3 on Dn, n > 2, such that S contains only double transpositions and double negators, then X is Hamilton-laceable when it is bipartite, or Hamilton-connected when it is not bipartite. Proof. The results of this theorem have been proved for n = 2 and n = 3 earlier. We proceed by induction on n. First consider the case that X is bipartite. As before, let H = (тьт2,.. .,Тп-1 ), where т1, т2,..., тп-1 are the double transpositions corresponding to the edges of a spanning tree of aux(S). We know there is a double transposition т' not contained in H from Lemma 5.1. Furthermore, we saw in the proof of Lemma 5.1 that edges between the subgraphs induced on the left cosets of H join the same left cosets as do the edges generated by a double negator т = (i i)(j j). For simplicity we work with т. As before, let A denote the elements f of (S) such that y(f ) is an even permutation and let B denote the other elements of the group. Clearly, A and B are the two parts of the bipartition of X . We know that I(L)H are the left cosets of H as L runs through all even cardinality subsets of {1,2,..., n}. If E1 and E2 are two subsets of the same even cardinality a and they have a — 1 elements in common, then there is an edge joining a vertex of I(E1)H and a vertex of I(E2)H. To see this let x, y be the two elements of E1AE2 (symmetric difference). Let f = a1a2 • • • an be an element of H so that ai G {x, x} and aj G {y, y}. Then we have I(E1)fT = I(E2)f which implies there is an edge between the left cosets I(E1)H and I(E2)H. Now form a quotient graph by contracting each left coset to a single vertex making two vertices adjacent if there is an edge joining vertices of the corresponding left cosets. From the preceding paragraph, we see that two vertices corresponding to left cosets I(E1)H and I(E1)H, |E1| = |E2|, are adjacent if E1 and E2 have all but one element in common. Thus, all left cosets corresponding to subsets of {1, 2,..., n} of the same cardinality k induce a subgraph containing the Johnson graph J(n, k). It is also easy to see that if E1 is a subset of E2 with |E11 = |E21 — 2, then there is an edge between the corresponding left cosets. Hence, the quotient graph contains a spanning subgraph isomorphic to QJ (n, C ), where C is the collection of all even cardinality subsets of {1, 2,..., n}. This graph is Hamilton-connected by Theorem 3.2. Thus, we may employ Lemma 3.1 to easily find Hamilton path in X from 123 • • • n to any vertex of B in any left coset different from H. If the target vertex v happens to lie in H, then there is a path from 123 • • • n to v spanning the vertices of H by induction. We then find any edge w1w2 in this path such that w1 and w2 have neighbors in different left cosets of H. We then use QJ (n, C — 0), which also is Hamilton-connected by Theorem 3.2, along with Lemma 3.1 to find a path Q from w1T' to w2T' spanning all the vertices of the remaining left cosets. We remove the edge w1w2 from the initial path spanning H and patch Q in to get a Hamilton path in X from 123 • • • n to v. This completes the bipartite case. Now assume that S contains the double negator т = (i i)(j j). Note that <^(f) and ¥>(f т) either are both even permutations or both odd permutations. Thus, an edge generated by a double negator either has both end vertices in A or both end vertices in B. There are two cases to consider: Either aux(S) has a spanning tree with a leaf k different from both i and j or there is no such spanning tree. We first consider the case that there is such a tree T. Without loss of generality we assume that n is the leaf of T different from i and j. In other words, n is fixed by т. Let S' be the subset of S containing all double transpositions and double negators that fix the element n. Because aux(S') contains a spanning tree and the double negator т, the components of Cay(Dn; S') have order 2n-2 (n — 1)! and are Hamilton-connected by induction. We now have exactly the same situation as in the proof of Theorem 4.3, namely, a subgraph each of whose components is composed of all the permutations whose last coordinate is constant. There is one component for each element of {1,1,... ,n,n}. The double transposition corresponding to the edge of T incident with n is then used to connect the components together exactly as was done in the proof of Theorem 4.3. Hence, X is Hamilton-connected. This leaves us with the other case, namely, there is no spanning tree with a leaf different from i and j. The preceding induction proof cannot be applied. It is not hard to see that this forces aux(S) to be a path whose end vertices are i and j. We maintain the same notation, that is, we let S' = {т1,т2,..., rn-1} be the double transpositions corresponding to the edges of the spanning tree, H = (S'}, X' be the subgraph Cay(Dn; S'), and Y the component of X' containing 123 • • • n. By Lemma 4.1, X' has 2n-1 components. The components are bipartite and Hamilton-laceable by Theorem 2.3. Moreover, all the components are isomorphic to Y. For each L С {1,2,..., n}, |L| even, the action of the involution I(L) on each permutation is to negate the entries in the coordinates corresponding to the elements of L. Hence, the 2n-1 components of Cay(Dn; S') are the subgraphs induced on the left cosets I (L)H as L runs through all subsets of {1, 2,... ,n} of even cardinality. As we saw earlier in this proof, the quotient graph of X obtained by contracting each left coset to a single vertex, deleting all loops, and replacing multiple edges by a single edge yields a spanning subgraph isomorphic to QJ(n, C), where C contains all even integers between 0 and n inclusive. We employ Theorem 3.2 frequently. We define the bipartition A and B as before. Because X is vertex-transitive, it suffices to find a Hamilton path in X from 123 • • • n to any other vertex v. We shall refer to v as the target vertex. Let v be a target vertex in the part A of any component I(L)H different from Y. Theorem 3.2 provides a Hamilton path in the quotient graph from the vertex corresponding to Y to the vertex corresponding to I(L)H. Since edges generated by т have both end vertices in either A or B, we use a (slightly) modified version of Lemma 3.1 to obtain a Hamilton path from 123 • • • n to v using the fact that there are an even number of components. Now let the target vertex v be in part B of Y. There is a path P from 123 • • • n to v spanning the vertices of Y by Theorem 2.3. There must be two successive vertices w1,w2 on P with neighbors y1,y2 in different components I(a,b)H and I(a,c)H for some a, b, c. Considering the graph QJ (n, C'), where C ' = {2,4,..., 2 |_n/2j}, Theorem 3.2 and Lemma 3.1 imply there is a path Q from y1 to y2 spanning all the vertices of the components corresponding to C'. We then obtain a Hamilton path from 123 • • • n to v by removing the edge w1w2, adding the edges y1w1 and y2w2, and adding the path Q. Thus, X has a Hamilton path from 123 • • • n to any vertex in part B of Y. To complete the proof of the theorem, we must find a way to circumvent the fact that the components are only Hamilton-laceable. Before introducing the trick we use, let's review the strategy we have used so far. We find a path Q spanning Y that starts at 123 • • • n and finishes at any vertex w of part B in Y. We choose w so that wt belongs to a left coset I (L)H not containing the target vertex v. We then use the fact that the graph QJ (n, C), where C contains all the even integers in {2, 3,4,..., n}, is Hamilton-connected so that we may apply Lemma 3.1 to find a path P from wt to v spanning all the left cosets of H distinct from H itself. We attach P to Q using the edge from w to wt to obtain a Hamilton path in X from 123 • • • n to v. Because the number of left cosets of H distinct from H is odd (that is, the number of vertices of Q(n, C) is odd), this works only for vertices in part A. The trick we are about to introduce is based on removing either one or three special vertices from the QJ(n, C) graph mentioned above. If we remove a single vertex corresponding to any 2-subset, or three vertices corresponding to three 2-subsets of the form {a, b}, {a, c}, {b, c}, or three 2-subsets of the form {a, d}, {b, d}, {c, d}, then the resulting subgraph of QJ (n, C ) remains Hamilton-connected. We leave this up to the reader and refer to [3] if guidance is required. The key is to prove that removing these three vertices from J(n, 2), n > 4, leaves a Hamilton-connected graph. When the target vertex v is in part B of some component I(L)H different from Y, choose a vertex w in part B of Y such that the neighbor wt of w belongs to a left coset I (a, b)H different from I (L)H. Then choose a path Q from 123 • • • n to w that spans Y. Suppose there is an edge xy on Q so that both хт and yT lie in the same left coset C, and C does not contain v and is different from I (a, b)H. We then remove the edge xy from Q and attach the edges to хт and yT. We now find a path from хт to yT spanning the vertices of C. This now gives us a path Q' from 123 • • • n to w spanning the two left cosets H and C. The graph QJ (n, C ) with a single vertex corresponding to a 2-subset removed is still Hamilton-connected. However, the number of left cosets left over is now even so that when we apply the strategy outlined above, we can reach a target vertex in part B as required. Alternatively, suppose there is an edge xy of Q so that хт and yT are in different left cosets C1 and C2, respectively, and a third left coset C3 so that C1, C2, C3 are all distinct from I (L)H and I (a, b)H. If there is a path from хт to yT spanning the vertices of the three left cosets C1, C2, C3, then we can replace the edge ху in Q with a path that spans the three left cosets, then we have a path from 123 • • • n to w spanning the four left cosets. The number of left cosets remaining is even and we can find a Hamilton path to a target vertex in any part B We need to show that one of the preceding conditions holds. If the target vertex v is in a left coset I(L)H such that ILI > 2, we don't concern ourselves with designating the set L. If ILI = 2, then we let the left coset containg v be I (a, c)H. Choose w in part B of Y so that wt e I (a, b)H, where b = c. Let Q be a path from 123 • • • n to w spanning the vertices of Y. Because n > 4, there is an element d distinct from a, b, c. If there is an edge ху of Q so that both хт and ут lie in any single one of the left cosets I (c, d)H, I (b, c)H, I (b, d)H, then the first condition above holds and there is a Hamilton path from 123 • • • n to v in X. If we cannot find an edge ху such that хт and ут lie in the same left coset, we need to examine aux(S) with more care. We know aux(S) is a path with end vertices i and j with the double negator т = (i i)(j j) joining the left cosets. Note that any double transposition in S alters at most one of the entries in coordinates i and j. Hence, if х and y are adjacent vertices of Y, then хт and ут either lie in the same left coset or lie in different left cosets I(L1 )H and I(L2)H such that IL11 = IL2I = 2, and L1 and L2 have one element in common. The number of edges from Y to a fixed coset I(L)H, where ILI = 2, is 2(n — 2)! which is at least 4 because n > 4. Hence, there is an internal vertex х of Q such that хт e I(c, d)H, where d is distinct from a, b and c. Consider the predecessor y of х on Q. If ут e I (c, d)H, we are done. If ут e I (a, d)H, then the three left cosets I (a, d)H, I (b, d)H, I (c, d)H do the job. If ут e I (b, d)H, then the three left cosets I (b, d)H, I (c, d)H, I (b, c)H do the job. If ут lies in I (b, c)H, we also easily find three left cosets that work. In fact, the only left coset that is bad for ут is I (a, c)H. But if ут lies in the left coset I (a, c)H, then the successor of x on Q, call it z, cannot have zt in I (a, c)H because this forces the edges yx followed by xz to be generated by the same double transposition because aux(S) is a path. However, double transpositions are involutions so that we cannot use them consecutively when generating distinct vertices. Thus, xt and zt must lie in left cosets satisfying one of the conditions above. This leaves us with the target vertex being in part A of Y as the only missing case. Suppose that v lies in part A of Y and vt g I (a, b)H. Let w be in part B of Y such that wt lies in I (a, b)H as well. Let Q be a path from 123 • • • n to w spanning the vertices of Y. The vertex v is an internal vertex of Q. Remove the edge between the predecessor u of v and v. This breaks Q into a subpath Q1 from 123 • • • n to u, and a subpath Q2 from v to w. Because u is adjacent to v and by changing the labels, if necessary, we may assume that ut G I (a, e)H for some e. If e = c, then find a vertex z internal to either Q1 or Q2 such that zt g I(b, d)H. Just as above, one of the two neighbors of z on Q allows us to use either one or three left cosets to augment either Q1 or Q2. We then cover all the remaining cosets using a path with wt and ut as the end vertices as before because there are an even number of cosets unused. We have the desired Hamilton path. If e = d a similar argument works. If e G {b, c, d} it is even simpler because we have a new symbol to work with. However, it is possible that e = b so that wt and ut lie in the same left coset I (a, b)H. We now have the situation that Q1 has been extended to part B of the left coset I (a, b)H via the edge to ut. Similarly, Q2 has been extended to part B of the same left coset via the edge to vt . Now choose a vertex u1 in part A of I (a, b)H whose i, j entries are disjoint from the i, j entries of ut. Then choose a path Q' from u1 to wt that spans the vertices of I (a, b)H. The successor u2 of ut then is adjacent to a vertex in a different left coset of H than is u1. This allows us to obtain a Hamilton path from 123 • • • n to v in X and completes the proof. □ Acknowledgements. The author wishes to thank Adam Piggot for his enlightening lessons about Coxeter groups. I also thank Eva Czabarka and Laszlo Szekele for bringing the connection between Cayley graphs on Sn i S2 and genome switching to my attention. References [1] S. Akers, D. Harel and B. Krishnamurthy, The star graph: attractive alternative to the n-cube, in: Interconnection Networks For High-Performance Parallel Computers, IEEE Computer Society Press, Los Alamitos, 1994, 145-152. [2] B. Alspach, The search for long paths and cycles in vertex-transitive graphs and digraphs, in: Kevin L. 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Discrete Math. 51 (1984), 293-304. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 55-67 Polarity graphs revisited Martin Bachraty Comenius University, Bratislava, Slovakia Jozef Siran Open University, Milton Keynes, U.K., and Slovak University of Technology, Bratislava, Slovakia Received 29 August 2013, accepted 2 March 2014, published online 22 April 2014 Abstract Polarity graphs, also known as Brown graphs, and their minor modifications are the largest currently known graphs of diameter 2 and a given maximum degree d such that d - 1 is a prime power larger than 5. In view of the recent interest in the degree-diameter problem restricted to vertex-transitive and Cayley graphs we investigate ways of turning the (non-regular) polarity graphs to large vertex-transitive graphs of diameter 2 and given degree. We review certain properties of polarity graphs, giving new and shorter proofs. Then we show that polarity graphs of maximum even degree d cannot be spanning subgraphs of vertex-transitive graphs of degree at most d + 2. If d - 1 is a power of 2, there are two large vertex-transitive induced subgraphs of the corresponding polarity graph, one of degree d - 1 and the other of degree d - 2. We show that the subgraphs of degree d - 1 cannot be extended to vertex-transitive graphs of diameter 2 by adding a relatively small non-edge orbital. On the positive side, we prove that the subgraphs of degree d - 2 can be extended to the largest currently known Cayley graphs of given degree and diameter 2 found by Siagiova and the second author [J. Combin. Theory Ser. B 102 (2012), 470-473]. Keywords: Graph, polarity graph, degree, diameter, automorphism, group, vertex-transitive graph, Cayley graph. Math. Subj. Class.: 05C25 E-mail addresses: mato4247@gmail.com (Martin Bachraty), j.siran@open.ac.uk (Jozef Siran) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 1 Introduction A graph of diameter 2 and maximum degree d can have at most d2 + 1 vertices. This can be seen by rooting the graph at a vertex of maximum degree and observing that the vertex set of the graph is the union of vertices at distance 0,1 and 2 from the root, giving the estimate 1 + d + d(d — 1) = d2 +1, also known as the Moore bound for diameter 2. Suchagraphof order d2 + 1 exists if and only if d =2,3,7 and possibly 57, by the classical result of [9]. Examples for the first three degrees - the pentagon, the Petersen graph and the Hoffman-Singleton graph - are unique, and existence of a graph of diameter 2, degree 57 and order 572 + 1 = 3250 is still an open problem. For all other values of d > 4 it is known [5] that the largest order of a graph of diameter 2 and maximum degree d is at most d2 — 1, but by [11] examples of graphs of that order are known only for d = 4, 5. Investigation of large graphs of given degree and diameter in general is part of the well known degree-diameter problem, surveyed in [11]. If d = q + 1 where q is prime power such that q > 7, the largest currently known order of a graph of diameter 2 and maximum degree d is d2 — d +1 if q is odd and d2 — d +2 if q is even. Examples of graphs of such order, missing the Moore bound only by d and d — 1, are the polarity graphs B(q) for odd q and their minor modifications for even q; both will be described in the next section. Extensions of polarity graphs by adding Hamilton cycles and maximum matchings taken from the complement were considered in [15] and give graphs of maximum degree d, diameter 2 and order at least d2 — 2d1525 for every sufficiently large d. This shows that the Moore bound can be met at least asymptotically for all sufficiently large degrees. The polarity graphs B(q) were first introduced in 1962 by Erdos and Renyi [6] and later in 1966 independently by Brown [3] (and considered again by Erdos, Renyi and Sos [7]) in connection with asymptotic determination of the largest number of edges in a graph of a given order without cycles of length four. The notation B(q) is derived from the fact that in the degree-diameter research community these graphs have also been known as Brown graphs after their second independent discoverer. Properties of polarity graphs have been studied in considerable detail in [12], including determination of the automorphism group of these graphs and their important subgraphs. Since polarity graphs are not regular, they cannot be vertex-transitive; nevertheless they have a relatively large automorphism group which has three orbits on vertices. The role of polarity graphs in the degree-diameter problem was realized later (cf. [11]) and their modification in the case when q is a power of two comes from [5] and [4]. In view of the recent advances in constructions of large vertex-transitive graphs with given degree and diameter [11] it is of interest to ask if one can modify polarity graphs to obtain large vertex-transitive graphs of a given degree and diameter 2 for an infinite set of degrees. We will consider modifications by inserting new edges into a polarity graph or into an induced subgraph of a polarity graph. It is unclear if extending polarity graphs by Hamilton cycles and maximum matchings [15] can ever produce vertex-transitive graphs. In Section 4 we show that it is not possible to extend a polarity graph B(q) for odd q > 37 to a vertex-transitive graph by increasing its maximum degree by two. Regarding subgraphs, the graph B(q) for q a power of 2 contains two large vertex-transitive induced subgraphs, one of degree q and order q2 and the other of degree q — 1 and order q(q — 1). We prove that the first subgraph cannot be extended to a vertex-transitive graph of diameter 2 by adding non-edge orbitals induced by the smallest transitive group of automorphisms of the graph. In contrast with this, the second subgraph is shown to be isomorphic to the Cayley graph discovered in [14], which is extendable to a Cayley graph of diameter 2 and degree d' + O(Vd') for d! = q — 1 and which shows that the Moore bound for diameter 2 can be approached at least asymptotically. This equips the construction of [14] with a strong geometric flavour. Presentation of these results is preceded in Section 2 by a description of polarity graphs and their properties and in Section 3 by a study of symmetry properties of polarity graphs. In the final Section 5 we discuss possible generalizations. 2 Polarity graphs and their structure Let q be a prime power and let F = GF(q) be the Galois field of order q. We let PG(2, q) denote the standard projective plane over F, with points represented by projective triples, that is, equivalence classes [a] of triples a = (aba2,a3) = (0,0,0) of elements of F, where two triples are equivalent if they are a non-zero multiple of each other. If a = (ai , a2, a3), we will simply write [a] = [ai, a2, a3 ]. The vertex set of the polarity graph B(q) is the set of all the q2 + q +1 points of PG(2, q), and two distinct vertices [a] and [b], where a = (ai, a2, a3) and b = (bi,b2, b3), are adjacent in B(q) if the corresponding triples are orthogonal, that is, if abT = aibi + a2b2 + a3b3 = 0. In the terminology of projective geometry this means that distinct vertices [a] and [b] are adjacent if and only if, in the projective plane PG(2, q), the point [a] lies on the line with homogeneous coordinates [b] and the point [b] lies on the line with homogeneous coordinates [a]. Parsons [12] derived a number of facts on the structure of polarity graphs. We will now give alternative proofs to some of the results of [12] we will need later. Our proofs are much shorter and are based on known facts about projective planes as presented in [8]. Let [a] = [ai, a2, a3] be a vertex of B (q). Identification of neighbours of [a] amounts to determining the solutions (xi ,x2,x3) of the linear equation axT = aixi + a2x2 + a3x3 = 0. This equation has q2-1 non-zero solutions that represent (q2 — 1)/(q-1) = q+1 distinct projective points, which are different from [a] if and only if aaT = a2 + a\ + a3 = 0. It follows that a vertex [a] has q or q +1 neighbours in B(q) according to whether aaT is equal to zero or not. The projective triples [a] of PG(2, q) such that aaT = 0 are precisely those lying on the quadric xxT = 0, which is non-degenerate (and hence a conic) if and only if q is odd. Accordingly, the vertices [a] such that aaT = 0 will be called quadric vertices. Lemma 2.1. The graph B(q) contains exactly q + 1 quadric vertices. Proof. If q is odd, this is Theorem 5.21(i) of [8]. If q is even, a vertex [ai, a2, a3] is quadric if and only if a2 + a2 + a3 = 0, which is in characteristic 2 equivalent to ai + a2 + a3 = 0, and there are exactly (q2 — 1)/(q — 1) = q + 1 such projective triples in this case. □ The vertex set of B(q) is thus a disjoint union of the set V of q2 vertices of degree q + 1 and the set W of q +1 quadric vertices, lying on the quadric xxT = 0. Let Vi be the subset of V comprising all vertices adjacent to at least one quadric vertex and let V2 = V\Vi. With this notation we now present further structural information on the polarity graphs B(q) which we will use later. Proposition 2.2. For every prime power q the graph B(q) has the following properties: (i) The set W ofquadric vertices is independent. (ii) Each pair of vertices of V (adjacent or not) are connected by a unique path of length 2, while no edge incident to a quadric vertex is contained in any triangle; in particular, B(q) has diameter 2. (iii) If q is odd, then every vertex of V1 is adjacent to exactly two quadric vertices, and |Vi| = q(q +1)/2, |V2| = q(q - 1)/2. (iv) If q is odd, then the subgraphs of B(q) induced by V1 and V2 are regular of degree (q — 1)/2 and (q + 1)/2, respectively. (v) If q is even, then |V1| = q2 and V2 is empty; moreover, V1 contains a vertex v adjacent to all quadric vertices and every vertex in V1 \{v} is adjacent to exactly one quadric vertex and the subgraph of B(q) induced by the set V1\{v} is regular of degree q. Proof. Let [a] = [a1,a2,a3] and [b] = [b1,b2,b3] be two distinct vertices of B(q), adjacent or not. Since the vectors a and b are linearly independent over F, the solution space of the linear system axT = 0, bxT = 0 has dimension one. It follows that no pair of quadric vertices can be adjacent and that every pair of distinct vertices are connected by exactly one path of length two, proving (i) and (ii). Note that (ii) also follows from the property of PG(2, q) that any two points lie on a unique line. Let q be odd. Invoking Chapters 7 and 8 of [8], the set W forms a conic and hence an oval. By Corollary 8.2 of [8] applied to the oval W, every vertex of V1 and V2 corresponds to a line of PG(2, q) containing exactly two points of W (a bisecant) or no point of W (an external line), respectively, and |V1| = q(q + 1)/2, |V2| = q(q — 1)/2, which proves (iii). Table 8.1 of [8] shows that a bisecant contains (q — 1)/2 points each lying on exactly two lines determined by projective coordinates corresponding to a vertex in W, while an external line contains (q + 1)/2 points each of which lies on no line determined by projective coordinates corresponding to a vertex in W. This exactly translates to (iv). If q is even, the q +1 vertices of W have the form [a1, a2, a3] with a1 + a2 + a3 =0. The vertex v = [1,1,1] adjacent to every vertex of W is, in the terminology of [8], the nucleus of W. By Corollary 8.8 of [8] on vertices different from the nucleus, every vertex of V1 is incident to exactly one vertex of W and V2 = 0, proving (v). □ We note that the authors of [5] and [4] observed that, for q even, one may extend the polarity graph B(q) by adding a vertex and making it incident to all vertices in W; the new graphs will still have diameter 2. 3 Polarity graphs and their automorphisms The automorphism group of B(q) was determined in [12]. Here we give a different and shorter proof, including a more detailed discussion on groups. The idea is to relate the polarity graphs B(q) to the point-line incidence graph of PG(2, q). We will represent the points and lines of PG(2, q) by projective triples (vectors) of F3 as in the case of vertices of B(q), except that points will be represented by row vectors and lines will be represented by column vectors (distinguished by the 'transpose' superscript). In this notation, a point [a] lies on a line [bT] in PG(2, q) if and only if abT = 0. The involution в on the union of the point set and the line set of PG(2, q) that interchanges [x] with [xT ] is the standard polarity of PG(2, q). The point-line incidence graph I(q) of PG(2, q) is the bipartite graph whose vertex set is the union of the point and line sets of PG(2, q), with a vertex [a] adjacent to a vertex [bT] if and only if the point [a] lies on the line [bT], that is, abT = 0. Observe that the standard polarity в is an automorphism of the bipartite graph I(q), interchanging its two vertex-parts. The fundamental theorem of projective geometry (see e.g. [8]) tells us that the subgroup of all automorphisms of the graph I(q) that fix each of its two vertex parts setwise is isomorphic to the extension PrL(3, q) of the 3-dimensional projective linear group PGL(3, q) over F by the group of Galois automorphisms of F over the prime field of F. Elements of PrL(3, q) are pairs (A, p), where A e PGL(3, q) can be identified with an invertible 3 X 3 matrix over F and p e Gal(F). The action of such an element (A, p) on vertices [x] and [yT] of I(q) is given by first applying A via the assignment [x] ^ [xA] and [yT] ^ [A-1yT] and then applying p to all elements of the resulting projective triples. We continue with a remark regarding orthogonal groups. By the 3-dimensional projective orthogonal group PGO(3, q) we mean the factor group of the subgroup of GL(3, q) consisting of orthogonal matrices by the centre of this subgroup (trivial if q is even and isomorphic to Z2 if q is odd). In characteristic 2 our definition is different from what appears to be a more usual way of introducing an orthogonal group in terms of preservation of a bilinear form and having an irreducible action on a vector space; nevertheless we hope that no confusion will arise. The obvious extension PrO(3, q) of PGO(3, q) by Gal(F) acts on B(q) as a group of automorphisms. Indeed, an element of PrO(3, q) can be identified with a pair (A, p) as above, but this time with A being a 3 x 3 orthogonal matrix, that is, such that AT = A-1, with the obvious identification of A with -A if q is odd. The action is simply given by (A, p)[x] = [p(xA)], and it preserves edges of B(q) since xyT = 0 is equivalent to (xA)(yA)T = 0 by orthogonality of A. We begin by showing that that there are no other automorphisms of B(q). Theorem 3.1. For every prime power q the automorphism group of the polarity graph B(q) is isomorphic to PrO(3, q). Proof. Every automorphism a of B(q) induces an automorphism a of I(q) given by a[x] = a[x] and a[xT] = (a[x])T for every projective triple [x]. By the Fundamental theorem of projective geometry and our earlier discussion, the automorphism a may be represented by an action of a pair (A, p) representing an element of PrL(3, q), given by a[x] = [p(xA)] and a[xT] = [p(A-1xT)] for all projective triples [x]. But since a[x] = a[x] and a[xT] = (a[x])T, we have (a[x])T = a[xT] and hence [p(xA)]T = [p(A-1xT)]. As this is valid for all pairs (A, p), we have [xA]T = [A-1xT] for all projective triples [x]. Letting x be successively equal to (1,0,0), (0,1,0), (0,0,1), (1,1,0) and (1,0,1) one concludes that A-1 = SAT for some non-zero S e F. Taking determinants yields 1 = |SATA| = S3|A|2 and this is an equation between elements of the (cyclic) multiplicative group F * of F. We will show that there is a y e F * such that y2 = S. This is obvious if q is even (since in such a case every element of F * has a unique square root), or if S = 1. If q is odd and S = 1, then 1 = S3|A|2 implies that a third power of S =1 is equal to 1 or to a second power of | A | 1 if |A| = 1. This is, in the cyclic group F * of even order q - 1, possible only if q - 1 is divisible by 3 and there is a 7 G F * such that S = 72 (and, in the second case, if |A| = 7-3). In all circumstances we therefore have a 7 G F * such that S = 72. The matrix AY = 7A is orthogonal, i.e., A-1 — Ay. Since all our calculations are done with projective triples, the action of the original automorphism a may equivalently be described by a[x\ = [ip(xAY)] and a[xT\ = [^>(A-1 xT)] for all projective triples [x\, where the pair (AY, ф) represents an element of РГО(3, q) as a subgroup of РГЬ(3, q). This shows that every automorphism of B(q) is induced by an element of РГО(3, q). Since this group acts on B(q) as we saw earlier, the result follows. □ It is known (cf. [8]) that PGO(3, q) = PGL(2, q) and РГО(3, q) = РГЬ(2, q) for every prime power q. Theorem 3.1 thus implies that if q = pn where p is a prime, then the graph B(q) has exactly nq(q2 - 1) automorphisms. The groups РГО(3, q) and PGO(3, q) obviously preserve the sets W, V1 and V2 and the sets W, {v} and V1\{v}, depending on whether q is odd or even; it is easy to show that these sets are, in fact, orbits of PGO(3, q) on the vertices of B(q). Corollary 7.15 of [8] tells us that the group PGO(3, q) is triply transitive on W. For odd q the analysis of [12] shows that PGO(3, q) acts arc-transitively on the subgraphs induced by the vertex set V1 and V2. We can say much more if q is even, extending the last result of Section 6 of [12]. Let B0(q) be the subgraph of B(q) induced by the set V0 = Vl\{v}. Theorem 3.2. If q is a power of 2, the automorphism group of the graph B0(q) is isomorphic to PrO(3, q). Moreover, if q > 4, the smallest subgroup of PrO(3, q) acting transitively on vertices of B0(q) is the group PGO(3, q), which also acts regularly on arcs of B0(q). In particular, B0(q) is a vertex-transitive non-Cayley graph if q > 2. Proof. For every w G W let Nw be the set of neighbours of w in B(q) distinct from v. Part (v) of Proposition 2.2 implies that the sets Nw (w g W) form a partition of the vertex set of B0 (q). In the subgraph B0 (q) the distance of any two vertices is greater than 2 if and only if the two vertices are in the same set Nw for some w g W .It follows that any automorphism of B0(q) preserves the partition {Nw; w g W} and hence extends to an automorphism of the entire polarity graph B(q). Consequently, by Theorem 3.1, the automorphism group of B0(q) is isomorphic to PrO(3, q); this was also noted in Section 6 of [12]. The rest of the proof will address the smallest group transitive on vertices of B0 (q). It is easy to check that B0(2) is isomorphic to a complete graph of order 3, admitting a regular action of a subgroup of PrO(3, q) = S3 isomorphic to Z3. From now on we therefore assume that q > 4. By the remark after Theorem 3.1 we know that PrO(3, q) = PrL(2, q), and for q = 2n with n > 2 we have PrL(2, q) = SL(2, q) x Zn, the split extension being defined by the natural action of Zn = Aut(GF(q)) on SL(2, q). In what follows we will, for simplicity of the arguments, identify the group PrO(3, q) with the group G = SL(2, q) x Zn. We also let K = SL(2, q) and note that K is normal in G. Suppose that H is a subgroup of G acting transitively on the vertex set of B0 (q). Letting H0 = H n K and observing that H0 is normal in H, from the second group isomorphism theorem we have H/H0 = HK/K. It follows that the order of H/H0 is a divisor of n. On the other hand, the transitivity assumption on H implies that the order of H is a multiple of q2 - 1, the number of vertices of B0(q). These two facts imply that |H0| = t(q2 - 1)/n for some positive integer t. Since H0 has now been identified with a subgroup of K = SL(2, q), the task reduces to identification of subgroups of the group SL(2, q) = PSL(2, q), q = 2n, of order t(q2 - 1 )/n. We will show that PSL(2, q) admits no such proper subgroup H0 if n > 2, using Dickson's classification of subgroups of PSL(2, q) for q = 2n as displayed in [13]. By this classification, subgroups of PSL(2, q) split into four classes: (a) cyclic and dihedral subgroups, (b) affine subgroups, (c) subgroups isomorphic to A4 or A5 for n even, and (d) subgroups of the form PSL(2,2m) where m is a divisor of n. We analyse the cases separately, recalling that throughout we assume q = 2n and n > 2. (a) Cyclic and dihedral subgroups. The largest order of a subgroup of PSL(2, q) that is cyclic or dihedral is 2(q + 1). It is easy to see that 2(q +1) < t(q2 - 1)/n for n > 3 and all t > 1; if n = 2 then t has to be even and then the same inequality holds. It follows that H0 cannot be cyclic or dihedral. (b) Affine subgroups. The largest order of an affine subgroup of PSL(2, q) is q(q - 1) and the second largest order of such a subgroup is - 1) if n is even. If q(q - 1) = t(q2 - 1)/n, then nq = t(q + 1) and q + 1 would have to divide n, a contradiction. For the second largest order, observe that - 1) < q +1 < (q2 - 1)/n, the last inequality being a consequence of n > q - 1. We conclude that H0 cannot be affine. (c) The groups A4, As. We may eliminate A4 since its order is 12 and the order of B0(22) is 15. Since the order of B0(24) is 28 -1, the only feasible value of n for H0 = A5 is n = 2. In this case, however, A5 = PSL(2,22) and so H0 would not be a proper subgroup of PSL(2,22). (d) Subgroups PSL(2, 2m) where m | n. Let H0 = PSL(2, 2m) for m a proper divisor of n. If n G {2,3}, then m = 1 and the order of PSL(2, 2) is too small for H0 to be transitive on vertices of B0(2n). If n > 4, then the largest order of such a subgroup H0 is at most ^J~q(q - 1). It is easy to check, however, that - 1) < (q2 - 1)/n for n > 4, giving a contradiction again. The above arguments show that, for q = 2n and n > 2, the smallest subgroup of PrO(3, q) = PrL(2, q) transitive on vertices of B0(q) is the group PGO(3, q) = PSL(2, q). It is a matter of routine to check that this group is, in fact, regular on the arc set of B0(q). Combining the two facts we conclude that B0(q) is an arc-transitive (and, of course, vertex-transitive) non-Cayley graph for all n > 2. □ 4 Vertex-transitive graphs from polarity graphs? Polarity graphs are, of course, not vertex-transitive. Being the largest currently known examples of graphs of maximum degree q + 1 and diameter 2, however, it is interesting to ask if one cannot add "a few" edges to a polarity graph to obtain a vertex-transitive graph. In [15] it was shown that it is impossible to construct a vertex-transitive graph of degree q+1 which would contain B(q) as a spanning subgraph. We extend this result to the degree q + 3 for odd q > 37. Theorem 4.1. For any odd prime power q > 37 there is no vertex-transitive graph of degree q + 3 which contains the polarity graph B(q) as a spanning subgraph. Proof. Let B = B(q) and let B' be a graph containing B as a spanning subgraph. Let E and E' be the edge set of B and B', respectively; edges of the set E and E'\E will be called old and new, respectively. Suppose now that B' is a vertex-transitive graph of degree q+3. Let u, v be vertices of B such that e = uv is a new edge and let N(u) and N(v) be the set of neighbours of u and v in B. From the fact that any two distinct non-adjacent vertices of B are joined by a unique path of length 2 it follows that there is a unique vertex, say, w, in N(u) n N(v). If |N(u)| = |N(v)|, the set of all old edges joining the set N(u)\{w} with the set N(v)\{w} forms a perfect matching between the two sets. If not, then |N(u) | and |N(v) | differ by one. Without loss of generality, if |N(u) | = |N(v) | + 1, then exactly one neighbour of u, say, t, is joined to w. In this case there is a perfect matching between the sets N(u)\{w, t} and N(v)\{w}. It follows that any new edge e = uv is contained in a set Se of quadrangles such that (1) |Se| > q - 1, and (2) any two quadrangles in Se share just the edge e and its end-vertices u, v. An edge e g E' will be called thick if there exists a set Se of quadrangles with the properties (1) and (2) above. Note that this definition does not require e to be new, but the fact we have derived above implies that every new edge is thick. By the assumed vertex-transitivity, every vertex of B' must be incident to the same number, say, t, of thick edges. Since the degree of B' is supposed to be q + 3, every vertex in W is incident to at least three thick edges, so that t > 3. If t = 3, the three thick edges are all new and every vertex in V would be incident to exactly two new thick edges and one old thick edge. This would mean that there is a perfect matching on V formed by old thick edges, which is impossible since | V | = q2 and q is odd. It follows that t > 4, every vertex in W is incident to at least one old thick edge and every vertex in V is incident to at least two old thick edges. Properties of B imply that for any vertex v g V2 the subgraph of B induced by the set N(v) is a perfect matching of (q + 1)/2 edges. Vertex-transitivity of B' then implies that for every vertex w g W the subgraph of B' induced by the set N'(w) of all q + 3 neighbours of w in B' contains a subset EW of (q + 1)/2 mutually independent edges. Note that since there were no edges in N(w) in the graph B, all the edges in E'w must be new. At most three edges of E'w join a vertex from N(w) with a vertex in N'(w)\N(w), and so the subgraph of B' induced by the set N(w) contains a subset Ew с E'w of least (q + 1)/2 - 3 = (q - 5)/2 new edges. Since any two neighbourhoods of vertices of W in the graph B intersect in exactly one vertex, the sets of new edges Ew, w g W, are mutually disjoint. For counting, imagine that every new edge consists of two new half-edges incident to the corresponding end-vertices. The q(q + 1)/2 vertices of V1, each incident with two new half-edges, are incident to a total of q(q + 1) new half-edges. At least (q + 1)(q - 5) of these are 'absorbed' by vertices of V1 since the sets Ew, w g W, are pairwise disjoint and N (w) с Vi for every w g W .It follows that there are at most (q + 1)q - (q + 1)(q - 5) = 5(q + 1) new edges that join vertices of V1 with vertices outside V1. In particular, there are at most 5(q + 1) new edges between V1 and V2. We know that every vertex w g W is incident to an old thick edge; let e = wv be such an edge, v g V1. Let Se be a set of quadrangles with the properties (1) and (2) introduced earlier. At most 3 quadrangles of Se can contain a new edge incident with w, at most 2 such quadrangles can contain a new edge incident with v, and since v has at most (q - 1)/2 + 2 neighbours from V1 in the graph B', at most (q - 1)/2 + 2 quadrangles in Se contain a new edge having both end-vertices in V1. Note that in each quadrangle containing three old edges the fourth edge must be new. Consequently, there are at least (q - 1) - ((q - 1)/2 + 7) = (q - 15)/2 quadrangles in Se containing at least one new edge joining a vertex in V1 with a vertex in V2. Since our considerations are valid for every vertex w g W, the neighbourhoods N (w) in the graph B intersect just in one vertex, and every vertex in V1 is incident to two new edges, we conclude that there are at least ((q + 1 )(q — 15)/2)/2 new edges joining V1 with V2. But we have seen earlier that the number of such edges is at most 5(q + 1). Thus, (q + 1)(q — 15)/4 < 5(q + 1), that is, q < 35, contrary to our assumption that q > 37. □ Another obvious method to create large vertex-transitive graphs from polarity graphs is to take a large vertex-transitive subgraph of B(q) and try to extend it to a vertex-transitive graph of diameter 2 by adding edges. The hottest candidate for this is the subgraph B0 (q) if q is a power of 2, which we have encountered in the previous section. Theorem 3.2, however, is bad news for adding edges to B0(q) to produce a vertex-transitive graph of diameter two. Namely, the most natural approach would be to take the smallest group H acting transitively on the set of vertices of B0(q) and identify the smallest possible number of H-orbits of non-adjacent pairs of vertices so that making these pairs adjacent would yield a graph of diameter 2 with B0(q) as a spanning subgraph. But by Theorem 3.2, for q > 4 the smallest such subgroup H is isomorphic to PGO(3, q), acting transitively and with vertex stabilisers of order q. It follows that an orbit furnishing new edges would increase the degree of the resulting graph by at least q. This would make the resulting graph uninteresting from the point of view of the degree-diameter problem. Before continuing let us comment on the isomorphism of the groups PGO(3, q) and PGL(2, q), mentioned after the proof of Theorem 3.1. In [8] an isomorphism of the two groups is given, induced by the quadratic form x" + x^3. While for odd q such a form is equivalent to x" + x" + x" which we have been using, this is not true for even q since for q a power of 2 the form x\ + x" + x" is degenerate. It is easy to check that, with respect to this form, all elements of PGO(3, q) for q even have the form 1 + a 1 + c 1 + a + c 1+b 1+d 1+b+d K1 + a + b 1 + c + d 1 + a + b + c + dj where a,b,c,d e F = GF(q) and ad + bc = 1, which implies that (a, b) = (0,0). Then, for even q, one may check that the mapping ф : PGO(3, q) ^ PGL(2, q) given by 1 + a 1 + c 1 + a + c \ 1+b 1+d 1+b+d i ^ (da 1 + a + b 1 + c+ d 1 + a + b + c+ d d b is a group isomorphism. We will use its inverse ф-1 in the proof of our last two result. In contrast with Theorem 3.2, there exists a subgroup of PGO(3, q) that is transitive on the set V* = Vo \ {[t, t, 1], t e GF(q)}; let B*(q) be the subgraph of Bo(q) induced by the set V*. Our last result shows that we can construct large Cayley graphs of diameter 2 and degree d = q + O(^q) by adding edges to B*(q). Since the order of B*(q) is q(q — 1) = d" — O(d3/"), the resulting graphs will be asymptotically approaching the Moore bound. Theorem 4.2. For every even prime power q there exists a Cayley graph of diameter 2 and degree q + O(^q) with B*(q) as a spanning subgraph. Proof. Let H be the subgroup of PGO(3, q) formed by all the matrices as in (4) for which a + b + c + d = 0. It is straightforward to check that |H| = q(q — 1) and that H acts regularly on the vertex set of the graph B*(q). It follows that B*(q) is a Cayley graph Cay(H, X) for the group H and some inverse-closed generating set X с H such that |X| = q - 1, which is the degree of B*(q). In order to create a Cayley graph of diameter 2 from B*(q) by adding O(^q) edges it is sufficient to show that one can extend the generating set X to a set X ' D X by adding O(^q) elements of H. Since we are dealing with a Cayley graph, it is sufficient to check distances from one particular vertex u, say, u = [1,0,0]. Compared with the graph B(q), in our new graph B*(q) the vertex u lost two neighbours, namely, v1 = [0,1,1] and v2 = [0,0,1]. We consider the effect caused by losing the two neighbours separately. Since uvi is not an edge of B* (q), we lost the paths of length 2 joining u with vertices in the subset U1 of B*(q) of the form [1,t,t], t G F, t = 0,1. Consider the subgroup H1 < H formed by the matrices Ф-1( J9 ), g G F*, where J = ( g 0 9 = U+g-1 g-1 It may be verified that H1 = F * and H1 acts regularly on the set U1 U {u}. Geometrically, H1 can be identified with the group of homologies (that is, central-axial collineations with a non-incident center-axis pair) with centre v1 and axis vv2. Since F * is Abelian (in fact, cyclic), there exists an inverse-closed set X1 of at most elements such that the Cayley graph Cay(H1, X1) with vertex set U1 U {u} has diameter 2. The effect of the missing edge uv2 is that we lost the paths of length 2 joining u with vertices in the subset U2 of B*(q) of the form [a + 1, a, 0], a G F*. Let now H2 < H be the subgroup of H consisting of the matrices ф-1(Ьа), a G F+, where a +1 a a a +1 Obviously, H2 = F + and H2 is easily seen to be a regular permutation group on the set U2 U {u}. From the point of view of geometry, H2 can be identified with the group of elations (central-axial collineations with an incident center-axis pair) with centre [1,1,0] and axis vv2. Again, it is well known that there exists an inverse-closed set X2 of at most [2yq] elements, making Cay(H2, X2) a graph of diameter 2 with vertex set U2 U {u}. It is now easy to check that the graph Cay(H, X') with X' = X U X1 U X2 has the required properties. □ Cayley graphs of diameter 2, order q(q - 1) and degree q + O(^q) for even q have been recently constructed in [14] as follows. For q a power of 2 and F = GF(q) consider the one-dimensional affine group Gq = AGL(1,q) ~ F + x F*, with elements written as pairs (g, a), g G F*, and a G F+ and with multiplication in the form (g, a)(h, b) = (gh, ah + b) for g, h G F * and a, b G F +. The set Yq = {(y2, y); y G F *} is an inverse-closed generating set of Gq. The graphs of [14] are formed by taking the Cayley graph Cay(Gq, Yq) and adding a suitable set of further O(^q) generators to Yq. Our last result shows that, quite surprisingly, the Cayley graphs Cay(Gq, Yq ) from [14] are isomorphic to the graphs B* (q). This gives the construction of [14] a strong geometric flavour. Theorem 4.3. If q is a power of 2, the graph Cay(Gq, Yq) is isomorphic to B*(q). Proof. We will use the notation introduced in the proof of Theorem 4.2, by which the graph B* (q) is isomorphic to the Cayley graph Cay (H, X ). It may be checked that every element of H can be written as a product ф-1( Jg La) with g g F * and a G F +. Let us identify this product with the ordered pair [g, a]. One easily verifies that in this identification the multiplication о of elements of H is represented in the form [g, a] o [h, b] = [gh, ah-2 + b]. A further composition of the mapping ф-1 ( JgLa) ^ [g, a] with ф : [g, a] ^ (g-2, a) establishes an isomorphism H = Gq. It remains to analyse the generating set X, which is uniquely determined by the elements of H that take a fixed vertex of B* (q) to all its neighbours; in what follows our fixed vertex will be u = [1,0,0]. Now, multiplication of the row vector (1,0,0) by the matrix ф-1( JgLa) yields the vector (1 + ag, 1 + g + ag, 1 + g). This means that the element of H encoded [g, a] takes the vertex u onto the vertex [1 + ag, 1 + g + ag, 1 + g] of B* (q). Since the neighbours of u in B*(q) have the form [0,1, s] for s G F, s = 1, the elements of H taking u to the neighbours of u are encoded by pairs [t-1, t] for t g F *. The generating set X of H thus consists, in our representation, of all the pairs [t-1, t], where t g F*. Composition with the mapping ф introduced earlier finally establishes the isomorphism from the Cayley graph Cay(H, X) onto the Cayley graph Cay(Gq, Yq) of [14]. □ 5 Remarks Adjacency in polarity graphs B(q) has been defined by means of the standard dot product. For odd q the standard dot product is just a special case of a symmetric non-singular bilinear form. What happens if one uses a more general form instead? Following [2], for an odd prime power q let Q be a non-singular quadratic form over F3 where F = FG(q), and let в(x, y) = Q(x + y) — Q(x) — Q(y) be the corresponding non-singular symmetric bilinear form. We now may let Bß(q) be the graph on the same vertex set as B(q), but with two distinct vertices [a] and [b] adjacent if в (a, b) = 0. It is known, however (see e.g. [2, 8]), that in dimension 3 and for odd q, all non-singular quadratic forms are equivalent and their equivalence is induced by linear transformations (i.e., by change of bases). It follows that for odd q all such graphs Bß(q) are isomorphic to B(q), with isomorphisms being provided by the corresponding linear transformations. Of course, such a correspondence between quadratic forms and bilinear forms fails in characteristic 2. Another way of generalizing polarity graphs is to define them on more general finite projective planes. Recall that a finite projective plane P is a collection of a finite number of points and lines such that every two points are incident with a unique line, every two lines are incident with a unique point, and there are four points no three of which are incident with a line. It is well known that for any such P there is an integer n such that any line is incident with precisely n + 1 points and, dually, any point is incident with exactly n +1 lines. One then speaks about a projective plane of order n, and it is then easy to show that the number of points and the number of lines are both equal to n2 + n +1. (An outstanding conjecture in finite geometry is that the order of a finite projective plane must be a prime power.) Suppose now that P has a polarity, that is, a bijection n from the point set onto the line set of P with the property that for every two points A and B, A is incident with n(B) if and only if B is incident with n(A). We may then define the generalized polarity graph Bpn with vertex set equal to the point set of P and with two distinct points A and B adjacent if A is incident with n(B). This graph obviously has diameter 2 by the properties of the projective plane. Observe that if P = PG(2, q) is the standard projective plane as introduced in Section 2 and if one considers the standard polarity n interchanging projective vectors with their transposes, then the graph Bpcoincides with the polarity graph B(q). It is of interest to point out that if P is a (general) finite projective plane of order n with a polarity n, then, by [1], the number m(n) of self-conjugate points (those incident with their n-images) satisfies m(n) > n +1, and if m(n) > n +1 then n is a square and every prime divisor of n divides m(n) — 1. Since the corresponding generalized polarity graph Bp has exactly m(n) vertices of degree n and all the remaining vertices have degree n + 1 it follows that for n = q, where q is an even power of a prime, the graph Bp,n need not be isomorphic to B(q). Investigation of such a generalization of polarity graphs may lead to interesting results. We conclude by commenting on vertex-transitive extensions of polarity graphs. In general, by a vertex-transitive closure of a graph G we will understand any vertex-transitive supergraph of G on the same vertex set. We may then define dvt(G) to be the smallest degree of a vertex-transitive closure of G. In this terminology, our Theorem 4.1 says that dvt(Bq) > q + 5 for any odd prime power q > 37. Determining or at least estimating dvt(G) for arbitrary graphs G appears to be an interesting problem on its own. Acknowledgement. The authors would like to thank an anonymous referee for carefully reading the manuscript, suggesting important improvements and providing geometric insight into the groups used in Section 4. The second author acknowledges support from the VEGA Research Grants 1/0871/11, 1/0065/13 and 1/0007/14, the APVV Research Grants 0223-10 and 0136-12, as well as the APVV support as part of the EUROCORES Programme EUROGIGA, project GREGAS, ESF-EC-0009-10, financed by the European Science Foundation. References [1] R. Baer, Polarities in finite projective planes, Bull. Amer. Math. Soc. 52 (1946), 77-93. [2] S. Ball and Z. Weiner, An Introduction to Finite Geometry, Preprint (2011). [3] W. G. Brown, On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966), 281-285. [4] C. Delorme, Examples of products giving large graphs with given degree and diameter, Discrete Applied Math. 37-38 (1992), 157-167. [5] P. Erdos, S. Fajtlowicz and A. J. Hoffman, Maximum degree in graphs of diameter 2, Networks 10 (1980), 87-90. [6] P. Erd6s and A. Renyi, On a problem in the theory of graphs, Magyar Tud. Akad. Mat. Kutató Int. Kozl. 7 (1962), 623-641. [7] P. Erd6s, A. Renyi and V. T. Sós, On a problem of graph theory, Studia Sci. Math. Hungar. 1 (1966), 215-235. [8] J. Hirschfeld, Projective geometries over finite fields, Oxford University Press, 1998. [9] A. J. Hoffman and R. R. Singleton, On Moore graphs with diameter 2 and 3, IBM J. Res. Develop. 4 (1960), 497-504. [10] B. Huppert, Endliche Gruppen 1, Springer, 1967. [11] M. Miller and J. Siran, Moore graphs and beyond: A survey of the degree-diameter problem, Electronic J. Combin. (2013), Dynamic survey DS14. [12] T. D. Parsons, Graphs from projective planes, Aequat. Math. 14 (1976), 167-189. [13] J. De Saedeleer and D. Leemans, On the rank two geometries of the groups PSL(2, q): part I, Ars Math. Contemp. 3 (2010), 177-192. [14] J. Siagiova and J. Siran, Approaching the Moore bound for diameter two by Cayley graphs, J. Combin. Theory Ser. B 102 (2012), 470-473. [15] J. Siran, J. Siagiova and M. Ždfmalova, Large graphs of diameter two and given degree, in: Proc. IWONT 2010, Univ. Politecnica de Catalunya, 2011, 347-359. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 69-82 Tight orientably-regular polytopes Marston Conder University of Auckland Auckland 1142, New Zealand Gabe Cunningham University ofMassachusetts Boston Boston, Massachusetts 02125, USA Received 9 October 2013, accepted 9 February 2014, published online 7 May 2014 Abstract It is known that every equivelar abstract polytope of type {p1,... ,pn-1} has at least 2p1 • • • pn-1 flags. Polytopes that attain this lower bound are called tight. Here we investigate the conditions under which there is a tight orientably-regular polytope of type {p1,... ,pn-1}. We show that it is necessary and sufficient that whenever pi is odd, both pi-1 and pi+1 (when defined) are even divisors of 2pi. Keywords: Abstract regular polytope, equivelar polytope, flat polytope, tight polytope. Math. Subj. Class.: 51M20, 52B15, 05E18 1 Introduction Abstract polytopes are ranked partially-ordered sets that resemble the face-lattice of a convex polytope in several key ways. Many discrete geometric objects can be viewed as an abstract polytope by considering their face-lattices, but there are also many new kinds of structures that have no immediate geometric analogue. A flag of an abstract polytope is a chain in the poset that contains one element of each rank. In many ways, it is more natural to work with the flags of a polytope rather than the faces themselves. For example, every automorphism (order-preserving bijection) of a polytope is completely determined by its effect on any single flag. Regular polytopes are those for which the automorphism group acts transitively on the set of flags. The automorphism group of a regular polytope is a quotient of some string Coxeter group [p1,... ,pn-1], and conversely, every sufficiently nice quotient of a string E-mail addresses: m.conder@auckland.ac.nz (Marston Conder), gabriel.cunningham@gmail.com (Gabe Cunningham) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Coxeter group appears as the automorphism group of a regular polytope. Indeed, it is actually possible to reconstruct a regular polytope from its automorphism group, so that much of the study of regular polytopes is purely group-theoretic. We say a regular polytope P is of type (or has Schlafli symbol) {pl,... ,pn-1} if [pi,... ,pn-1] is the minimal string Coxeter group that covers the automorphism group of P, in a way that p1,... ,pn-1 are the orders of the relevant generators. There is an equivalent formulation of this property that is entirely combinatorial, and hence it is possible to define a Schlafli symbol for many non-regular polytopes, including chiral polytopes (see [10]) and other two-orbit polytopes (see [7]). Any polytope with a well-defined Schlafli symbol is said to be equivelar. In [3], the first author determined the smallest regular polytope (by number of flags) in each rank. To begin with, he showed that every regular polytope of type {p1,..., pn-1} has at least 2p1 • • • pn-1 flags. Polytopes that meet this lower bound are called tight. He then exhibited a family of tight polytopes, one in each rank, of type {4,..., 4}. Using properties of the automorphism groups of regular polytopes, he showed that each polytope was the smallest regular polytope in rank n > 9, and that in smaller ranks, the minimum was also attained by a tight polytope (with type or dual type {3}, {3,4}, {4,3,4}, {3, 6, 3,4}, {4, 3,6,3,4}, {3, 6,3,6,3,4} or {4,3, 6, 3,6,3,4}, respectively). The second author showed in [5] that the bound on the number of flags extended to any equivelar polytope, regardless of regularity. Accordingly, it makes sense to extend the definition of tight polytopes to include any polytope of type {p1,... ,pn-1} with 2p1 • • • pn-1 flags. An alternate formulation was proved as well, showing that an equivelar polytope is tight if and only if every face is incident with all faces (if any) that are 2 ranks higher. Tightness is a restrictive property, and not every Schlafli symbol is possible for a tight polytope. In order for there to be a tight polytope of type {p1,... ,pn-1}, it is necessary that no two adjacent values pi and pi+1 are odd. Theorem 5.1 in [5] shows that this condition is sufficient in rank 3. In higher ranks, the question of sufficiency is still open. Constructing non-regular polytopes in high ranks is difficult. In order to determine which Schlafli symbols are possible for a tight polytope, it is helpful to begin by considering only regular polytopes. If every pi is even, then Theorem 5.3 in [3] and Theorem 6.3 in [5] show that there is a tight regular polytope of type {p1,... ,pn-1}. Also the computational data from [2, 6] led the second author to conjecture that if p is odd and q > 2p, there is no tight regular polyhedron of type {p, q}. Although we are currently unable to prove this conjecture, we can show that for tight orientably-regular polyhedra, if p is odd then q must divide 2p. Moreover, we are able to prove the following generalisation in higher ranks: Theorem 1.1. There is a tight orientably-regular polytope of type {p1,... ,pn-1} if and only if each of the integers pi-1 and pi+1 (when defined) is an even divisor of 2pi whenever pi is odd, for 1 < i < n. 2 Background Our background information is mostly taken from [8, Chs. 2, 3, 4], with a few small additions. 2.1 Definition of a polytope Let P be a ranked partially-ordered set, the elements of which are called faces, and suppose that the faces of P range in rank from -1 to n. We call each face of rank j a j-face, and we say that two faces are incident if they are comparable. We also call the 0-faces, 1-faces and (n - 1)-faces the vertices, edges and facets of P, respectively. A flag is a maximal chain in P. We say that two flags are adjacent if they differ in exactly one face, and that they are j-adjacent if they differ only in their j -faces. If F and G are faces of P such that F < G, then the section G/F consists of those faces H such that F < H < G. If F is a j-face and G is a k-face, then we say that the rank of the section G/F is k-j — 1. If removing G and F from the Hasse diagram of G/F leaves us with a connected graph, then we say that G/F is connected. That is, for any two faces H and H' in G/F (other than F and G themselves), there is a sequence of faces H = Ho, H i,..., H k = H ' such that F < Hj < G for 0 < i < k and the faces Hj_ i and H are incident for 1 < i < k. By convention, we also define all sections of rank at most 1 to be connected. We say that P is an (abstract) polytope of rank n, or briefly, an n-polytope, if it satisfies the following four properties: (a) There is a unique greatest face Fn of rank n, and a unique least face F_ i of rank — 1. (b) Each flag has n+2 faces. (c) Every section is connected. (d) Every section of rank 1 is a diamond — that is, whenever F is a (j — 1)-face and G is a (j + 1)-face for some j, with F < G, there are exactly two j-faces H with F < H < G. Condition (d) is known as the diamond condition. Note that this condition ensures that for 0 < j < n, every flag Ф has a unique j-adjacent flag, which we denote by Ф^. In ranks — 1, 0, and 1, there is a unique polytope up to isomorphism. Abstract polytopes of rank 2 are also called abstract polygons, and for each 2 < p < x>, there is a unique abstract polygon with p vertices and p edges, denoted by {p}. If F is a j-face and G is a k-face of a polytope with F < G, then the section G/F itself is a (k — j — 1)-polytope. We may identify a face F with the section F/F_ i, and call the section Fn/F the co-face at F. The co-face at a vertex F0 is also called a vertex-figure at Fo. If P is an n-polytope, F is an (i — 2)-face of P, and G is an (i + 1)-face of P with F < G, then the section G/F is an abstract polygon. If it happens that for 1 < i < n, each such section is (isomorphic to) the same polygon {pj, no matter which (i—2)-face F and incident (i+1)-face G we choose, then we say that P has Schlafli symbol {pi,... ,pn-i}, or that P is of type {pi,... ,pn-i}. Also when this happens, we say that P is equivelar. All sections of an equivelar polytope are themselves equivelar polytopes. In particular, if P is an equivelar polytope of type {pi,... ,pn-i}, then all its facets are equivelar polytopes of type {pi,... ,pn_2}, and all its vertex-figures are equivelar polytopes of type {p2, . . . ,pn_i}. Next, let P and Q be two polytopes of the same rank. A surjective function 7 : P ^ Q is called a covering if it preserves incidence of faces, ranks of faces, and adjacency of flags. If there exists such a covering 7 : P ^ Q, then we say that P covers Q. The dual of a polytope P is the polytope obtained by reversing the partial order. If P is an equivelar polytope of type {pi,... ,pn-i}, then the dual of P is an equivelar polytope of type {pn_i,... ,pi}. 2.2 Regularity For polytopes P and Q, an isomorphism from P to Q is an incidence- and rank-preserving bijection. By connectedness and the diamond condition, every polytope isomorphism is uniquely determined by its effect on a given flag. An isomorphism from P to itself is an automorphism of P, and the group of all automorphisms of P is denoted by Г(Р). We will denote the identity automorphism by e. We say that P is regular if the natural action of r(P) on the flags of P is transitive (and hence regular, in the sense of being sharply-transitive). For convex polytopes, this definition is equivalent to any of the usual definitions of regularity. Now let P be any regular polytope, and choose a flag Ф, which we call a base flag. Then the automorphism group r(P) is generated by the abstract reflections p0, ... , pn-1, where pi maps Ф to the unique flag Ф® that is i-adjacent to Ф. These generators satisfy p2 = e for all i, and (pipj)2 = e for all i and j such that |i - j| > 2. Every regular polytope is equivelar, and if its Schlafli symbol is {p1,... ,pn-1}, then the order of each pi-1pi is pi. Note that if P is a regular polytope of type {p1,... ,pn-1}, then r(P) is a quotient of the string Coxeter group [p1,... ,pn-1], which is the abstract group generated by n elements x0,..., xn-1 subject to the defining relations x"2 = 1, (xi-1xi)pi = 1, and (xixj)2 = 1 whenever |i — j| > 2. Next, if Г is any group generated by elements p0,..., pn-1, we define Г/ = (pi | i g I} for each subset I of the index set {0,1,..., n — 1}. If Г is the automorphism group r(P) of a regular polytope P, then these subgroups satisfy the following condition, known as the intersection condition: Г/ n Tj = T/nj for all I,J С {0,1,..., n — 1}. (2.1) More generally, if Г is any group generated by elements p0, ... , pn-1 of order 2 such that (pipj)2 = 1 whenever |i — j| > 2, then we say that Г is a string group generated by involutions, and abbreviate this to say that Г is an sggi. If the sggi Г also satisfies the intersection condition (2.1) given above, then we call Г a string C-group. There is a natural way of building a regular polytope P (Г) from a string C-group Г such that Г^ (Г)) = Г and P ^(P )) = P. In particular, the i-faces of P (Г) are taken to be the cosets of the subgroup Г = (pj | j = i}, with incidence of faces Г^ and Г ф given by Г^ < Г j ф if and only if i < j and Г^ n Г ф = 0. This construction is also easily applied when Г is any sggi (not necessarily a string C-group), but in that case, the resulting poset is not always a polytope. The following theory from [8] helps us determine when a given sggi is a string C-group: Proposition 2.1. Let Г = (p0,..., pn-1} be an sggi, and Л = (Ao,..., An-1} a string C-group. If there is a homomorphism п : Г ^ Л sending each ai to Ai, and if n is one-to-one on the subgroup (p0,..., pn-2} or the subgroup (p1,..., pn-1}, then Г is a string C-group. Proposition 2.2. Let Г = (p0,..., pn-1} be an sggi. If both (p0,..., pn-2} and (p1,..., pn-1} are string C-groups, and (po,..., pn-2} n (pb ..., pn-1} С (pb ..., pn-2}, then Г is a string C-group. Given a regular n-polytope P with automorphism group Г = (p0,..., pn_ 1}, we define the abstract rotations o\,..., m each i-face of P is (k, m)-flat, and for j < k each co-j-face of P is (k —j — 1, m— j — 1)-flat. Next, we consider tightness. An equivelar polytope P of type {p1,... ,pn-1} has at least 2p1 • • • pn-1 flags, by [5, Proposition 3.3]. Whenever P has exactly this number of flags, we say that P is tight. It is clear that P is tight if and only if its dual is tight, and that in a tight polytope, every section of rank 3 or more is tight. Also we will need the following, taken from [5, Theorem 4.4]: Theorem 2.4. Let n > 3 and let P be an equivelar n-polytope. Then P is tight if and only if it is (i, i + 2)-flatfor 0 < i < n — 3. Later in this paper we will build polytopes inductively, and for that, the following approach is useful. We say that the regular n-polytope P has the flat amalgamation property (or FAP) with respect to its k-faces, if adding the relations pi = e for i > k to r(P) yields a presentation for (p0,..., pk-1}. Similarly, we say that P has the FAP with respect to its co-k-faces if adding the relations pi = e for i < k yields a presentation for (pfc + b . . . , pn_1}. We will also use the following, taken from [8, Theorem 4F9]: Theorem 2.5. Suppose m, n > 2, and 0 < k < m — 2 where k > m — n. Let P1 be a regular m-polytope, and let P2 be a regular n-polytope such that the co-k-faces of P1 are isomorphic to the (m — k — l)-faces of P2. Also suppose that P1 has the FAP with respect to its co-k-faces, and that P2 has the FAP with respect to its (m — k-l)-faces. Then there exists a regular (k+n+l)-polytope P such that P is (k, m)-flat, and the m-faces of P are isomorphic to Pi, while the co-k-faces of P are isomorphic to P2. Furthermore, P has the FAP with respect to its m-faces and its co-k-faces. 3 Tight orientably-regular polyhedra We now consider the values of p and q for which there is a tight orientably-regular polyhedron of type {p, q}. By [5, Proposition 3.5], there are no tight polyhedra of type {p, q} when p and q are both odd. Also by [3, Theorem 5.3] and [5, Theorem 6.3], if p and q are both even then there exists a tight orientably-regular polyhedron whose automorphism group is the quotient of the Coxeter group [p, q] obtained by adding the extra relation (x0x1x2x1 )2 = 1. (Note that this is the group of the polyhedron {p, q | 2} described in [8, p. 196]; see [9] for more information on this and related polyhedra.) Indeed, for even p and q, there can often be multiple (non-isomorphic) tight polyhedra; for example, there are two of type {4, 8} that are non-isomorphic (see [2]). When p is odd and q is even (or vice-versa), the situation is more complicated. Evidence from [2] and [6] led the second author to conjecture that there are no tight regular polyhedra of type {p, q} if p is odd and q > 2p (see [5]). We will show that this is true in the orientably-regular case. In fact, we will prove something stronger, namely that q must divide 2p. We start by showing that if p is odd and q is an even divisor of 2p, then there is a tight orientably-regular polyhedron of type {p, q}. To do this, we define r(p, q) as the group (P0,P1,P21 Po2,Pi2,P22, (poPi)p, (pip2)q, (P0P2)2, (P0P1P2P1P2)2 ), which is obtainable by adding one extra relator to the Coxeter group [p, q]. (Note that this is the group of the polyhedron {p, q} ,2 described in [8, p. 196].) Theorem 3.1. Let p > 3 be odd, and let q be an even divisor of 2p. Then there is a tight orientably-regular polyhedron P of type {p, q} such that Г (P) = r(p, q). Proof. Let r(p, q) = (p0, p1, p2). In light of the construction in Section 2.2, all we need to do is show that r(p, q) is a string C-group of order 2pq, in which the order of p0p1 is p and the order of p1 p2 is q. First, note that the element w = (p1p2)2 generates a cyclic normal subgroup N of r(p, q), since each of p1 and p2 conjugates w to its inverse, and the extra relation (p0p1p2p1p2)2 = 1 implies that p0 does the same. Factoring out N gives quotient r(p, 2), in which the extra relation (p0p1p2p1p2)2 = 1 is redundant. In fact r(p, 2) is isomorphic to the string Coxeter group [p, 2], which is an extension of the dihedral group of order 2p, and has order 4p. In particular, r(p, q) covers r(p, 2) = [p, 2], and it follows that p0p1 has order p (rather than some proper divisor of p). Also the cover from r(p, q) to r(p, 2) is one-to-one on (p0, p1), and so by Proposition 2.1, we find that r(p, q) is a string C-group. Next, we observe that the dihedral group Dq = ( y1, y21 y12, y22, (y1 y2 )q ) is a quotient of r(p, q), via an epimorphism taking p1 ^ y1, p2 ^ y2 and p0 ^ (y1y2)p-1y1. (Note that the defining relations for r(p, q) are satisfied by their images in Dq, since (y1y2)p-1y1 has order 2, the order of (y1y2)p-1 divides p (as p — 1 is even and q divides 2p), the order of (У1У2)Р-1У1У2 = (У1У2)Р divides 2 (as q divides 2p), and (У1У2)Р-1У2У1У2 has order 2.) In particular, the image of p1p2 is y1y2, which has order q, and hence p1p2 has order q. Finally, |r(p,q)| = \T{p,q)/N||N| = |Г(р,2)||N| = 4p(q/2) = 2pq, since N = We will show that in fact, the only tight orientably-regular polyhedra of type {p, q} with p odd are those given in Theorem 3.1. We proceed with the help of a simple lemma. Lemma 3.2. Let P be an orientably-regular polyhedron of type {p, q}, with p odd, and with automorphism group Г (P ) generated by the reflections P0,P1,P2. If u = (p1p2)2 = a2 generates a normal subgroup of Г+ (P), then u is central, and q divides 2p. Proof. For simplicity, let x = a1 = p0p1 and y = a2 = p1p2, so that xy = p0p2 and hence xp = yq = (xy)2 = 1, and also u = y2. By hypothesis, (y2) is normal, and so xy2x-1 = y2k for some k. It follows that y2 = xpy2x-p = y2kP and that y2 = (xy)2y2(xy)-2 = y2k , and therefore 2 = 2k2 = 2kp mod q. Then also 2 = 2k2kp-2 = 2kp-2 mod q, and by induction 2 = 2kp = 2kp-2 = • • • = 2k mod q, since p is odd. Thus xy2x-1 = y2k = y2, and so u = y2 is central. Moreover, since y2 commutes with x (which has order p) and xy = y-1x-1, we find that y2p = xpy2p = (xy2)p = (y-1x-1y)p = y-1x-py = 1, and so 2p is a multiple of q. □ We also utilise a connection between tight polyhedra and regular Cayley maps, as is explained in [4]. Specifically, suppose that the finite group G is generated by two non-involutory elements x and y such that xy has order 2, and that G can be written as AY where Y = (y) is core-free in G (that is, Y contains no non-trivial normal subgroup of G), and A is a subgroup of G such that A n Y = {1}. Then G is the group Г+(М) of orientation-preserving automorphisms group of a regular Cayley map M for the group A. Furthermore, this map M is reflexible if and only if G admits an automorphism taking x ^ xy2 (= y-1x-1y) and y ^ y-1. Theorem 3.3. Let p > 3 be odd. If P is a tight orientably-regular polyhedron of type {p, q}, then q is an even divisor of 2p, and Г (P ) is isomorphic to r(p, q). Proof. Let G = Г+ (P), and let a1 = p0p1 and a2 = p1p2 be its standard generators. Also take F = (a1) and V = (a2), which are the stabilisers in r+(P) of a 2-face and incident vertex of P. Then F n V = (e) since P is a polytope, and G = FV since P is tight. Now, let N be the core of V in G (which is the largest normal subgroup of G contained in V), and let G_ = G/N, V = V/N_and F = FN/N. Then G = VF, and V n F is trivial, and also V is core-free. Thus G is the orientation-preserving automorphism group of a regular Cayley map M for the cyclic group F. Furthermore, since P is an orientably-regular polyhedron, the group G = r+(P) has an automorphism taking a1 ^ and a2 ^ a-1, and G has the analogous property. Hence M is reflexible. On the other hand, by [4, Theorem 3.7] we know that the only reflexible regular Cayley map for a cyclic group of odd order p is the equatorial map on the sphere, with p vertices of valence 2. Thus |V | = 2, and so q = \V \ is even, and N = (a2). In particular, (a2) is a normal subgroup of r+(P), and hence by Lemma 3.2 we also find that q divides 2p, and that a2 is central. But a"2 = (p1p2)2 is inverted under conjugation by p1, and now centralised by a1 = p0p1, and therefore also inverted under conjugation by p0. Hence the relation (p0p1p2p1p2)2 = e holds in r(P), so r(P) is a quotient of r(p, q). Then finally, since P is tight we have |r(P)| = 2pq = |r(p, q)|, and it follows ((p1p2)2) has order q/2. □ that r(P) = r(p, q). □ Combining Theorem 3.1 with Theorem 3.3 and [5, Theorem 6.3], we can now draw the following conclusion: Theorem 3.4. There is a tight orientably-regular polyhedron of type {p, q} if and only if one of the following is true : (a) p and q are both even, or (b) p is odd and q is an even divisor of 2p, or (c) q is odd and p is an even divisor of 2q. 4 Tight orientably-regular polytopes in higher ranks Now that we know which Schlafli symbols appear among tight orientably-regular polyhe-dra, we can proceed to classify the tight orientably-regular polytopes of arbitrary rank. We will say that the (n- 1)-tuple (pi,... ,pn-1) is admissible if each of pj_i andpi+1 (when defined) is an even divisor of 2pi whenever pi is odd. Theorem 4.1. If P is a tight orientably-regular polytope of type {p1,... ,pn-1}, then the (n-1)-tuple (p1,... ,pn-1) is admissible. Proof. If P is tight and orientably-regular, then by [5, Proposition 3.8] we know that all of its sections of rank 3 are tight and orientably-regular. Hence in particular, if pi is odd then the sections of P of type {pj-1,pj} and {pj,pj+1} are tight and orientably-regular. The rest now follows from Theorem 3.3. □ We will prove that this necessary condition is also sufficient, which will then complete the proof of Theorem 1.1. We do this by constructing the automorphism group of a tight orientably-regular polytope of the given type. Let (p1,... ,pn-1) be an admissible (n-l)-tuple. Then we define the group r(p1,..., pn-1) to be the quotient of the string Coxeter group [p1,... ,pn-1] obtained by adding n-2 extra relations r1 = ■ ■ ■ = rn-2 = 1, where Note that if n = 3, this definition of r(p1,p2) coincides with the one in the previous section. For n > 4, the group r(p1,... ,pn-1) is the amalgamation of r(p1,... ,pn-2) with r(p2,... ,pn-1) in the obvious way, subject to the extra relation (x0xn-1)2 = 1. Also let P(p1,... ,pn-1 ) be the poset obtained from r(p1,... ,pn-1), using the construction in Section 2.2. We will show that r(p1,... ,pn-1) is a string C-group of order 2p1p2.. .pn-1, and then since every relator of r(p1,... ,pn-1) has even length, it follows that P(p1,... ,pn-1) is a tight orientably-regular polytope of type {p1,... ,pn-1}. We start by considering the order of r(p 1,..., pn -1 ). Proposition 4.2. Let pi be even. Then every element of r(p1,... ,pn-1) either commutes with (xi-1xi)2 or inverts it by conjugation. In particular, the square of every element of r(p1,... ,pn-1) commutes with (xi-1xi)2. ri = :+1) -1) 2 2 if pi and pi+1 are both even, or if pi is odd and pi+1 is even, or if pi is even and pi+1 is odd. Proof. Let ш = (xi-1xi)2. If j < i — 3 or j > i + 2, then xj commutes with both xi-1 and xi, and so commutes with ш. Also it is clear that xi-1 and xi both conjugate ш to ш-1, so it remains to consider only xi-2 and xi+1. Now since pi is even, the relator ri-1 is either (xi-2xi-1xixi-1)2 or (xi-2xi-1xixi-1xi)2. In the first case, xi-2 commutes with xi-1xixi-1 and hence with (xi-1xixi-1)xi = ш, while in the second case, we have ^-2ш)2 = 1 and so xi-2 conjugates ш to ш-1. Similarly, the relator ri is (xi-1xixi+1xi)2 or (xi+1xixi-1xixi-1)2, and in these two cases we find that ii+1wxj+1 = ш or ш-1, respectively. Thus every generator xj of r(p1,... ,pn-1) either commutes with (xi-1xi)2 or inverts it by conjugation, and it follows that the same is true for every element of r(p1,... ,pn-1). The rest follows easily. □ Proposition 4.3. Let (p1,... ,pn-1) be an admissible (n — 1)-tuple with the property that for every i strictly between 1 and n—1, either pi = 2 or pi-1 = pi+1 = 2. Then yi = xi-1xi has order pi for all i, and |r(p1,... ,pn-1)| = 2p1 • • • pn-1. Proof. We use induction on n, together with the observation that if pj = 2, then 1 = (xj-1xj)2, so that xj-1 commutes with xj, and therefore (x0,..., xj-1) centralises (xj ,...,x„-1). First, if p1 = 2, then r(pbp2,... ,pn-1) = r(2,p2,... ,pn-1) = (xo) xT(p2,... ,pn-1), and so |r(pb ... ,p„-1)| = 2|r(p2,... ,pn-1 )| = 4p2 • • • pn-1 = 2p1p2 ••• pn-1. Otherwise p2 = 2 and r(pbp2,... ,p„-1) = r(pb 2,p3,... ,pn-0 = r(p1) x r(p3,...,p„-1 ), and therefore |r(p1,... ,pn-1)| = |r(pO| |r(p3,... ,p„-1)| = 2p1 2p3 • • • pn-1 = 2p1p2 • • • pn-1. The claim about the orders of the elements yi = xi-1xi follows easily by induction as well. □ Lemma 4.4. Let qi be the order of xi-1xi in r(p1,... ,pn-1), for 1 < i < n. Then qi = pi whenever pi is odd, and also |r(p1,... ,pn-1)| = 2q1 • • • qn-1, which divides 2p1 • • • pn-1. Proof. Let ki = pi when pi is odd, or 2 when pi is even. Then since ki divides pi for all i, there exists an epimorphism n : r(p1,... ,pn-1) ^ Г(к1,..., kn-1). Also the (n— 1)-tuple (k1,..., kn-1) is admissible, and indeed ki-1 and ki+1 are both 2 whenever ki is odd (since pi-1 andpi+1 are both even whenever pi is odd). Thus (k1,..., kn-1) satisfies the hypotheses of Proposition 4.3, and so |Г(к1,..., kn-1) | = 2k1 • • • kn-1. Moreover, Proposition 4.3 tells us that when pi is odd, the order of the image of xi in r(k1,..., kn-1) is ki = pi, and so qi = pi; on the other hand, if pi is even, then the order of the image of xi in Г(к1,..., kn-1 ) is ki = 2, and so qi is even in that case. Now the kernel of the epimorphism n is the smallest normal subgroup of r(p1,..., pn-1) containing the elements (xi-1xi)2 for those i such that pi is even. By Proposition 4.2, however, the subgroup N generated by these elements is normal in r(p1,..., pn-1), and abelian. Hence in particular, N = ker n, and also by the intersection condition, |N | is the product of the numbers qi/2 over all i for which pi is even. Thus |r(p1, . . . ,pn-1)| = 2q1 ••• qn-1. □ In order to use Theorem 2.5 to build our tight regular polytopes recursively, we need two more observations. The first concerns the flat amalgamation property (FAP): Proposition 4.5. f p2 is even, then P(p1,..., pn-1 ) has the FAP with respect to its 2-faces, and if pn-2 is even, then P(p1,... ,pn-1) has the FAP with respect to its co-(n — 3)-faces. Proof. Let p2 be even, and consider the effect of killing the generators xi of r(p1,..., pn-1), for i > 2 (that is, by adding the relations xi = 1 to the presentation for r(p1,..., pn-1)). Each of the relators r3,..., rn-2 contains only generators xi with i > 2, so becomes redundant, and may be removed. The relator r2 reduces to x2 or x4, while r1 reduces to (x0x1)2, which is equivalent to x2, and hence all of these become redundant too. Thus adding the relations xi = 1 to r(p1,... ,pn-1) has the same effect as adding the relations xi = 1 to the string Coxeter group [p1,... ,pn-1]. It is easy to see that this gives the quotient group with presentation ( x0,x1 | x^,x2, (x0x1)pi}, which is the automorphism group of the 2-faces of P(p1,... ,pn-1). Thus P(p1,... ,pn-1) has the FAP with respect to its 2-faces. The second claim can be proved by a dual argument. □ Proposition 4.6. Let P be an equivelar n-polytope with tight m-faces and tight co-k-faces, where m > k + 3. Then P is tight. Proof. Since the m-faces are tight, they are (i, i+2)-flat for 0 < i < m—3, by Theorem 2.4, and then by Proposition 2.3, the polytope P is (i, i+2)-flat for 0 < i < m—3. Similarly, the co-k-faces are (i, i+2)-flat for 0 < i < n—k—4, and P is (i, i + 2)-flat for k+1 < i < n—3. Finally, since m > k + 3, we see that P is (i, i + 2)-flat for 0 < i < n — 3, and again Theorem 2.4 applies, to show that P is tight. □ We can now prove the following. Theorem 4.7. Let (p1,... ,pn-1) be an admissible (n—1)-tuple, with n > 4. Also suppose that pi-1 and pi+1 are both even, for some i (with 2 < i < n—2). IfP(p1,... ,pi) isatight orientably-regular polytope of type {p1,... ,pi}, and P (pi,... ,pn-1) is a tight orientably-regular polytope of type {pi,... ,pn-1}, then P (p1,... ,pn-1) isa tight orientably-regular polytope of type {pb ... ,pn-1}. Proof. Let P1 = P(p1,... ,pi) and P2 = P(pi,... ,pn-1), which by hypothesis are tight orientably-regular polytopes of the appropriate types. Since pi-1 and pi+1 are even, Proposition 4.5 tells us that P1 has the FAP with respect to its co-(i — 2)-faces, and P2 has the FAP with respect to its 2-faces. Then by Theorem 2.5, there exists a regular polytope P with (i+1)-faces isomorphic to P1 and co-(i—2)-faces isomorphic to P2. Moreover, since P1 and P2 are both tight, Proposition 4.6 implies that P is also tight, and then since P is of type {p1,... ,pn-1}, we find that |r(P)| = 2p1 • • • pn-1. But also the (i + 1)-faces of P are isomorphic to P1, and the co-(i — 2)-faces are isomorphic to P2, and so the standard generators of r(P) must satisfy all the relations of r(p1,... ,pn-1 ). In particular, r(P) is a quotient of r(p1,... ,pn-1), and so |r(p1,... ,pn-1)| > 2p1 • • • pn-1. On the other hand, |r(pb ... ,pn-1)| < 2p1 ••• pn-1 by Lemma 4.4. Thus |r(p1,... ,pn-1)| = 2p1 • • • pn-1, and hence also r(P) = r(p1,... ,pn-1), and P = P(p1,... ,pn-1). Thus P(p1,... ,pn-1) is a tight polytope of type {p1,... ,pn-1}, and finally, since all the defining relations of r(p1,... ,pn-1) have even length, we find that P(p1,... ,pn-1) is orientably-regular. □ Note that the above theorem helps us deal with a large number of possibilities, once we have enough 'building blocks' in place. We are assuming that the (n — 1)-tuple (p1,..., pn-1) is admissible, so that pi-1 and pi+1 are even divisors of 2pi whenever pi is odd. Now suppose that n > 6. If p2 and p4 are both even, then Theorem 4.7 will apply, and if not, then one of them is odd, say p2, in which case p1 and p3 must both be even, and again Theorem 4.7 will apply. Hence this leaves us with just a few cases to verify, namely admissible (n— 1)-tuples (pi,... ,pn-i) with n = 4 or 5 for which there is no i such that pi_i and pj+i are both even. The only such cases are as follows: • n = 4, with pi odd, p2 even and p3 even, or dually, pi even, p2 even and p3 odd, • n = 4, with pi odd, p2 even and p3 odd, • n = 5, with pi odd, p2 even, p3 even and p4 odd. We start with the cases where n = 4: Proposition 4.8. Ifpi isodd, p2 is an even divisor of 2pi, and p3 > 2, then r(pi,p2,p3) is a string C-group, and P (pi, p2, p3 ) is a tight orientably-regular polytope of type {pi, q, p3 }, for some even q dividing p2. Proof. Let Г = r(pi,p2,p3) = (x0,xi,x2,^3), and let Г = r(pi, 2,p3) = (y0,yi,y2, y3), where yj = xj is the image of xj under the natural epimorphism n : r(pi,p2,p3) ^ r(pi, 2,p3), for 0 < i < 3. By Proposition 4.3, we know that r(pi, 2,p3) = [pi, 2,p3]. Also the subgroup (x0, xi, x2) of Г covers [pi, 2], and this cover is one-to-one on (x0, xi), so (x0,xi,x2) is a string C-group, by Proposition 2.1. A similar argument shows that (xi,x2,x3) is a string C-group. Now the intersection of these two string C-groups is (xi, X2), since the intersection of their images in Г is (y0, yi, У2) n (yi, У2, У3) = (yi, У2), and the kernel of n is ((xix2)2). Hence by Proposition 2.2, T(pbp2,p3) is a string C-group, and the rest follows easily from Lemma 4.4. □ Lemma 4.9. If pj = 2 for some i, then in Г^1, ... ,pn_i) = (x0,..., xn_i), we have (x0,..., xj) = Г^1, ... ,pj) and (xj_i,..., xn_i) = Г^,... ,pn_i). Proof. First consider Л = (x0,..., xj). This is obtainable as a quotient of Г^1, ... ,pj) by adding extra relations. But also it can be obtained from the group ... ,pn_i) by killing the unwanted generators xj+i,..., xn_i. (Note that for i +1 < j < n — 1, the relation rj = 1 and all relations of the form (pjpk)m = 1 become redundant and may be removed, and the same holds for the relations rj = 1 and rj_i = 1 since the assumption that pj = 2 implies that [ xj—i, xj] — (xj—i, x j)2 = 1 and hence that x0,..., xj_i commute with xj,..., xn_i.) It follows that Л = Г^1, ... ,pj). Also (xj_i,..., xn_i) = Г^,..., pn_ i ), by the dual argument. □ Theorem 4.10. If pi is odd, p2 is an even divisor of 2pi, andp3 is even, then P(pi,p2,p3) is a tight orientably-regular polytope of type {pi ,p2,p3}. Proof. First, the group ^^2^3) = (x0, xi, x2,x3) covers Г(pl,p2, 2) = (y0,yi,y2, y3), say, since p3 is even. Also by Lemma 4.9 we know that (y0, yi, y2) is isomorphic to ^pbp^, and hence in particular, yiy2 has order p2. It follows that the order of xix2 is also p2, and then the conclusion follows from Proposition 4.8. □ Theorem 4.11. If pi and p3 are odd, and p2 is an even divisor of both 2pi and 2p3, then P (pi,p2,p3) is a tight orientably-regular polytope of type {pi,p2,p3}. Proof. Under the given assumptions, the group T(pbp2,p3) is obtained from the string Coxeter group [pi,p2,p3] = (x0,xi,x2,x3) by adding the two extra relations (x0xix2xix2)2 = 1 and (x3x2xix2xi)2 = 1. These imply that the element ш = (xix2)2 is inverted under conjugation by each of x0 and x3, and hence by all the xi. Now let yi = xi_1xi for 1 < i < 3. These elements generate the orientation-preserving subgroup Г+(р1,р2,Рз), and they all centralise w = y22. It follows that Г+(р1 ,p2,p3) has presentation ( У1,У2,Уз 1 ypi, У22, Уз 3, (У1У2)2, (У2У3)2, (У1У2У3)2, [У1,У22], [У3,У22]}. We now exhibit a permutation representation of this group on the Cartesian product Zpi X Zp2, by letting each yi induce the permutation ni, where (j,k)n2 = (j,k+1) for all (j,k), It is easy to see that n1 and n2 have orders p1 and p2 respectively (since p2 divides 2p1), and that the order of n3 divides p2 /2 and hence divides p3. It is also easy to verify that they satisfy the other defining relations for r+(p1,p2,p3), and thus we do have a permutation representation. In particular, since n2 has order p2, so does y2 = x1x2, and again the conclusion follows from Proposition 4.8. □ We now handle the remaining case. Theorem 4.12. If p1 and p4 are odd, p2 is an even divisor of 2p1, and p3 is an even divisor of 2p4, then P (p1,p2,p3,p4) is a tight orientably-regular polytope of type {p1,... ,p4}. Proof. Take Г = r(pb .. .,p4) = (xo,... ,x4}, and Л = r(p1,p2, 2,p4) = (yo,.. .,У4}. Then Г covers Л, and this induces a cover from (x0,..., x3} to (y0,..., y3}, which is isomorphic to r(p1,p2, 2) by Lemma 4.9. Similarly we have a cover from (x0, x1, x2} to (y0, y1, y2}, which is isomorphic to r(p1,p2). But on the other hand, (x0, x1, x2} is a quotient of r(p1,p2), and hence these two groups are isomorphic. In particular, the cover from (x0,..., x3} to r(p1,p2, 2) is one-to-one on the facets, so (x0,..., x3} is a string C-group. By a dual argument, (x1,..., x4} is also a string C-group. Next, let Д = r(p1, 2, 2,p4) = (z0,..., z4}, and let n be the covering homomorphism from Г to Д. The kernel of n is the subgroup generated by (x1x2)2 and (x2x3)2, since the defining relations for Г = T(p^ ... ,p4) imply that these two elements are centralised or inverted under conjugation by each generator xi. In particular, ker n C (x1,x2,x3}. As also the intersection of (z0,..., z3} and (z1,..., z4} in Д is (z1, z2, z3}, it follows that intersection of (x0,..., x3} and (x1,..., x4} is (x1, x2, x3}. Hence Г is a string C-group. Now (x0, x1, x2} = Г(p1 ,p2), and by Theorem 3.1 we know the polytope P(p1,p2) has type {p1 ,p2}. Similarly (x2,x3,x4} = Г(pз,p4), and P(p3,p4) has type {p3,p4}. It follows that P(p1,p2,p3,p4) is an orientably-regular polytope of type {p1,... ,p4}. In particular, the order of xi_ 1xi is pi (for 1 < i < 4), and so by Lemma 4.4, P(p1, p2, p3, p4 ) is tight. □ This gives us all the building blocks we need. With the help of Theorem 4.7, we now know that P(p1,... ,pn-1) is a tight orientably-regular polytope of type {p1,... ,pn-1} whenever (p1,... ,pn-1) is admissible, and the proof of Theorem 1.1 is complete. 5 Tight non-orientably-regular polytopes We have not yet been able to completely characterise the Schlafli symbols of tight, non-orientably-regular polytopes, but we have made some partial progress. For example, we can easily find an infinite family of tight, non-orientably-regular polyhedra. Theorem 5.1. For every odd positive integer k, there exists a non-orientably-regular tight polyhedron of type {3k, 4}, with automorphism group A(k) having presentation ( P0,P1,P2 1 Po2, P2, P2 , (P0P1)3k, (P0P2^ (P1P2)4,P0P1 P2P1P0P1P2P1P2 ). Proof. We note that Л(1) is the automorphism group of the hemi-octahedron (of type {3,4}), and that Л^) covers Л(1), for every k. Hence in each Л^), the order of p1p2 is 4 (and not 1 or 2). Next, because the covering is one-to-one on the vertex-figures, it follows from Proposition 2.1 that Л^) is a string C-group. Also Л^) has a relation of odd length, and so it must be the automorphism group of a non-orientably-regular polyhedron. Now let N be the subgroup generated by the involutions p2 and p1p2p1. Since their product has order 2, this is a Klein 4-group. Moreover, N is normalised by p1 and p2, and also by p0 since p0 centralises p2 and the final relation in the definition of Л^) gives (p1p2p1)Po = p1p2p1p2. It is now easy to see that N is the normal closure of (p2). The quotient Л^/N is isomorphic to (p0,p1 |Po,Pi, (p0p1)3k ), with the final relator for Л^) becoming trivial, and so Л^/N is dihedral of order 6k. In particular, this shows that p0p1 has order 3k (and that ^(k)| = ^(k)/N||N| = 24k). □ The computational data that we have on polytopes with up to 2000 flags (obtained with the help of Magma [1]) suggests that these polyhedra are the only tight non-orientably-regular polyhedra of type {p, q} with p odd. Using Theorem 5.1, it is possible to build tight, non-orientably-regular polytopes in much the same way as we did in Theorem 4.7. In particular, the regular polytope with automorphism group Л^) has the FAP with respect to its 2-faces, and its dual has the FAP with respect to its vertex-figures (co-0-faces). Then by Theorem 2.5, we know there are tight, non-orientably-regular polytopes of type {4,3k, 4} for each odd k, and of type {4, 3k, r} for each odd k and each even r dividing 3k. It is possible to continue in this fashion, building tight non-orientably-regular polytopes of every rank. Finally, just as with orientably-regular polytopes, there are some kinds of Schlafli symbol for which no examples can be constructed using Theorem 2.5. In fact (and in contrast with the situation for orientably-regular polytopes), there seem to be no tight non-orientably-regular polytopes of some of these types at all. For example, there are no tight regular polytopes of type {3,4, r} with r > 3 and with 2000 flags or fewer, but the reason for this is not clear. References [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I, The user language, J. Symbolic Comput. 24 (1997), 235-265. [2] M. Conder, Lists of all regular polytopes with up to 2000 flags, http://www.math. auckland.ac.nz/~conder. [3] M. Conder, The smallest regular polytopes of any given rank, Adv. Math. 236 (2013), 92-110. [4] M. D. E. Conder and T. W. Tucker, Regular Cayley maps for cyclic groups, Trans. Amer. Math. Soc. 366 (2014), 3585-3609. [5] G. Cunningham, Minimal equivelar polytopes, Ars Math. Contemp. 7 (2014), 299-315. [6] M. I. Hartley, An atlas of small regular abstract polytopes, Period. Math. Hung. 53 (2006), 149-156. [7] I. Hubard, Two-orbit polyhedra from groups, Eur. J. Comb. 31 (2010), 943-960. [8] P. McMullen and E. Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge University Press, Cambridge, 2002. [9] E. Schulte, Amalgamation of regular incidence-polytopes, Proc. London Math. Soc. 3 (1988), 303-328. [10] E. Schulte and A. Ivic Weiss, Chiral polytopes, Applied geometry and discrete mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 493-516. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 83-94 Reachability relations, transitive digraphs and groups Aleksander MalniC * University of Ljubljana, Faculty of Education, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia Primož PotoCnik University of Ljubljana, Faculty of Mathematics and Physics Jadranska 19, 1000 Ljubljana, Slovenia IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia Norbert Seifterf Montanuniversität Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria Primož Spari * University of Ljubljana, Faculty of Education, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia Received 28 August 2013, accepted 9 March 2014, published online 7 May 2014 In [6] it was shown that properties of digraphs such as growth, property Z, and number of ends are reflected by the properties of certain reachability relations defined on the vertices of the corresponding digraphs. In this paper we study these relations in connection with certain properties of automorphism groups of transitive digraphs. In particular, one of the main results shows that if a transitive digraph admits a nilpotent subgroup of automorphisms with finitely many orbits, then its nilpotency class and the number of orbits are closely related to particular properties of reachability relations defined on the digraphs in question. The obtained results have interesting implications for Cayley digraphs of certain types of groups such as torsion-free groups of polynomial growth. * Supported in part by "ARRS - Agencija za znanost Republike Slovenije", program no. P1-0285. t Corresponding author. Supported by ÖAD - WTZ (Österreich-Slowenien, project no. SI 20/2009. * Supported in part by "ARRS - Agencija za znanost Republike Slovenije", program no. P1-0285. Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Keywords: Cayley digraph, reachability relation. Math. Subj. Class.: 05C20, 05C25 1 Introduction and preliminaries In [2], highly arc-transitive digraphs were considered from several different viewpoints, leading to - besides many nice results - a number of interesting problems. One of these problems, which remained open for a very long time and was finally settled in [4], concerned a certain reachability relation defined on the edges of digraphs. A subset of the authors of this paper also worked on this 'reachability problem' [5] and several other questions concerning highly-arc-transitive digraphs. In [6], as an offspring of our considerations, we became interested in reachability relations defined on vertices rather than edges, which we review in the sequel. A digraph is an ordered pair D = (V (D), E (D)), where V (D) is the vertex-set and E (D) С V (D) X V (D) is the edge-set. Note that a digraph can have loops (v, v) as well as pairs of 'oppositely directed' edges of the form (u, v) and (v, u). We also emphasize that with this definition our digraphs are always simple in the sense that between two vertices there can be at most one edge in each direction. Digraphs considered in this paper are connected in the sense that their underlying undirected graphs are connected. By Aut(D) we denote the automorphism group of a digraph D. We say that D is transitive if some H С Aut(D) acts transitively on the vertices of D. Also, if g G Aut(D), then g v denotes the image of v g V (D) under g and H v denotes the orbit of v under some subset H С Aut(D). To make sure that no ambiguity arises, we explicitely define Cayley digraphs as they are understood in this paper. The Cayley digraph Cay(G,S) of a group G with respect to a generating set S has the group G as its vertex set and the edges are given by right multiplication by the generators: E (C ay (G, S)) = {(g,gs)|s G S}. If Cay(G,S ) is defined in this way, then G acts regularly on Cay(G, S) by left multiplication. A walk W = (v0, £1,v1,..., £n, vn) from v0 to vn of length n > 0 (denoted by | W|) is a sequence of n + 1 (not necessarily pairwise distinct) vertices v0, v1,.. .,vn G V (D), and n indicators £1,£2,... ,£n G {1, -1} such that for all j G {1,2,..., n} we have £j = 1 ^ (vj-i,vj) G E(W), £j = -1 ^ (vj,vj-i) G E(W). W is called a closed walk if v0 = vn. Intuitively, a walk is a traversal in the digraph from vertex to vertex along edges, where indicators 1 and -1 tell whether the traversal respects the direction of edges or not. The formal definition of a walk as above has been chosen in order to make proofs unambiguous. If the vertices of a walk W are pairwise different then W is called a path. A walk (or a path) is directed if its indicators are all equal to 1 or to -1, and is alternating if the values of the indicators alternate. Let W = (v0, £1, v1,..., £n, vn) be a walk. We let the inverse walk of W be W-1 = (vn, —£n, vn-1,..., —£1, v0). Moreover, for 0 < i < j < n, the subsequence iWj = (vi,£i+1, . . . ,£j, vj) E-mail addresses: aleksander.malnic@pef.uni-lj.si (Aleksander Malnic), primoz.potocnik@fmf.uni-lj.si (PrimoZ Potocnik), seifter@unileoben.ac.at (Norbert Seifter), primoz.sparl@pef.uni-lj.si (PrimoZ Sparl) of W is called a subwalk. Furthermore, let W ' = (u0,51,u1,... ,5m,um) be a walk such that u0 = vn. Then the concatenation of W and W' is the walk W • W ' = (vo, £1, vi,..., Vn-1, £n, uo, öi,ui,..., 6m, um) of length n + m. We now introduce two families of reachability relations defined on vertices of a digraph. Let W = (v0, £1, v1,..., £n, vn) be a walk. The weight of the walk W is defined as Z (W ) = £1 + £2 + ... + £n. Let k be a nonnegative integer. We say that a vertex u g V (D) is R+-related to a vertex v g V (D), in symbols uR+v, if there exists a walk W from u to v such that Z(W) = 0, and that for every 0 < j < |W| we have Z(0Wj) g [0, k]. For a given pair of vertices u, v, the set of all such walks is denoted by R+[u, v]. Analogously we say that u is R--related to v, in symbols uR-v, if there exists a walk W such that Z(W) = 0, and that for every 0 < j < |W| we have Z(0Wj) g [-k, 0]. For a given pair of vertices u, v, the set of all such walks is denoted by R- [u, v]. Note that R+ and R- are equivalence relations. Their equivalence classes are denoted by R+(v) and R-(v), v g V (D), respectively. If D is transitive, then the equivalence classes of R+ (and similarly of R-) form an imprimitivity system for Aut(D). Observe that the sequences (R+)heZ+ and (R-)heZ+ are ascending: for all k we have R+ C R++1 and R- C R-+1. Their respective unions R+ = U R+ and R- = U R- hez+ hez+ are thus also equivalence relations, and their equivalence classes form imprimitivity systems for Aut(D) whenever D is transitive. As was shown in [6], R+ = R+ holds whenever R+ = R++1. In this case, the smallest nonnegative integer k such that R+ = R+ holds is called the exponent exp+(D) of D. If R+ = R+ for all k, then we set exp+(D) = то. The exponent exp-(D) is defined analogously. We say that the relation R+ (R+, R-, R-) is universal if uR+v (uR+v, uR-v, uR-v) holds for any pair u, v G V (D). We mention (see [6]) that all of the above relations are universal, provided that the digraph in question is connected and has a loop at every vertex. In [6] it was also shown that properties of the two sequences of equivalence relations (R+)heZ+ and (R-)heZ+ are strongly related to other properties of digraphs such as having property Z, the number of ends, growth conditions and vertex degree. Furthermore, in [8] the relations Ra,b were studied, where a is a non-positive integer or a = -то and b is a non-negative integer or b = то. We say that a vertex u is Ra,b-related to a vertex v if there exists a 0-weighted walk from u to v such that every subwalk starting at u has weight in the interval [a, b]. The distance distD (u, v) between vertices u and v in a connected digraph D is the length of a shortest path from u to v in the underlying undirected graph. The growth function fD (v, n), n > 0, with respect to some v G V (D) is given by fD (v,n) = |{u G V (D) | distD (v, u) < n}|. If D is transitive, then this function does not depend on a particular vertex v e V (D). In this case we denote it by fD (n). We say that a transitive digraph D has polynomial growth if there are positive constants c and d such that f D (n) < cnd holds for all n > 0. The digraph D has exponential growth if there is a constant c > 1 such that fD (n) > cn holds for all n > 0. If the growth function of a digraph D grows faster than any polynomial but D does not have exponential growth, then we say that D has intermediate growth. In the case of polynomial growth it can be shown that there always exist constants ci, c2 and an integer d > 1 such that ci nd < fD (n) < C2nd holds for all n > 0. We call this integer d the growth degree of D. We remark that the definitions concerning growth coincide with the usual definitions in the context of undirected graphs. Let D be a digraph and let т be a partition of the vertex set of D. The quotient digraph DT of D with respect to т is the digraph with vertex set т and (uT, vT ) e E (D) if and only if there exist vertices u e uT and v e vT such that (u,v) e E (D). If W = (v0, e1, v1, e2,..., £„, vn) is a walk in D, then the quotient walk WT of W is defined to be the walk W = ((v0)T,e1, (v1)T,e2,...,en, (vn)T). Note that for every j, 0 < j < |W| = |WT|, we have Z(0Wj) = Z(0(WTj). We emphasize that we consider these quotient digraphs as simple digraphs in the sense that if there are several edges in the same direction between two sets in т, then the quotient digraph contains exactly one directed edge between the respective vertices. But of course these quotient graphs might contain loops if there is an edge (u, v) e E (D) for some u e vT. Let G be a group acting transitively on D and let H be a normal subgroup of G. Then the orbits of H on V(D) give rise to an imprimitivity system т of G on V(D). The respective quotient digraph DT is usually denoted by DH. As mentioned above, if D is transitive, then R+ and R- give rise to imprimitivity systems of Aut(D) on D. The respective quotient digraphs are denoted by D/R+ and D/R- and can be described easily (see e. g. [8]). The digraph D/R+ either is (1) a finite directed cycle or (2) a two-way infinite directed line or (3) an infinite regular directed tree with indegree 1 and outdegree > 1. Considering R- the first two possibilities are the same, but if D/R- is neither of these digraphs, then it is a regular tree with outdegree 1 and indegree > 1 . 2 Motivation and main result The aim of this paper is to investigate the interplay between properties of groups and properties of reachability relations in their Cayley digraphs. For example, as a consequence of the last paragraph of the previous section, we immediately see that the quotient digraphs with respect to R+ of Cayley digraphs of finitely generated groups with polynomial or intermediate growth are either finite directed cycles or directed lines. Further, from [6, Theorem 4.12] we know that a finitely generated group G has exponential growth if for at least one Cayley digraph D of G, at least one of the exponents exp+(D) or exp-(D) is infinite. Additionally, by Gromov's important result [3], a finitely generated group has polynomial growth if and only if it contains a normal nilpotent subgroup of finite index. Hence the following question arises naturally: What can be said about properties of our reachability relations in Cayley digraphs of finitely generated groups with polynomial growth? In fact we carry out our considerations by assuming that nilpotent groups act with finitely many orbits on digraphs. The results for Cayley digraphs of groups with polynomial growth are then obtained as corollaries. The main result of this paper is the following theorem. To avoid ambiguity, we recall the definition of nilpotent groups: For a group G = G0, let Gi+1 = [G0, Gi] for i > 0. If G = G0 > G1 > ... > Gr > Gr+1 = 1 then we say that G is nilpotent of class r. Theorem 2.1. Let a group G act transitively on a connected digraph D, and let N < G be a normal nilpotent subgroup of class r acting with m orbits on D, where 1 < m < то. Then exp+(D) = exp-(D) < m(r + 2) — 1. Although we are mainly interested in properties of our relations in Cayley digraphs of finitely generated groups, we emphasize that - with the exception of those results explicitely formulated for finitely generated groups - we never assume that the graphs in consideration are locally finite. 3 Auxiliary results In this section we prove several results which will be useful for our main investigations, carried out in Section 4. Lemma 3.1. Let D be a digraph with minimal in- and outdegree at least 1 and let к > 1 be an integer. Then for any two vertices u,v G V (D) we have that uR+v if and only if there exists a walk W G R+[u,v] that is a concatenation of walks of the form (u0,1, u1,1,..., 1, uk, —1, uk+1, —1,..., —1, u2k). An analogous result holds for the relation R Proof. We prove the assertion for the relation R+ and leave the analogous proof for R- to the reader. To this end suppose uR+v and let W' = (u0, e1, u1, e2, u2, e3,..., e„, un) G R+[u, v]. Observe that, since the minimal in- and outdegrees of D are at least 1, there is a directed walk of any prescribed positive or negative weight starting at any vertex of D. A walk W g R+ [u, v], as described in the statement of the lemma, can now be obtained from W' inductively by inserting a concatenation of such a directed walk of appropriate length with its inverse at each vertex ui for which ei = ei+1 and Z(oW/) is not 0 or к. □ For a group G, a positive integer к, and subsets S, T C G let ST = {st|s g S, t g T}, Sk = and S-k = S-1 •• ■ S-1,. kk Corollary 3.2. Let D = Cay(G, S) be a Cayley digraph of a group G with respect to the generating set S. Then for any integer к > 1 and any g G G we have that R+(g) = gR+ (1) = g(SkS-k> and R-(g) = gR-(1) = g(S-kSk>. Proof. We prove the assertions for R+ and leave the analogous proof for Rk to the reader. The fact that R+(g) = gR+(1) is obvious since G has a natural left regular action on D while R+(1) = (SkS-k) follows fromLemma 3.1. □ Lemma 3.3. Let D be a digraph and let т be a partition of the vertex set of D. Suppose that for each u,v G V (D) with (uT, vT ) G E(DT ) there exist u' G uT and v' G vT such that (u, v') and (u', v) are arcs of D. Then for each к > 1 and each u,v G V (D) we have that uT R+vT if and only if there exists some w G vT such that uR+w. An analogous result holds for the relation R-k. Proof. We prove the result for R+ and leave the analogous proof for R- to the reader. Let к > 1 and let u,v G V (D). Suppose first that for some w g vT we have that uR+w and let W G R+ [u, w]. Since vT = wT, the walk WT is contained in R+[uT, vT]. This proves one implication. Suppose now that uTR+vT and let W = (uT,e1,x1,e2,... ,xn,en+1,vT) G R+[uT, vT]. Then by assumption one can successively find representatives xH G xH and w G vT such that W = (u, £1,x1 ,e2,x2,e3,..., xn, £n+i, w) G R+[u, w]. □ Remark 3.4. Observe that the condition of the above lemma is satisfied if т consists of the orbits of some group acting on D. Lemma 3.5. Let a group G act transitively on a digraph D and let H bea normal subgroup of G such that each of its subgroups is normal in G. Then exp+(D) < exp+(DH ) +1 and expk(D) < expk(DH) + 1. Proof. We prove the result for exp+(D). The proof for expk(D) is analogous and is left to the reader. If exp+(DH) = то, there is nothing to prove. We may thus assume that exp+( D h ) = к for some integer к > 0. To show that exp+(D) < к +1 let u G V (D) and v G R+(u) be arbitrary. Consider the equivalence class B = R++1(u) and the H-orbit Hu. Note that both of these sets are blocks of imprimitivity for the action of G on V(D). Let K be the setwise stabilizer in H of the set B. Note that the K-orbit of u is K u = H u n B and is thus a block of imprimitivity for G. Moreover, by assumption on H the subgroup K is normal in G, and so the block system generated by the block K u coincides with the block system given by the orbits of K. Consequently, any two vertices within the same H-orbit are R++i related if and only if they belong to the same K-orbit. We first show that exp+(DK) < к. If this is not the case, then there exists Kw G V (DK ) suchthatK w G R++1(K u) \ R+(K u). By Lemma 3.3 there exists w' G K w such that uR++1w'. Moreover, since exp+(DH) = к there exists z G Hw' = Hw such that uR+z. But then zR++1w', and so Kz = Kw' = Kw, implying that Kw G R+(Ku), a contradiction. Hence exp+(DK ) < к. But then K v G R+(K u) = R++1(K u) in D k , and by Lemma 3.3 there exists some x G Kv such that uR++1x. Since x G Kv we have that xR++1v, and so v,R'++1v holds. Since u and v were arbitrary subject to the condition that uR+v, this shows that exp+(D) < к +1. □ Lemma 3.6. Let a group G act transitively on a digraph D with finite exponents exp+(D) and expk(D). Furthermore, let т denote the imprimitivity system of G on V(D) which is induced by the equivalence classes with respect to R+ or R . Then every g e G which leaves invariant at least one block of т leaves invariant all blocks of т. Proof. Since the exponents exp+(D) and exp-(D) are both finite, [6, Corollary 3.5] implies that R+ = R-, and so the discussion from the last paragraph of the first section implies that DT is a finite cycle or the two-way infinite directed line. Hence, the only automorphism of DT which fixes a vertex is the identity. On the other hand, every automorphism g e G which leaves invariant a block of т induces an automorphism of DT fixing a vertex of DT, and the result follows. □ 4 R+ and R- in transitive digraphs We start with a simple observation concerning Cayley digraphs of abelian groups. Proposition 4.1. Let G be an abelian group acting transitively on a digraph D. Then exp +(D) = exp-(D) = 1. Proof. Since G is abelian, D is a Cayley graph of G. Then Corollary 3.2 implies that R+ = R- and [6, Corollary 3.4] implies that R+ = R+ = R- = R-, as claimed. □ We now generalise this result to nilpotent groups. Theorem 4.2. Let G be a nilpotent group of class r acting transitively on a digraph D. Then exp+(D) = exp-(D) < r +1. Proof. We first show that exp+ (D) < r + 1. The proof is carried out by induction on r. If r = 0, then G is an abelian group and Proposition 4.1 applies. Suppose now that r > 1. As G(r+1) = 1, we have that H = G(r) is contained in the center of G, and so each of its subgroups is normal in G. Hence Lemma 3.5 implies that exp+(D) < exp(DH) + 1. Now, the quotient group G/H is a nilpotent group of class r - 1 and acts transitively on the quotient digraph DH. By induction hypothesis we thus have that exp+(DH) < r. Consequently, exp+(D) < r + 1, as claimed. The fact that exp-(D) < r +1 follows by analogous arguments. Then [6, Corollary 3.5], implies that exp+ (D) = exp-(D). □ The next example shows that the bound from the above theorem is tight, that is, for every positive integer r there exists a nilpotent group G of class r and a digraph D on which G acts transitively such that exp+(D) = r + 1 = exp- (D) holds. Example 4.3. Already for the smallest nonabelian finitely generated nilpotent group, the dihedral group D8 of order 8 (of nilpotency class 1), this is the case. Let us write D8 = (/, ai,a2 |f2 = al = a2 = 1,/ai/-1 = aia2,/a2 = a2/, aia2 = a2ai). Then for the Cayley digraph D = Cay(D8, {/, /ai}) we clearly have that exp+(D) = exp-(D) = 2. In fact, this example happens to be the smallest member of the following infinite family. Let n > 1 be an integer and let Gn be the semidirect product of the elementary abelian group zn by the cyclic group Z2n-i generated by Gn = (/, ab a2,..., an), where / is of order 2n-\ the aj are involutions commuting with each other and /a/-i = ajai+i holds for all i, 1 < i < n, while / and an commute. One can verify that for S = {/, /aia2 • • • an} we have that (S®S-i) = (ai; a2,..., až) holds for all i, 1 < i < n, and so Corollary 3.2 implies that exp+(Cay(Gn, S)) = n. Moreover, it can be verified that Gn is nilpotent of class n - 1. Indeed, we have that G(i) = (ai+i, ai+2,..., an) holds for each i, 1 < i < n - 1, and of course then G(n) = 1. The Cayley graph Cay(Gn, S) thus attains the bound from the above theorem. We shall now see, that the above theorem cannot be generalized to solvable groups. Example 4.4. The lamplighter group L is the wreath product Z2 l Z. The standard representation for L is (a, t|a2, [tmat_m,tnat_n],m,n G Z). If we consider the Cayley digraph of L with respect to the generating set S = {t, at}, then this Cayley digraph is the horocyclic product of two directed trees with indegree 1 and outdegree 2. In this digraph R+ = R+ clearly holds for all k G Z+. This shows that for solvable groups we cannot expect a result like Theorem 4.2. As was shown in [6], a connected, locally finite, transitive digraph D has exponential growth if at least one of the exponents exp+ (D) or exp_(D) is infinite. Hence these exponents must be finite if a connected, locally finite, transitive digraph D does not have exponential growth. So the question arises if we can find a bound on exp+ (D) and exp_ (D) which depends on the growth rate of D or on certain properties of groups acting transitively on D. In the sequel we show that this is indeed possible. We first consider the case where a digraph D allows a transitive action of a group G containing a normal abelian subgroup, acting with finitely many orbits on D, thereby obtaining a tight bound for exp+ (D) and exp_ (D). We then explore a more general situation where a transitive group G contains a normal nilpotent subgroup acting with finitely many orbits on D. We start by proving two auxiliary results. Lemma 4.5. Let D be a connected digraph, and let G be a transitive subgroup of Aut(D) having a normal subgroup H < G with m, 1 < m < то, orbits on D. If for some (and hence every) u G V (D) the set R+(u) is contained in Hu, then the following hold: (i) For every v G V (D) the set R+(v) is contained in H v. (ii) The quotient digraph DH is a directed cycle. Proof. Observe that if m =1 there is nothing to prove, so we may assume m > 2. To prove (i) we show that R+ (v) C H v for all v G V (D) and all k. We do that by induction on k. The base of induction (k = 1) holds by assumption. Let now k > 1 and suppose that R+ (v) C H v holds for all j < k. Pick an arbitrary vertex v G V (D) and let w G R++ i(v). Let v = v0, w = vn and choose a walk W = (v0,1, vb ..., vn-1, -1, vn) G R++1 [v, w]. Suppose first that for all i, 0 < i < n, we have that Z(0Wj) > 0. In this case v1R+vn_1, and so induction hypothesis implies that vn-1 G Hv1, that is, vn-1 = hv1 for some h G H. Then (hv0, vn-1) G E(D), and so hv0R+vn. Then, by assumption, we have that hv0 g hvn, and so v g hw (recall that v = v0 and w = vn). Suppose now that 0 < i1 < i2 < • • • < it = n are such that Z(0Wj. ) = 0. By the above argument v^ G Hv, vi2 G Hvix,... , w G Hvit-1. Hence v G Hw, which proves (i). We now prove (ii). Let Hv be an H-orbit. Since D is connected and H has at least two orbits which are blocks of imprimitivity for G, there exists an H-orbitH w = H v such that (H w,H v) G E(DH ). It follows that there exists a vertex w' g h w with (w', v) G E (D). Consequently, the quotient digraph DH must have indegree one (for otherwise we obtain a vertex x G Hw which is R+-related to w'). Since DH is finite, it is a simple directed cycle. □ Lemma 4.6. Let D be a digraph, and let G be a transitive subgroup of Aut(D) having an abelian normal subgroup H 2. Let Д = H u for some u G V (D). We first construct an auxiliary digraph D* with vertex set Д and an edge (w, v) whenever there exists a directed path of length m in D from w to v. The restriction of H on Д acts regularly on Д. The digraph D* thus is a Cayley digraph of an abelian group (possibly disconnected). Therefore exp+(D*) < 1 by Proposition 4.1. Now, let vR+w for some v, w G V (D) and let us show that in this case vR+ w holds. By definition of R+ we have that vR+w holds for some integer k. Then Lemma 3.1 implies that there exists a walk in R+[v, w] which is a concatenation of walks of the form W = (v0,1, vi, 1,..., 1, vk, -1, vk_i, -1,..., -1, v2k). By transitivity it suffices to prove that v0R+v2k. Let t, r with 0 < r < m be the integers such that k = tm + r. By Lemma 4.5 the vertices vo, vm, v2m,..., vtm and v2k, v2k_m, v2k_2m,..., v2k_tm all belong to the H-orbit Hv0. Hence v2k = hv0, vtm = hl v0 and v2k_tm = h2v0 for some h, hi, h2 G H. Now, 0Wtm • (hlh-1 ((2k_tm)W2k)) is a walk from v0 to x = hlh--1 v2k = hlh--1hv0. As H is abelian, x = hh- h1 v0. Therefore, hh- (tmW2k_tm) g R+[x, v2k], and so r < m implies that v0R+ v2k if and only if v0R+x, that is, we can assume r = 0. Since v0 G Hv, we have v0 = ho v for some h0 g H. It follows that the walk W corresponds to a walk W* G R+[h0, hih_1hh0] in D*. Since exp+(D*) < 1, the walk W* can be replaced by a walk in R+[h0, h1h_1hh0], implying that W can be replaced by a walk in R+ [v0, v2k]. Therefore, R+ C R+, implying that R+ = R+. Analogously, it can be shown that R_ = Rm. Then [6, Corollary 3.5] completes the proof. □ To prove the next theorem we need the following result from [6]. Proposition 4.7. ([6], Proposition 3.11) Let D bea digraph, let т be the set of equivalence classes of R+, and let u G V (D). Then, for any v G V (D) and any k > 2 we have that uR+v if and only if uT R+_1vT. An analogous assertion holds for R_ when taking the quotient with respect to R_. Theorem 4.8. Let D be a digraph and let G < Aut(D) be a transitive subgroup having an abelian normal subgroup H acting with m, 1 < m < то, orbits on V(D). Then exp+(D) = exp_(D) < m. Proof. We prove that exp+(D) < m and leave the analogous proof for exp_(D) to the reader. We proceed by induction on m. If m = 1, then the result follows from Proposition 4.1. Suppose the assertion holds for all n < m, m > 2, and suppose that H has m orbits on D. If for some u G V (D) the set R+(u) is contained in H u, then Lemma 4.6 applies. Assume now that the equivalence classes with respect to R+ are not contained in the orbits of H and consider the quotient digraph D/R+. Let K be the kernel of the action of G on D/R+ and let N = HK/K = H/(H n K) be the induced faithful action of H on D/R+. Observe that, since the R1-equivalence classes are not fully contained in the H -orbits, N acts with at most m orbits on D/R+. By induction hypothesis (note that N is an abelian normal subgroup of G/K) we have that exp+(D/R+) < у. By Proposition 4.7 it follows thatexp+(D) = exp+(D/R+) + 1 < < m. Analogously it can be shown that exp-(D) < m. Then again [6, Corollary 3.5] completes the proof. □ Proof of Theorem 2.1 Let a group G act transitively on a connected digraph D, and let N < G be a normal nilpotent subgroup of class r acting with m orbits on D, where 1 < m < то. We first prove that exp+(D) < m(r + 2) — 1. The proof is done by induction on m. If m =1, then the result holds by Theorem 4.2. If m > 2 we distinguish two cases, depending on the structure of DN. Case 1. Dn is not isomorphic to a directed cycle on m > 2 vertices. In this case Lemma 4.5 implies that, for any v e V (D), the set R+ (v) is not completely contained in one orbit of N. Let т denote the imprimitivity system of G on D consisting of the equivalence classes with respect to R+. Then the permutation group GT, induced by the action of G on т, acts transitively on DT. Furthermore, NT acts with at most у orbits. In addition NT is nilpotent of class at most r. Then, by induction hypothesis, exp+ (DT) < у (r + 2) — 1 holds and the result follows by Proposition 4.7. Case 2. DN is isomorphic to a directed cycle C = (c1,..., cm) on m > 2 vertices. Let O1,..., Om denote the orbits of N on V(D) which correspond to the vertices c1,..., cm e Dn . Then of course there is no edge in D which connects two vertices which are both contained in the same orbit. Furthermore, all edges of D are directed from Oi to Oi+1, 1 < i < m, where indices are taken modulo m. Then for every v e Oj, 1 < i < m, R+(v) C Oi holds. Of course exp+(D) < m — 1 holds if R+-1 (v) = Oi for some v e Oi and some i, 1 < i < m. Hence we only have to consider the case when R+-1(v) is properly contained in Oi for every i, 1 < i < m, and every vertex v e Oi. By BM i e I, we denote the equivalence classes of Ry-1 on O1. For v e O1, let P (v) denote the set of all directed paths starting at v and containing exactly one vertex from each orbit Oi, 1 < i < m. Since DN is isomorphic to a directed cycle with m vertices and N acts transitively on each of its orbits, P (v) = 0 for all v e O1. Furthermore, for i e I let S be the subdigraph of D induced by the vertices of the union U veB P (v). Note that since the sets Bt are different equivalence classes with respect to R^-1, the digraphs i e I, are pairwise disjoint. We first define Pm as the set of all directed paths P — (v1,..., vm+1) in D where v j e Oj for 1 < j < m and vm+1 e O1. Analogously we define P-m as the set of all inverses of the paths in Pm. Furthermore, let denote the set of all walks which are contained in R^-1 [u, v] for some vertices u, v e O1. Let v1, v2 e O1 now satisfy v1R+v2. If v1 and v2 are both contained in one and the same set BM i e I, then of course v1R+-1v2 holds. Now let v1 e Bt1 and v2 e Bl2, i1 = i2. Then there is a walk W e R+ [v1, v2] which is the concatenation of finitely many paths and walks which are contained in Pm, P-m or Ry-1. Let D' now be the digraph with vertex set I with (i1, i2) e E(D') whenever there exists a path P e Pm with origin in Bt1 and terminal vertex in Bl2. Observe that, in general, the digraph D' might not be locally finite. Nevertheless, the restriction of N to O1 induces a transitive group acting on D' which is nilpotent of class at most r. Thus Theorem 4.2 implies that exp+ (D') < r + 1. Observe that, by Lemma 3.1, we can assume that the walk W is the concatenation of t paths from Pm, followed by a walk in and then t paths from P-m, for some non- negative integer t. Let u0, u1,..., u2t+1 be the vertices of W, contained in O1, given in the order they are met when traversing W. Thus u0, u1,..., ut-1 are the origins of the paths from Pm while the vertices ut+1, ut+2,..., u2t are the origins of the paths from P-m. The walk W thus naturally gives rise to the walk W' = (i0, i1,..., it-1, it+1, it+2,..., i2t+1) in D', where for each i we have that ui G B4 (observe that ut G Blt+1 = Blt). Of course W' G R+[i0, i2t+1], and so exp+(D') < r +1 implies that there is a walk W' g R++1[i0, i2t+1]. Since the sets Bt. are equivalence classes of the relation R^-1 on D it is now clear that this walk gives rise to some walk in R^+^i. [v1, v2]. Since exp- (D) < m(r + 2) — 1 holds by similar arguments, [6, Corollary 3.5] implies that exp+(D) = exp-(D). □ Corollary 4.9. Let G be a finitely generated group, let N be a normal nilpotent subgroup of finite index m in G and let D denote a Cayley digraph of G with respect to some finite generating set S. Then exp+ (D) = exp-(D) < m(r + 2) — 1 holds, where r is the nilpotency class of N. It is natural to ask if this bound is tight. All examples we know in fact satisfy the inequality exp+(D) < m(r + 1). We thus pose the following problem. Problem 4.10. Is it true that exp+(D) = exp-(D) < m(r + 1) holds for the Cayley digraphs of groups described in Corollary 4.9? For Cayley digraphs D of finitely generated torsion-free groups G with polynomial growth we even obtain bounds for exp+ (D) and exp-(D) which only depend on the growth degree. To formulate the result we first have to consider GL(n, Z). Theorem 4.11. (see e.g. [7]) The orders of the finite subgroups of GL(n, Z) are bounded by some function g(n) ofn alone. Theorem 4.12. (see e.g. [7]) Let G be a finitely generated torsion-free group with polynomial growth of degree d. Then G contains a normal nilpotent subgroup of class less than V2d and index at most g(d), where g(d) is the function of Theorem 4.11. Corollary 4.13. Let G be a finitely generated torsion-free group with polynomial growth of degree d. Then for any Cayley digraph D of G, exp+ (D) and exp- (D) are bounded by g(d)(V2d + 2) — 1, where g(d) is the function of Theorem 4.11 We conclude the paper with the following observations. Let G < Aut(D) act transitively on a digraph D with finite exponents exp+ (D) and exp- (D). Then Lemma 3.6 implies that the equivalence classes of the relation R+ = R- are orbits of a normal subgroup of G. Thus, if this relation is not universal and if the digraph has indegree or outdegree at least 2, then this normal subgroup of G is proper and not trivial. As a consequence, if G is simple, the relation R+ = R- is universal on D. As was already mentioned above it was shown in [6, Theorem 4.12] that a connected infinite locally finite transitive digraph D has exponential growth if at least one of the exponents exp+(D) or exp-(D) is infinite. At this point we recall the following problem from combinatorial group theory (see e.g. [1]), which was originally posed by R. I. Grigorchuk. Problem 4.14. Does every finitely generated infinite simple group have exponential growth? The following proposition then allows to formulate a conjecture which closely relates this problem to reachability relations. Proposition 4.15. If a finitely generated infinite simple group G does not have exponential growth, then for every finite generating set S of G there is a finite integer kS > 1, such that R+ = R- is universal in C (G, S). Proof. Follows immediately from [6, Theorem 4.12] and Lemma 3.6. □ Conjecture 4.16. Let G be a finitely generated infinite group. Then there is a finite generating set S of G such that for the Cayley digraph D of G with respect to S one of the following holds: • At least one of the exponents exp+(D) or exp-(D) is infinite and hence D has exponential growth. • Both, exp+(D) and exp-(D) are finite and the reachability relations R+ and Rare not universal on D. Observe that by Proposition 4.15 the validity of this conjecture would provide a positive answer to Grigorchuk's problem. References [1] G. Baumslag, A. G. Myasnikov and V. Shpilrain, Open problems in combinatorial group theory. Second edition, Combinatorial and geometric group theory, S. Cleary, R. Gilman, A. G. Myasnikov, V. Shpilrain, eds., Contemporary Mathematics 296, AMS 2002. [2] P. J. Cameron, C. E. Praeger and N. C. Wormald, Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 13 (1993), 377-396. [3] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci. Publ. Math. 53 (1981), 53-78. [4] M. DeVos, B. Mohar and R. Samal, Reachability in highly arc-transitive digraphs, arXiv:1110. 2945. [5] A. Malnic, D. Marušic, R. Möller, N. Seifter, V. Trofimov and B. Zgrablic, Highly arc transitive digraphs: reachability, topological groups, Europ. J. Combinatorics 26 (2005), 19-28. [6] A. Malnic, D. Marušic, N. Seifter, P. Sparl and B. Zgrablic, Reachability relations in digraphs, Europ. J. Combin. 29 (2008), 1566-1581. [7] A. Mann, How groups grow, LMS Lecture Notes Ser. 395, Cambridge Uni. Press, 2012. [8] N. Seifter and V. I. Trofimov, Reachability relations and the structure of transitive digraphs, Electron. J. Combin. 16 (2009), R26. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 95-113 The clone cover Aleksander MalniC University of Ljubljana, Faculty of Education, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia Tomaž Pisanski University of Ljubljana, Faculty of Mathematics and Physics, Jadranska 19, 1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia Arjana Žitnik University of Ljubljana, Faculty of Mathematics and Physics, Jadranska 19, 1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia Received 21 July 2013, accepted 30 April 2014, published online 16 September 2014 Each finite graph on n vertices determines a special (n - 1)-fold covering graph that we call the clone cover. Several equivalent definitions and basic properties about this remarkable construction are presented. In particular, we show that for k > 2, the clone cover of a k-connected graph is k-connected, the clone cover of a planar graph is planar and the clone cover of a hamiltonian graph is hamiltonian. As for symmetry properties, in most cases we also understand the structure of the automorphism groups of these covers. A particularly nice property is that every automorphism of the base graph lifts to an automorphism of its clone cover. We also show that the covering projection from the clone cover onto its corresponding 2-connected base graph is never a regular covering, except when the base graph is a cycle. Keywords: Covering projection, canonical cover, regular cover, automorphisms. Math. Subj. Class.: 57M10, 20B25, 05E18, 05C75, 05C76 * Dedicated to Dragan Marušic at the occasion of his 60th birthday. E-mail addresses: aleksander.malnic@pef.uni-lj.si (Aleksander Malnic), tomaz.pisanski@fmf.uni-lj.si (Tomaž Pisanski), arjana.žitnik@fmf.uni-lj.si (Arjana Žitnik) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 1 Introduction Some coverings are "natural" or canonical in the sense that they are determined by the graph itself. A typical example is the universal cover, which is a tree, usually, an infinite tree [6]. Another such example is the so-called canonical double cover, or the Kronecker cover. It can be described as the tensor product of the graph in question by K2 [5]. And there is also the trivial cover with the identity mapping as the covering projection. In this paper we describe another canonical cover, an (n - 1)-fold covering of a graph on n vertices, called the clone cover.1 The clone cover of a graph X will be denoted by Clone(X). We present four equivalent definitions and several basic properties of this canonical covering graph. An application in mathematical chemistry can be found in [3]. As usual in the theory of covering graphs we will always assume that X is connected. First, we study graph-theoretical properties such as connectedness, genus, and hamil-tonicity. It turns out that the clone cover of a planar graph is also planar, the clone cover of a k-connected graph for k > 2 is also k-connected, and the clone cover of a hamil-tonian graph is also hamiltonian. All these properties are far from being guaranteed for general covering graphs. In the second part of the paper we study automorphisms of such covers. Each automorphism of a graph X lifts to an automorphism of Clone(X), and moreover, the automorphism group of X embeds isomorphically in the automorphism group of Clone(X). This also is not true for general coverings. In most cases the covering projection Clone(X) ^ X is irregular, with trivial group of covering transformations. Finally, there is a natural quotient projection Clone(X) ^ X, called contraction, that is different from the covering projection. This enables us to determine the full automorphism group of Clone(X ) for certain classes of 2-connected graphs X. 2 Preliminaries In this section we review some basic definitions and elementary properties of covering graphs. The most frequent descriptions and constructions of coverings use voltage graphs. These were first introduced by Gross and Tucker and popularized in their classic text [4]. In this paper a slightly different but equivalent approach is taken, following [9]. There are two differences in the approaches. While [4] requires a choice of directions of edges in the base graph, the approach in [9] mantains the base graph as completely undirected. The other advantage of [9] is that the base graph may also be a pregraph, that is, a graph with pending semi-edges. Pregraphs, however, will not be used in this paper. A graph X is a quadruple X = (V, S, i, r) where V is a finite set of vertices, S is a finite set of arcs, i is a mapping S ^ V, specifying the initial vertex of each arc, while the reversal involution r : S ^ S is an involution without fixed points. The terminal vertex of an arc is then specified by the mapping t : S ^ V, t(s) = i(r(s)). An arc s and its reverse r(s) form an edge with endvertices i(s) and i(r(s)). Two vertices are adjacent if they are the endvertices of a common edge. If every edge of the graph has two distinct endvertices and no two edges have the same endvertices, the graph is simple. We consider only simple graphs, with at least one edge, to avoid trivialities. We will use the following notation. The set of vertices of a graph X will be denoted by V(X), the set of its arcs by S(X), and the set of its edges by E(X). In a simple graph every edge is uniquely determined by its endvertices. Therefore we will denote an edge with endvertices u and v by {u, v}. An arc with initial vertex u and terminal vertex v will 1 Note that this construction was previously called TheCover. be denoted by [u, v], or more briefly by uv. The set of vertices, adjacent to a vertex v of X, will be denoted by N(v). Let X and Y be graphs. A mapping p : Y ^ X that takes vertices to vertices and arcs to arcs is called a homomorphism if p(i(s)) = i(p(s)) and p(r(s)) = r(p(s)) for every s G S .A surjective homomorphism p : Y ^ X is called a covering projection if the set N(v) is mapped bijectively onto the set N(p(v)), for each vertex v g V(Y). The graph X is usually referred to as the base graph and Y as the covering graph. We call p-1(u) the fiber over u G V (X ), and p-1(s) the fiber over s G S (X ). We will assume that X is connected. This implies that all the fibers are of the same size. By [4, Theorem 2.4.5], every covering graph can be obtained as follows. Let L be a finite (labeling) set and X a finite connected simple graph. Let т : X ^ Sym(L) be a permutation voltage assignment on X, defined by т(s) G Sym(L) for each arc s in X, and satisfying the condition т(r(s)) = т-1(s). The graph X together with the assignment т is called a permutation voltage graph (X, т). To a permutation voltage graph (X, т) we associate a derived graph Y = CovT(X), with vertex set V(Y) = V(X) x L, arc set S(Y) = S(X) x L, and mappings i,r satisfying i(s,j) = (i(s),j) forany (s,j) G S(Y) and r(sj) = (r(s)jT(s)) forany (s,j) G S(Y). In other words, with each arc uv g S(X) and each j g L we associate the arc (uv, j) = [(u,j), (v,jT("'v))] from (u,j) G V(y) to (v,jT(uv)) G V(Y). Note that its reverse arc is (vu, jT(uv)) = [(v, jT("v)), (u, j)], from (v, jT(uv)) to (u, j). Hence these opposite arcs form an edge "over" the edge {u,v}. Therefore the graph Y is a covering graph over the base graph X, with the natural covering projection p : Y ^ X taking a vertex (u, j) G V(Y) to u G V(X) and an arc (uv, j) in Y to the arc uv in X. 3 Constructions Let X be a connected graph on n > 2 vertices. We begin by constructing a canonical n-fold covering graph of X, where we use V(X) as the labeling set L. Let т : X ^ Sym(V(X)) be a permutation voltage assignment on X defined by the transposition т(uv) = (u, v) G Sym(V(X)) for each arc uv in X. The associated covering graph is denoted by Cov(X). The vertex set of Cov(X) is V(X) x V(X) while E(X) x V(X) is the edge set. The edge set can be naturally partitioned into three subsets, namely, the subset of • diagonal edges {(u, u), (v, v)}, • connecting edges {(u, v), (v, u)}, u = v, and • inner edges {(u,w), (v, w)}, w = u, v. We call this partition the fundamental edge partition. The three different types of edges in the fiber over one edge are shown in Figure 1. Example 3.1. Let us consider the graph K2,3. The voltage assignment in Figure 2 determines the 5-fold covering graph Cov(K2,3) in Figure 3. Figure 1: The lift of an edge. Figure 2: A voltage graph K2,3 for Cov(K2,3). 3.1 The construction of Clone(X) via Cov(X) Let X be a connected graph on n > 2 vertices. The following proposition carves Clone(X) out of the auxiliary graph Cov(X). Proposition 3.2. Cov(X) is a disjoint union of two covering graphs of X. One is isomorphic to Covid (X) = X. Proof. The subgraph of Cov(X) induced by the diagonal vertices {(v, v), v g X} (and diagonal edges) is isomorphic to X, and the restriction of the covering projection to it gives Covid (X). □ The subgraph of Cov(X) that does not contain the diagonal vertices is called the clone cover and denoted by Clone(X). Clearly, Clone(X) is an (n — 1)-fold covering graph over X. We call the subgraph of Clone(X), spanned by the vertices {(v, i); v G V(X)\i}, the i-th layer of Clone(X). Example 3.3. Let X be the cycle on n vertices for n > 3. Then Clone(X) is the cycle on n(n — 1) vertices. (a) (b) Figure 3: Cov(K2,3) has two components: (a) Covid(K2,3), (b) Clone(K2,3). 3.2 A direct permutation voltage graph construction This construction depends on the choice of the base vertex b of X. The permutation voltages are taken from Sym(V(X) - {b}). They are defined as follows. The permutation voltages on arcs incident with the vertex b are equal to the identity while the voltages of arcs uv not involving b are, as before, equal to the transposition (u, v). The corresponding covering graph is denoted, for the time being, by Cloned(X, b). 3.3 Combinatorial Construction Let X be a connected graph on n vertices. We define the graph Clonec(X) as follows. The vertex set W of Clonec(X ) consists of all n(n-1) pairs of vertices (u, v) G V (X ) x V (X ) with u = v. There are two sets of edges. Each edge {u, v} from X gives rise to the edge {(u, v), (v,u)} in Clonec(X) (these will correspond to the connecting edges). For each w G V(X), different from u and v, we get in total (n - 2) (inner) edges {(u, w), (v, w)}. It is not hard to show that the projection from W to V(X) defined by (u, v) ^ u is an (n - 1)-fold covering projection. 3.4 Graphical Construction Let Xv denote the graph X with vertex v removed. The graph Cloneg (X ) is obtained from the collection of n vertex-deleted subgraphs Xv = X - v by adding, for each edge {u, v} of X, an edge joining the vertex u in Xv to the vertex v in X„; see Figure 4. (a) (b) Figure 4: (a) The graph X and one of its vertices u. Cloneg (X) is obtained in such a way that each vertex u is replaced by a vertex deleted subgraph Xu. In (b), this is shown for the vertex u. The edges of Cloneg (X) can be naturally partitioned (or colored) into two classes: the edges belonging to each vertex-deleted subgraph Xv, and the connecting edges. Each edge of X lifts to one connecting edge and (n - 2) original edges. Example 3.4. Figure 5 shows the graphical construction of Clone(K2,3). Proposition 3.5. Let X be a 2-connected graph. Then X is a minor of Cloneg (X). Proof. If X is 2-connected, then for every vertex u g V(X) the vertex-deleted subgraph X„ is connected. If for each u we contract the edges of the copy of X„ from Cloneg (X), (a) K2,3 (b) Clone(K2,3). Figure 5: The graphical construction of Clone(K2,3). then this copy of Xu is contracted to a single vertex and the resulting graph is isomorphic to X. Hence X is a minor of Cloneg (X). □ 3.5 Equivalence of the four constructions Here we prove that the above four definitions are equivalent. Theorem 3.6. The covers Clone(X), Cloned(X, b), Clonec(X), and Cloneg(X) are isomorphic. Proof. It is easy to see that Clone(X) and Clonec(X) are isomorphic since they have the same vertex set and the same edge set. Also the mapping that sends the vertex (Xu, v) of Cloneg (X) to the vertex (v, u) in Clone(X) is an isomorphism. To finish the proof we show that Clonec (X) and Cloned (X, b) are isomorphic. Define the mapping ^ : V(Clonec(X)) ^ V(Cloned(X, b)) by v(u v) = /(u,v) if v = b, ^(u,v) \ (u,u) if v = b. This is obviously bijective. The edges of the form {(u, b), (v, b)} are mapped to the edges of the form {(u, u), (v, v)}, while all other edges of Clonec(X) are mapped to the edges with the same labels in Cloned (X, b). This shows that ^ is an isomorphism. In particular, the choice of the vertex b in Cloned (X) = Cloned (X, b) is irrelevant. □ 3.6 Lifts of cycles Recall from general theory [4] that any voltage assignment can be naturally extended from arcs to walks by successively multiplying voltages of arcs encountered along the walk. The voltage of a walk actually tells how this walk lifts to the corresponding covering graph. We are particularly interested in how a given cycle of the base graph lifts. Clearly, a cycle lifts to a collection of cycles. Theorem 3.7. [4, Theorem 2.4.3] Consider a covering projection p : X ^ X arising from a permutation voltage assignment in Sn on X. If C is a cycle of length k in X whose voltage has cycle structure (ci,..., cn ), then the preimage of C in the derived graph has ci + • • • + cn components, consisting of exactly Cj cycles of length k j, for j = 1,..., n. Let X be a graph on n vertices, with the voltage assignment т(uv) = (u, v) G S (V (X )) for each arc uv in X. Recall that this assignment gives rise to the covering graph Cov(X) which consists of an isomorphic copy of X, and Clone(X). In this particular setting it is easy to see that the voltage of a directed cycle C = v0v1... vmv0 in X, rooted at v0, is then (vm, vm_i,..., vi)(vo) G Sym(V(X)). The following proposition is therefore a direct consequence of Theorem 3.7 . Proposition 3.8. Let X bea connected graph. A k-cycle in X based at u lifts in Clone(X ) to one "long" cycle of length k(k — 1) based at (u, v), where v = u is any vertex in the cycle, and n — k "short" cycles of length k based at (u, v) where v is any vertex not in the cycle. Corollary 3.9. Let X be a connected graph on n vertices. If X contains a cycle of length k < n then also Clone(X ) contains a cycle of length k. 4 Graph-theoretical properties A natural problem to consider is the impact of a given graph invariant of a graph such as girth, connectivity or diameter, on its clone cover. Some invariants are easy to determine. For instance, girth is a well-known graph invariant measuring the lengh of the shortest cycle in a graph. Any connected graph that is not a cycle has the same girth as its clone cover by Corollary 3.9, and the girth of the clone cover of a cycle on n vertices is n(n — 1) by Proposition 3.8. In this section some other graph invarants are studied. 4.1 Connectivity The graph Clone(X) can be connected or disconnected, with an easy test for connectivity. Recall that a block of a graph X is a maximal connected subgraph of X without a cutvertex. If X contains no cut-vertex, then X itself is called a block. Theorem4.1. Let X be a connected graph. Then Clone(X ) is connected if and only if X is a block. Moreover, if X is k-connected, where k > 2, then Clone(X) is also k-connected. Proof. In this proof we will use the graphical construction of Clone. Suppose X has a cut-vertex v, and let the vertices v1 and v2 be in different blocks of X. Then the vertices (XV1 ,v2) and (Xv2,v1) are in different components of Clone(X), since every path between them would pass through Xv, and in Xv there is no edge between the vertices of the blocks of v1 and v2. Therefore Clone(X) is not connected. If X is a block that is not 2-conneced, then it is isomorphic to the complete graph on two vertices. So Clone(X) is isomorphic to X and hence connected. Suppose now that X is k-connected, where k > 2. We will prove that Clone(X) is k-connected (and therefore also connected). By the global version of Menger's theorem, it is enough to prove that for any two distinct vertices in Clone(X) there exist k internally disjoint paths between them. Note that each of the subgraphs Xu of Clone(X) is connected since X is 2-connected. We use the following notation. Let P = u1u2... ut beapathin X. Then P(už, ...,uj ) denotes the part of P between the vertices u and uj for 1 < i < j < t. Let u g V(X) be distinct from u1, u2,..., ut. By Pu(u1, u2,..., ut), or more briefly, by Pu, we denote the path (Xu, u1)(Xu, u2)... (Xu, ut) in Clone(X) that is contained in Xu. We denote a path in Clone(X) between the vertices (XUl, u2) and (XUt, ut-1) of the form (XUl ,U2)(Xu2 ,U1 ) . . . (XU2 ,U3)(XU3 ,U2) . . . (Xut-2 ,Ut-1 )(Xut-i ,Ut-2) . . . (Xut-i ,Ut)(Xut ,Ut-1 ) by pP(u1 , u2 ,..., ut), or more briefly, by P. The walks in the same copy of any vertex deleted subgraph can be arbitrary paths. Let (XU, v) and (Xw, z) be two distinct vertices of Clone(X). We now construct k internally disjoint paths between them. We distinguish two cases. Case 1. Suppose u = w. Let P1,..., Pk be k internally disjoint paths between v and z in X. If none of them contains u, we have k disjoint paths between (XU, v) and (XU, z) in Clone(X) which are all contained in XU. If one of the paths, say P1, contains u, we have only k — 1 internally disjoint paths contained in XU. We now construct another path that will also use other vertex-deleted subgraphs. Let P = P1 = vu1... utuut+1... usz. Let Q be a path between u and ut+1 in X that does not contain ut, and let R be a path between ut and u in X that does not contain ut+1. Since X is 2-connected, such paths exist. Then Pu (v,U1, . . . ,Ut)(Xut ,u)Qut (Xut+1 ,Ut)RUt+i (Xu ,Ut+1 )Pu (Ut+1 . . . , Ut, z) is a path between (XU, v) and (XU, z) that is internally disjoint from each of P2,..., Pk. Case 2. Suppose u = w. Let P4 = uu\ ... ut., w be k internally disjoint paths between u and w in X. By Dirac's Fan Lemma (see, for example, [10, Theorem 4.2.23]), there exist internally disjoint paths from v to u1,..., uk in X, say Q1,..., Qk. Similarly, there exist internally disjoint paths from uj ,..., uk to z in X, say R1,..., Rk. Suppose first that u is not contained in any of the paths Ql or Rl. Then for i = 1,..., k the paths S 4 = Q U P г (u,U1 ,...,uti ,w)RR w are k internally disjoint paths between (XU, v) and (Xw, z); see the top path in Figure 6. Figure 6: Two of internally disjoint paths in Clone(X). Suppose now that u belongs to Qj for some j. Let Q = Qj = vv1... vs uuj. If vs belongs to Pj, we just interchange the roles of uj and vs and construct as before. Otherwise, we may assume that vs does not belong to any of the paths P4 for i = j. We can do this since if for some i the vertex vs belongs to P4, so P4 is of the form uu\ ... vs ... ut. w, then we may replace P4 by the path uvs... ut w that is also internally disjoint from the other k — 1 paths between u and w in X. We define = QU (v, v!,..., vs)(Xvs, u)(X„., u{)(Xuj, vs)... (XU, u2)P4(u22, . . . , uj , w)RW. Note that the subgraph XVs is not used by any of the other paths Si for i = j, so Sj is internally disjoint with them; see the bottom path in Figure 6. If u belongs to some Re, we modify Se in a similar way as above. Again, we have k internally disjoint paths between (Xu,v) and (Xw,z). □ If X is connected, every block of X is either a maximal 2-connected subgraph or a bridge. Different blocks can have at most one vertex in common, which is then a cut-vertex of X. Therefore every edge lies in a unique block, and X is the union of its blocks. We denote by Xi ®v X2, or just Xi ф X2, the union of two graphs with a common vertex v. We denote by X1 U X2 the disjoint union of two graphs. Lemma 4.2. Let X = B фv C be composed of two blocks B and C with a common vertex v. Let {u1,u2,..., up, v} be the vertex set of B and let {w1,w2,..., wq, v} be the vertex set of C. Then Clone(X ) is isomorphic to the following graph: (Clone(B) ©(v,Ul) C ©(v,«2) C ф • • • 0(v,«p) C)U (Clone(C) ф( B ф( v,W2 ) B ф v,Wq ) B). Proof. As in the proof of Theorem 4.1 we see that Clone(B) and Clone(C) are connected and lie in different components of Clone(X). The claim follows from the fact that every vertex deleted subgraph Xui contains a copy of C, and every vertex deleted subgraph Xw. contains a copy of B. □ Corollary 4.3. Let X be a connected graph and let B1,B2,... ,Bk be the blocks of X. In other words, X = B1 ф B2 ф • • • ф Bk. Let Ci consist of the blocks of X different from i,for i = 1,..., k. Then Clone(X ) is isomorphic to the following graph (with blocks attached at appropriate vertices): (Clone(B1) ф*-1 (C1)) U (Clone(B2) ф^1 (C2)) U • • • U (Clone(Bfc) ф*-1 (Cfc)). In particular, the number of components of Clone(X ) is equal to the number of blocks of X. 4.2 Bipartiteness Any covering graph of a bipartite graph is obviously bipartite. The graph Clone(Cn) is the cycle Cn(n-1), hence it is bipartite. It turns out that odd cycles are the only non-bipartite graphs for which the clone cover is bipartite. Proposition 4.4. Let X be a graph that is not a cycle. Then Clone(X ) is bipartite if and only if X is bipartite. Proof. Suppose X is not bipartite, and let C be an odd cycle in X as short as possible. Since X is not a cycle, there exists a vertex v of X not in C. In Clone(X) there is a copy of C in the layer corresponding to v, thus also Clone(X) is not bipartite. Conversely, suppose that Clone(X) is not bipartite. Then it contains an odd cycle. By Proposition 3.8, this must come from a cycle of the same odd length in X (since k(k - 1) is even for all k), and therefore X is not bipartite. □ 4.3 Hamiltonicity Although the cycle structure of the clone covers is fairly well understood, no complete characterization of hamiltonian clone covers is known. However, there is a simple sufficient condition for the base graph to have a hamiltonian clone cover. By Proposition 3.8, a Hamilton cycle (of length n) in the base graph X lifts to one cycle of length n(n — 1) in Clone(X). The cycle of length n(n — 1) is a Hamilton cycle in Clone(X). We record this formally. Theorem 4.5. Let X be a hamiltonian graph. Then Clone(X) is hamiltonian. We only have a partial converse of this result. However, no examples are known of a 2-connected non-hamiltonian graph for which the clone cover is hamiltonian. Proposition 4.6. Let X bea non-hamiltonian graph of minimal degree at most three. Then Clone(X ) is also non-hamiltonian. Proof. Let v be a vertex of degree at most three in X, and suppose that Clone(X) has a Hamilton cycle H. Denote by Hv the subgraph of H restricted to Xv. Since Xv is connected to the rest of the graph by at most three edges, Hv forms a Hamilton path in Xv. Denote the vertices of degree 1 of this path by (Xv, u) and (Xv, w). Then u and w are neighbors of v in X. By adding edges {v, u} and {v, w} to the projection of Hv on X, we obtain a Hamilton cycle in X .A contradiction. □ Proposition 4.7. Let X be a graph and let v G V(X ) be a vertex of degree k such that X\N(v) has more than k components (X is not hamiltonian). Then Clone(X) is also not hamiltonian. Proof. Let N (v) = {vi, v2,..., vk} and let U = {(v, vi), (v, v2),..., (v,vk )}. Then Clone(X)\U has more than k components since Xv\U has at least k components and is not connected to the rest of the graph. Therefore Clone(X ) is not hamiltonian. □ 4.4 Planarity Recall the graphical construction of Clone(X): the graph Clone(X) can be obtained from the graph X by "replacing" each vertex v by Xv = X — v. Using this fact, we make the following observations. Theorem 4.8. Let X be a graph. Then Clone(X ) is planar if and only if X is planar. Proof. Let X be a planar graph and let Y be a planar embedding of X. We choose an orientation of the plane. Let u be a vertex of X and let Yu be an embedding of X such that u lies on the outer face with the cyclic order of the neighbors of each vertex reversed with regard to Y. Then the order of the neighbors of u along the outer face of Yu — u is the same as the order of the neighbors of u in Y. Therefore it is possible to replace u in Y by the graph Yu — u, and connect each of the neighbors of u in Y by the corresponding neighbor of u in Yu such that this replacement yields a plane graph again. Doing this for each vertex of Y we obtain a planar embedding of Clone(X ). Conversely, if X is not planar, then it contains a copy of K3 3 or K5 as a minor. If X is 2-connected, then it is a minor of Clone(X) by Proposition 3.5, and therefore also Clone(X) contains a copy of K3,3 or K5 as a minor. If X is not 2-connected, then Clone(X) contains each block of X as a subgraph by Corollary 4.3. In this case, at least one block of X is not planar and therefore also Clone(X) is not planar. □ Similarly we can give an upper bound on the genus of Clone(X) in terms of genus of Theorem 4.9. Let X be a graph on n vertices. The following bound holds for the genus j: 7(Clone(X)) < (n + 1)y(X). The same inequality holds for the non-orientable genus. Proof. Let m denote the number of edges of X and let g = y (X). Let f denote the number of faces in the genus embedding of X, and let f be the number od faces of length i, for i > 3. Suppose first that all the faces of X are cycles. Recall that the voltage of a cycle of length k is a cycle of length k — 1 in Sym(V(X)) = Sn fixing n — k +1 symbols. By Proposition 3.8, such a cycle lifts to one cycle of length k(k — 1) and n — k cycles of length k in Clone(X). Take an embedding of Clone(X)) in which the cyclic order of the edges around each vertex is the same as in the genus embedding of X. Then a face of X lifts to a face of Clone(X)). Denote by n', m', f', g' the number of vertices, number of edges, number of faces, and the genus of this embedding of Clone(X)), respectively. Then n' = (n — 1)n, m' = (n — 1) m, and f ' = ^ (n — i + 1)fi = ^ (n + 1)fi — ^ ifi = (n + 1)f — 2m. i> 3 i> 3 i> 3 Now we can compute g': g' = (2 + m' — n' — f')/2 = (2 + m(n — 1) — n(n — 1) — f (n +1)+2m)/2 = (2n + 2 + m(n +1) — n(n +1) — f (n + 1))/2 = (n + 1)g. If a face of X of length k is not a cycle, it lifts to more than n — k + 1 faces, which makes the genus of such an embedding of Clone(X) even smaller than (n + 1)g. In any case we have 7(Clone(X)) < (n + 1)g. The same reasoning holds also for the nonorientable embeddings. □ Example 4.10. The genus of K3 3, which is a graph on 6 vertices, is 1. By Theorem 4.9, the genus of Clone(K3,3) is at most 7; see Figure 7. 5 Algebraic properties A reasonable assumption when studying algebraic properties of covering graphs, or indeed graphs in general, is to restrict our considerations to connected covering graphs - which in our case translates to requiring that the base graphs are at least 2-connected, in view of Theorem 4.1 and the assumption that the base graph is not K2. There are two different kinds of automorphisms of a covering graph: the ones that are lifts of some automorphism of the base graph, and the ones that are not. Along these lines we consider certain structural properties of the automorphism group of Clone(X ), edge-and vertex-transitivity of Clone(X), and regularity of the covering projection Clone(X) ^ X. Automorphisms that respect the fundamental edge partition, see Subsection 5.2 below, will play a significant role in this context. 5.1 Lifts of automorphisms along the covering projection Certain automorphisms of a covering graph can be studied in terms of automorphisms of the base graph. Such automorphisms arise as lifts of automorphisms, a concept we shall now define. Let p : X ^ X be a covering projection of graphs, and let f be an automorphism of X. We say that f lifts if there exists an automorphism f of X, a lift of f, such that the following diagram X —^ X X —- X is commutative; in other words, f o p = p o f. Observe that a lift of an automorphism maps bijectively fibers over vertices (resp. edges) to fibers over vertices (resp. edges), in particular, any lift of f maps the fiber over a vertex u g V (X ) to the fiber over the vertex f(u). Suppose that all the elements of a subgroup G < Aut(X ) have a lift. Then the lifts of all automorphisms from G form a subgroup of Aut(X) which we denote by G. In particular, the lift of the trivial group is known as the group of covering transformations and is denoted by CT(p). Further, if G lifts, then there exists an epimorphism pG : G ^ G with CT(p) as its kernel. Hence G is an extension of CT(p) by G, and the set of all lifts of a given f g Aut(X) is a coset of CT(p) in G. As a final opening remark in this section, recall from general theory that CT(p) acts semiregularly on the covering graph X whenever X is connected; that is, CT(p) acts without fixed points both on vertices and on arcs of X. Moreover, each lift is uniquely determined by the mapping of a single vertex. For a background on lifting automorphisms in terms of voltages we refer the reader to [8]. In general, not every automorphism of the base graph X lifts. This is not the case with the natural covering projection p : Clone(X) ^ X, p = prx : (u, v) ^ u. To this end let us introduce, for each automorphism f of the graph X, a mapping f : Clone(X) ^ Clone(X ) which we call the diagonal mapping, defined by /: (u,i) ^ (f(u),f(i)). Theorem 5.1. Let f be an automorphism of X. Then the map f is an automorphism of p p Clone(X) and is a lift of f. Proof. Obviously, f is well defined since i = u implies f (i) = f (u), and moreover, it is bijective on the vertex set of Clone(X). We will show that f maps arcs to arcs, so it is an automorphism. Let uv be an arc in X, and let a be an arc in Clone(X) from (u, i) to (v, i(u,v)). The vertex (u, i) is mapped by f to (f (u), f (i)). Since f (i(u'v)) = f (i)(f (u)f (v)) and the vertex (v, i(u'v)) is mapped by f to the vertex (f (v), f (i(u'v)) = (f (v), f (i)(f(u)f (v))), it follows that f (a) is an arc in Clone(X), as required. Let (u, i) be a vertex from Clone(X ). Then f o p(u, i) = f (u) = p(f (u), f (i)) = p o f(u, i). Let a be an arc in Clone(X) from (u, i) to (v, i(u,v)). Then f o p(a) = f ([u,v]) = [f (u),f (v)] = p([(f (u),f (i)), (f (v)f (i)(u'v))j) = p o fa). Thus, f is an automorphism and clearly a lift of f. □ In view of the above theorem, Clone(X) inherits all the symmetries of X in the sense that there is a natural injection of Aut(X ) into Aut(Clone(X )) taking f ^ f. This injection is actually a group homomorphism, as we shall see shortly. For convenience we denote by A = Aut(X ) the full automorphism group of X and A its lift. Proposition 5.2. The set A of all diagonal mappings is a subgroup of Aut(Clone(X)), isomorphic to A, and A = CT(p) x A. Proof. The set A is clearly a complete system of coset representatives of CT(p) within A since for each f g A there is only one diagonal mapping. Moreover, from the definition of the diagonal mapping it easily follows that fg = fg . Hence A is a complement to CT(p) within A, and so A = CT(p) x A, as required. □ From now we shall be explicitly assuming that the base graph X is 2-connected, as already anticipated at the beginning of this section. The simplest base graphs of this kind are the cycles. Proposition 5.3. Let p : Clone(Cn) ^ Cn be the covering projection for the n-cycle Cn, for n > 3. Then CT(p) is isomorphic to Zn-1, and Aut(Clone(Cn)) = Dn(n-1) = Zn-i x Dn. Proof. Recall that Clone(Cn) = Cn(n-1) which has the automorphism group isomorphic to the dihedral group Dn(n-1) of order 2n(n — 1). The subgroup of Aut(Clone(Cn)) generated by the n-step rotation of Cn(n-1) is isomorphic to Zn-1. It is easy to see that each of these automorphisms is a covering transformation. Since A = Aut(Cn) is isomorphic to the dihedral group of order 2n, it follows that A = CT(p) x A has order 2n(n — 1). Hence Aut(Clone(Cn)) = A. □ Clone(Cn) appears to be rather special, since every vertex deleted subgraph of a cycle is acyclic. In all other cases where the clone cover is connected, the group of covering transformations is trivial, and this fact has strong impact on symmetry properties of Clone. Theorem 5.4. Let X bea 2-connected graph that is not a cycle. Then CT(p) is trivial, and hence the lifted group is equal to the group of diagonal mappings - that is, A = A. Proof. From the assumptions it easily follows that in X there exist two distinct vertices, connected with three internally disjoint paths P, Q, Д. Let C be the cycle formed by the paths P and Q, and let C be the cycle formed by P and Д. Let k and k' be the lengths of C and C', respectively. Then C lifts to one k(k - 1) cycle and to n - k cycles of length k, while C' lifts to one k'(k' — 1) cycle and n — k' cycles of length k', where n = |V(X)|. Denote the k(k — 1) cycle over C by C and the k'(k' — 1) cycle over C' by C". Let now f G CT(p) be a covering transformation. Since f permutes the cycles over C we have f (Č) = C Similarly, f (C") = C '. Denote the union of C and C' by Y = C U C', and let Y be the connected component of the preimage p-1 ( Y) containing C U C. Note that У = Clone(Y). Moreover, f (У) = У. Further, the restriction of f to Y is a covering transformation of the projection Clone(Y) ^ Y. Since CT(p) acts without fixed points, it is enough to show that the group of covering transformations of the projection Clone(Y) ^ Y is trivial. To formally prove the above assertion we shall actually prove that the group of covering transformations of the auxiliary covering p : Cov(Y) ^ Y (which is isomorphic to that of Clone(Y) ^ Y) must be trivial. Let V(Y) = {uo, u1..., us, xs+1,..., xk-1, ys+1,..., Vk'-1} be the vertex set of Y = C U C', and let C = uou1... usxs+1... xk- 1uo and C' = uou1... usys+1... yk'-1u0 be the corresponding directed cycles, rooted at u0. The first cycle has voltage a = (xk-1, ... ,xs+1,us,... ,u1) whileß = (yk'-1,... ,ys+1,us,... ,u1) is the voltage of the second one. From general theory [8, Corollary 7.3] it easily follows that the group of covering transformations of a connected cover given by permutation voltages is isomorphic to the centralizer in the symmetric group of the subgroup generated by the voltages of all closed walks at a chosen vertex. Hence in our case CT(p) is isomorphic to the centralizer of a and ß in Sym(V(Y)). Let т commute with both a and в. If we represent these permutations graphically as a colored digraph on the vertex set V (Y), then т corresponds to a color- and direction-preserving automorphism of this digraph. From the structure of the above colored 'permutation digraph', it is now immediate that т must be trivial, as required. □ Recall that a cover is regular if the fiber-preserving automorphisms act transitively on each fiber. The three canonical covers mentioned in the introduction, namely the universal cover, the Kronecker cover, and the trivial cover, are all regular covers for all base graphs. However, the clone cover is in most cases an irregular covering. Theorem 5.5. Let X be a 2-connected graph. Then Clone(X ) ^ X is not a regular covering projection unless X = Cn. Proof. By Theorem 5.4 we know that the group of covering transformations is trivial, except when X = Cn. This completes the proof. □ In particular, the graph Clone(Kn), n > 4, is an irregular cover of Kn. 5.2 Automorphisms that respect the fundamental edge partition We will say that an automorphism of Clone(X) respects the fundamental edge partition if it takes inner edges to inner edges, and connecting edges to connecting edges. From the graphical construction of Clone(X ) it follows that inner edges can be naturally partitioned into layers - which are nothing but the vertex deleted subgraphs of X. Consequently, the property of preserving the edge partition is equivalent to requiring that layers are mapped to layers, at least when Clone(X) is connected. Proposition 5.6. Let X be a 2-connected graph. An automorphism of Clone(X) respects the fundamental edge partition if and only if it maps layers to layers. Proof. Let f be an automorphism of Clone(X). Suppose f maps layers to layers. Then it maps inner edges to inner edges. Hence it must also map connecting edges to connecting edges, and must therefore respect the fundamental edge partition. Conversely, suppose that f respects the fundamental edge partition. Then it maps inner edges to inner edges. Since X is 2-connected, every layer of Clone(X), which is just a vertex-deleted subgraph of X, is connected. Take two vertices (u, v) and (w, v) from the same layer of Clone(X). Then there exists a path between them, consisting only of inner edges. But then also f (u, v) and f (w, v) are connected by a path consisting only of inner edges. Therefore f (u, v) and f (w, v) are in the same layer of Clone(X). This shows that f takes layers to layers. □ Note that all automorphisms that respect the fundamental edge partition form a subgroup in Aut(Clone(X)) which we denote E. We are now going to explicitly describe the structure of this group whenever Clone(X) is connected. To start with, note that there is a natural mapping contr : Clone(X ) ^ X, called contraction, defined by collapsing each vertex-deleted subgraph to its corresponding vertex v. To put it differently, contraction is in fact the projection contr = pr2 : (u, v) ^ v onto the second coordinate - in contrast with the covering projection which is the projection onto the first coordinate. Let now f and f be automorphisms of X and Clone(X ), respectively, such that the following diagram Clone(X ) —— Clone(X ) contr contr XfX is commutative. We then say that f lifts and that f projects along the contraction. In a similar fashion we speak about lifting and projecting groups. In view of Proposition 5.6 we have the following obvious characterization of the fundamental edge partition preserving subgroup. Proposition 5.7. Let X be a 2-connected graph. Then E is precisely the subgroup of automorphisms of Clone(X ) that projects along the contraction. Moreover, the contraction induces a group homomorphism E ^ Aut(X). We have already remarked that Clone(Cn) is rather special for several reasons. Apart from the fact that its group of covering transformations is not trivial, it is also true that it does not respect the fundamental edge partition, in view of the next result and Theorem 5.2. Theorem 5.8. Let X be a 2-connected graph and let A = Aut(X). Then the maximal subgroup in the lifted group A that respects the fundamental edge partition is the group A of diagonal mappings. In particular, the induced homomorphism E ^ A is surjective. Proof. Choose an arc uv in X and its corresponding unique connecting arc [(u, v), (v, u)] in Clone(X ). If f g A preserves the fundamental edge partition, then it must map [(u, v), (v,u)] to the unique connecting arc over [f (u),f (v)], that is, to [(f (u),f (v)), (f (v),f (u))]. It follows that f (u,v) = (f (u),f (v)) = f(u, v), where f is the diagonal mapping. Since the covering graph is connected, a lift of f G Aut(X) is uniquely determined by the mapping of a single vertex. Hence f = f. The final statement obviously holds since each f g A lifts to f G E. This completes the proof. □ In order to identify the group E, we need another definition. For a vertex v g V(X), let An (v) < Av denote the subgroup in the stabilizer of v fixing all vertices in the neighborhood N(v) point-wise. For each f g AN(v) let fЙ(х i) = f(x,i) if i = v f (x,i) \ (f(x),v) if i = v. Clearly, f N is an automorphism of Clone(X ) that preserves the fundamental edge partition; its projection along the contraction is the identity automorphism of X, but fN does not project along the covering projection; see below. Let aN(v) = {fN | f G AN(v)}. Note that AN(v) = An(v). We are now ready to identify the fundamental edge partition preserving subgroup E. Theorem 5.9. Let X be a 2-connected graph. Denote its automorphism group Aut(X ) by A, and let A be the group of diagonal mappings of Clone(X ). Then E is the internal semi-direct product Uvev(X)aN(v) * A. Proof. For each vertex v g V = V(X), a typical element of the Cartesian product of groups nveV AN (v) has the form nveV f|, where fv G AN (v). Note further that nveVAN(v) indeed exists as a group of automorphisms of Clone(X). Since each automorphism fv projects to the identity along the contraction, we know that nveVAN(v) is contained in the kernel K of the homomorphism E ^ A induced by the contraction. Conversely, let f G E be in the kernel K. Then f fixes each layer set-wise, and its restriction to the v-th layer induces an automorphism fv of the vertex deleted subgraph Xv. It follows that f can be written as the product f = nveVfj, with commuting factors. Hence f is an element of the Cartesian product of subgroups AN(v), v g V, and so the kernel is precisely this group. Since the homomorphism E ^ A is surjective, E is an extension of its kernel K by A. Observe that the diagonal mapping / g A projects to f both along the covering projection and along the contraction. This means that the group A is a system of coset representatives of K, and so E is a semi-direct product of nveV AN(v) by A. □ 5.3 The full automorphism group of Clone Let X be a 2-connected graph. We have seen that A = Aut(X) embeds in Aut(Clone(X)) as the group of diagonal mappings A. Now Clone(X) may have other automorphisms, for two reasons. Firstly, the group of covering transformations is not always trivial; this happens only when X is a cycle, and in that case the lifted group is in fact the full automorphism group of Clone(X). Secondly, the fundamental edge partition preserving subgroup E may be larger than A In view of Theorem 5.9, the latter happens if and only if there exists a nontrivial automorphism of X fixing a vertex and its neighboring vertices point-wise. Example 5.10 provides an instance of such a case. Example 5.10. The automorphism group of K2 3 is S2 x S3 and has 12 elements. On the other hand, Aut(Clone(K2 3)) has additional automorphisms of order two, and has 96 elements; see Figure 3. In fact, the full automorphism group is equal to E, which is by Theorem 5.9 isomorphic to Z3 x (S3 x Z2). The case when when the full automorphism group of Clone(X) is equal to E is particularly interesting, since then the group Aut(Clone(X)) can be determined using Theorem 5.9. Moreover, the fact that every automorphism respects the fundamental edge-partition has strong impact on the transitivity properties of Clone(X ). Theorem 5.11. Let X be a 2-connectedgraph such that every automorphism of Clone(X) respects the fundamental edge partition. Then the following hold. (a) Clone(X) is not edge-transitive. (b) Clone(X) is not vertex-transitive unless X = Kn,for some n > 4. Proof. If every automorphism of Clone(X ) respects the fundamental edge partition, then no inner edge can be mapped by an automorphism to a connecting edge, and vice versa. Therefore Clone(X) is not edge-transitive. Also in this case, Clone(X) can be vertex-transitive only if all the vertex-deleted subgraphs of X are vertex-transitive, and isomorphic to each other. This only happens when X is a complete graph on more than 3 vertices. □ In certain cases it is easy to show that any automorphism of Clone(X) preserves the fundamental edge partition. Proposition 5.12. Let X be a 2-connected graph of girth g such that each edge of each vertex deleted subgraph is contained in a cycle of length at most 2g — 1. Then each automorphism of Clone(X ) preserves the fundamental edge partition. Proof. Since a layer naturally corresponds to a vertex deleted subgraph of X, each inner edge of Clone(X) belongs to a cycle of length at most 2g — 1. On the other hand, no connecting edge belongs to such a cycle. The conclusion follows. □ There are several families of graphs which fulfill the conditions of Proposition 5.12. We state the following without proof. Proposition 5.13. • The complete graph Kn for n > 4 has girth 3. Each vertex deleted subgraph of Kn is isomorphic to Kn-1; any edge of a complete graph on at least 3 vertices is contained in a 3-cycle. • A hypercube Qn for n > 3 has girth 4 and every edge in any vertex-deleted subgraph of Qn lies in a 4-cycle. • A cartesian product of 2-connected graphs has girth at most 4 and every edge in any vertex-deleted subgraph of such a graph lies in a 4-cycle. • A generalized Petersen graph G(n, 2) for n > 8 has girth 5 and every edge in any vertex-deleted subgraph lies in a 5-cycle or an 8-cycle. Moreover, the above families of graphs are all 2-connected. Since several other generalized Petersen graphs also fulfill the conditions of Proposition 5.12, it would be interesting to characterize them. Such a characterization has to take into account the girth of every single member of the family of generalized Petersen graphs, which, in turn, was calculated in [1]. On the other hand, many interesting families of 2-connected graphs do not satisfy the condition of Proposition 5.12, yet their clone covers still have the required property, for instance, the wheel graphs Wn, n > 6. In contrast, the clone cover of a cycle is different: the fundamental edge partition preserving subgroup has index n - 1 in the automorphism group of Clone(Cn). We believe that this is the only exception. Conjecture 5.14. Let X be a 2-connected graph that is not a cycle. Then any automorphism of Clone(X ) preserves the fundamental edge partition. 6 Concluding remarks Note that Clone(Kn ) has yet another nice description, namely, as the line graph of the first subdivision of Kn. Along these lines, Clone(Kn) was considered in [7], where it was shown that with few exceptions, Clone(Kn) is the only m-sheeted covering graph of Kn, for m < n - 1, such that the full automorphism group of Kn has a lift. Figure 8: Clone(K5) is not a regular cover. It is vertex- but not edge-transitive. On the other hand, Clone(K5 ) was studied in connection with a graph-theoretical interpretation of the Jahn-Teller effect [2]. In order to clarify the role of degenerate eigenvalues (that is, eigenvalues having higher multiplicities) in the Jahn-Teller distortion, graphs with symmetry group S5 were sought [3]. It turned out that both K5 and Clone(K5) have their automorphism group isomorphic to S5. Our paper gives, among other results, a theoretical background for this result. As noted in Proposition 5.13, the conditions of Proposition 5.12 are satisfied by Kn for n > 4. Hence according to Theorem 5.9, Aut(Kn) is isomorphic to Aut(Clone(Kn)). Figure 8 depicts the graph Clone(K5), which was used in [3]. Acknowledgements The authors would like to thank both referees for reading the manuscript carefully and for their detailed comments which helped to improve the presentation of the paper. The authors were supported in part by 'Agencija za raziskovalno dejavnost Republike Slovenije', Grants P1-0285, J1-4021, J1-4010 (AM); P1-0294, L1-4292 (TP); P1-0294 (AŽ); and by European Science Foundation, Eurocores Eurogiga - GReGAS, N1-0011. References [1] M. Boben, T. Pisanski and A. Žitnik, I-graphs and the corresponding configurations, J. Combin. Designs 13 (2005), 406-424. [2] A. Ceulemans, E. Lijnen, P. W. Fowler, R. B. Mallion and T. Pisanski, Graph theory and the Jahn-Teller theorem, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), 971-989. [3] A. Ceulemans, E. Lijnen, P. W. Fowler, R. B. Mallion and T. Pisanski, S5 graphs as model systems for icosahedral Jahn-Teller problems, Theor. Chem. Acc. 131 (2012), article 1246 (10 pages). [4] J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley-Interscience, New York, 1987. [5] W. Imrich and T. Pisanski, Multiple Kronecker coverings of graphs, Europ. J. Combin. 29 (2008), 1116-1122. [6] F. T. Leighton, Finite common coverings of graphs, J. Combin. Theory ser. B 33 (1982), 231238. [7] A. Malnic and D. Marušic, Imprimitive groups and graph coverings, in: D. Jungnickel, S. A. Vanstone (eds), Coding Theory, Design Theory, Group Theory, Proc. Marshall Hall Conf., J. Wiley and Sons, New York, 1993, pp. 227-235. [8] A. Malnic, R. Nedela and M. Skoviera, Lifting graph automorphisms by voltage assignments, Europ. J. Combin. 21 (2000), 927-947. [9] T. Pisanski, A classification of cubic bicirculants, Discrete Math. 307 (2007), 567-578. [10] D. B. West, Introduction to Graph Theory, Second Edition, Pearson Education, 2002. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 115-131 Unstable graphs: A fresh outlook via TF-automorphisms Josef Lauri University of Malta, Department of Mathematics, Malta Russell Mizzi University of Malta, Department of Mathematics, Malta Raffaele Scapellato * Dipartimento di Matematica, Politecnico di Milano, Italy Received 4 September 2013, accepted 30 June 2014, published online 21 September 2014 In this paper, we first establish the very close link between stability of graphs, a concept first introduced by Marušic, Scapellato and Zagaglia Salvi and studied most notably by Surowski and Wilson, and two-fold automorphisms. The concept of two-fold isomorphisms, as far as we know, first appeared in Zelinka's work on isotopies of digraphs and later studied formally by the authors with a greater emphasis on undirected graphs. We then turn our attention to the stability of graphs which have every edge on a triangle, but with the fresh outlook provided by TF-automorphisms. Amongst such graphs are strongly regular graphs with certain parameters. The advantages of this fresh outlook are highlighted when we ultimately present a method of constructing and generating unstable graphs with large diameter having every edge lying on a triangle. This was a rather surprising outcome. Keywords: Graphs, canonical double covers, two-fold isomorphisms. Math. Subj. Class.: 05C25, 05C20, 05C76 * Corresponding author. E-mail addresses: josef.lauri@um.edu.mt (Josef Lauri), russell.mizzi@um.edu.mt (Russell Mizzi), raffaele.scapellato@polimi.it (Raffaele Scapellato) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 1 General Introduction and Notation Let G and H be simple graphs, that is, undirected and without loops or multiple edges. Consider the edge {u, v} to be the set of arcs {(u, v), (v, u)}. A two-fold isomorphism or TF-isomorphism from G to H is apair of bijections a, ß: V (G) ^ V (H ) such that (u, v) is an arc of G if and only if ((a(u), ß(v)) is an arc of H. When such a pair of bijections exist, we say that G and H are TF-isomorphic and the TF-isomorphism is denoted by (a, ß). The inverse of (a, ß), that is, (a-1, ß-1) is a TF-isomorphism from H to G. Furthermore, if (a1, ß1) and (a2, ß2) are both TF-isomorphisms from G to H then so is (a1a2, ß1ß2). When a = ß, the TF-isomorphism can be identified with the isomorphism a. 1 1 7 4 H G 2 6 2 6 7 5 3 3 5 4 Figure 1: G and H are two non-isomorphic TF-isomorphic graphs. The two graphs G, H in Figure 1, which have the same vertex set V (G) = V (H), are non-isomorphic and yet they are TF-isomorphic. In fact (a, ß) where a =(2 5) and ß = (1 4)(3 6) is a TF-isomorphism from G to H. The concept was first studied by Zelinka [15, 16] in the the context of isotopy of digraphs. We shall extend it further to the case of mixed graphs (see next section). Some graph properties are preserved by a TF-isomorphism. Such is the case with the degree sequence, as illustrated by Figure 1. We also know that two graphs are TF-isomorphic if and only if that they have isomorphic canonical double covers [4]. Alternating paths or A-trails, which we shall define in full below, are invariant under TF-isomorphism. For instance, the alternating path 5 —> 6 <— 1 —> 2 in G is mapped by (a, ß) to the similarly alternating path 2 —> 3 <— 1 —> 2 which we shall later be calling "semi-closed". 2 Notation A mixed graph is a pair G = (V(G), A(G)) where V(G) is a set and A(G) is a set of ordered pairs of elements of V(G). The elements of V(G) are called vertices and the elements of A(G) are called arcs. When referring to an arc (u, v), we say that u is adjacent to v and v is adjacent from u. Sometimes we use u —> v to represent an arc (u, v) G V(G). The vertex u is the start-vertex and v is the end-vertex of a given arc (u, v). An arc of the form (u, u) is called a loop. A mixed graph cannot contain multiple arcs, that is, it cannot contain the arc (u, v) more than once. A mixed graph G is called bipartite if there is a partition of V(G) into two sets X and Y, which we call colour classes, such that for each arc (u, v) of G the set {u, v} intersects both X and Y. A set S of arcs is self-paired if, whenever (u, v) G S, (v, u) is also in S. If S = {(u, v), (v, u)}, then we consider S to be the unordered pair {u, v}; this unordered pair is called an edge. It is useful to consider two special cases of mixed graphs. A graph is a mixed graph without loops whose arc-set is self-paired. The edge set of a graph is denoted by E(G). A digraph is a mixed graph with no loops in which no set of arcs is self-paired. The inverse G' of a mixed graph G is obtained from G by reversing all its arcs, that is V (G') =V (G) and (v, u) is an arc of G' if and only if (u, v) is an arc of G. A digraph G may therefore be characterised as a mixed graph for which A(G) and A(G') are disjoint. Given a mixed graph G and a vertex v g V (G), we define the in-neighbourhood Nin(v) by Nin(v) = {x G V(G)|(x,v) G A(G)}. Similarly we define the out-neighbourhood N0„t(v) by Nout(v) = {x G V(G)|(v, x) G A(G)}. The in-degree Pi„(v) of a vertex v is defined by pin(v) = |Nin(v)| and the out-degree pout(v) of a vertex v is defined by Pout (v) = |Nout(v) |. When G is a graph, these notions reduce to the usual neighbourhood N (v) = Ni „(v) = Nout(v) and degree p(v) = Pi„(v) = Pout(v). Let G be a graph and let v g V (G). Let N (v) be the neighbourhood of v. We say that G is vertex-determining if N(x) = N(y) for any two distinct vertices x and y of G [8]. A set P of arcs of G is called a trail of length k if its elements can be ordered in a sequence ab a2,..., ak such that each ai has a common vertex with ai+i for all i = 1, ..., k — 1. If u is the vertex of a1, that is not in a2 and v is the vertex of ak which is not in ak-1, then we say that P joins u and v; u is called the first vertex of P and v is called the last vertex with respect to the sequence a1, a2, ..., ak. If, whenever ai = (x, y), either ai+1 = (x, z) or ai+1 = (z, y) for some new vertex z, P is called an alternating trail or A-trail. If the first vertex u and the start-vertex v of an A-trail P are different, then P is said to be open. If they are equal then we have to distinguish between two cases. When the number of arcs is even then P is called closed while when the number of arcs is odd then P is called semi-closed. Note that if P is semi-closed then either (i) a1 = (u, x) for some vertex x and ak = (y, u) for some vertex y or (ii) a1 = (x, u) and ak = (u, y). If P is closed then either a1 = (u, x) and ak = (u, y) or a1 = (x, u) and ak = (y, u). Observe also that the choice of the first (equal to the last) vertex for a closed A-trail is not unique but depends on the ordering of the arcs. However, this choice is unique for semi-closed A-trails as this simple argument shows: Suppose P is semi-closed and the arcs of P are ordered such that u is the unique (in that ordering) first and last vertex, that is, it is the unique vertex such as the first and the last arcs in the ordering in P do not alternate in direction at the meeting point u. Therefore, it is easy to see that both pin(u) and pout(u) (degrees taken in P as a subgraph induced by its arcs) are odd whereas any other vertex v in the trail has both pin(v) and pout(v) even. This is because, in the given ordering, arcs have to alternate in direction at v and therefore in-arcs of the form (x, v) are paired with out-arcs of the form (v, y). Therefore, in no ordering of the arcs of P can u be anything but the only vertex at which the first and last arcs do not alternate. The same argument holds for open A-trails. Any other graph theoretical terms which we use are standard and can be found in any graph theory textbook such as [1]. For information on automorphism groups, the reader is referred to [7]. Let G and H be two mixed graphs and suppose that a, ß are bijections from V(G) to V(H). The concept of TF-isomorphisms may be extended from graphs to mixed graphs. In fact, we can say that the pair (a, ß) is a two-fold isomorphism (or TF-isomorphism) from G to H if the following holds: (u, v) is an arc of G if and only if (a(u),ß(v)) is an arc of H. We then say that G and H are TF-isomorphic and write G =TF H. Note that when a = ß the pair (a, ß) is a TF-isomorphism if and only if a itself is an isomorphism. If a = ß, then the given TF-isomorphism (a, ß) is essentially different from a usual isomorphism and hence we call (a, ß) a non-trivial TF-isomorphism. If (a, ß) is a non-trivial TF-isomorphism from a mixed graph G to a mixed graph H, the bijections a and ß need not necessarily be isomorphisms from G to H. This is illustrated by the graphs in Figure 1, examples found in [5], and also others presented below. When G = H, (a, ß) is said to be a TF-automorphism and it is again called nontrivial if a = ß. The set of all TF-automorphisms of G with multiplication defined by (a,ß)(j,ö) = (aj,ßö) is a subgroup of SV(G) x SV(G) and it is called the two-fold automorphism group of G and is denoted by AutTF(G). Note that if we identify an automorphism a with the TF-automorphism (a, a), then Aut(G) C AutTF(G). When a graph has no non-trivial TF-automorphisms, Aut(G) =AutTF(G). It is possible for an asymmetric graph G, that is a graph with |Aut(G)| = 1, to have non-trivial TF-automorphisms. This was one of our main results in [5]. The main theme of this paper is stability of graphs, an idea introduced by Marušic et al. [8] and studied extensively by others, most notably by Wilson [14] andSurowski [12], [13]. Let G be a graph and let CDC(G) be its canonical double cover or duplex. This means that V(CDC(G)) = V(G) x Z2 and {(u, 0), (v, 1)} and {(u, 1), (v, 0)} are edges of CDC(G) if and only if {u, v} is an edge of G. One may think of the second entry in the notation used for vertices of CDC(G), that is 0 or 1 as colours. For the reader familiar with products of graphs we wish to add that CDC(G) is the direct product G x K2 of G by K2. Recall that the graph CDC(G) is bipartite and we may denote its colour classes by V0 = V x {0} and Vi = V x {1} containing vertices of the type (u, 0) and (u, 1) respectively. A graph is said to be unstable if Aut(G) x Z2 is aproper subgroup of the set Aut(CDC(G)). The elements of Aut(CDC(G)) \ Aut(G) x Z2 will be called unexpected automorphisms of CDC(G). In other words, a graph G is unstable if at least one element of Aut(CDC(G)) is not a lifting of some element of Aut(G). In this paper, we shall investigate the relationship between the stability of the graph G and its two-fold automorphism group AutTF(G). 3 Unstable Graphs and TF-automorphisms Consider Aut(CDC(G)). Now, let S be the set-wise stabiliser of Vo in Aut(CDC(G)), which of course coincides with the set-wise stabiliser of V1. Note that every a e S also fixes Vi set-wise. We will show that it is the structure of S which essentially determines whether CDC(G) has unexpected automorphisms which cannot be lifted from automorphisms of G. The following result, which is based on [9], Lemma 2.1, implies that these unexpected automorphisms of CDC(G) arise if the action of a on V0 is not mirrored by its action of Vi . Lemma 3.1. Let f : S ^ Sym(V) x Sym(V) be defined by f : a ^ (a,ß) where (a(v), 0) = a(v, 0) and (ß(v), 1) = a(v, 1), that is a, ß extract from a its action on V0 and V1 respectively. Then: 1. f is a group homomorphism; 2. f is injective and therefore f : S ^ f (S) is a group automorphism; 3. f (S) = {(a, ß) e Sym(V ) x Sym(V ) : x is adjacent to y in G if and only if a(x) is adjacent to ß (y) in G} that is, f (S) = AutTF(G) that is, (a, ß) (the ordered pair of separate actions of a on the two classes) is a TF-automorphism of G. Proof. The fact that f is a group homomorphism, that is, that f is a structure preserving map from S to Sym(V) x Sym(V) follows immediately from the definition since for any a1, a2 e S where f (ai) = (ab ßi) and f (a2) = (a2, ß2), f (ai)f (a 2) = (aißi)(a2ß2) = (aia2, ßiß2) = f (aia2). This map is clearly injective and therefore f : S ^ f (S) is a group automorphism. Consider an arc ((u, 0), (v, 1)). Since a e S C Aut(CDC(G)), (a(u), 0), (a(v), 1)) is also an arc of CDC(G). By definition, this arc may be denoted by ((a(u), 0), (ß(v), 1)) and, following the definition of CDC(G), it exists if and only if (a(u),ß(v)) is an arc of G. Hence f maps elements of S to (a, ß) which clearly take arcs of G to arcs of G. This implies that (a, ß) is a TF-automorphism of G and hence f (S) CAutTF(G). Conversely, let (a,ß) e AutTF(G). Define a by a(v, 0) = (a(v), 0) and a(v, 1) = (ß(v), 1)), then f (a) = (a,ß). Hence, AutTF(G) C f (S). Therefore f (S)= AutTF(g). □ Theorem 3.2. Let G be a graph. Then Aut(CDC(G)) = AutTF(G) x Z^. Furthermore, G is unstable if and only if it has a non-trivial TF-automorphism. Proof. From Lemma 3.1, f (S) = AutTF(G) which must have index 2 in Aut (CDC(G)). The permutation S(v,e) ^ (v,e + 1) is an automorphism of CDC(G) and S e f (S). Then Aut(CDC(G)) is generated by f (S) and S. Furthermore, f (S) П (S) = id and f (S) < Aut (CDc(g)) being of index 2. Since Aut(CDC(G)) = AutTF(G) x Z2, G is stable if and only if AutTF(G) = Aut(G). □ As shown in [5], Proposition 3.1, if (a, ß) G AutTF(G) then (7,7-1) G AutTF(G) (where 7 = aß-1). This means that for any edge {x, y} of G, {y(x), 7-1(y)} is also an edge. A permutation 7 of V(G) with this property is called an anti-automorphism. Such maps possess intriguing applications to the study of cancellation of graphs in direct products with arbitrary bipartite graphs, that is, the characterisation of those graphs G for which G X C ~ H X C implies G ~ H, whenever C is a bipartite graph (see [2], Chapter 9). The second part of Theorem 3.2 could be rephrased as follows: "G is unstable if and only if it has an anti-automorphism of order different from 2". Note that the existence of an anti-automorphisms of order 2 does not imply instability since such a map corresponds to a trivial TF-automorphisms. Notice that AutTF(G) is the edge set of the graph factorial G! introduced in [2] in the wake of investigations of anti-automorphisms. G! shares many properties with factorials of natural numbers and is interesting per se. As CDC(G) = G x K2 one could thus rephrase Theorem 3.2 in the language of [2] as Aut(G x K2) = E(G!) x Z2. Figure 2: A stable and unstable graph which are TF-isomorphic. It is natural to ask whether it can happen that a stable graph is TF-isomorphic to an unstable one. The answer is yes and an example is shown in Figure 2 with the Petersen graph being stable and the other graph which is TF-isomorphic to it being unstable. Both graphs have the same bipartite canonical double cover since they are TF-isomorphic. If loops are allowed, then the smallest pair of TF-isomorphic graphs, only one of which is stable is given by G = K3 and H the path of three vertices with a loop at each of the end-vertices. 2 1 3 4 5 Porcu, motivated by the quest of finding pairs of co-spectral graphs, was the first to study non-isomorphic graphs having the same canonical double cover [11], but his work was overlooked by several mathematicians interested in these questions. Porcu was the first author to study the example given in Figure 1. The same example was re-discovered much later in [3]. Pacco and Scapellato [10], answering a question raised by Porcu, proved a result giving the number of graphs having a given bipartite graph B as canonical double covering terms of involutions in Aut(B). A later extension of their result appeared in [2], Theorem 9.15. We should point out here (as noted by Surowski in [12]) that if a graph G is stable, that is, Aut(CDC(G)) = Aut(G) x Z2, then the semi-direct product must be a direct product because Aut(G) is normal in Aut(CDC(G)) since it has index 2 and also Z2 is normal since its generator commutes with every element of Aut(CDC(G)), by stability. In fact, as Surowski comments, the stability of G is equivalent to the centrality of Z2 in Aut(CDC(G)), which is the lift of the identity in Aut(G). (a,0)* s(a,1) (b,0)/ \(b,1) (c,0)) ^(c,1) (d,0) S \(d,1) CDC(G) Figure 3: An example used to show how a TF-automorphism (a, ß) of G can be obtained from an automorphism a of CDC(G). It is worth noting that the ideas explored in the proof of Lemma 3.1 may be used to extract TF-automorphisms of a graph G from automorphisms of CDC(G) which fix the colour classes. In fact, let a be such an automorphism. Define the permutations a and ß of V(G) as follows: a(x) = y if and only if a(x, 0) = (y, 0) and ß(x) = y if and only if a(x, 1) = (y, 1). Then (a, ß) is a TF-automorphism of G. We remark that a and ß are not necessarily automorphisms of G as we shall show in the example shown in Figure 3. The automorphism a is chosen so that it fixes one component of CDC(G) whilst being an automorphism of the other component. In order to have a more concise representation, we denote vertices of CDC(G) of the form (u, 0), that is, elements of the colour class V0 by u0 and similarly denote vertices of the form (u, 1) in V1 by ui. Using this notation, a = (a0)(b1)(c0)(d1)(e)(a1 e1)(b0 d0)(c1). The permutations a and ß of G are extracted from a as described in the proof of Lemma 3.1. For instance, to obtain a, we restrict the action of a to the elements of V0, that is, those vertices of the form (v, 0) or v0 when using the new notation and then drop the subscript. Similarly, the permutation ß is obtained from the action of a restricted to V1. Therefore, a = (a)(c)(e)(b d) and ß = (b)(d)(a e)(c). Note that neither a nor ß is an automorphism of G, but (a, ß) is a TF-automorphism of G which in turn can be lifted to the unexpected automorphism a of CDC(G). This example illustrates Lemma 3.1 since the graph G is unstable and has a non-trivial TF-automorphism. The result of Lemma 3.1 and the subsequent example lead us to other questions regarding the nature of the permutations a and ß which, as discussed in the preceding example given in Figure 3, may not be automorphisms of G. If (a, id) is a non-trivial TF-automorphism of a graph G, then G is not vertex-determining. In fact, since a = id then a(u) = v for some u = v and the TF-automorphism (a, id) fixes the neighbours of u and takes u to v. Hence u and v must have the same neighbourhood set, which implies that G is not vertex-determining. We shall use this idea to prove some results below. An alternative way of looking at this is to consider Lemma 3.1 and to note that a graph G is stable if and only if given a(v, 0) = (a(v), 0), there exists no ß = a such that (ß(v), 1)) = a(v, 0). Hence, as implied by Theorem 3.2 a graph G is stable if and only if f (£) С where is the diagonal group of (a, ß), a, ß automorphisms of G, with a = ß. Proposition 3.3. If (a, ß ) is a non-trivial TF-automorphism of a graph G but a and ß are automorphisms of CG, then G is not vertex-determining. Proof. Since a is an automorphism of G and (a, ß) is a TF-automorphism, the group AutTF(G) must also contain (a, ß)(a-1,a-1) = (id, ßa-1 ). Since aß-1 = id, let u be a vertex such that v = aß-1 (u) is different from u. Then for each neighbour w of u the arc (w, u) is taken to the arc (w, v), so that N(u) is contained in N(v) and vice versa. Therefore u and v have the same neighbourhood, so G is not vertex-determining. □ Proposition 3.4. If (a, ß) is a non-trivial TF-automorphism of a graph G and a, ß have a different order, then G is not vertex-determining. Proof. Let (a, ß) be an element of AutTF(G) with the orders of a and ß being p and q respectively and assume without loss of generality that p < q. Since AutTF(G) is a group, (a, ß)p = (ap, ßp) = (id, ßp) must also be in AutTF(G). The same argument used in the proof of Proposition 3.3 holds since ßp = id. Hence G is not vertex-determining. □ Proposition 3.3 and Proposition 3.4 are equivalent to their counterparts in [8] in which they are stated in terms of adjacency matrices. In [8], it is shown that if a graph G is unstable but vertex-determining and (a, ß) is a non-trivial TF-automorphism of G, then a and ß must not be automorphisms of G and must have the same order. This then gives us information about automorphisms a of CDC(G) which are liftings of TF-automorphisms of G. 4 Triangles In this section we shall study the behaviour of a non-trivial TF-automorphism of a graph G acting on a subgraph of G isomorphic to K3 with the intent of obtaining information regarding the stability of graphs which have triangles as a basic characteristic of their structure. Strongly regular graphs in which every pair of adjacent vertices has a common neighbour are an example. The stability of such graphs has been studied by Surowski [12]. We believe that this section is interesting because it is a source of simple examples of unstable graphs and also because a detailed analysis of what happens to triangles can throw more light on TF-automorphisms of graphs. Proposition 4.1. Let (a, ß) be a TF-isomorphism from a graph G to a graph G. The action of (a,ß) on some subgraph H = K3 yields either (a) a closed A-trail of length 6 with no repeated vertices, (b) a pair of oriented A-connected triangles with exactly one common vertex, (c) a pair of oriented triangles with exactly two common vertices or (d) an undirected triangle as illustrated in Figure 4(a),(b),(c) and (d) respectively. Proof. Let V(H) = {1, 2, 3}. The semi-closed A-trials 1 —> 2 <— 3 —> 1 and 1 <— 2 —> 3 <— 1 of H are taken by (a, ß) to a(1) —> ß(2) <— a(3) —> ß(1) and ß(1) <— a(2) —> ß(3) <— a(1), respectively; together, these two A-trails form a closed A-trail consisting of six arcs. Consider the possible equalities a(v) = ß(v) for v e V (H ). Up to a reordering of the vertices, we have the following four cases, corresponding to the four cases stated above: (a) a(v) = ß(v) for all v. (b) a(v) = ß(v) for v = 1, 3 but a(2) = ß(2). (c) a(v) = ß(v) for v = 1 but a(v) = ß(v) for v = 2, 3. (d) a(v) = ß(v) for all v. For each case, the corresponding graph is shown in Figure 4. □ In general, when a TF-automorphism (a, ß) acts on the arcs of G it maps any triangle H into another triangle if a(x) = ß(x) for every vertex x of the triangle or it fits one of the other configurations described by Proposition 4.1. If a graph G in which every edge lies in a triangle is unstable, then it must have a non-trivial TF-automorphism which follows one of the configurations illustrated in Figure 4(a), (b) and (c). Proposition 4.2. Let (a, ß) be a TF-isomorphism from G to G'. When the TF-isomorphism acting on K3, a subgraph of G, yields a closed A-trail of length 6 with no repeated vertices as shown in Figure 4(a), either two triangles with no common vertex or a triangle and a closed A-trail with 6 arcs are mapped to a subgraph isomorphic to C6. In the cases when the TF-isomorphism acting on a K3 yields the images illustrated in Figure 4 (b), (c), the pair of triangles which are either mapped to two triangles with exactly one common vertex or to two triangles with exactly one common edge must have a common vertex. Proof. Refer to Figure 4. In the case illustrated in Figure 4(a) the arcs of one closed A-trail P of length 6 can be the co-domain the arcs of a triangle H. The A-trail P' obtained by reversing the arcs of P can be the co-domain of another triangle K. We claim that H and K are vertex disjoint. In fact, suppose not and assume that the two triangles have a common vertex u. The pair of vertices a(u) and ß (u) where a (u) = ß(u) are in both P and P ' and this is contradiction as the in-degree of a(u) and similarly the out-degree of ß(u) must be zero and this makes it impossible to identify arcs of P with arcs of P' to form the edges of a C6. Figure 5 shows an example where setting a(3) = ß(5), ß(2) = a(6), a(1) = ß(4), ß(3) = a(5), a(2) = ß(6), ß(1) = a(4) would be one way of associating one directed C6 with the other so that the alternating connected circuits form an undirected C6. The other possibility is illustrated in Figure 6. In this example ß(1) = a(4), a(2) = ß(5), ß(3) = a(6), a(1) = ß(7), ß(2) = a(8) and a(3) = ß(9) so that a closed A-trail of length 6 covering a K3 is mapped to a closed A-trail of length 6 covering half of the arcs of an undirected C6 whilst the rest of the arcs come from an A-trail of length 6 covering half of the arcs of another subgraph isomorphic to C6. The K3 and the C6 in the domain of the TF-isomorphism cannot have a common vertex and the proof is analogous to the one concerning the former case. The proof for the remaining cases may be carried over along the same lines as the above. Refer to Figure 7. We observe that the A-trail P1 described by ß(3) <— a(2) —> ß(1), the A-trail P2 described by a(1) —> ß(2) <— a(3) and the A-trails P{ and P2' obtained by reversing the arcs of P1 and P2 respectively would imply by the preservation of A-trails, that in the pre-image of the subgraph, there are four A-trails passing through the vertex 2. This is only possible if the triangles in the pre-image have a common vertex. □ Figure 8 shows an example to illustrate the result of Proposition 4.2 where a(1) = ß (4), ß(3) = a(5), a(3) = ß(5) and ß(1) = a(4). Figure 8 shows the smallest unstable graphs which have a TF-automorphism which takes a triangle to a pair of directed triangles with a common vertex as illustrated in Figure 4(b). Figure 9 shows the smallest graph which has a nontrivial TF-automorphism which maps a triangle to the mixed graph illustrated in Figure 4(c). 5 Unstable graphs of arbitrarily large diameter In this section, we present a method of constructing unstable graphs of an arbitrarily high diameter. If H, K are graphs, let [H, K] be the graph whose vertex set is the union V (H )U V (K ) and whose edge set is the union of E(H), E(K) plus the edges of the complete bipartite graph with classes V(H) and V(K). More generally, if Hj are graphs, where i G Zm, let G = [Ho, H1, ..., Hm-1] be the graph whose vertex set is the union of all V(Hj) and whose edge set is the union of all E([Hj, H(j+1)]). In other words, G contains all vertices and edges of the graphs Hj, plus all edges of the complete bipartite graph connecting two consecutive Hj's. Now, assume that none of the Hj has isolated vertices. Let (aj, ßj) : Hj ^ Hj+1 be TF-isomorphisms as i runs over Zm. Assume that the product (ao,ßo)(abß1)...(am-bßm-1) = (id, id). Note that the latter assumption is not a restriction, because one can always take (a0, ß0) as the inverse of the product of the remaining TF-isomorphisms. Theorem 5.1. With the above assumptions, let G = [Ho, H1, ..., Hm-1]. Define two permutations a, ß of V (G) as follows. For v G V (G), let i be such that v G V (Hi); then set a(v) = ai(v) and ß(v) = ßi(v). Then the following hold: 1. (a, ß) is a TF-automorphism of G; 2. if m > 4, diam G = (m — e)/2, where e = 0 if m is even and e = 1 if m is odd; 3. each edge of G belongs to a triangle. Proof. First note that if (u, v) is an arc of G and both u, v belong to the same Hi, then the image of (u, v) is an arc of Hi+1, hence of G, because (a, ß) acts like (ai, ßi) in Hi. If u, v do not belong to the same Hi, then they belong to consecutive graphs, say Hi, Hi+1, so a(u) and ß(u) belong to the consecutive graphs Hi+1, Hi+2, and are adjacent because all the arcs between these two graphs belong to G. This proves (1). Concerning distance, a path from u in H0 to v in Hs, where s < (m — e)/2, must pass through all graphs H1, H2, ..., Hs-1 or else Hm-1, Hm-2, ..., Hs+1. Since such a path can be found, d(u, v) = s. For two vertices in, say, Hi and Hj with i = j, the same argument shows that d(u, v) cannot exceed (m — e)/2. Finally, if u, v lie in the same Hi, they have a common neighbour in Hi+1, then d(u, v) is less or equal than 2 (regardless to their distance within Hi). This proves (2). What about triangles? If {u, v} is an edge of some Hi then letting w g Hi+1 the vertex w is adjacent to both u and v, hence {u, v, w} is a triangle. If {u, v} is an edge of some [Hi, Hi+1], say with u G Hi and v G Hi+1, take any neighbour w of u in Hi (recall the assumption about no isolated vertices) and get the triangle {u, v, w}. This proves (3). □ Until now, we did not mention that the concerned TF-isomorphisms are non-trivial, so all the above would work fine for the case of isomorphisms too. But adding the hypothesis that at least one of them is non-trivial, the obtained graph G has a non-trivial TF-isomorphism, namely (a, ß) as described above. Theorem 5.1 shows that there are unstable graphs of arbitrarily high diameter, where each edge belongs to a triangle. Surowski [12, 13] proved various results concerning graph stability. In [12], Proposition 2.1, he claims that if G is a connected graph of diameter d > 4 in which every edge lies in a triangle, then G is stable. However, by taking m > 7 in Theorem 5.1 we get infinitely many counterexamples to this claim by taking all the Hi isomorphic to the same vertex-determining bipartite graph, because such a vertex-transitive graph is unstable, therefore one can find a non-trivial TF-isomorphism from Hi to Hi+1, and since these Hi are isomorphic, the resulting graph G is vertex-determining, has diameter k > 4, and is unstable. One of these counterexamples is illustrated in Figure 10. We detected one possible flaw in Surowski's proof. It is claimed in [12] that whenever an automorphism of CDC(G) fixes (v, 1) it also fixes (v, —1). We have not seen a proof of this result. Besides, in our last example, G has a non-trivial fixed-point-free TF-automorphism, which implies that CDC(G) has a fixed-point-free automorphism that fixes the colour classes. This claim is also used in [12] Proposition 2.2 which states that if G is a strongly regular graph with k > ^ = Л > 1, then G is stable. Hence, we believe that at this point, the stability of strongly regular graphs with these parameters requires further investigation. 6 Concluding Remarks The use of TF-isomorphisms in the study of stability of graphs provides a fresh outlook which allows us to view facts within a more concrete framework and also provides tools to obtain new results. For instance, we can investigate the structure of the given graph without actually requiring to lift the graph to its canonical double cover, but only having to reason within the original graph. Furthermore, the insights that we already have about TF-isomorphisms of graphs may be considered to be new tools added to a limited toolkit. In particular, let us mention the idea of graph invariants under the action of TF-isomorphisms, such as A-trails, a topic which we have started to study in [6]. To be able to find out how the subgraphs of a graph are related to other subgraphs within the graph itself in the case of unstable graphs fills a gap in our understanding of graph stability and using TF-isomorphisms appears to be a promising approach in this sense. We believe that this paper substantiates these claims. Furthermore, it motivates us to carry out further investigations. Some pending questions such as those concerning the stability of certain strongly regular graphs have already been indicated. The study of how TF-isomorphisms act on common subgraphs such as triangles is another useful lead. Nevertheless, the more ambitious aim would be the classification of unstable graphs in terms of the types of TF-automorphisms which they admit. Acknowledgement The authors would like to thank two anonymous referees for their careful reading of the submitted paper and their detailed comments which enabled us to improve significantly the final version of the paper. Figure 4: The configurations of possible images of a triangle under the action of a TF-automorphism as described in Proposition 4.1. / \ Figure 7: An example to illustrate the result of Proposition 4.2. a(3) = ß(5) a(4) = ß(1) a(3) = ß(5) a(4) = ß(1) Figure 8: The smallest unstable graphs where a triangle is taken to a two directed triangles sharing a vertex. 1 a(4) = ß(1) 3 a(2) = ß(2) a(3) = ß(3) a(1) = ß(4) 3 1 5 4 2 Figure 9: The smallest unstable graph which has a TF-automorphism taking a triangle to the mixed graph as illustrated in Figure 4(c). Figure 10: A graph constructed using Theorem 5.1. References [1] A. Bondy and U. Murty, Graph Theory (Graduate Texts in Mathematics), Springer, 2008. [2] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, Second Edition, Discrete Mathematics and Its Applications, Taylor & Francis, 2011. [3] W. Imrich and T. Pisanski, Multiple Kronecker covering graphs, Eur. J. Comb. 29 (2008), 1116— 1122, doi:10.1016/j.ejc.2007.07.001. [4] J. Lauri, R. Mižži and R. Scapellato, Two-fold orbital digraphs and other constructions., International J. of Pure and Applied Math. 1 (2004), 63-93. [5] J. Lauri, R. Mižži and R. Scapellato, Two-fold automorphisms of graphs., Australasian J. Combinatorics. 49 (2011), 165-176. [6] J. Lauri, R. Mižži and R. Scapellato, A generalisation of isomorphisms with applications., Preprint (2014). [7] J. Lauri and R. Scapellato, Topics in Graph Automorphisms and Reconstruction, Cambridge University Press, 2003. [8] D. Marušic, R. Scapellato and N. Zagaglia Salvi, A characterižation of particular symmetric (0,1)-matrices, Linear Algebra Appl. 119 (1989), 153-162, doi:10.1016/0024-3795(89) 90075-X. [9] D. Marušic, R. Scapellato and N. Zagaglia Salvi, Generaližed Cayley graphs, Discrete Mathematics 102 (1992), 279 - 285, doi:http://dx.doi.org/10.1016/0012-365X(92)90121-U. [10] W. Pacco and R. Scapellato, Digraphs having the same canonical double covering, Discrete Math. (1997), 291-296. [11] L. Porcu, Sul raddoppio di un grafo, Att. Ist. Lombardo (Rend. Sci.) A (1976), 353-360. [12] D. B. Surowski, Stability of arc-transitive graphs, J. Graph Theory 38 (2001), 95-110, doi: 10.1002/jgt.1026. [13] D. B. Surowski, Automorphism groups of certain unstable graphs, Math. Slovaca 53 (2003), 215-232. [14] S. Wilson, Unexpected symmetries in unstable graphs, J. Combin. Theory Ser. B 98 (2008), 359-383, doi:10.1016/j.jctb.2007.08.001. [15] B. Zelinka, The group of autotopies of a digraph, Czechoslovak Mathematical Journal 21 (1971), 619-624. [16] B. Zelinka, Isotopy of digraphs, Czechoslovak Mathematical Journal 22 (1972), 353-360. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 133-148 A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two Dedicated to Dragan Marusic on the occasion of his 60th birthday Primož PotoCnik * Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia affiliated also with: IAM, University of Primorska, Muzejski trg 2, SI-6000 Koper, Slovenia and with: IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia Pablo Spiga Dipartimento di Matematica Pura e Applicata, University of Milano-Bicocca, Via Cozzi 53, 20126 Milano, Italy Gabriel Verretf Centre for Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia and FAMNIT, University of Primorska, Glagoljaška 8, SI-6000 Koper, Slovenia Received 18 October 2013, accepted 6 June 2014, published online 22 September 2014 Abstract A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a 1 -arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed. Keywords: Graph, digraph, edge-transitive, vertex-transitive, arc-transitive, half-arc-transitive. Math. Subj. Class.: 05E18, 20B25 * Supported by Slovenian Research Agency, projects L1-4292 and P1-0222. t Supported by UWA as part of the Australian Research Council grant DE130101001. E-mail addresses: primož.potocnik@fmf.uni-lj.si (Primož Potocnik), pablo.spiga@unimib.it (Pablo Spiga), gabriel.verret@uwa.edu.au (Gabriel Verret) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 1 Introduction Recall that a graph Г is called 1 -arc-transitive provided that its automorphism group Aut(r) acts transitively on its edge-set Е(Г) and on its vertex-set V(T) but intransitively on its arc-set А(Г). More generally, if G is a subgroup of Aut(r) such that G acts transitively on Е(Г) and V(r) but intransitively on А(Г), then G is said to act 1 -arc-transitively on Г and we say that Г is (G, 1 )-arc-transitive. To shorten notation, we shall say that a 2-arc-transitive graph is a HAT and that a graph admitting a 1 -arc-transitive group of automorphisms is a GHAT. Clearly, any HAT is also a GHAT. Conversely, a GHAT is either a HAT or arc-transitive. The history of GHATs goes back to Tutte who, in his 1966 paper [38, 7.35, p.59], proved that every GHAT is of even valence and asked whether HATs exist at all. The first examples of HATs were discovered a few years later by Bouwer [6]. After a short break, interest in GHATs picked up again in the 90s, largely due to a series of influential papers of Marušic concerning the GHATs of valence 4 (see [1, 17, 21, 24], to list a few). For a nice survey of the topic, we refer the reader to [16], and for an overview of some more recent results, see [13, 22]. To shorten notation further, we shall say that a connected GHAT (HAT, respectively) of valence 4 is a 4-GHAT (4-HAT, respectively). The main result of this paper is a compilation of a complete list of all 4-GHATs with at most 1000 vertices. This improves an unpublished work [23], where all 4-HATs of order up to 869 vertices (with the exception of order 512) with the vertex-stabiliser of order 2 were determined. Our result was obtained indirectly using an intimate relation between 4-GHATs and connected arc-transitive asymmetric digraphs of in- and out-valence 2 (we shall call such digraphs 2-ATDs for short) - see Section 2.2 for details on this relationship. These results can be succinctly summarised as follows: Theorem 1.1. There are precisely 26457 pairwise non-isomorphic 2-ATDs on at most 1000 vertices, and precisely 11941 4-GHATs on at most 1000 vertices, of which 8695 are arc-transitive and 3246 are 2 -arc-transitive. The actual lists of (di)graphs, together with a spreadsheet (in a "comma separated values" format) with some graph theoretical invariants, is available at [28]. The rest of this section is devoted to some interesting facts gleaned from these lists. All the relevant definitions that are omitted here can be found in Section 2. In Section 3, we explain how the lists were computed and present the theoretical background which assures that the computations were exhaustive. In Section 4, information about the format of the files available on [28] is given. We now proceed with a few comments on the census of 4-HATs. By a vertex-stabiliser of a vertex-transitive graph or digraph Г, we mean the stabiliser of a vertex in Aut(r). Even though it is known that a vertex-stabiliser of a 4-HAT can be arbitrarily large (see [18]), not many examples of 4-HATs with vertex-stabilisers of order larger than 2 were known, and all known examples had a very large number of vertices. Recently, Conder and Sparl (see also [8]) discovered a 4-HAT on 256 vertices with vertex-stabiliser of order 4 and proved that this is the smallest such example. This fact is confirmed by our census; in fact, the following theorem can be deduced from the census. Theorem 1.2. Amongst the 3246 4-HATs on at most 1000 vertices, there are seventeen with vertex-stabiliser of order 4, three with vertex-stabiliser of order 8, and none with larger vertex-stabilisers. The smallest 4-HAT with vertex-stabiliser of order 4 has order 256 and the smallest two with vertex-stabilisers of order 8 have 768 vertices; the third 4-HAT with vertex-stabiliser of order 8 has 896 vertices. Another curiosity about 4-HATs is that those with a non-abelian vertex-stabiliser tend to be very rare (at least amongst the "small" graphs). The first known 4-HAT with a non-abelian vertex-stabiliser was discovered by Conder and Marusic (see [7]) and has 10752 vertices. Further examples of 4-HATs with non-abelian vertex-stabilisers were discovered recently (see [8]), including one with a vertex-stabiliser of order 16. However, the one on 10752 vertices remains the smallest known example. Using our list, the following fact is easily checked. Theorem 1.3. Every 4-HAT with a non-abelian vertex-stabiliser has more than 1000 vertices. In fact, there are strong indications that the graph on 10752 vertices discovered by Conder and Marusic is the smallest 4-HAT with a non-abelian vertex-stabiliser. We will call a 4-HAT with a non-solvable automorphism group a non-solvable 4-HAT. The first known non-solvable 4-HAT was constructed by Marusic and Xu [24]; and its order is 7!/2. An infinite family of non-solvable 4-HATs were constructed later by Malnic and Marusic [15]. The smallest member of this family has an even larger order, namely 11!/2. To the best of our knowledge, no smaller non-solvable 4-HATs appeared in literature. Perhaps surprisingly, small examples of non-solvable 4-HATs seem not to be too rare, as can be checked from our census (as well as from the unpublished work [23]). Theorem 1.4. There are thirty-two non-solvable 4-HATs with at most 1000 vertices. The smallest one, named HAT[480,44], has order 480, girth 5, radius 5, attachment number 2, alter-exponent 2, and alter-perimeter 1. It is non-Cayley and non-bipartite. (The terms radius, attachment number, alter-exponent, and alter-perimeter appearing in the statement of Theorem 1.4 are defined in Sections 4.2 and 4.3.) Let us now continue with a few comments on the census of 2-ATDs. All the undefined notions mentioned in the theorems below are explained in Sections 2, 4.2 and 4.3. It is not surprising that, apart from the generalised wreath digraphs (see Section 2.3 for the definition), very few of the 2-ATDs on at most 1000 vertices are 2-arc-transitive. In fact, the following can be deduced from the census. Theorem 1.5. Out of the 26457 2-ATDs on at most 1000 vertices, 961 are generalised wreath digraphs. Of the remaining 25496, only 1199 are 2-arc-transitive (the smallest having order 18), only 255 are 3-arc-transitive (the smallest having order 42), only 61 are 4-arc-transitive (the smallest having order 90), and only 6 are 5-arc-transitive (the smallest two having order 640); none of them is 6-arc-transitive. Note that the non-existence of a 6-arc-transitive non-generalised-wreath 2-ATD on at most 1000 vertices follows from a more general result (see Corollary 3.2). Recall that there is no 4-HAT on at most 1000 vertices with a non-abelian vertex-stabiliser (Theorem 1.3). Consequently (see Section 2.2), every 2-ATD on at most 1000 vertices with a non-abelian vertex-stabiliser has an arc-transitive underlying graph; and there are indeed such examples. In fact, the following holds (see Section 2.1 for the definition of self-opposite). Theorem 1.6. There are precisely forty-five 2-ATDs on at most 1000 vertices with a nonabelian vertex-stabiliser. They are all self-opposite, at least 3-arc-transitive, have non-solvable automorphism groups, and radius 3. The smallest of these digraphs has order 42, and the smallest that is 4-arc-transitive has order 90. There are no 5-arc-transitive 2-ATDs with a non-abelian vertex-stabiliser and order at most 1000. If a 2-ATD is self-opposite, then the isomorphism between the digraph and its opposite digraph is an automorphism of the underlying graph, making the underlying graph arc-transitive. Hence, self-opposite 2-ATDs always yield arc-transitive 4-GHATs. However, the converse is not always true: there are 2-ATDs that are not self-opposite, but have an arc-transitive underlying graph. In this case, the index of the automorphism group of the 2-ATD in the automorphism group of its underlying graph must be larger than 2 (for otherwise the former would be normal in the latter and thus any automorphism of the underlying graph would either preserve the arc-set of the digraph, or map it to the arc-set of the opposite digraph). It is perhaps surprising that there are not many small examples of such behaviour. Theorem 1.7. There are precisely fifty-two 2-ATDs on at most 1000 vertices that are not self-opposite but have an arc-transitive underlying graph. The smallest two have order 21. None of these digraphs is 2-arc-transitive. The index of the automorphism group of these digraphs in the automorphism group of the underlying graphs is always 8. We finish this section by mentioning an earlier attempt of Stephen Wilson and the first author of this paper to compile a census of all small edge-transitive graphs of valence 4, and thus, in particular, of all 4-GHATs; see [31]. The results of this paper confirm that the list given in [31] contains all 4-GHATs of order at most 100. 2 Notation and definitions 2.1 Digraphs and graphs A digraph is an ordered pair (V, A) where V is a finite non-empty set and A С V x V is a binary relation on V. We say that (V, A) is asymmetric if A is asymmetric, and we say that (V, A) is a graph if A is irreflexive and symmetric. If Г = (V, A) is a digraph, then we shall refer to the set V and the relation A as the vertex-set and the arc-set of Г, and denote them by V(r) and А(Г), respectively. Members of V and A are called vertices and arcs, respectively. If (u, v) is an arc of a digraph Г, then u is called the tail, and v the head of (u, v). If Г is a graph, then the unordered pair {u, v} is called an edge of Г and the set of all edges of Г is denoted Е(Г). If Г is a digraph, then the opposite digraph Горр has vertex-set V^) and arc-set {(v, u) : (u, v) G А(Г)}. The underlying graph of Г is the graph with vertex-set V^) and with arc-set А(Г) U А(Горр). A digraph is called connected provided that its underlying graph is connected. Let v be a vertex of a digraph Г. Then the out-neighbourhood of v in Г, denoted by Г+^), is the set of all vertices u of Г such that (v, u) G А(Г), and similarly, the in-neighbourhood T-(v) is defined as the set of all vertices u of Г such that (u, v) G А(Г). Further, we let val+(v) = |Г+ (v) | and val-(v) = |Г-(v)| be the out-valence and invalerne of Г, respectively. If there exists an integer r such that val+(v) = val-(v) = r for every v g V^), then we say that Г is regular of valence r, or simply that Г is an r-valent digraph. An s-arc of a digraph Г is an (s + 1)-tuple (v0, vb..., vs) of vertices of Г, such that (vi-i, vj) is an arc of Г for every i e {1,..., s} and vi-1 = vi+1 for every i e {1,..., s — 1}. If x = (v0, v1,..., vs) is an s-arc of Г, then every s-arc of the form (v1, v2,..., vs, w) is called a successor of x. An automorphism of a digraph Г is a permutation of V(r) which preserves the arc-set А(Г). Let G be a subgroup of the full automorphism group Aut(r) of Г. We say that Г is G-vertex-transitive or G-arc-transitive provided that G acts transitively on V^) or А(Г), respectively. Similarly, we say that Г is (G, s)-arc-transitive if G acts transitively on the set of s-arcs of Г. If Г is a graph, we say that it is G-edge-transitive provided that G acts transitively on Е(Г). When G = Аи^Г), the prefix G in the above notations is usually omitted. If Г is a digraph and v e V^), then a v-shunt is an automorphism of Г which maps v to an out-neighbour of v. 2.2 From 4-GHATs to 2-ATDs and back If Г is a connected 4-valent (G, 2 )-arc-transitive graph, then G has two paired orbits on the arc-set of Г, each orbit having the property that each vertex of Г is the head of precisely two arcs, and also the tail of precisely two arcs of the orbit. By taking any of these two orbits as an arc-set of a digraph on the same vertex-set, one thus obtains a 2-ATD whose underlying graph is Г, and admitting G as an arc-transitive group of automorphisms. Conversely, the underlying graph of a G-arc-transitive 2-ATD is a (G, 2 )-arc-transitive 4-GHAT. In this sense the study of 4-GHATs is equivalent to the study of 2-ATDs. In Section 3, we explain how a complete list of all 2-ATDs on at most 1000 vertices was obtained. The above discussion shows how this yields a complete list of all 4-GHATs on at most 1000 vertices. 2.3 Generalised wreath digraphs Let n be an integer with n > 3, let V = Zn x Z2, and let A = {((i, a), (i + 1, b)) : i e Zn, a, b e Z2}. The asymmetric digraph (V, A) is called a wreath digraph and denoted by W n. If Г is a digraph and r is a positive integer, then the r-th partial line digraph of Г, denoted Plr (Г), is the digraph with vertex-set equal to the set of r-arcs of Г and with (x, y) being an arc of Plr (Г) whenever y is a successor of x. If r = 0, then we let Plr (Г) = Г. Let r be a positive integer. The (r — 1)-th partial line digraph Plr-1(Wn) of the wreath digraph Wn is denoted by W(n, r) and called a generalised wreath digraph. Generalised wreath digraphs were first introduced in [32], where W(n, r) was denoted Cn(2, r). It was proved there that Aut(W(n, r)) = C2 i Cn = Cn x Cn and that Aut(W(n, r)) acts transitively on the (n—r)-arcs but not on the (n — r +1)-arcs of W(n, r) [32, Theorem 2.8]. In particular, W (n, r) is arc-transitive if and only if n > r + 1. Note that |V(W (n, r))| = n2r, and thus |Aut(WV(n,r))v | = n2n/n2r = 2n-r. The underlying graph of a generalised wreath digraph will be called a generalised wreath graph. 2.4 Coset digraphs Let G be a group generated by a core-free subgroup H and an element g with g-1 e HgH. One can construct the coset digraph, denoted Cos(G, H, g), whose vertex-set is the set G/H of right cosets of H in G, and where (Hx, Hy) is an arc if and only if yx-1 e HgH. Note that the condition g-1 e HgH guarantees that the arc-set is an asymmetric relation. Moreover, since G = (H, g), the digraph Cos(G, H, g) is connected. The digraph Cos(G, H, g) is G-arc-transitive (with G acting upon G/H by right multiplication), and hence Cos(G, H, g) is a G-arc-transitive and G-vertex-transitive digraph with g being a v-shunt (where v = H • 1 e G/H). On the other hand, it is folklore that every such graph arises as a coset digraph. Lemma 2.1. If Г is a connected G-arc-transitive and G-vertex-transitive digraph, v is a vertex of Г, and g is a v-shunt contained in G, then Г = Cos(G, Gv ,g). 3 Constructing the census If Г is a G-vertex-transitive digraph with n vertices, then |G| = n|Gv |. If one wants to use the coset digraph construction to obtain all 2-ATDs on n vertices, one thus needs to consider all groups G of order n|Gv | that can act as arc-transitive groups of 2-ATDs. In order for this approach to be practical, two issues must be resolved: First, one must get some control over |G| and thus over |Gv |. (Recall that in W(n, r), |Gv | can grow exponentially with |V(W(n, r))|, as n ^ ж and r is fixed). Second, one must obtain enough structural information about G to be able to construct all possibilities. Fortunately, both of these issues were resolved successfully. The problem of bounding |Gv | was resolved in a recent paper [37] and details can be found in Section 3.1. The second problem was dealt with in [19], and later, in greater generality in [30] (both of these papers rely heavily on a group-theoretical result of Glauberman [12]); the summary of relevant results is given in Section 3.2. 3.1 Bounding the order of the vertex-stabiliser The crucial result that made our compilation of a complete census of all small 2-ATDs possible is Theorem 3.1, stated below, which shows that the generalised wreath digraphs (defined in Section 2.3) are very special in the sense of having large vertex-stabilisers. In fact, together with the correspondence described in Section 2.2, [37, Theorem 9.2] has the following corollary: Theorem 3.1. Let Г be a G-arc-transitive 2-ATD on at most m vertices and let t be the largest integer such that m > t2t+2. Then one of the following occurs: 1. Г = W(n, r) for some n > 3 and 1 < r < n — 1, 2. |Gv |< max{16, 2t}, 3. (Г, G) appears in the last line of [37, Table 1]. In particular, |V(^| = 8100. The following is an easy corollary: Corollary 3.2. Let Г be a G-arc-transitive 2-ATD on at most 1000 vertices. Then either |Gv | < 32 or Г = W(n, r) for some n > 3 and 1 < r < n — 1. 3.2 Structure of the vertex-stabiliser Definition 3.3. Let s and a be positive integers satisfying | s < a < s, and let c be a function assigning a value citj e {0,1} to each pair of integers i, j with a < j < s - 1 and 1 < i < 2a - 2s + j + 1. Let AS,a be the group generated by {x0, x1,..., xs—1, g} and subject to the defining relations: • /т»2 — yv»2 — — yv»2 — i • xo — Xi — • • • — Xs_i — 1, • xg — xi+1 for i e {0,1,..., s - 2}; • if j < a, then [x0, Xj] — 1; • if j > a, then [xo, xj] — xs—a xs—a+1 • • • r+2a+j+1,j. Furthermore, let As,a be the family of all groups As,a for some c. It was proved in [19] (see also [30]) that every group G acting arc-transitively on a 2-ATD is isomorphic to a quotient of some As,a. More precisely, the following can be deduced from [19] or [30]. Theorem 3.4. Let Г be a G-arc-transitive 2-ATD, let v e V(T) and let s be the largest integer such that G acts transitively on the set of s-arcs of Г. Then there exists an integer a satisfying § s < a < s, a function c as in Definition 3.3, and an epimorphism p : As,a — G, which maps the group (x0,..., xs—1} isomorphically onto Gv and the generator g to some v-shunt in G. In particular, |Gv | — 2s. In this case, we will say that (Г, G) is of type Acs a, and call the group As a the universal group of the pair (Г, G). For s, a, and a function c satisfying the conditions of Definition 3.3, let c' be the function defined by ci,j — c2a—2s+j+2 — i,j. The relationship between c and c! can be visualised as follows: if one fixes the index j and views the function i i—у ci,j as the sequence [c1,j, c2j-,..., c2a—2s+j+1jj], then the sequence for c' is obtained by reversing the one for c. If G — As,a then we denote the opposite type As,a by Gopp. Observe that if (Г, G) is of type G, then (Горр, G) is of typeGopp. A class of groups, obtained from As,a by taking only one group in each pair {G, Gopp}, G e As,a, will be denoted A^a. (Note that some groups (5 might have the property that G — Gopp.) In view of Corollary 3.2, we shall be mainly interested in the universal groups As,a with s < 5 (as, excluding generalised wreath digraphs, these are the only types of 2-ATDs of order at most 1000). We list the relevant classes Ased for s < 5 explicitly in Table 1. Groups in Ased, for a fixed s will be named by As, where i will be a positive integer, where groups with larger a will be indexed with lower i. Also, the generators x0, x1, x2, x3, and x4 will be denoted a, b, c, d, and e, respectively. 3.3 The algorithm and its implementation We now have all the tools required to present a practical algorithm that takes an integer m as input and returns a complete list of all 2-ATDs on at most m vertices (see Algorithm 1). It is based on the fact that every such digraph can be obtained as a coset digraph of some group G (see Lemma 2.1), and that G is in fact an epimorphic image of some group As,a (see Theorem 3.4) with Gv and the shunt being the corresponding images of (x0,..., xs—1} and g in As,a. Moreover, if Г is not a generalised wreath digraph or the exceptional digraph on 8100 vertices mentioned in part 3 of Theorem 3.1, then the parameter s satisfies s2s+2 < m, and name G A (a, g | a2) A2 (a,b,g | a2, b2, agb, [a,b]) A3 (a,b, c,g | a2,b2, c2, agb,bgc, [a,b], [a,c]) A3 (a, b, c, g | a2, b2, c2, agb,bgc, [a,b], [a,c]b) A4 (a, b, c, d, g | a2, b2, c2, d2, agb, bgc, cgd, [a, b], [a, c], [a, d]) A4 (a, b, c, d, g | a2, b2, c2, d2, agb, bgc, cgd, [a, b], [a, c], [a, d]b) A4 (a, b, c, d, g | a2,b2, c2, d2, agb, bgc, cgd, [a, b], [a, c], [a, d]bc) Ai (a, b, c, d, e, g | a2, b2, c2, d2, e2, d2, agb,bgc, cgd,dge, [a,b], [a,c], [a,d], [a, e]) Ai (a, b, c, d, e, g | a2, b2, c2, d2, e2, d2, agb, bgc, cgd, dge, [a, b], [a, c], [a, d], [a, e]b) Ai (a, b, c, d, e, g | a2, b2, c2, d2, e2, d2, agb, bgc, cgd, dge, [a, b], [a, c], [a, d], [a, e]c) Ai (a, b, c, d, e, g | a2, b2, c2, d2, e2, d2, agb, bgc, cgd, dge, [a, b], [a, c], [a, d], [a, e]bc) Ai (a, b, c, d, e, g | a2,b2, c2, d2, e2, d2, agb,bgc, cgd,dge, [a,b], [a,c], [a,d], [a,e]bd) Ai (a, b, c, d, e, g | a2, b2, c2, d2, e2, d2, agb, bgc, cgd, dge, [a, b], [a, c], [a, d], [a, e]bcd) Table 1: Universal groups of 2-ATDs with |Gv | < 32 the order of the epimorphic image G is bounded by 2s m (see Theorem 3.1). The algorithm thus basically boils down to the task of finding normal subgroups of bounded index in the finitely presented groups AS,a. Practical implementations of this algorithm have several limitations. First, the best known algorithm for finding normal subgroups of low index in a finitely presented group is an algorithm due to Firth and Holt [11]. The only publicly available implementation is the LowlndexNormalSubgroups routine in Magma [5] and the most recent version allows one to compute only the normal subgroups of index at most 5 • 105; hence only automorphisms groups of order 5 • 105 can possibly be obtained in this way. More importantly, even when only normal subgroups of relatively small index need to be computed, some finitely presented groups are computationally difficult. For example, finding all normal subgroups of index at most 2048 of the group A} = C2 * seems to represent a considerable challenge for the LowlndexNormalSubgroups routine in Magma. In order to overcome this problem, we have used a recently computed catalogue of all (2, *)-groups of order at most 6000 [29], where by a (2, *)-group we mean any group generated by an involution x and one other element g. Since A} is a (2, *)-group and every non-cyclic quotient of a (2, *)-group is also a (2, *)-group, this catalogue can be used to obtain all the quotients of A} of order up to 6000. Consequently, all 2-ATDs admitting an arc-regular group of automorphisms of order at most 3000 can be obtained. Similarly, since A} is also a (2, *)-group, we can use this catalogue to obtain all the 2-ATDs of order at most 1500 admitting an arc-transitive group G with |Gv | =4. It should be mentioned that the concept of a (2, *) -group is equivalent to that of a rotary map, that can be described as groups generated by two elements the product of which is Algorithm 1 2-ATDs on at most m vertices. Require: positive integer m Ensure: D = {Г : Г is 2-ATD, |V(Г)| < m} Let t be the largest integer such that m > t2t+2 ; Let D be the list of all arc-transitive generalised wreath digraphs on at most m vertices; If m > 8100, add to D the exceptional digraph Г on 8100 vertices, mentioned in part 3 of Theorem 3.1; for s G {1,..., max{4, t}} do for a G {[ f si, f § s! + 1,...,s} do for G G A^d do Let N be the set of all normal subgroups of G of index at most 2s m; for N gN do Let G := G/N and let p : G ^ G be the quotient projection; Let H := p((xo,... ,Xs_i)); if H is core-free in G and |H| = 2s and p(g)-1 G Hp(g)H then Let C := cos(G, H, p); for Г g {C, C°PP} do if Г is not isomorphism to any of the digraphs in D then add Г to the list D; end if end for end if end for end for end for end for an involution. A catalogue of all (2, *)-groups of order at most 2000 could thus be derived from Conder's catalogue of rotary maps with at most 1000 edges [9]. Conversely, the catalogue of (2, *)-group of order up to 6000 [29] extends the list in [9] up to 3000 edges in the orientable case and to 1500 in the non-orientable case. Like A} and A}, the groups with (x0,..., xs_}) abelian (namely those with a = s and Oi,j = 0 for all i, j) are also computationally very difficult. One can make the task easier by dividing it into cases, where the order of g is fixed in each case. Since g represents a shunt, it can be proved that its order cannot exceed the order of the digraph (see, for example, [27, Lemma 13]). Cases can then be run in parallel on a multi-core computer. 4 The census and accompanying data Using Algorithm 1, we found that there are exactly 26457 2-ATDs of order up to 1000. Following the recipe explained in Section 2.2, we have also computed all the 4-GHATs, which we split in two lists: 4-HATs and arc-transitive 4-GHATs. The data about these graphs, together with Magma code that generates them, is available on-line at [28]. The package contains ten files. The file "Census-ATD-1k-README.txt" is a text file containing information similar to the information in this section. The remaining nine files come in groups of three, one group for each of the three lists (2-ATDs, arc-transitive 4-GHATs, 4-HATs). In each group, there is a *.mgm file, a *.txt file and a *.csv file. The *.mgm file contains Magma code that generates the corresponding digraphs. After loading the file in Magma, a double sequence is generated (named either ATD, GHAT, or HAT, depending on the file). The length of each double sequence is 1000 and the n-th component of the sequence is the sequence of all the corresponding digraphs of order n, with the exception of the generalised wreath digraphs. Thus, ATD[32,2] will return the second of the four non-generalised-wreath 2-ATDs on 32 vertices (the ordering of the digraphs in the sequence ATD[32] is arbitrary). In order to include the generalised wreath digraphs into the corresponding sequence, one can call the procedure AddGWD( ^atd,gwd) in the case of the 2-ATDs, or AddGWG(~GHAT,GWG) in the case of the 4-GHATs (note that a generalised wreath graph is never 2-arc-transitive). The *.txt file contains the list of neighbours of each digraph. This file is needed when the *.mgm file is loaded into Magma, but, being an ASCII file, it can be used also by other computer systems to reconstruct the digraphs. For the details of the format, see the "README" file. Finally, the *.csv file is a "comma separated values" file representing a spreadsheet containing some precomputed graph invariants. We shall first introduce some of these invariants and then discuss each *.csv separately. 4.1 Walks and cycles Let Г be a digraph. A walk of length n in Г is an (n + 1)-tuple (v0, v},..., vn) of vertices of Г such that, for any i G {1,... n}, either (vi_i, vi) or (vi, vi_i) is an arc of Г. The walk is closed if v0 = vn and simple if the vertices vi are pairwise distinct (with the possible exception of the first and the last vertex when the walk is closed). A closed simple walk in Г is called a cyclet. The inverse of a cyclet (v0,..., vn_}, v0) is the cyclet (v0, vn_},..., v}, v0), and a cyclet (v0,..., vn_}, v0) is said to be a shift of a cyclet (u0,..., un_1,u0) provided that there exists k g Zn such that ui = vi+k for all i G Zn. Two cyclets W and U are said to be congruent provided that W is a shift of either U or the inverse of U. The relations of "being a shift of" and "being congruent to" are clearly equivalence relations, and their equivalence classes are called oriented cycles and cycles, respectively. With a slight abuse of terminology, we shall sometimes identify a (oriented) cycle with any of its representatives. 4.2 Alter-equivalence, alter-exponent, alter-perimeter, and alter-sequence Let Г be an asymmetric digraph. The signature of a walk W = (v0, v1,..., vn) is an n-tuple (e1,e2,..., e„), where = 1 if (vj_i, vj) is an arc of Г, and = -1 otherwise. The signature of a walk W will be denoted by a(W). The sum of all the integers in a(W) is called the sum of the walk W and denoted by s(W ); similarly, the kth partial sum sk(W ) is the sum of the initial walk (v0, v1,..., vk) of length k. By convention, we let s0(W) = 0. The tolerance of a walk W of length n, denoted T(W), is the set {sk(W) : k G {0,1,..., n}}. Observe that the tolerance of a walk is always an interval of integers containing 0. Let t be a positive integer or то. We say that two vertices u and v of Г are alter-equivalent with tolerance t if there is a walk from u to v with sum 0 and tolerance contained in [0, t]; we shall then write uAtv. The equivalence class of At containing a vertex v will be denoted by At (v). Since we assume that Г is a finite digraph, there exists an integer e > 0 such that Ae = Ae+1 (and then Ae = ATO). The smallest such integer is called the alter-exponent of Г and denoted by exp(r). The number of equivalence classes of is called the alter-perimeter of Г. The name originates from the fact that the quotient digraph of Г with respect to is either a directed cycle or the complete graph K2 or the graph K1 with one vertex. If e is the alter-exponent of a (vertex-transitive) digraph Г, then the finite sequence [|A1 (v) |, |A2 (v) |,..., |Ae(v)|] is called the alter-sequence of Г. Several interesting properties of the alter-exponent can be proved (see [20] for example). For example, if Г is connected and G-vertex-transitive, then ехр(Г) is the smallest positive integer e such that the setwise stabiliser GAe(v) is normal in G. The group GAe(v) is the group generated by all vertex-stabilisers in G and G/GAe(v) is a cyclic group. All notions defined in this section for digraphs generalise to half-arc-transitive graphs, where instead of the graph one of the two natural arc-transitive digraphs are considered. As was shown in [20], all the parameters defined here remain the same if instead of a digraph, its opposite digraph is considered. The notions defined in this section were later generalised in the context of infinite digraphs [14]. 4.3 Alternating cycles - radius and attachment number A walk W in an asymmetric digraph is called alternating if its tolerance is either [0,1] or [-1,0] (that is, if the signs in its signature alternate). Similarly, a cycle is called alternating provided that any (and thus every) of its representatives is an alternating walk. This notion was introduced in [17] and used to classify the so-called tightly attached 4-GHATs and 4-HATs of odd radius. The concept of alternating cycles was explored further in a number of papers on 4-HATs (see for example [21, 34]). Let Г be a 2-ATD, let C be the set of all alternating cycles of Г, and let G = Aut^). The set C is clearly preserved by the action of G upon the cycles of Г. Moreover, since Г is arc-transitive, G acts transitively on C. In particular, all the alternating cycles of Г are of equal length. Half of the length of an alternating cycle is called the radius of Г. Since Г is 2-valent, every vertex of Г belongs to precisely two alternating cycles. It thus follows from vertex-transitivity of Г that any (unordered) pair of intersecting cycles can be mapped to any other such pair, implying that there exists a constant a such that any two cycles meet either in 0 or in a vertices. The parameter a is then called the attachment number of Г. In general, the attachment number divides the length of the alternating cycle (twice the radius), and there are digraphs where a equals this length; they were classified in [17, Proposition 2.4], where it was shown that their underlying graphs are always arc-transitive. A 2-valent asymmetric digraph with attachment number a is called tightly attached if a equals the radius, is called antipodally attached if a = 2, and is called loosely attached if a =1. Note that tightly attached 2-ATDs are precisely those with alter-exponent 1. 4.4 Consistent cycles Let Г be a graph and let G < Аи^Г). A (oriented) cycle C in a graph Г is called G-consistent provided that there exists g e G that preserves C and acts upon it as a 1-step rotation. A G-orbit of G-consistent oriented cycles is said to be symmetric if it contains the inverse of any (and thus each) of its members, and is chiral otherwise. Consistent oriented cycles were first introduced by Conway in a public lecture [10] (see also [3,25, 26]). Conway's original result states that in an arc-transitive graph of valence d, the automorphism group of the graph has exactly d — 1 orbits on the set of oriented cycles. In particular, if Г is 4-valent and G-arc-transitive, then there are precisely three G-orbits of G-consistent oriented cycles. Since chiral orbits of G-consistent cycles come in pairs of mutually inverse oriented cycles, this implies that there must be at least one symmetric orbit, while the other two are either both chiral or both symmetric. Conway's result was generalised in [4] to the case of 1 -arc-transitive graphs by showing that if Г is a 4-valent (G, 1 )-arc-transitive graph, then there are precisely four G-orbits of G-consistent oriented cycles, all of them chiral. These four orbits of oriented cycles thus constitute precisely two G-orbits of G-consistent (non-oriented) cycles. 4.5 Metacirculants A weak metacirculant is a graph whose automorphism group contains a vertex-transitive metacyclic group G, generated by p and a, such that the cyclic group (p) is semiregular on the vertex-set of the graph, and is normal in G. This notion was introduced by Marusic and Sparl [22] and generalises that of a metacirculant introduced by Alspach and Parsons [2]. Metacirculants admitting 2-arc-transitive groups of automorphisms were first investigated in [33]. Recently, the interesting problem of classifying all 4-HATs that are weak metacirculants was considered in [22, 35, 36]. Such 4-HATs fall into four (not necessarily disjoint) classes (called Class I, Class II, Class III, and Class IV), depending on the structure of the quotient by the orbits of the semiregular element p. For a precise definition of the class of a 4-HAT weak metacirculant see, for example, [35, Section 2]. Since a given 4-HAT may admit several vertex-transitive metacyclic groups, a fixed graph can fall into several of these four classes. Several interesting facts about 4-HAT (weak) metacirculants are known. For example, tightly attached 4-HATs are precisely the 4-HATs that are weak metacirculants of Class I. 4.6 The data on 2-ATDs The "Census-ATD-1k-data.csv" file concerns 2-ATDs. Each line of the file represents one of the digraphs in the census, and has 19 fields described below. Since this file is in "csv" format, every occurrence of a comma in a field is substituted with a semicolon. • Name: the name of the digraph (for example, ATD[32,2]); • | v| : the order of the digraph; • SelfOpp: contains "yes" if the digraph is isomorphic to its opposite digraph and "no" otherwise; • Opp: the name of the opposite digraph (the same as "Name" if the digraph is self-opposite); • isUndAT: "yes" if the underlying graph is arc-transitive, "no" otherwise; • UndGrph: the name of the underlying graph, as given in the files "Census-HAT-lk-data.csv" and "Census-GHAT-1k-data.csv" - if the underlying graph is generalized wreath, then this is indicated by, say, "GWD(m,k)" where m and k are the defining parameters. • s: the largest integer s, such that the digraph is s-arc-transitive; • GvAb: "Ab" if the vertex-stabiliser in the automorphism group of the digraph is abelian, otherwise "n-Ab"; • |Tv:Gv|: the index of the automorphism group G of the digraph in the smallest arc-transitive group T of the underlying graph that contains G - if there's no such group T,then 0; • |Av:Gv|: the index of the automorphism group of the digraph in the automorphism group of the underlying graph; • Solv: this field contains "solve" if the automorphism group of the digraph is solvable and "n-solv" otherwise; • Rad: the radius, that is, half of the length of an alternating cycle; • AtNo: the attachment number, that is, the size of the intersection of two intersecting alternating cycles; • AtTy: the attachment type, that is: "loose" if the attachment number is 1, "antipodal" if 2, and "tight" if equal to the radius, otherwise "—"; • |AltCyc|: the number of alternating cycles; • AltExp: the alter-exponent; • AltPer: the alter-perimeter; • AltSeq: the alter-sequence; • IsGWD: "yes" if the digraph is generalized wreath, and "no" otherwise. 4.7 The data on arc-transitive 4-GHATs The "Census-GHAT-1k-data.csv" file concerns arc-transitive 4-GHATs . Each line of the file represents one of the graphs in the census, and has nine fields, described below. Note, however, that the file does not contain the generalised wreath graphs. • Name: the name of the graph (for example GHAT[9,1]); • | v| : the order of the graph; • gir: the girth (length of a shortest cycle) of the graph; • bip: this field contains "b" if the graph is bipartite and "nb" otherwise; • CayTy: this field contains "Circ" if the graph is a circulant (that is, a Cayley graph on a cyclic group), "AbCay" if the graph is Cayley graph on an abelian group, but not a circulant, and "Cay" if it is a Cayley but not on an abelian group - it contains "n-Cay" otherwise; • | Av | : the order of the vertex-stabiliser in the automorphism group of the graph; • |Gv |: a sequence of the orders of vertex-stabilisers of the maximal half-arc-transitive subgroups of the automorphism group - up to conjugacy in the automorphism group; • solv: this field contains "solve" if the automorphism group of the graph is solvable and "n-solv" otherwise; • [|ConCyc|]: the sequence of the lengths of A-consistent oriented cycles of the graph (one cycle per each A-orbit, where A is the automorphism group of the graph) - the symbols "c" and "s" indicate whether the corresponding cycle is chiral or symmetric - for example, [4c; 4c; 10s] means there are two chiral orbits of A-consistent cycles, both containing cycles of length 4, and one orbit of symmetric consistent cycles, containing cycles of length 10. 4.8 The data on 4-HATs The "Census-HAT-1k-data.csv" file concerns 4-HATs. Each line of the file represents one of the graphs in the census, and has 16 fields. The fileds |v|, gir, bip, and Solv are as in Section 4.7, and the fields Rad, AtNo, AtTy, AltExp, AltPer and AltSeq are as in Section 4.6. The remaining fileds as follows: • Name: the name of the graph (for example HAT[27,1]); • IsCay: this field contains "Cay" if the graph is Cayley and "n-Cay" otherwise; • |Gv |: the order of the vertex-stabiliser in the automorphism group of the graph; • CCa: the length of a shortest consistent cycle; • CCb: the length of a longest consistent cycle; • MetaCircCl: "{}" if the graph is not a meta-circulant; otherwise a set of classes of meta-circulants that represents the graph. References [1] B. Alspach, D. Marušic, L. Nowitz, Constructing graphs which are 1 -transitive, J. Austral. Math. Soc. Ser. A 56 (1994), 391-402. [2] B. Alspach, T. D. Parsons, A construction for vertex-transitive graphs Canad. J. Math. 34 (1982), 307-318. [3] N. 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Sparl, A list of 1 -arc-transitive tetravalent graphs with the vertex-stabiliser of order 2, unpublished work, available upon request from P. Sparl, primoz.sparl@pef.uni-lj.si. [24] D. Marušic, M.-Y. Xu, 2-transitive graph of valency 4 with a nonsolvable group of automorphisms, J. Graph Theory 25 (1997), 133-138. [25] S. Miklavic, A note on a conjecture on consistent cycles, Ars Math. Contemp. 6 (2013), 389392. [26] S. Miklavic, P. Potocnik, S. Wilson, Overlap in consistent cycles, J. Graph Theory 55 (2007), 55-71. [27] P. Potocnik, P. Spiga, G. Verret, Cubic vertex-transitive graphs on up to 1280 vertices, J. Symbolic Comput. 50 (2013), 465-477. [28] P. Potocnik, P. Spiga, G. Verret, Census of 2-valent arc-transitive digraphs, http://www. fmf.uni-lj.si/~potocnik/work.htm, accessed October 2013. [29] P. Potocnik, P. Spiga, G. Verret, Census of (2, *)-groups, http://www.fmf.uni-lj.si/ -potocnik/work.htm, accessed October 2013. [30] P. Potocnik, G. Verret, On the vertex-stabiliser in arc-transitive digraphs, J. Combin. Theory Ser. B. 100 (2010), 497-509. [31] P. Potocnik, S. Wilson, Census of tetravalent edge-transitive graphs, https://jan.ucc. nau.edu/~swilson/C4Site/BigTable.html, accessed October 2013. [32] C. E. Praeger, Highly arc transitive digraphs, European J. Combin. 10 (1989), 281-292. [33] M. Sajna, Half-transitivity of some metacirculants, Discrete Math. 185 (1998), 117-136. [34] P. Sparl, A classification of tightly attached half-arc-transitive graphs of valency 4, J. Combin. Theory Ser. B 98 (2008), 1076-1108. [35] P. Sparl, On the classification of quartic half-arc-transitive metacirculants, Discrete Math. 309 (2009), 2271-2283. [36] P. Sparl, Almost all quartic half-arc-transitive weak metacirculants of Class II are of Class IV, Discrete Math. 310 (2010), 1737-1742. [37] P. Spiga, G. Verret, On the order of vertex-stabilisers in vertex-transitive graphs with local group Cp x Cp or Cp; C2,http://arxiv.org/pdf/1311.4308.pdf. [38] W. T. Tutte, Connectivity in Graphs, University of Toronto Press, Toronto (1966). /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 149-162 Some recent discoveries about half-arc-transitive graphs Dedicated to Dragan Marusic on the occasion of his 60th birthday Marston D. E. Conder * Mathematics Department, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Primož PotoCnik t Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia and IAM, University ofPrimorska, Muzejski trg 2, 6000 Koper, Slovenia Primož Spari * Faculty of Education, University of Ljubljana, Kardeljeva ploščad 16, 1000 Ljubljana, Slovenia and Institute for Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia and IAM, University ofPrimorska, Muzejski trg 2, 6000 Koper, Slovenia Received 11 October 2013, accepted 5 June 2014, published online 22 September 2014 Abstract We present some new discoveries about graphs that are half-arc-transitive (that is, vertex- and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest *The first author was supported by a James Cook Fellowship and a Marsden Fund grant (UOA1015) from the Royal Society of New Zealand t The second author was supported by the Slovenian Research Agency, projects L1-4292 and P1-0222. * The third author was supported in part by the Slovenian Research Agency, projects J1-4010 and J1-4021 and program P1-0285. ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser D4 (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse. Keywords: Graph, edge-transitive, vertex-transitive, half-arc-transitive. Math. Subj. Class.: 05E18, 20B25 1 Introduction A graph X is said to be vertex-transitive, edge-transitive, or arc-transitive, if its automorphism group Aut X acts transitively on the set V(X) of all vertices of X, the set E(X) of all edges of X, or the set A(X) of all arcs (ordered pairs of adjacent vertices) of X, respectively. An arc-transitive graph is also called symmetric. The graph X is called half-arc-transitive if it is vertex-transitive and edge-transitive but not arc-transitive. (A related class of graphs consists of those which are regular and edge-transitive but not vertex-transitive. Any such graph is called semi-symmetric. Semi-symmetric 3-valent graphs will be the topic of another forthcoming paper by the first two authors.) Every half-arc-transitive graph X is regular, with even valency — indeed Aut X has two orbits on arcs, with half the arcs emanating from any vertex v lying in each orbit. The smallest possible valency of a half-arc-transitive graph is 4, and this is the valency of the smallest half-arc-transitive graph, a graph on 27 vertices constructed independently by Doyle [7] and Holt [10]. Furthermore, examples exists for every even valency greater than 2. This was proved (in answer to a question by Tutte [21]) by Bouwer [2], who constructed a family of examples of valency 2k for all k > 2, with the property that the stabiliser in Aut X of every vertex v induces the symmetric group Sk on the neighbourhood of v, acting with two orbits of length k. (The latter property was not explicitly mentioned by Bouwer, but may be easily deduced from his construction.) For valency greater than 4, the vertex-stabilisers in Bouwer's examples are non-abelian. In contrast, for quite some time all known examples of 4-valent half-arc-transitive graphs had vertex-stabilisers that are abelian, or more precisely, elementary abelian 2-groups. The first known example of a 4-valent half-arc-transitive graph with non-abelian vertex-stabiliser was found in 1999 by Conder and Marušic, who produced an example of order 10752 from a transitive permutation group of degree 32 and order 86016, with dihedral point stabiliser (of order 8) having a non-self-paired sub-orbit; see [4]. Until recently, however, this was the only known example, and no examples were known of 4-valent half-arc-transitive graphs with vertex-stabiliser that is neither abelian nor dihedral. In this paper, we exhibit a number of new examples with particular properties. In Section 3 we give the smallest half-arc-transitive 4-valent graph (on 256 vertices) with vertex-stabiliser of order 4, and also the two smallest half-arc-transitive 4-valent graphs (on 768 vertices) with vertex-stabiliser of order 8. The fact that the former has E-mail addresses: m.conder@auckland.ac.nz (Marston D. E. Conder), primoz.potocnik@fmf.uni-lj.si (Primož Potočnik), primoz.sparl@pef.uni-lj.si (Primož Sparl) order 256 disproves the conjecture by Feng, Kwak, Xu and Zhou [8] that every 4-valent half-arc-transitive graph of prime-power order has vertex stabilisers of order 2. Then in Section 4 we describe two new half-arc-transitive 4-valent graphs with vertex-stabiliser D4 (of order 8), and in Section 5 we produce the first known half-arc-transitive 4-valent graph with vertex-stabiliser that is neither abelian nor dihedral. (In this last example, the vertex-stabiliser is isomorphic to D4 x C2, of order 16.) In Section 6 we use half-arc-transitive group actions to provide an answer to a recent question of Charles Delorme [5] about 2-arc-transitive digraphs that are not isomorphic to their reverse. The reverse Rev(D) of a digraph D is obtained by reversing all the arcs of D, and then D is called self-reverse if Rev(D) is isomorphic to D. Also a 2-arc in a digraph D is a directed walk (v0, v1, v2) of length 2 on three vertices, and then D is called arc-transitive or 2-arc-transitive if its automorphism group Aut D is transitive on the set of all arcs in D or the set of all 2-arcs in D, respectively. In [5, §5.2], Delorme asked this question: Do finite digraphs exist that are vertex-transitive, arc-transitive and 2-arc-transitive, but not self-reverse? We answer this question in the affirmative. Finally, we make some concluding remarks and pose some further questions in Section 7. 2 Further background Before proceeding, we provide some more background information. Throughout this paper, graphs are assumed to be finite and simple, and unless otherwise specified, also undirected and connected. For any graph (or digraph) X, we let V(X), E(X) and A(X) be the vertex-set, edge-set, and arc-set of X, respectively, and we let Aut X be the automorphism group of X. As mentioned earlier, we say that X is vertex-transitive, edge-transitive, or arc-transitive, if AutX is transitive on V(X), E(X) or A(X), respectively, and we say that the graph X is half-arc-transitive if it is vertex- and edge-transitive but not arc-transitive. More generally, if G is any group of automorphisms of X (that is, any subgroup of Aut X), then G is said to be vertex-transitive, edge-transitive or arc-transitive on X if G acts transitively on V(X), E(X) or A(X), respectively, and G is half-arc-transitive on X if G is vertex- and edge-transitive but not arc-transitive on X. In the latter case, we also say that X is (G, 1 )-arc-transitive, or (G, 1, H)-arc-transitive when it needs to be stressed that the stabiliser Gv of a given vertex v is isomorphic to a particular subgroup H of G. Next, we repeat an explanation given in [14] of a connection between half-arc-transitive group actions and transitive permutation groups with a non-self-paired sub-orbit. Let G be a transitive permutation group acting on a set V. An orbital of G is an orbit of the natural action of G on the Cartesian product V x V, and a sub-orbit of G on V is an orbit of the stabiliser Gv of a given point v g V. For any given point v g V there is a 1-to-1 correspondence between the set of all orbitals of G and the set of all sub-orbits of G on V (with regard to v), with the orbital containing the pair (v, w) g V x V corresponding to the orbit of Gv containing w. In particular, the diagonal orbital {(w, w) : w g V} corresponds to the trivial sub-orbit {v}. For any sub-orbit W of G (with regard to v), let Дw be the corresponding orbital of G, which contains the pair (v, w) for each w g W. Then the orbital digraph X(G, V; W) of (G, V) relative to W is the digraph (or oriented graph) with vertex-set V and arc-set Д w. The underlying undirected graph, with orientations of all arcs ignored, is denoted by X* (G, V; W). The paired orbital of a given orbital Д is the orbital Д' = {(v, w) : (w, v) G Д}. The orbital Д is said to be self-paired if Д' = Д, and non-self-paired otherwise; in the latter case Д П Д' = 0. This notion of pairing also carries over to sub-orbits in a natural way. It is important to note that for a non-self-paired sub-orbit W of G, the orbital digraph X(G, V; W) is an oriented graph, while the underlying undirected graph X*(G, V; W) admits a half-arc-transitive action of G. In the special case where V is the set (G : H) of all (right) cosets of a subgroup H of G, and W is a non-self-paired sub-orbit of the action of G on V (by right multiplication), the graph X*(G, V; W) is (G, 2,H)-arc-transitive, with valency 2|W|. This graph might or might not be half-arc-transitive, depending on whether it admits any additional automorphisms that reverse an arc. The example in [4] began with H being dihedral of order 8, and had G as its full automorphism group. For some further references on half-arc-transitive graphs (which are often referred to simply as half-transitive graphs), see the survey paper by Marušic [15], and for the theory of permutation groups, see [6, 22]. 3 The smallest half-arc-transitive 4-valent graphs with vertex-stabilisers of order 4 and 8 In this section we present the smallest 4-valent half-arc-transitive graphs for which the vertex-stabilisers have order 4 and 8, respectively. These graphs have 256 and 768 vertices, and will also appear in a recently computed census of all 4-valent half-arc-transitive graphs of order at most 1000 (see [18]). 3.1 Stabiliser of order 4 If G is the automorphism group of a half-arc-transitive 4-valent graph such that the stabiliser H in G of a vertex v has order 4, then H is isomorphic to the Klein group V4, and so is generated by two commuting involutions p and q. At the same time, G is generated by H and an automorphism a mapping v to an out-neigbour w of v (in one of the two corresponding orbital digraphs). Generators p and q of H can be chosen in such a way that q = a-1pa. In addition, since H is the stabiliser in a transitive permutation group G, the core of H in G is trivial. Conversely, given a group G generated by two commuting involutions p and q, and an element a such that q = a-1 pa, where p and q generate a core-free subgroup H of G, one can construct the orbital digraph X = X* (G, V; W), where V is the coset-space (G : H), with G acting upon V by right-multiplication, and W is the sub-orbit of this action containing the coset Ha. It can be seen that such a sub-orbit is necessarily non-self-paired, implying that X is 4-valent and admits G as a half-arc-transitive group of automorphisms. In particular, if X admits no additional automorphisms, then X is a half-arc-transitive graph with valency 4 and vertex-stabiliser V4. This happens quite frequently, although not for small orders. Candidates for G of order up to 1023 can be checked quite easily using the small groups database in Magma [1], but none of them has the required properties. Order 1024 is more challenging by this approach, since there are 49 487 365 422 groups of order 1024, and they are not easily available. Instead, we can use an algorithm for finding all normal subgroups of given index in a finitely-presented group, as described in [3], or better still, [9]. The latter version is implemented in Magma [1] as the LowlndexNormalSubgroups procedure. We can apply this to the finitely-presented group F = {p,q,a | p2 = q2 = (pq)2 = a-1paq =1}, and for every normal subgroup N found, consider the quotient F/N as a candidate for G. It turns out there are 3102 normal subgroups of index up to 1024 in F, and remarkably, four of them give good candidates. The quotient via one such normal subgroup is the group G used in the following: Theorem 3.1. Let G be the group with presentation {p, q, a |p2 = q2 = (pq)2 = a-1paq = a8 = (pa-2qa2)2 = a3qapa-1pqa-2qa-1 q =1}, and let W be the sub-orbit with regard to H = {p, q} containing the coset Ha. Then G has order 1024, and X *(G, H ; W ) is a 4-valent half-arc-transitive graph of order 256, girth 8 and diameter 8, with automorphism group G, and vertex-stabiliser Gv isomorphic to V4. Indeed the sub-orbit W containing the coset Ha is {Ha, Hap}, of size 2, and is not self-paired, its paired sub-orbit being {Ha-1, Ha-1q}. The graph X = X *(G, H ; W ) can easily be constructed with the help of Magma, and the fact that Aut X = G verified using the function AutomorphismGroup. As mentioned in the first section, the fact that the order of this graph is 256 = 28 shows that the conjecture by Feng, Kwak, Xu and Zhou [8] (that every 4-valent half-arc-transitive graph of prime-power order has vertex stabilisers of order 2) is false. By inspecting normal subgroups of G = Aut X, one finds that G has an elementary abelian normal subgroup K of order 16, and another normal subgroup J isomorphic to C4 x C4 X C2. With respect to these two normal subgroups, X is an abelian regular cover of two smaller graphs, namely the Rose-window graph RW8(6, 5), as defined in [23, §5], and the "doubled" 8-cycle DC8 (that is, the (multi)graph obtained from the directed cycle C8 by doubling each edge). Accordingly, the graph X has at least two alternative constructions as a covering graph. Theorem 3.2. The graph X = X *(G, H ; W ) in Theorem 3.1 is an abelian regular cover of the Rose-window graph RW8(6, 5), with covering group K = C24, and also an abelian regular cover of the doubled cycle DC8, with covering group J = C4 x C4 x C2. In fact, the Rose-Window graph RW8(6,5) is isomorphic to the Cartesian product C4 □ C4 of two cycles of length 4, and moreover, the quotient projection X ^ X/K is a covering projection, and G/K is the largest subgroup of Aut (X/K) acting half-arc-transitively on X/K. (For background information on quotients and covering projections of graphs, see [12, 20] for example.) The construction of X as an abelian regular cover of DC8 can be described as follows. Let J = {a,b, c | a4 = b4 = c2 = [a, b] = [a, c] = [b, c] = 1}, which is abelian of order 32, and isomorphic to C4xC4 xC2, and let p be the automorphism of J of order 8 that takes a to b, and b to ac, and c to a2c, or equivalently, is given by ip(aibjck) = aj+2kbicJ+k for 0 < i,j < 3 and 0 < k < 1. Now label the vertices of DC8 with the elements of Z8 in a natural way, and label the edges with the elements of {ei : i e Z8} U {fi : i e Z8} in such a way that the two edges incident to both i and i + 1 are labelled ei and fi, for each i e Z8. We can now describe a certain voltage assignment of the voltages from J to the set of the arcs of DC8. (For the terminology on graph coverings via voltage assignments, see [13].) For each i e Z8, we put the trivial voltage 1 on all of the arcs corresponding to the edge fi, and put the voltage ^>i(a) on the arc corresponding to ei (directed from i to i + 1). It can easily be verified that the covering graph obtained in this way is a 4-valent half-arc-transitive graph of order 256, with vertex stabilisers of order 4, as above. Similarly, the other three normal subgroups giving good candidates for G all give rise to half-arc-transitive graphs that are isomorphic to X. Let us mention also that 4-valent half-arc-transitive graphs with vertex stabilisers of order 4 have been known to exist for some time. In fact, an infinite family of such graphs was constructed in each of the papers [11] and [16]. These graphs, however, are very large. The smallest graph in the first family has order ЦТ. > g • ю13, and the smallest one in the second family has order 9 979 200. 3.2 Stabiliser of order 8 As was shown in [17] (see also [19]), if the vertex-stabiliser H = Gv in a half-arc-transitive group of automorphisms G of a connected tetravalent graph X has order 8, then H is either elementary abelian or dihedral. Moreover, as in the case where |H | = 4, the group G is generated by H and any element a of G that maps the vertex v to an out-neighbour of v. Using these observations we can prove the following: Theorem 3.3. Let X be a 4-valent half-arc-transitive graph with automorphism group G, and suppose the stabiliser H = Gv of a vertex v has order 8. (a) If H is dihedral, then G is a quotient of the finitely generated group 03,1 = (p, q,r,a | p2 = q2 = r2 = [p, q] = [q, r] = 1, [p, r] = q, q = a-1pa, r = a-1qa) of order 8|V (X )|. There is no such graph X of order up to 768. (b) If H is abelian, then G is a quotient of the finitely generated group 03,0 = (p, q,r,a | p2 = q2 = r2 = [p, q] = [p, r] = [q, r] = 1, q = a-1 pa, r = a-1qa) of order 8|V (X )|. There is no such graph of order less than 768, but there are two (non-isomorphic) examples of order 768, say X1 and X2, which arise from orbital digraphs for quotients of G3,0 with respect to the additional relator sets Ri = {a12, (ra2pa-2)2, a-1pa-4rqpa6pqa-1, a-1rapa-2pa-1rpapa2papa-1} and R2 = {a12, (ra2pa-2)2,rapa-1ra2qpa-4ra2}, respectively. Proof. (a) If H is dihedral, then the generators p, q, r of H = Gv can be chosen so that they satisfy the relations p2 = q2 = r2 = [p, q] = [q, r] = 1 and [p, r] = (pr)2 = q, and the automorphism a (moving v to an out-neighbour of v) chosen so that q = a-1pa and r = a-1qa. Thus G is a quotient of G3,1, of order 8|V(X)|. Inspection of the normal subgroups of index at most 8 • 768 = 6144 in G3,1, and the corresponding orbital digraphs, shows there are no such X of order up to 768 (with H = Gv dihedral of order 8). (b) If H is abelian, then G is a quotient of G3,0, by a similar argument to that in part (a). Computation of normal subgroups of index at most 6144 in G3 0 shows there is no such X of order less than 768 (with H = Gv abelian of order 8), but there are two (non-isomorphic) examples which arise from normal subgroups of index 768, namely the normal closures of the sets R1 and R2 as given. In each case, it is easy to check using Magma that the corresponding quotient of G3,0 is the full automorphism group of the graph. □ Theorem 3.4. The graphs X1 and X2 in Theorem 3.3 are non-isomorphic regular covers of the doubled cycle DC3, as well as elementary abelian regular covers of the so-called Hill Capping HC (Q3) of the cube Q3 (see [24]). Proof. The automorphism group of Xi has a normal subgroup K1 of index 24 and order 256, generated by x = apa-1r, y = a3qra3rq = a6[a3, rq], z = a3pqra3rqp = a6[a3, rqp] and u = a3, all of which have order 4, with x, y and z generating an abelian normal subgroup of order 64, and xu = xy2z2, yu = x2y-1z2 and zu = z-1. The centre and derived subgroup of K1 are the elementary abelian 2-subgroups {x2,y2, z2,u2} and {x2,y2, z2}, of orders 16 and 8 respectively. (In particular, K1 has nilpotency class 2.) This subgroup K1 acts semi-regularly on V (X1), and the quotient graph X1/K1 is isomorphic to the doubled cycle DC3. It follows that X1 can be reconstructed as a regular cover of DC3, with covering group K1. This can be achieved as follows. First, label the vertices of DC3 with the elements of Z3, and label the edges of DC3 with elements of the set [ei : i e Z3} U [fi : i e Z3}, so that ei and fi are the two edges incident to both i and i + 1 (for each i e Z3). Also let ei and fi be the corresponding arcs from i to i +1, for i e Z3. Then the graph X1 is isomorphic to the cover of DC3 obtained from the voltage assignment p on DC3 defined by p(eo) = p(e2) = 1, p(e{) = u, p(fo) = x, p(f) = uxy and pf) = (yz)-1. Similarly, the automorphism group of X2 has a normal subgroup K2 of index 24 and order 256, and nilpotency class 2, acting semi-regularly on V(X2). The quotient graph X2/K2 is again isomorphic to the doubled cycle DC3, and hence X2 can be reconstructed as a regular cover of DC3, with covering group K2 (not isomorphic to K1). Finally, each of Aut X1 and Aut X2 contains an elementary abelian normal subgroup K of order 16, with respect to which the quotient graphs X1 /K and X2/K are both isomorphic to HC (Q3 ). Hence X1 and X2 can be reconstructed as elementary abelian covers of HC(Q3). □ Here we note that the graph HC(Q3 ) happens to be the unique tetravalent arc-transitive graph of order 48, girth 4, and diameter 6. 4 Two new half-arc-transitive graphs with vertex-stabiliser D4 The first (and until recently, the only) known example of a half-arc-transitive 4-valent graph with non-abelian vertex-stabiliser is described in [4]. This graph has 10752 vertices, and its automorphism group G of order 86 016 is generated by two elements a and b of orders 8 and 24, which satisfy the (defining) relations a8 = (ab-1)2 = a-2 bab-2 ab = (ab3ab2ab2)2 = a-3ba2b-3a2b = (a2ba2babab)2 = a3b2a2b2aba3bababab2 = 1. The graph is the underlying graph of the orbital digraph X (G, V; W ) where V is the coset space (G : H) for the subgroup H generated by p = a-1b and q = a-1pa and r = a-1qa, and W is the non-self-paired sub-orbit {Ha, Hb}. In response to a comment made by Dragan MarusiC about this graph being somewhat unique, in a lecture at a workshop at the Fields Institute in October 2011, the first author decided to look for more examples. Somewhat surprisingly, it turns out there is another example on 10752 vertices (with vertex-stabiliser D4), not isomorphic to the first. Also there exists an example on 21 870 vertices, with similar properties. Theorem 4.1. There are at least two non-isomorphic half-arc-transitive 4-valent graphs of order 10 752 with non-abelian vertex-stabiliser of order 8. The automorphism group of the new one of order 10 752 is a different group G of order 86 016, generated by two elements a and b of orders 16 and 12 which satisfy the (defining) relations a16 = b12 = (ab-1)2 = a-2bab-2ab = (ab3ab2ab2)2 = a-3b2ab-3ab2 = (a2b2 abab2 ab)2 = a5b3a5bab = 1. Again if V is the coset space (G : H) for the subgroup H generated by p = a-1b and q = a-1pa and r = a-1qa, and W is the non-self-paired sub-orbit {Ha, Hb}, then the 4-valent underlying graph of the orbital digraph X (G, V; W) is half-arc-transitive, but it is not isomorphic to the first example. Theorem 4.2. There exists at least one half-arc-transitive 4-valent graph of order 21 870 with non-abelian vertex-stabiliser of order 8. The automorphism group of the one we found has order 174 960, and is generated by two elements a and b of orders 8 and 24 which satisfy the (defining) relations a8 = (ab-1)2 = a-2bab-2ab = a-3ba2b-3 a2b = (a3b)5 = a3b-1a-3b2ababab2ab2a2b = 1. Both of these new examples were found with the help of Magma [1], in the same way as the first one in [4]. Also Magma can be used to verify that the full automorphism group of the graph is as stated, in each case. 5 A half-arc-transitive 4-valent graph with a non-abelian and non-dihedral vertex-stabiliser In his 2011 lecture at the Fields Institute (mentioned in the previous section), Dragan Marusic made the observation that no half-arc-transitive 4-valent graph was known with vertex-stabiliser that is neither abelian nor dihedral. We provide an example here. We begin with an arc-transitive 4-valent graph on 90 vertices, which can be constructed from a group G of order 1440, generated by two elements c and d subject to the relations c8 = d10 = (cd)6 = (cd-1)2 = (cd2)4 = c-2dcd-2cd = c3d-1 c-2dcdc-2d-1 = d2 c-3d-1c4d-1c-3 d2 = 1. Let H be the subgroup generated by p = c-1d and conjugates q = c-1pc, r = c-1qc and s = c-1rc. Then H is isomorphic to the direct product D4 x C2, or order 16. If V is the coset space (G : H), and W is the non-self-paired sub-orbit {Hc, Hd}, then the 4-valent underlying graph X of the orbital digraph X (G, V, W) is arc-transitive, but admits a half-arc-transitive action of G. (In fact, its full automorphism group has order 2880.) The half-arc-transitive action of G on X lifts to the action of a larger group G on a regular cover of X, which we will prove is half-arc-transitive. The group we take is the transitive permutation group G of degree 60, generated by the elements a = (1, 2)(3, 4, 6, 8,12,17, 21, 27, 36,44, 53, 60, 59, 52, 47, 37, 45, 35, 26, 20,16,11, 7, 5) (9,13,18, 23, 30, 40, 51, 48, 57, 58, 50, 39, 29, 22, 28, 38, 34, 25, 33, 24,19,15,10,14) (31, 41, 49, 42)(32, 43)(46, 55, 54, 56) and b = (1, 2, 5, 3,14, 6, 8, 29,17,47, 55,46, 37, 36,48, 53, 60, 30, 52, 21, 56, 54, 27, 45, 38, 26, 20,19,11, 7)(4, 9,13,42, 31, 50, 39,12, 22, 51, 44, 57, 58, 41, 49,18, 23, 59,40, 28, 35, 34, 25, 43, 32, 33, 24,16,15,10). This group G is imprimitive, with blocks of sizes 3, 6 and 30. The action of G on the blocks of size 3 (which are {1,55,56} and its images) gives an epimorphism from G to the group G above, with elementary abelian kernel K of order 310. Let p = a-1b, q = a-1pa, r = a-1qa and s = a-1ra. These elements satisfy the relations p2 = q2 = r2 = s2 = b-1pbq = b-1qbr = b-1pqrbs = (rs)2 = (qs)2 = 1, as well as pa-1b = a-1paq = a-1qar = a-1ras = 1 and others. The subgroup H generated by p, q, r and s has order 16, and just like H above, is isomorphic to D4 x C2. Theorem 5.1. With the notation above, let X be the underlying graph of the orbital digraph X(G, V; W), where V is the coset space (G : H) and W is the sub-orbit {Ha, Hb}. Then X is a 4-valent half-arc-transitive graph of order 90 • 310, with automorphism group G, and vertex-stabiliser H = Gv = D4 x C2. In fact, X is a regular cover of X, and the action of G on X projects to the action of G on X. Proof. The sub-orbit W = {Ha, Hb} is non-self-paired, and so X is 4-valent, and G acts half-arc-transitively on X, with vertex-stabiliser H = D4 x C2. The group G has order 1440 • 310, and so the order of X is as given. This order makes X too large to construct and analyse easily using Magma, but nevertheless we can study the permutation representation of G on the right coset space V closely enough to prove that G is the full automorphism group of X. We will not provide all details, but we explain most of the argument below. First, let '1' be the vertex H in X, and let x1, x2, x3 and x4 be the four neighbours of 1 (which are the cosets Ha, Hb, Ha-1 (= Hb-1) and Ha-1s). Then by vertex-transitivity, the edges of X are images of the edges {1, xj under the action of elements of G. We can use that fact to find all vertices within a given small distance from vertex 1. The numbers of vertices at distances 0 to 9 from vertex 1 are 1, 4, 12, 36, 108, 324, 972, 2 916, 8 748 and 26 050, respectively. It turns out that no two vertices at distance 8 from vertex 1 are adjacent, and that no three such vertices have a common neighbor at distance 9 from vertex 1. It follows that the girth of X is 18, and that there are 3 • 8748 - 26050 = 194 girth cycles (of length 18) in X containing the vertex 1. Moreover, we find easily that a given 1-arc with initial vertex 1 lies in 97 girth cycles, a given 2-arc with initial vertex 1 lies in 31 or 35 girth cycles, a given 3-arc with initial vertex 1 lies in 10, 11 or 13 girth cycles, a given 4-arc with initial vertex 1 lies in 3, 4 or 7 girth cycles, and a given 5-arc with initial vertex 1 lies in 1, 2, 3 or 5 girth cycles. In fact, just one of the 2-arcs extending a given 1-arc (1, xi) lies in 35 girth cycles, with the other two lying in 31 . Now if a 2-arc of the form (u, v, w) lies in exactly 35 girth cycles, let us call the vertex w the 'twin' of vertex u at vertex v. This gives a 'twinning' (or pairing) of neighbours at each vertex in the usual way, but an important point is that this is a property of X (rather than just the action of G). Also it shows that the stabiliser in Aut X of any vertex of X is a 2-group. Label the neighbours of vertex 1 so that jxi, x3} and {x2, x4} are pairs of twins, and for each xi, let yi be the twin of vertex 1 at xi, and let zi and wi be the other two (twin) neighbours of xi. Also let us calla 5-arc 'special' if it lies in a unique girth cycle. If a 5-arc (u, xi, 1, xj ,v) with middle vertex 1 is special, then the analysis shows that {xi, xj } is not a twin pair. Note that Aut X must permute the special 5-arcs among themselves. Next, let B be the ball of radius 2 centred at vertex 1, and consider the pointwise stabiliser of B in AutX. If the automorphism h fixes every vertex in B, then for every 2-arc of the form (1, xi, v), we see that h fixes the twin of xi at vertex w, and for every special 5-arc (u, xi, 1, xj, v) with middle vertex 1, also h fixes every vertex in the unique girth cycle that contains it. These two observations are enough to show that h fixes every vertex at distance 3 from vertex 1. Then by induction and connectedness, the automorphism h is trivial. Hence every automorphism of X is completely determined by its effect on B. Now consider the permutations induced by each of p, q, r, s and a on B. We can choose the labels xi, yi, zi and wi of the 16 vertices of B \ {1} such that • p induces (x1, x3)(y1, y3)(z1, z3)(w1, w3)(z4, w4), and fixes all other vertices of B, • q induces (z1, w1)(z3, w3), and fixes all other vertices of B, • r induces (z2, w2)(z4, w4), and and fixes all other vertices of B, • s induces (x2, x4)(y2, y4)(z2, z4)(w2, w4)(z3, w3), and fixes all other vertices of B, and similarly, • a induces a permutation of the form (1, x1, z1,..., z2, x2)(x3, w2,..., y2)(x4, y1,..., w2) fe ..o^ ...x^ ...x^ ...x^ ...x^...).... Next, suppose there is some automorphism g of X that fixes vertex 1 and all its neighbours xi, and the twins yi of vertex 1 at those neighbours, but does not lie in the subgroup generated by q and r. Note that this automorphism g would have to fix or swap each pair {zi, wi}. Multiplying by q or r or qr if necessary, we can suppose that g fixes also the vertices z1, w1, z2 and w2, and so g induces either (z3, w3) or (z4, w4) or (z3, w3)(z4, w4) on the 2-ball B. Then in particular, by considering the permutations induced by various elements on the 2-ball B, and the fact that the point wise stabiliser of B is trivial, we deduce that g centralises q and r, and conjugates p to either p or pq, and conjugates s to s or sr. Similarly, g conjugates a to something of the form ah, where h fixes every vertex in B with the possible exception of z3, w3, z4 and w4, and in particular, it follows that h commutes with each of q and r, and h conjugates p to p or pq, and conjugates s to s or sr, as well. But q = p°, and so if g conjugates p to pq, we find that q = qg = (p°)g = (pg)°s = (pq)ah = (qr)h = qr, which forces r to be trivial, contradiction. Thus pg = p. Since p induces (x1, x3)(y1, y3)(z1, z3)(w1, w3)(z4, w4), it follows that g fixes z3 and w3, and hence g induces (z4, w4) on B. Thus every automorphism of X that fixes all the vertices of the 2-ball B apart from (possibly) z3, w3, z4 and w4, must fix z3 and w3, and so equals the identity or g. In particular, h =1 or g, and so g conjugates a to either a or ag. But also we know that g centralizes each of p, q and r, and conjugates s to sr, and hence sr = sg = (r°)g = (rg)°s = rah = sh. Thus h conjugates s to sr as well, and so h cannot be trivial, so h = g. In particular, g conjugates a to ag. But that means g-1ag = ag, which forces g-1 to be trivial, contradiction. Hence no such g exists. Next, for the moment, suppose that X is half-arc-transitive, but G is not the full automorphism group of X. Then there must be some automorphism g of X fixing the vertex 1 and its neighbours xi (and their twins уД but not lying in G. By what we just showed above, however, this is impossible. Hence if G = Aut X, then X must be arc-transitive, and Aut X contains G as a subgroup of index 2. In particular, G is normal in Aut X. Finally, suppose X is arc-transitive, and let t be any automorphism of X taking the arc (1, x1 ) to the arc (1, x2). Then by the 'twinning' property of X, we know that t must also take x3 to x4, and so t induces either (x 1, x2, x3, x4 ) or (x 1, x2 )(x3, x4 ) on the neighbours of 1. Multiplying by p if necessary, we may assume that t induces the double transposition (x1, x2)(x3, x4) on the neighbours of 1, and hence also that t induces (y1, y2)(y3, y4) on their 'twins'. Then since t swaps x1 with x2 and swaps y1 with y2, it must swap the set {z1,w1} of the other two neighbours of vertex x1 with the set {z2,w2} of the other two neighbours of vertex x2, and it follows that t induces either (z1, z2)(w1, w2) or (z1, z2, w1, w2) or (z1, w2, w1, z2) on those four vertices. Similarly, t induces either (z3, Z4)(w3, w4) or (z3, Z4, w3, w4) or (z3, w4, w3, Z4) on {z3, w3, Z4, w4}. This gives 16 possibilities for the effect of t on the 2-ball B, but since we can multiply t by q or r or qr (all of which fix the vertices 1, xi and у i for all i), we may reduce the possibilities for t to these four: t1 = (x1,x2)(x3,x4 )(y1,y2)(y3,y4)(z1,z2)(z3,z4)(w1,w2 )(w3,w4), t2 = (x1,x2)(x3,x4)(y1,y2)(y3,y4)(z1,z2,w1,w2)(z3, z4)(w3,w4), t3 = (x1, x2)(x3, x4)(y1, y2)(y3, y4)(z1, z2)(z3, w4)(w1, w2)(w3, z4), t4 = (x1, x2)(x3, x4)(y1, y2)(y3, y4)(z1, z2, w1, w2)(z3, w4)(w3, z4). In each case, it is easy to check that the candidate for t conjugates the permutation induced by q on the 2-ball to the permutation induced by r, and vice versa. Hence t-1qt = r, and t-1rt = q. Similarly, we find that t-1pt = s and t-1st = p when t = t1, while t-1pt = s and t-1st = pq when t = t2, and t-1pt = rs and t-1st = pq when t = t3, and t-1pt = rs and t-1st = p when t = t4. Now if t = t1, we find that atat-1 fixes all the vertices 1, xi and yi, and so by our earlier observations, atat-1 lies in the subgroup generated by q and r. Since also atat-1 fixes z2 and w2, we deduce that atat-1 = 1 or q, so tat-1 = a-1 or a-1q. Rearranging these (and using the fact that t-1 qt = r), we find that also t-1at = a-1 or a-1q. But a MAGMA computation shows there is no automorphism of the group G taking a to a-1, and p to s (and q to r, etc.), and also there is no automorphism of G taking a to a-1q, and p to s (and q to r, etc.). Hence t = t1. Similarly, if t = t2, we find that t-1at = sa-1 or sra-1, which upon rearrangement gives t-1 at = a-1r or a-1rq, but another MAGMA computation shows there is no automorphism of G taking a to a-1r or a-1rq, and p to s (and q to r, etc.). Hence t = t2. If t = t3, then t-1at = a-1 or a-1q, as in the case of t1. But if t conjugates a to a-1, and conjugates p to rs (as t3 does), then t conjugates q = pa to (rs)a 1 = arsa-1 = qr, rather than r, contradiction. Similarly, if t conjugates a to a-1q, and conjugates p to rs (as t3 does), then t conjugates q to rq, again a contradiction. Hence t = t3. Similarly, if t = t4, then t-1at = a-1r or a-1rq, but both of these possibilities give contradictions when combined with the facts that t conjugates p and q to rs and r, so t = t4. Thus no such t exists, and so X is half-arc-transitive, as claimed. □ 6 Answer to a question of Delorme We begin this section with the following: Proposition 6.1. Let X be a half-arc-transitive graph of valence 2k, and let G = Aut X. Then if A is one of the two orbits of G on the arcs of X, then the digraph D with vertex-set V(X) and arc-set A is regular, and admits G as a group of automorphisms, but D is not isomorphic to its reverse. Proof. First, D has out-valence k and in-valence k, and obviously G acts on D as a group of automorphisms. If D were isomorphic to its reverse digraph Rev(D), then the isomorphism from D to Rev(D) would be an automorphism of X not contained in G, which is a contradiction. Hence D is not self-reverse. □ Next, recall that an s-arc in a digraph D is a sequence (v0, v1,..., vs) of s + 1 vertices such that any two consecutive vertices vi-1 and vž form an arc (vž-1, vž).) We note that if k (the in- and out-valence of D) is 2, and s is the largest positive integer such that G acts transitively on the s-arcs of D, then it is well known that |Gv | = 2s (see for example [19]). As a consequence of these things, we have an answer to Delorme's question in [5]: Theorem 6.2. Let X be the unique half-arc-transitive 4-valent graph of order 256, with automorphism group G of order 1024 and vertex-stabiliser Gv = V4. Then the corresponding digraph D is vertex-transitive, arc-transitive and 2-arc-transitive, but is not self-reverse. Moreover, this is the smallest 4-valent digraph with these properties. The example provided by Theorem 6.2 is the smallest such digraph that comes from a half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, but in principle, a 2-arc-transitive non-self-reverse digraph could also come from an arc-transitive 4-valent graph X that admits a half-arc-transitive group action. (Indeed there is an arc-transitive 4-valent graph Y on 21 vertices with vertex-stabiliser D4 such that Aut Y contains a half arc-transitive subgroup G with Gv = Z2, such that the corresponding orbital digraph D = X (G, V ; W ) is not self-reverse; but this example is not 2-arc-transitive.) An inspection of the database of all such graphs of small order [18], however, shows that there is no such graph with fewer than 256 vertices, and so the above example of order 256 is the smallest. On the other hand, there are infinitely many such examples, since there are infinitely many half-arc-transitive 4-valent graphs with vertex-stabiliser V4; see [11, 16]. In fact, there is an infinite family of half-arc-transitive covers of the smallest example given above. With the help of the Reidemeister-Schreier process (implemented as the Rewrite command in Magma [1]), it can be shown that the kernel of the epimorphism from the group (p, q, a | p2 = q2 = (pq)2 = a-1paq = a8 = (pa-2qa2)2 = 1} to the group of order 1024 above has abelianisation Z23 ф Z10. It follows that for every odd positive integer m, there is a half-arc-transitive regular cover of order 256m10, with abelian covering group K = Zm0. 7 Final remarks In this paper, we have described the smallest 4-valent half-arc-transitive graphs with vertex-stabilisers of order 4 and 8, respectively. In each case, the vertex-stabiliser is abelian. It is also known that for any positive integer s, there is at least one 4-valent half-arc-transitive graph with abelian vertex-stabilisers of order 2s ; see [16]. The graphs constructed in [16], however, have very large orders. Hence the following question arises naturally. Question 7.1. What is the order of a smallest 4-valent half-arc-transitive graph with abelian vertex-stabiliser of order 2s, for each s > 4? Next, we have also found anew 4-valent half-arc-transitive graph with vertex-stabilisers isomorphic to the dihedral group D4, having the same order as the smallest known such graph (on 10 752 vertices). It is not yet known, however, if these two graphs are the smallest such examples. Hence we pose the following question. Question 7.2. Is there a 4-valent half-arc-transitive graph of order less than 10 752, with non-abelian vertex-stabilisers of order 8? We have also constructed the first known example of a 4-valent half-arc-transitive graph with vertex-stabilisers that are non-abelian and non-dihedral. This has order 5 314 410, with vertex-stabilisers of order 16. We conclude the paper with the following two questions. Question 7.3. Is there a 4-valent half-arc-transitive graph of order less than 5 314410, with non-abelian, non-dihedral vertex-stabilisers? Question 7.4. Does there exist a 4-valent half-arc-transitive graph with non-abelian vertex-stabilisers of order 2s, for every s > 3? References [1] W. Bosma, J. Cannon and C. Playoust, The Magma Algebra System I: The User Language, J. Symbolic Comput. 24 (1997), 235-265. [2] I. Z. Bouwer, Vertex and edge transitive, but not 1-transitive, graphs. Canad. Math. Bull. 13 (1970), 231-237. [3] M. Conder and P. Dobcsanyi, Applications and adaptations of the low index subgroups procedure, Math. Comp. 74 (2005), 485-497. [4] M. D. E. Conder and D. Marušic, A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer, J. Combin. Theory Ser. B 88 (2003), 67-76. [5] C. Delorme, Cayley digraphs and graphs, European J. Combin. 34 (2013), 1307-1315. [6] J. D. Dixon and B. Mortimer, Permutation Groups, Springer-Verlag, New York, 1996. [7] P. G. Doyle, On transitive graphs, Senior Thesis, Harvard College, 1976. [8] Y. Q. Feng, J. H. Kwak, M. Y. Xu and J. X. 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Wilson, A census of edge-transitive tetravalent graphs, http://jan.ucc.nau.edu/ ~swilson/C4Site/BigTable.html. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 163-175 A note on m-factorizations of complete multigraphs arising from designs* * Gyorgy Kiss f Department of Geometry andMTA-ELTE GAC Research Group Eötvös Lorand University 1117 Budapest, Pazmany s. 1/c, Hungary Christian Rubio-Montiel * Instituto de Matemàticas Universidad Nacional Autonoma de México Ciudad Universitaria, 04510, D.F., Mexico Received 7 September 2013, accepted 8 May 2014, published online 28 September 2014 Abstract Some new infinite families of simple, indecomposable m-factorizations of the complete multigraph ЛKv are presented. Most of the constructions come from finite geometries. Keywords: Graph factorization, affine and projective spaces, spread. Math. Subj. Class.: 05C70, 51E23 1 Introduction The complete multigraph ЛК has v vertices and Л edges joining each pair of vertices. An m-factor of the complete multigraph ЛК is a set of pairwise vertex-disjoint m-regular subgraphs, which induce a partition of the vertices. An m-factorization of ЛК-ц is a set of pairwise edge-disjoint m-factors such that these m-factors induce a partition of the edges. An m-factorization is called simple if the m-factors are pairwise distinct. Furthermore, an m-factorization of ЛК-ц is decomposable if there exist positive integers and such *The research was supported by the Mexican-Hungarian Intergovernmental Scientific and Technological Cooperation Project, Grant No. TÉT 10-1-2011-0471. t Author was supported by the Hungarian National Foundation for Scientific Research, Grant Nos. K 72537 and K 81310. ^Author was partially supported by CONACyT of Mexico, Grant Nos. 166306 and 178395; and PAPIIT of Mexico, Grant No. IN101912. E-mail addresses: kissgy@cs.elte.hu (Gyorgy Kiss), christian@matem.unam.mx (Christian Rubio-Montiel) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ that pi + p2 = A and the factorization is the union of the m-factorizations piKv and p2Kv, otherwise it is called indecomposable. There is no direct correspondence between simplicity and indecomposability. Many papers deal with m-factorizations of graphs andmultigraphs. This is an interesting problem in its own right, but it is motivated by several applications, too. In particular if m = 1, then a one-factorization of Kv corresponds to a schedule of a round robin tournament. For a comprehensive survey on one-factorizations we refer to [29]. A special case of 2-factorizations is the famous Oberwolfach problem, see e.g. [2, 8]. Several authors investigated 3-factorizations of AKv with a certain automorphism group, see e.g. [1, 24]. In general, decompositions of AKv is also a widely studied problem, see e.g. [12, 13, 18, 27]. As m increases, the structure of an arbitrary m-factor of AKv can be much more complicated and the existence problem becomes much more difficult. In this paper we restrict ourselves to construct factorizations in which all factors are regular graphs of degree m whose connected components are complete graphs on (m +1) vertices. In the case m =1 an indecomposable one-factorization of AK2n is denoted by IOF(2n, A). Only a few conditions on the parameters are known: if IOF(2n, A) exists, then A < 1 • 3 • ... • (2n — 3) [4]; each IOF(2n, A) can be embedded in a simple IOF(2s, A), provided that A < 2n < s [16]. Six infinite classes of indecomposable one-factorizations have been constructed so far, namely a simple IOF(2n, n — 1) when 2n — 1 is a prime [16], IOF(2(A + p), A) where A > 2 and p is the smallest prime wich does not divide A [3] (an improvement of this result can be found in [15]), a simple IOF(2h + 2, 2) where h is a positive integer [28], IOF(q2 + 1, q — 1) where q is an odd prime number [26], a simple IOF(q2 + 1, q +1) for any odd prime power q [25], and a simple IOF(q2, q) for any even prime power q [25]. Most of these constructions arise from finite geometry. The aim of this paper is to construct new simple and indecomposable m-factorizations of AKv for different values of m, A and v. In Section 2 we recall the basic combinatorial properties of designs and the geometric properties of finite affine and projective spaces. We also describe a general construction method of m-factorizations which is based on spreads of block designs. In Sections 3 and 4 affine spaces and projective spaces, respectively, are the key objects. We present several new multigraph factorizations using subspaces, subgeometries and other configurations of these structures. 2 Preliminaries In this section we collect some concepts and results from design theory. For a detailed introduction to block designs we refer to [14]. 2.1 Designs Let v, b, k, r and A be positive integers with v > 1. Let D = (P, B, I) be a triple consisting of a set P of v distinct objects, called points of D, a set B of b distinct objects, called blocks of D, and an incidence relation I, a subset of P x B. We say that x is incident with y (or y is incident with x) if and only if the ordered pair (x, y) is in I. D is called a 2 — (v, b, k, r, A) design if it satisfies the following axioms. (a) Each block of D is incident with exactly k distinct points of D. (b) Each point of D is incident with exactly r distinct blocks of D. (c) If x and y are distinct points of D, then there are exactly A blocks of D incident with both x and y. A 2 - (v, b, k, r, A) design is called a balanced incomplete block design and is denoted by (v, k, A)-design, too. The parameters of a 2 - (v, b, k, r, A) design are not all independent. The two basic equations connecting them are the following: vr = bk and r(k - 1) = A(v - 1). (2.1) These necessary conditions are not sufficient, for example no 2 - (43,43, 7,7,1) design exists. 2.2 Resolvability A resolution class (or, a parallel class) of a (v, k, A)-design is a partition of the point-set of the design into blocks. In general, an f -resolution class of a design is a collection of blocks, which together contain every point of the design exactly f times. A resolution of a design is a partition of the block-set of the design into r resolution classes. A (v, k, A)-design with a resolution is called resolvable. Necessary conditions for the existence of a resolvable (v, k, A)-design are A(v -1) = 0 (mod (k - 1)), v = 0 (mod k) and b > v + r - 1, (see [9]). Let D = (P, B, I) be a (v, k, A)-design, where P = {pbp2,... ,pv} is the set of its points and B = {B1,B2,..., Bb} is the set of its blocks. Identify the points of D with the vertices of the complete multigraph AKv. Then in the natural way, the set of points of each block of D induces in AKv a subgraph isomorphic to Kk. For Bj g B, let Gj be the subgraph of AKv induced by Bj. Then it follows from the properties of D that a resolution class of D gives a (k - 1)-factor of AKv and a resolution of D gives a (k - 1)-factorization of AKv. Hence we get the following well-known fact. Lemma 2.1 (Basic Construction). The existence a resolvable (v, k, A)-design is equivalent to the existence of a (k - 1)-factorization of the complete multigraph AKv. 2.3 Projective and affine spaces Most of our factorizations come from finite geometries. In this subsection we collect the basic properties of these objects. For a more detailed introduction we refer to the book of Hirschfeld [22]. Let Vn+1 be an (n + 1)-dimensional vector space over the finite field of q elements, GF(q). The n-dimensionalprojective space PG(n, q) is the geometry whose k-dimensional subspaces for k = 0,1,..., n are the (k + 1)-dimensional subspaces of Vn+1 with the zero deleted. A k-dimensional subspace of PG(n, q) is called k-space. In particular subspaces of dimension zero, one and two are respectively a point, a line and a plane, while a subspace of dimension n - 1 is called a hyperplane. The relation ~ x ~ y ^ 3 0 = a G GF(q) : x = ay is an equivalence relation on the elements of Vn+1 \ 0 whose equivalence classes are the points of PG(n, q). Let v = (v0, v1,..., vn) bea vector in Vn+1 \ 0. The equivalence class of v is denoted by [v]. The homogeneous coordinates of the point represented by [v] are (vo : v1 : ... : v„). Hence two (n + 1)-tuples (xo : X1 : ... : x„) and (yo : y1 : ... : y„) represent the same point of PG(n, q) if and only if there exists 0 = a G GF(q) such that xj = ayj holds for i = 0,1,..., n. A k-space contains those points whose representing vectors x satisfy the equation xA = 0, where A is an (n +1) x (n — k) matrix of rank n — k with entries in GF(q). In particular a hyperplane contains those points whose homogeneous coordinates (x0 : x1 : ... : xn) satisfy a linear equation uoxo + U1X1 + • • • + UnXn = 0 where u G GF(q) and (u0, u1,..., un) = 0. The basic combinatorial properties of PG(n, q) can be described by the q-nomial coefficients. [k]q equals to the number of k-dimensional subspaces in an n-dimensional vector space over GF(q), hence it is defined as n]_ (qn — 1)(qn — q) ... (qn — qk-1) k q := (qk — 1)(qk — q)... (qk — qk—1) ' The proof of the following proposition is straightforward. Proposition 2.2. • The number of k-dimensional subspaces in PG(n, q) is [k+i] q. • The number of k-dimensional subspaces of PG(n, q) through a given d-dimensional (d < k) subspace in PG(n, q) is [n-d . • In particular the number of k-dimensional subspaces of PG(n, q) through two distinct points in PG(n, q) is [n-i] q. If is any hyperplane of PG(n, q), then the n-dimensional affine space over GF(q) is AG(n, q) = PG(n, q) \ The subspaces of AG(n, q) are the subspaces of PG(n, q) with the points of deleted in each case. The hyperplane is called the hyperplane at infinity of AG(n, q), and for k = 0,1,..., n — 2 the k-dimensional subspaces in are called the k-spaces at infinity of AG(n, q). Let 1 < d < n be an integer. Two d-spaces of AG(n, q) are called parallel, if the corresponding d-spaces of PG(n, q) intersect in the same (d — 1)-space. The parallelism is an equivalence relation on the set of d-spaces of AG(n, q). As a straightforward corollary of Proposition 2.2 we get the following. Proposition 2.3. In AG(n, q) each equivalence class of parallel d-spaces contains qn-d subspaces. Projective and affine spaces provide examples of designs. Example 2.4. Let i < n be positive integers. The projective space PG(n, q) can be considered as a 2-design D = (P, B, I), where P is the set of points of PG(n, q), B is the set of i-spaces of PG(n, q) and I is the set theoretical inclusion. The parameters of D are - - q^11-1 b = rn+11 , k = , r = |"n] and Л = fn -11 . U+1J q' q— 1 ' L^q U— 1J q q-1 ' Li+1J q' q-1 ' L^q Example 2.5. Let i < n be positive integers. The affine space AG(n, q) can be considered as a 2-design D = (P, B, I), where P is the set of points of AG(n, q), B is the set of i-spaces of AG(n, q) and I is the set theoretical inclusion. The parameters of D are v = q b = qn-i tn]q, k = qj, r = tn]q and Л = tn-!]( n In the rest of this paper Examples 2.4 and 2.5 will be denoted by PG(i)(n, q) and by AG (i)(n, q), respectively. We will use the terminology from geometry. An i-spread, S j, of PG(n, q) (or of AG(n, q)) is a set of pairwise disjoint i-dimensional subspaces which gives a partition of the points of the geometry. In general, an f -fold i-spread, Sf, is a set of i-dimensional subspaces such that every point of the geometry is contained in exactly f subspaces of SJ. An i-packing, Pj, of PG(n, q) (or of AG(n, q)) is a set of spreads such that each i-dimensional subspace of the geometry is contained in exactly one of the spreads in Pj, i.e., the spreads give a partition of the i-dimensional subspaces of the geometry. The i-spreads, f -fold i-spreads and i-packings induce a resolution class, an f-resolution class and a resolution in PG(i) (n, q) (or in AG(i) (n, q)), respectively. It is easy to construct spreads and packings in AG(i) (n, q), because each parallel class of i-spaces is an i-spread. The situation is much more complicated in PG(i)(n, q). There are only a few constructions of spreads. The following theorem summarizes the known existence conditions. Theorem 2.6 ([22], Theorems 4.1 and 4.16). • There exists an i-spread in PG(i) (n, q) if and only if (i + 1) | (n + 1). • Suppose that i, I and n are positive integers such that (l +1)| gcd(i +1, n +1). Then there exists an f -fold i-spread in PG(i)(n, q), where f = (qi+1 — 1)/(q1+1 — 1). There exist several different 1-spreads (line spreads) in PG(1)(3,q). We briefly mention two types. Let l2 and l3 be three skew lines in PG(3, q). The set of the q + 1 transversals of l2 and l3 is called regulus and it is denoted by R(^1, 4 ). The classical construction of a line spread comes from a pencil of hyperbolic quadrics (see e.g. [20], Lemma 17.1.1) and it has the property that if it contains any three lines of a regulus R(^1, 4), then it contains each of the q + 1 lines of R(^1, ^3). This type of spread is called regular. A line spread in PG(3, q) is called aregular, if it contains no regulus. An example of an aregular spread can be found in [20], Lemma 17.3.3. 3 Factorizations arising from affine spaces In this section, we investigate the spreads and packings of AG(n, q) and the corresponding factorizations of multigraphs. In each case we apply Lemma 2.1, so we identify the points of AG(n, q) with the vertices of the complete multigraph. Theorem 3.1. Let q be a prime power, i < n be positive integers and Ai = [n-11]q. Then there exists a simple (qj — 1 )-factorization of Ai Kqn. is decomposable if and only if there exists an f-fold (i — 1)-spread in PG(i-1)(n — 1, q) for some 1 < f < Ai. Proof. Consider the n-dimensional affine space as AG(n, q) = PG(n, q) \ where is isomorphic to PG(n — 1, q). Take the design D = AG(i)(n, q) and apply Lemma 2.1. If nj-1 is an (i — 1)-space of , then the set of the qn-i parallel affine i-spaces through nj-1 is an i-spread of D. This spread induces a (qj — 1)-factor Fj for j e {1,..., r}. If п-1, П2-1,..., ng-1 are distinct (i — 1)-spaces of and they form an f-fold spread, then f = (g(qj — 1))/(qn — 1), and the union of the corresponding (qj — 1)-factors Fj, for j = 1,2,..., g, gives a (qj — 1)-factorization of f Kqn. Distinct (i — 1)-spaces of obviously define distinct (дг — 1)-factors, so this factorization is simple. In particular if we consider all (i — 1)-spaces of then n , f = q n qi — 1 n—1 i i q qn — 1 i—1 hence the union of the corresponding factors gives a simple (ql — 1)-factorization Fi of AjKqn . Suppose that F% is decomposable, then there exist two positive integers and p2 such that + p2 = A i and Fi can be written as the union Fi = F1 U F2; F1 and F2 are (ql — 1)-factorizations of p1Kqn and p2Kqn, respectively, having no (qi — 1)-factors in common, since Fi is simple. For h = 1,2, the relation ph(q2 ) = (q2)qn-i|Fh| holds, hence ph(qn — 1) = (qi — 1)|Fh|. Without loss of generality we can set F1 = uf1=1Fj with fi = (pi (qn — 1))/(qi — 1), and F2 = Fi \Fi, f2 = |F2|. Let u1 and u2 be two affine points and let w be the point at infinity of the line u1u2. Since Fh is a factorization of phKqn, there are exactly ph factors of Fh containing the edge [u1, u2], say Fi , Fj ,..., Fj . The edge [u1, u2] belongs the Fi if and only if w G П J 1 J 2 J f-bfo J s for every 1 < s < ph. This happens if and only if Uf= 1П 1 exactly ph times, which means that Uf=1n! 1 is a ph-fold spread in H 1 contains each point of for every h = 1, 2. It is thus proved that if Fi is decomposable, then PG(i f -fold spread for some 1 < f < Ai. _1)(n — 1, q) posesses an Vice versa, suppose that there exists a p1-fold spread in PG(i 1) (n — 1, q) for some 1 < p1 < Ai. Let F1 = Uf= 1Fj be a p1-fold spread in Then |F1| = f1 = p1(qn — 1)/(qi — 1). Let T be the set of all (i — 1)-dimensional subspaces in and let F2 = T\F1. Then |T| = ["tq, hence |f2| — p1(qn — 1)/(qi — 1) n—1 i1 p1 qn — 1 qi — 1 , so if p2 = — p1, then F2 is a p2-fold spread in and 1 < p2 < Ai holds. As we have already seen, Fh defines a (qi — 1)-factorization of phKqn for h = 1, 2. Then Fi = F1 U F2, because p1 + p2 = Ai. Hence the (qi — 1)-factorization Fi of AiKqn is decomposable. □ A g q q q Corollary 3.2. If gcd(i, n) > 1 then the (qi — 1)-factorization Fi of AiKqn is decomposable. Proof. Let 1 < I + 1 be a divisor of gcd(i, n). Then it follows from Theorem 2.6 that there exists an (qi — 1)/(q1+1 — 1)-fold spread in so Fi is decomposable. □ To decide the decomposability of Fi in the cases gcd(i, n) = 1 is a hard problem in general. We prove its indecomposability in the following important case. Theorem 3.3. The (qn-1 — 1)-factorization Fn-1 of (qn-1 — 1)/(q — 1)Kqn is indecomposable. Proof. It is enough to prove that if ug=1n" 2 is an /-fold (n - 2)-spread in then ug=1n"-2 consists of all (n - 2)-dimensional subspaces of because this implies / = An-1, so the statement follows from Theorem 3.1 J " ing of the point-subspace pairs p e nj 2 in gives Each nj 2 contains exactly (qn 1 - 1 )/(q - 1) points, thus the standard double count- hence qn-1 - 1 qn - 1 g-7— = f-^ q - 1 q - 1 f = g(qn-1 - 1) qn - 1 But gcd(qn - 1, qn-1 - 1) = q - 1 and / is an integer, so g > (qn - 1 )/(q - 1) which implies g = (qn - 1 )/(q - 1), hence / = An-1. □ In particular if n = 2, we get the following. Corollary 3.4. If q is a prime power then there exists a simple and indecomposable (q -1)-factorization of Kq2. If q = 2r then each (ql - 1)-factor in F® is the vertex-disjoint union of 2r-® complete graphs on 2® vertices. It is well-known that these graphs can be partitioned into one-factors in many ways (but not in all the ways, it was proved by Hartman and Rosa [19], that there is no cyclic one-factorization of K2i for i > 3), hence Theorem 3.1 implies several one-factorizations of A®K2r. Each of the one-factorizations arising from F® is simple, because distinct (i - 1)-dimensional subspaces define distinct (q® - 1)-factors of F®, and the one-factors of A®Kqn arising from distinct (q® - 1)-factors of F® are distinct, because they are the union of qn-® one-factors on q® vertices of a connected component. There are both decomposable and indecomposable one-factorizations among these examples. We show it in the smallest case q = 2, n = 3. Let F2 be the 3-factorization of 3Kg induced by AG(3,2). Let PG(3,2) = AG(3, 2) U Then is isomorphic to the Fano plane. Let its points be 0,1, 2,3,4, 5 and 6 such that for j = 0,1,..., 6, the triples Lj = ( j, j +1, j + 3) form the lines of the plane, where the addition is taken modulo 7. Now the 3-factors of F2 can be described in the following way. Let a be a fixed point in AG(3,2). Then Lj defines a 3-factor Fj whose connected components are complete graphs K2i = K4. Let Lj,a be the complete graph containing a, and let Lj a be the other component of Fj. defines one-factors and a one-factorization of K8 in the following obvious way. The edge joining two points of AG(3, 2), say b and c, belong to the one-factor Gs if and only if b, c and s are collinear points in PG(3,2). Then G — U^—gGs is a one-factorization of K8. We can define a decomposable one-factorization of 3K8 in the following way. Take Lja and L ja and let s e Lj be any point. Then Gs gives a one-factor of Lja and a one-factor of Lja. Hence Gj = UseLjGs is the union of three one-factors of 3K8, and G' = u6=0Gj is a one-factorization of 3k8. In there are three lines through the point s, hence G ' contains each one-factor Gs three times. Thus G' is decomposable, because it is obviously the union of three copies of G. But we can define an indecomposable one-factorization, too. Let L j be a line in take Lj,a and Lj,a and let M j be the one-factor which contains the following pairs of points in AG(3, 2) : ' - (b, c) if b,c G Lj,a and b, c, j are collinear in PG(3, 2). - (b, c) if b, c G Lj,a and b, c, j + 1 are collinear in PG(3,2). Let M j be the one-factor which contains the following pairs of points in AG(3, 2) : - (b, c) if b, c G Lj,a and b, c, j + 1 are collinear in PG(3,2). - (b, c) if b, c G L j,a and b, c, j + 3 are collinear in PG(3,2). Finally let M j be the one-factor which contains the following pairs of points in AG(3,2) : - (b, c) if b, c G Lj,a and b, c, j + 3 are collinear in PG(3,2). - (b, c) if b, c G L ja and b, c, j are collinear in PG(3, 2). Then M j = uf=1 Mj is a union of three one-factors of 3K8, and M = U j=0Mj is a one-factorization of 3K8. Suppose that this one-factorization is decomposable. Then it contains a one-factorization E of K8. E is the union of seven one-factors. We may assume without loss of generality, that M0j belongs to E. It contains an edge through a, let it be (a, b), and a pair (c, d) for which the lines ab and cd are parallel lines in AG(3,2). There are two more lines in the parallel class of ab, say ef and gh. It follows from the definition of the one-factors that exactly one of them contains the pairs (e, f ) and (a, b), another one contains the pairs (e, f ) and (c, d), and a third one contains the pairs (e, f ) and (g, h). But E contains each pair exactly once, hence it must contain the one-factor containing the pairs (e, f ) and (g, h). But this is a one-factor of type M0, where t = 1. Hence E contains M0 where t = 2 or 3. If we repeat the previous argument, we get that E must contain M0 for 1 = l = t, too. Thus E is the union of triples of type Mj, t = 1, 2,3, but this is a contradiction, because E consists of seven one-factors. 4 Factorizations arising from projective spaces There are two basic types of partitioning the point-set of finite projective spaces. Both types give factorizations of some multigraphs. In this section we discuss these constructions. 4.1 Spreads consisting of subspaces It is easy to construct spreads in PG(i) (n, q), Theorem 2.6 gives a necessary and sufficient existence condition. Packings are much more complicated objects. Only a few packings in PG(1)( n, q) have been constructed so far. In each case of the known packings either n or q satisfies some conditions. Theorem 4.1 (Beutelspacher, [6]). Let 1 < k be an integer and let n = 2k — 1. Then there exists a packing in PG(1)( n, q) . Theorem 4.2 (Baker, [5]). Let 1 < k be an integer. Then there exists a packing in PG(1)(2k - 1, 2). Applying the Basic Construction Lemma, we get the following existence theorems. a — 1 Corollary 4.3. Let q be a prime power, 1 < k be an integer and v = aa-1 . Then there exists a q-factorization of Kv induced by a line-packing in PG(2k — 1, q). Corollary 4.4. Let 1 < k be an integer and v = qq-1 . There exists a 2-factorization K, induced by a line-packing in PG(2k — 1, 2). If k = 2 then Corollary 4.4 gives a solution of Kirkman's fifteen schoolgirls problem, which was first posed in 1850 (for the history of the problem we refer to [7]), while Corollary 4.3 gives a solution of the generalised problem in the case of (q2 + 1)(q + 1) schoolgirls. The complete classification of packings in PG(i)(n, q) is known only in the case i = 1, n = 3 and q = 2. There are 240 projectively distinct packings of lines in PG(3,2) (see [20], Subsection 17.5). If gcd(q + 1, 3) = 3, then there is a construction of aregular spreads in PG(1)(3,q) due to Bruen and Hirschfeld [11] which is completly different from the constructions of Theorems 4.1 and 4.2. It is based on the geometric properties of twisted cubics. A normal rational curve of order 3 in PG(3, q) is called twisted cubic. It is known that a twisted cubic is projectively equivalent to the set of points {(t3 : t2 : t : 1) : t G GF(q)} U {(1 : 0 : 0 : 0)}. In [20] it was shown that there exist aregular spreads given by a twisted cubic. For a detailed description of twisted cubics and the proofs of the following theorems we refer to [20], Section 21. Theorem 4.5. Let Gq be the group of projectivities in PG(3, q) fixing a twisted cubic C. Then • Gq = PGL(2, q) and it acts triply transitively on the points of C. • If q > 5 then the number of twisted cubics in PG(3, q) is q5(q4 — 1)(q3 — 1). Theorem 4.6. Let C be a twisted cubic in PG(3, q). If gcd(q +1,3) =3, then there exists a spread in PG(1) (3, q) induced by C. Using the spreads associated to twisted cubics and the Basic Construction Lemma, we get the following multigraph factorization. Theorem 4.7. Let q > 5 be a prime power, Л = q5(q4 — 1)(q—1) and v = q3 + q2 + q +1. If gcd(q + 1, 3) = 3, then there exists a simple q-factorization of ЛК induced by the set of twisted cubics in PG(3, q). Proof. Let C be the set of twisted cubics in PG(3, q). For C g C let LC be the spread in PG(1) (3, q) induced by C. If t is a line and q denotes the number of twisted cubics C with the property that t belongs to LC, then it follows from Theorem 4.5 that q does not depend on t. Hence |{twisted cubics in PG(3, q)}| x |{lines in a spread of PG(3, q)}| Q = |{lines in PG(3, q)}| _q5(q4 — 1)(q3 — 1) x (q2 + 1) _ 5, 4 (q2 + 1)(q2 + q + 1) q5 (q4 — 1)(q — 1). Thus C induces a |C|-fold spread in PG(1)(3, q). Éach spread LC induces a q-factor in Kv, therefore the Basic Construction Lemma gives that U LC is a q-factorization of ЛК. Ce с Any two distinct twisted cubics define different spreads, hence the factorization is simple by definition. □ 4.2 Constructions from subgeometries If the order of the base field is not prime, then projective spaces can be partitioned by subgeometries. Let 1 < k be an integer. Since GF(q) is a subfield of GF(qk), so PG(n, q) is naturally embedded into PG(n, qk) if the coordinate system is fixed. Any PG(n, q) embedded into PG(n, qk) is called a subgeometry. Using cyclic projectivities one can prove that any PG(n, qk) can be partitioned by subgeometries PG(n, q). For a detailed description of cyclic projectivities, subgeometries, and the proofs of the following three theorems we refer to [22], Section 4. Theorem 4.8 ([22], Lemma 4.20). Let s(n, q, qk) denote the number of subgeometries PG(n, q) in PG(n, qk). Then n+1 s(n,q,qk) = q(n+1)(k-1) Ц qki-1 q*-1 . Theorem 4.9 ([22], Theorem 4.29). PG(n, qk) can be partitioned into 0(n, q, qk) = ((qq'°-1)(-rt1++11q-11)) disjoint subgeometries PG(n, q) if and only if gcd(k, n + 1) = 1. Theorem 4.10 ([22], Theorem 4.35). Suppose that gcd(k, n +1) = 1. Let p0(n, q, qk) denote the number of projectivities which act cyclically on a PG(n, q) of PG(n, qk ) such that determine different partitions. Then n П (qki -1) po(n,q,qk) = qk(n+1) i=1 +, . n+1 Any given subgeometry PG(n, q) is contained in n П (q* - 1) , , (n+1) i=1 Po(n,q) = q( 2 )-—:— n+1 of these partitions. We can consider the partitions of the point-set of PG(n, qk) by subgeometries PG(n,q). Each partition of PG(n, qk) into subgeometries PG(n, q) defines a (q(q,_-1)) -factor k ( n+1)_1 of , with v — —j—. Each projectivity which acts cyclically on a PG(n, q) defines a -factorizations of the corresponding complete multigraph. Theorem 4.11. Let q be a prime power, 1 < k and n be positive integers for which ( n+1 ) k n_1 gcd(k, n + 1) = 1 holds. Let A = q( q2k-)1((nq+-)1()q(^n1-1) П (qki - 1) and v = ^k--1. i=1 Then there exist a simple ( q qq - ) -factorization of AKv induced by the set of those projec- q-1. tivities which act cyclically on a PG(n, q) of PG(n, qk) such that they determine different partitions. Proof. It follows from Theorem 4.8 that the number Se of subgeometries PG(n, q) through two points of PG(n, qk) is s(n, q, qk) X |{pointsin PG(n, q)}| X (|{pointsin PG(n, q)}|- 1) e = |{points in PG(n, qk)}| X (|{points in PG(n,qk)}| - 1) = q(n+1)(k —1)(qk - 1)qki - 1 = qk-1(q - 1) \\qj - 1 . Each cyclic projectivity determines different partitions, hence it determines different factors. Thus A = Se X p0(n, q). □ We cannot decide the decomposability of the factorization construted in the previous theorem in general, but we can prove the existence of indecomposable factorizations in some cases. To do this we need the following result from number theory. Lemma 4.12 ([22], Lemma 4.24). If r, s and x are positive integers with x > 1, then (Xr-1)(XX-1) is an integer if and only if gcd(r, s) = 1. We apply it in a particular case. Proposition 4.13. Let q be a prime power, 1 < k and n be positive integers for which / kn -i n -i \ k(n + 1) -i gcd(k, n + 1) = 1 and gcd(k, n) = 1 hold. Let d = gcd f , ^-—t) , v = q qk —i and m = q qq—j1. Suppose that F is an m-factorization of AKv for some A such that each factor is the disjoint union of 0(n, q, qk) complete graphs on (qn+1 - 1)/(q - 1) vertices. qn 1 k 1 qkn 1 If f denotes the number of m-factors in F then d(q—^ divides A and q 1 d(qk-1) divides f. q Proof. The standard double counting gives A X (2) = (m2+1) X 0(n,q,qk) X f, qkn 1 qn 1 qn 1 thus A X q 1 dqqk-1) = f X d(q—i). Because of Lemma 4.12, divides A, hence qk-1 divides f. □ As a direct corollary of the previous proposition we get the following result about the indecomposibility of the factorizations constructed in Theorem 4.11. Theorem 4.14. Let q be a prime power, 1 < k and n be positive integers for which / kn_л n_-i \ k(n + 1)_-i gcd(k, n + 1) = 1 and gcd(k, n) = 1 hold. Let d = gcd f 1, 3q-T)°' - q-1 qk —1 , q— 1 J , qk — 1 _ Til /7Х-) fbi /1 V/Ì ЛГ/ pf /f P 7 Т-И n 7/7 /im Si in Л/^ЛШ p q—1 qn 1 and m = q qq—1 . Then there exist a simple and indecomposable m-factorization of AKv ( n + 1 ) k n_1 where A = t dg—) for some t in {1,..., d ^ЛП+У П (qkì - 1)}. ì=1 Acknowledgement The authors are grateful to the anonymous reviewers for their detailed and helpful comments and suggestions. References [1] P. Adams, D. Bryant and B. Maenhaut, Cube factorizations of complete graphs, J. Combin. Des. 12 (2004), 381-388. [2] B. Alspach, The Oberwolfach problem, in: C. J. Colbourn and J. H. Dinitz (eds), CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, 394-395. [3] D. Archdeacon and J. H. Dinitz, Constructing indecomposable 1-factorizations of the complete multigraph, Discrete Math. 92 (1991), 9-19. [4] A. H. 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Sonnino, 1-factorizations of complete multigraphs arising from finite geometry, J. Combin. Theory Ser. A 93 (2001), 385-390. [27] B. R. Smith, Cycle decompositions of complete multigraphs, J. Combin. Des. 18 (2010), 85-93. [28] A. Sonnino, One-factorizations of complete multigraphs arising from maximal (k; n)-arcs in PG(2, 2h), Discrete Math. 231 (2001), 447-451. [29] W. D. Wallis, One-factorizations, Mathematics and its Applications, 390, Kluwer Academic Publishers Group, Dordrecht, 1997. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 177-194 Regular embeddings of cycles with multiple edges revisited Dedicated to Dragan Marusic on the occasion of his 60th birthday Kan Hu Faculty of Natural Sciences, Matej Bel University, 974 01 Banska Bystrica, Slovakia and Institute of Mathematics, Slovak Academy of Sciences, 975 49 Banska Bystrica, Slovakia and School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, Zhejiang 316000, People's Republic of China Roman Nedela Faculty of Natural Sciences, Matej Bel University, 974 01 Banska Bystrica, Slovakia and Institute ofMathematics, Slovak Academy ofSciences, 975 49 Banska Bystrica, Slovakia Martin Skoviera Department of Computer Science, Comenius University, 842 15 Bratislava, Slovakia Naer Wang Faculty of Natural Sciences, Matej Bel University, 974 01 Banska Bystrica, Slovakia and School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, Zhejiang 316000, People's Republic of China Received 10 March 2014, accepted 13 May 2014, published online 28 September 2014 ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract Regular embeddings of cycles with multiple edges have been reappearing in the literature for quite some time, both in and outside topological graph theory. The present paper aims to draw a complete picture of these maps by providing a detailed description, classification, and enumeration of regular embeddings of cycles with multiple edges on both orientable and non-orientable surfaces. Most of the results have been known in one form or another, but here they are presented from a unique viewpoint based on finite group theory. Our approach brings additional information about both the maps and their automorphism groups, and also gives extra insight into their relationships. Keywords: Regular embedding, multiple edge, Holder's Theorem, Möbius map. Math. Subj. Class.: 20B25, 05C10 1 Introduction Classification of all regular embeddings of a given graph on orientable or non-orientable surfaces has been addressed by many researchers in topological graph theory. On the abstract level, the classification problem was solved by Gardiner et al. where graphs which underlie a regular map were characterised by means of a condition requiring the existence of certain subgroups in the graph automorphism group [7]. This condition allows one to identify all existing map automorphism groups within the graph automorphism group and subsequently to determine all regular embeddings of the graph that have a specified subgroup as its automorphism group. Nevertheless, practical application of the condition depends on understanding the structure of the automorphism group of the graph and therefore has serious limitations. At present, a complete classification is known for only a few infinite classes of graphs, most notably, for complete graphs [8, 9, 27], complete bipartite graphs [11, 22], hypercubes [4, 14, 21], and for several others basic classes of graphs (see for example [6, 29]). In this paper we focus on regular maps whose underlying graph is a cycle with multiple edges; for brevity we call such maps multicyclic. Multicyclic maps can be regarded as combinations of two well understood families of maps: spherical embeddings of cycles and dipole maps. The study of these maps has a fairly long history which exceeds the context of the proper topological graph theory [1, 3, 5, 17]. In early papers, these maps typically occur in the dual form as maps where each face meets precisely two others; in [26] such maps are called bicontactual. An important special case, regular maps with two faces, was extensively discussed by Coxeter and Moser in their celebrated book [5], mentioning much older works of Brahana [2] and Threlfall [24]. In full generality, bicontactual regular maps were first considered in 1985 by Wilson [26]. Using a geometric approach depending on tracing map diagrams Wilson derived a classification of bicontactual maps on both orientable and non-orientable surfaces. Unfortunately, his result does not immediately translate via surface duality to regular embeddings of cycles with multiple edges, and even the basic information such as the orientability character or the number of regular E-mail addresses: kanhu@savbb.sk (Kan Hu), nedela@savbb.sk (Roman Nedela), skoviera@dcs.fmph.uniba.sk (Martin Skoviera), naerwang@savbb.sk (Naer Wang) embeddings for a given length and multiplicity of a cycle is hard to extract. In 1989, Wilson [27, Theorem 3] proved that a regular embedding of any graph with multiple edges is either a totally branched covering over a regular embedding of the corresponding simple graph, or a cantankerous map, a regular map with edge-multiplicity 2 where every 2-cycle is orientation-reversing. He went on to show that (among other things) an even cycle has a regular embedding with every multiplicity while an odd cycle has a regular embedding with every odd multiplicity but with no even multiplicity. Cantankerous maps, under a more appropriate term Möbius regular maps, were again studied by Li and Siran in [15, 16] within the context of maps with an unfaithful action of the automorphism group on the vertex set. With the help of general results about unfaithful maps they produced a classification of all multicyclic regular maps on both orientable and non-orientable surfaces [15, Propositions 7 and 8]. In particular, they proved that the doubled n-cycle admits a Möbius regular embedding if and only if n is divisible by 3. Nevertheless, neither the enumeration of isomorphism classes of the maps nor orientable regularity were treated in their papers. To complete the history of classification of multicyclic regular maps we should mention an unpublished work of Skoviera and Zlatos [23] where a general framework for the study of regular embeddings of graphs with multiplicity m based on Zm-valuations was developed. In a subsequent work [22], this theory was applied to deriving a classification of orientably regular embeddings of multicycles. Although the latter classification enables enumeration, no information about the automorphism groups of the maps was given. The present paper intends to complete the picture of multicyclic regular maps by providing a detailed description, classification, and enumeration of multicyclic maps on both orientable and non-orientable surfaces along with additional information about the automorphism groups of maps and relationships between them. In contrast to the majority of previous papers on this topic we deal with both orientably regular maps (where map automorphisms are necessarily orientation-preserving) and with regular maps (where map automorphisms include reflections and the maps may also lie on non-orientable surfaces). Our interest in these maps is substantiated by the fact that large classes of regular maps have multicyclic quotients and that multicyclic maps can often serve as important extremal examples [1, 3, 17]. The following two theorems, stated in a simplified enumerative form, are our main results. More detailed statements can be found in the subsequent sections. Throughout the paper сПт) denotes the graph resulting from the cycle Cn of length n by replacing each edge with m parallel edges. Theorem 1.1. Let p be the number of orientably regular embeddings of the graph сПт) where n > 3 and m > 2, and let p,(m) denote the number of solutions of the congruence e2 = 1 (mod m). Then (i) p = 0 if n is odd and m is even, (ii) p =1 if both n and m are odd, (iii) p = p,(m) if n is even and m is odd, (iv) p = 2p,(m) if n = 0 (mod 4) and m is even, (v) p = 2^(2m) if n = 2 (mod 4) and m is even. Furthermore, all these embeddings are reflexible. Theorem 1.2. Let q be the number of non-orientable regular embeddings of the graph сПт) where n > 3 and m > 2. Then (i) q =1 if both n and m are odd, in which case the antipodal double cover of the map is an orientably regular embedding of c2(m) corresponding to the solution e = —1 of the congruence e2 = 1 (mod m) listed in item (iii) of Theorem 1.1, (ii) q =1 if n = 0 (mod 3) and m = 2, in which case the map is a Möbius map, and (iii) q = 0 in all other cases. Theorem 1.2 has an interesting corollary which strengthens a result of Wilson [26] and shows that there exist no regular maps with nilpotent automorphism group on non-orientable surfaces of genus greater than 1. In contrast, nilpotent regular maps on orientable surfaces of arbitrarily large genus are abundant; for further information see [18]. Theorem 1.3. There exist no non-orientable regular maps whose automorphism group is nilpotent except the dihedral embeddings of bouquets of 2n circles into the projective plane and their duals, embeddings of cycles of length 2n, for n > 1. 2 Orientably regular embeddings of It is well-known that every d-valent orientably regular map M can be identified with a triple (G; x, y) where G is a finite group and {x, y} is a generating set for G with xd = y2 = 1; see, for example [10, 21]. Such a triple is called an algebraic orientably regular map. Elements of the group represent darts of M, that is, edges endowed with an orientation. The right translation g ^ gx, g g G, by the generator x corresponds to the rotation of the map, the permutation that cyclically permutes darts directed away from vertices consistently with the orientation of the surface. The translation g ^ gy, g g G, by the generator y corresponds to the dart-reversing involution, which switches the direction of each dart to the opposite direction. The vertices, edges, and faces of M are in a one-to-one correspondence with the left cosets of (x), (y), and (xy), respectively, and the incidence between the objects corresponds to non-empty intersection of cosets. A map homomorphism (G; x, y) ^ (G'; x', y') between algebraic maps (G; x, y) and (G'; x',y') is a group homomorphism G ^ G' that takes x to x' and y to y'. In topological terms, a map homomorphism corresponds to an orientation-preserving covering projection of maps, possibly branched over vertices, face-centres, and free ends of semiedges (where the branching index must be 2). It follows that two algebraic maps (G; x, y) and (G; x',y') represent isomorphic orientably regular maps if and only if there is an automorphism of G taking x to x' and y to y'. Each automorphism of the map M = (G; x, y) corresponds to a left translation g ^ ag, g g G, where a is a fixed element of G. In particular, the group Aut+(M) of all orientation preserving automorphisms of M is isomorphic to G. The map automorphism corresponding to the generator x generates a cyclic vertex-stabiliser in the automorphism group of (G; x,y), while y generates the edge-stabiliser, which is necessarily of order two. An orientably regular map M = (G; x, y) is reflexible if it is isomorphic to its mirror image M-1 = (G; x-1, y); otherwise (G; x, y) is chiral. In what follows, we often identify the group G that underlies an algebraic map M = (G; x, y) with its left regular representation, and it should be easy for the reader to see from the context which notion is in use. Before proceeding to the classification of orientably regular embeddings of cycles with multiple edges we present a general result about orientably regular embeddings of graphs with multiple edges. For a non-trivial simple graph X let X(m) denote the graph arising from X by replacing each edge with m parallel edges. To avoid trivial cases, the multiplicity m of every graph X(m) will always be at least 2. We show that every orientably regular embedding of X(m) determines two orientably regular maps, an orientably regular embedding of X and a regular embedding of the dipole graph Dm with multiplicity m, which consists of two vertices and m parallel edges joining them. In this context it may be useful to recall a result from [19] and [18] that every orientably regular embedding of Dm is isomorphic to a map D(m, e) arising from the metacyclic group G(m, e) given by the presentation G(m, e) = (x, y | xm = y2 = 1, yxy = xe). where e2 = 1 (mod m). Moreover, two dipole maps D(m, e) and D(m, e') are isomorphic if and only if e = e' (mod m). We are now ready for the result about the structure of orientably regular maps with multiple edges. Theorem 2.1. Let M = (G; x, y) be a regular map of valency d with underlying graph X(m) of order at least 2. Set A = (xd/m) and B = (xd/m, y). Then: (i) The group A is a normal subgroup of G, the map M/A = (G/A; xA,yA) is a regular embedding of X, and the natural projection M ^ M/A is a map homo-morphism bijective on the vertices. (ii) M' = (B; xd/m, y) is a dipole map isomorphic to D(m, e) for some integer e such that e2 = 1 (mod m). (iii) G = (x,y | xkm = y2 = 1,yxk y = xek,... ), where e2 = 1 (mod m) and k = d/m is the valency of X. In particular, the multiplicity m of M is the largest positive divisor q of d such that (xd/q ) < G. (iv) If M is not bipartite, then e = 1 (mod m). Proof. By our assumption, M contains neither loops nor semiedges. Since any two vertices of M are joined by m parallel edges and G acts regularly on the darts of M, the subgroup A = (xk ) fixes two vertices and acts regularly on the set of edges joining them. Applying the regularity again, A fixes all the vertices of M pointwise. In particular, A is a normal subgroup of G. The natural projection M = (G; x, y) ^ (G/A; xA, yA) = M/A is a map homomorphism which is bijective on the vertices. It follows that the underlying graph of M/A is X. This proves (i). By definition, y transposes a pair of adjacent vertices. It follows that M' = (B; xk, y) is an orientably regular map with two vertices and m parallel edges. Since B contains the cyclic group A as a subgroup of index 2, B is a metacyclic group with presentation B = (xk, y | (xk)m = y2 = 1, (xk)y = xek) where e2 = 1 (mod m) (see [19]). It follows that G has a presentation as stated in (iii). In particular, we see that the multiplicity m of M is the largest positive divisor q of d such that (xd/q) < G. This proves (ii) and (iii). To finish the proof, assume that M is non-bipartite. Thus there exists a relation w(x, y) = xai yx°2y .. . x°2r+1 y = 1 where y appears an odd number of times, say 2r + 1 times. Then xd/m = xd/mw(x,y) = w(x,y)(xd/m)e2r+1 = xde/m, and hence e = 1 (mod m), as required. □ Now we proceed to multicyclic regular maps. We start by introducing a family of orientably regular maps C(n, m; e, f ) as follows. Let C (n, m; e, f ) = (G; x, y) where G = G(n, m; e, f ) is a group given by the presentation G(n, m; e, f ) = (x, y | x2m = y2 = 1, y-1x2y = (x2)e, (xy)n = (x2)f >. (2.1) The parameters m and n are positive integers, n > 3, and e, f G Zm. We now show that every orientably regular embedding of сПт) is isomorphic to one of the maps C (n, m; e, f ) for suitable integers e and f, and classify these maps up to isomorphism. Theorem 2.2. The graph сПт), with n > 3 and m > 2, has an orientably regular embedding for each n and m, unless n is odd and m is even. Every such embedding is reflexible and is isomorphic to one of the maps C(n, m; e, f ) where e and f are as follows: (i) If both n and m are odd, then e =1 and f = (n + m)/2 (mod m). In particular, there is only one orientably regular embedding in this case. (ii) If n = 0 (mod 4) and m is odd, then e2 = 1 (mod m) and f = (e + 1)n/4 (mod m). (iii) If n = 2 (mod 4) and m is odd, then e2 = 1 (mod m) and f = ((e +1)n + 2m)/4 (mod m) for even e, and e2 = 1 (mod 2m) and f = (e +1)n/4 (mod m) for odd e. (iv) If n = 0 (mod 4) and m is even, then e2 = 1 (mod m) and f = (e + 1)n/4 (mod m) or f = ((e + 1)n + 2m)/4 (mod m). (v) If n = 2 (mod 4) and m is even, then e2 = 1 (mod 2m) and f = (e + 1)n/4 (mod m) or f = ((e + 1)n + 2m)/4 (mod m). Two such embeddings C(n, m; e, f ) and C(n, m; e', f ') are isomorphic if and only if e = e' (mod m) and f = f ' (mod m). Reflexible orientably regular embeddings of CÌm) were previously classified in [15, Proposition 7] leaving the possibility for the existence chiral maps open. It follows from Theorem 2.2 that no chiral embeddings of C(m) exist and therefore the two families coincide. Our proof of Theorem 2.2 uses a classical result of Heilder concerning the structure of metacyclic groups (see Zassenhaus [28, p. 99]). Theorem 2.3 (Holder's Theorem). Every extension of a cyclic group of order m > 2 by a cyclic group of order n > 2 is determined by two integers e and f satisfying the congruences en = 1 (mod m) and f (e — 1) = 0 (mod m), and is isomorphic to the group G(e, f ) with presentation G(e,f ) = (a,b | am = 1,bn = af ,b-1ab = ae). Furthermore, the extension determined by e and f is equivalent to that determined by e' and f ' if and only if e = e' (mod m) and f = f ' (mod m). Proof of Theorem 2.2. Let M = (G; x, y) be an orientably regular embedding of the graph CÌm). By Theorem 2.1(i), A = (x2) < G and M/A = (G/A; xA,yA) is an orientably regular embedding of the simple cycle Cn, where G/A = (x,y | X2 = y2 = (xy)n = 1). By Theorem 2.1(iii), the group G has the presentation (2.1) for some e,f G Zm where e2 = 1 (mod m). (2.2) Set a = x2 and b = xy. Then K = (a, b) is a metacyclic group with presentation (a,b | am = 1,bn = af ,ab = ae). (2.3) We apply Holder's Theorem to conclude that e and f satisfy the congruences f (e - 1) = 0 (mod m) (2.4) and en = 1 (mod m). (2.5) From the presentation of G we deduce that ay = ae and by = (xy)y = yx = yx2x-1 = x2eyx-1 = aeb-1. For brevity, denote s = J2 n- e1. Then af (2=4) aef = (af )y = (bn)y = (by)n = (aeb-1)n = asb-n = as-f, whence 2f = s (mod m). (2.6) In the above proof we see that G = K x (y), and hence G has an alternative presentation G = (x,y la = x2,b = xy, am = 1,bn = af ,ab = ae,y2 = 1,ay = ae,by = aeb-1). (2.7) Conversely, given a group G defined by (2.1) (or equivalently by (2.7)) with the parameters n, m, e, and f satisfying (2.2), (2.4), (2.5) and (2.6), we see that |G| = |K x (y)| = 2|K|, and from Holder's Theorem we get that |G| = 2mn. By Theorem 2.1, the map (G; x, y) corresponds to an orientably regular embedding of C.m). Recall that two embeddings C (n, m; e, f ) = (G(n, m; e, f ); x, y) and C (n, m; e', f ') = (G(n,m; e',f '); x',y') are isomorphic if and only if the assignment x ^ x', y ^ y' extends to a group isomorphism. Routine calculations show that this occurs if and only if e = e' (mod m) and f = f ' (mod m). For the maps (G(n,m; e, f ); x,y) and (G(n, m; e, f ); x-1 , y) the latter condition is clearly satisfied, which immediately implies that each of the maps C(n, m; e, f ) is reflexible. To obtain more details on these embeddings we need to solve the system of congruences (2.2), (2.4), (2.5) and (2.6). First notice that if n is odd and m is even, then e is odd. According to (2.2) we have e2 = 1 (mod m), and therefore s = X™—)1 e1 = (n — 1)(e + 1)/2 +1. However, 4|(n - 1)(e +1), so s is odd, violating (2.6). In other words, if n is odd and m is even, c4m) does not admit any orientably regular embedding. Our discussion now splits into five cases dealing with the remaining conditions on n and m, each corresponding to an item of Theorem 2.2. Case (i). Both n and m are odd. Theorem 2.1 (iv) yields that e = 1 (mod m). By substituting e =1 into (2.6) we get 2f = n (mod m), which implies that 2f = (n + m) (mod m) and consequently 2(f — (n + m)/2) = 0 (mod m). Since m is odd, we infer that f = (n + m)/2 (mod m), and Case (i) is done. Now we deal with the cases where n is even. Using (2.2) we get s = J2n—o el = (e + 1)n/2. Substituting for s into (2.6) we obtain Case (ii). n = 0 (mod 4) and m is odd. Since n = 0 (mod 4), from (2.8) we get f = (e + 1)n/4 (mod m). By substituting f into (2.4) we obtain Therefore e and f satisfy (2.4), and Case (ii) is done. Case (iii). n = 2 (mod 4) and m is odd. If e + 1 is even, then f = (e + 1)n/4 (mod m). By substituting f into (2.4) we obtain n(e2 — 1)/4 = 0 (mod m). Since n = 2 (mod 4), we get e2 — 1 = 0 (mod 2). If we combine this with (2.2), we get e2 = 1 (mod 2m). If e+1 is odd, then (e+ 1)n/2 is odd, and hence (e+ 1)n/2+m is even. We may rewrite (2.8) in the form 2f = (e + 1)n/2 + m (mod m), and obtain f = ((e + 1)n + 2m)/4 (mod m). By substituting f into (2.4) we further get 2f = (e + 1)n/2 (mod m). (2.8) f(e — 1) = (e2 — 1)n/4 = 0 (mod m). 2 (2.9) n(e2 — 1)/2 + m(e — 1) = 0 (mod 2m). Since m is odd, by the Chinese Remainder Theorem, this is equivalent to (mod 2). (2.10a) (2.10b) By applying (2.2) we may conclude that (2.10a) holds. Since n/2, m, e +1, and e — 1 are all odd, we see that (2.10b) holds, too. Hence (2.4) is satisfied by e and f exactly when the conditions in the statement are satisfied. This completes Case (iii). Case (iv). n = 0 (mod 4) and m is even. In this case (2.8) has two solutions f = (e + 1)n/4 and f = (e + 1)n/4 + m/2 in Zm. If we insert them into (2.4), we see that (2.4) is satisfied, and Case (iv) is complete. Case (v). n = 2 (mod 4) and m is even. (2.8) has two solutions in Zm, namely f = (e + 1)n/4 or f = (e + 1)n/4 + m/2. It remains to show e2 = 1 (mod 2m). By substitution of f into (2.4) we get (e2 - 1)n/4 = 0 (mod m). (2.11) By the assumption, we may set m = 2r m0 where r > 1 and m0 is odd. A combination of (2.2) and (2.11) yields the system je2 - 1 = 0 (mod 2rm0), je2 - 1 = 0 (mod 2r+1(m0/h)), where h = gcd(m,n/2). By the assumption, n/2 is odd, so h = gcd(m0,n/2). By the Chinese Remainder Theorem, the above system is equivalent to the system fe2 - 1 = 0 (mod 2r+1), le2 - 1 = 0 (mod m0). We now apply the Chinese Remainder Theorem once again and get e2 = 1 (mod 2m). This completes Case (v) as well as the proof of Theorem 2.2. □ The next corollary determines the basic parameters of the maps C (n, m; e, f ). Recall that the type of a regular or orientably regular map M is the symbol {p, q} where p is the face-size and q is the vertex-valency of M. Corollary 2.4. The map C (n, m; e, f ) has type {nm/h, 2m} and its genus is n(m-1)/2 -(h - 1), where h = gcd(f, m). Proof. To determine the type and the genus of C(n, m; e, f ) we need to determine the order of the element xy. Since (xy)n = (x2)f and x2 has order m, we see that the order of xy is nm/h where h = gcd(f, m). It follows that the map has type {nm/h, 2m}. Since |G(n, m; e, f )| = 2mn, the numbers of vertices, edges, and faces of C (n, m; e, f ) are n, mn, and 2h, respectively. Therefore, by the Euler-Poincare Formula, the map has genus n(m - 1)/2 - (h - 1), as claimed. □ Remark 2.5. Let A(g) denote the order of a largest group of conformal automorphisms of a compact Riemann surface of genus g. Accola [1] and MacLachlan [17] independently proved that 8(g + 1) < A(g) < 84(g - 1) for g > 2 and there are infinitely many integers g > 2 for which the equality A(g) = 8(g + 1) holds. If we take n = 4, e = -1, and f = 0 in Corollary 2.4, we get that the genus of C(4, m; -1,0) is g = m - 1 with the automorphism group G of order |G| = 8(g + 1), the lower bound of A(g). 3 Non-orientable regular embeddings of C(m) As in the orientable case, regular maps on non-orientable surfaces can be represented in a purely algebraic manner [12]. Every regular map M on a closed surface, and hence every regular map on a non-orientable surface, may be identified with a quadruple (G; l, r, t) where G is a finite group and {l,r,t} is a generating set for G with l2 = r2 = t2 = (lt)2 = 1, where the elements l, r, t and It are all nontrivial. Such a quadruple is called an algebraic regular map. Elements of G represent flags of M, pairwise incident triples of the form (v, e, f ) where v is a vertex, e is an edge and f is a face of M. The right translations of G by l, r, and t correspond to the longitudinal, rotary, and the transversal involution of M, respectively. The longitudinal involution fixes e and f of each flag (v, e, f ) while interchanging the end-vertices of e. The rotary involution fixes v and f of (v, e, f ) while interchanging the two edges sharing the same corner of f at v. The transversal involution fixes v and e of (v, e, f ) while interchanging the two faces incident with e. The vertices, edges, and faces of M are in a one-to-one correspondence with the letf cosets of the subgroups (r, t), (l, t), and (l, r), and the incidence between the objects corresponds to non-empty intersection of cosets. Two algebraic maps (G; l,r, t) and (G; l', r',t') represent isomorphic regular maps if and only if there is an automorphism of G taking l to l', r to r', and t to t'. Each automorphism of the map M = (G; l, r, t) corresponds to a left translation g ^ ag, g G G, where a is a fixed element of G. In particular, the group Aut(M) of all automorphisms of M is isomorphic to G. The underlying surface of a regular map (G; l, r, t) need not be non-orientable, nevertheless, the criterion of orientability is easy: a regular map (G; l, r, t) is orientable if and only if the even-word subgroup G+ = (rt, tl) has index 2 in G. Thus, if M = (G; l, r, t) is non-orientable, then G+ = G, and the triple M = (G; x, y) with x = rt and y = tl represents an orientably regular map such that Aut+ (M) = G = Aut(M). The orientably regular map M is known as the antipodal double cover over M; conversely, M is said to be a halved non-orientable quotient of M. An orientably regular map is called antipodal if it admits a halved non-orientable quotient. Observe that the involution t g G plays the role of a reflection of M, since xl = x-1 and yf' = y-1 = y. In general, an inner reflection of an orientably regular map N = (G; x, y) is any element g G G satisfying the following conditions: Orientably regular maps admitting inner reflections are called algebraically antipodal. It is proved in [20, Theorem 7.5] that an algebraically antipodal orientably regular map is antipodal with the exception of spherical dipole maps D(m, -1) where m is odd, their duals, and regular maps with a single vertex and valency at most 2. More precisely, if N = (G; x, y) is an orientably regular map and g is an inner reflection of N, then with the exception of the maps just mentioned, the map M = Ng = (G; xg, yg, g) is a halved quotient of N. Although the antipodal double cover over a non-orientable regular map is uniquely determined, the same is not true for halved quotients: an antipodal regular map may have different halved quotients corresponding to different inner reflections [25]. However, con- (3.1a) (3.1b) (3.1c) ditions (3.1a)-(3.1c) imply that if g1 and g2 are two inner reflections, then there exists a central involution z such that g2 = zg1. In particular, the number of inner reflections of an antipodal map equals the number of central involutions (including the identity). Before moving on to the classification of non-orientable multicyclic regular maps it will be useful to recall that non-orientable regular maps with multiple edges occur in two varieties: either every pair of parallel edges forms an orientation-preserving cycle or there exists a pair of parallel edges forming an orientation-reversing cycle. By regularity, in the latter case every edge must be involved in such a cycle. Following [16], we call such maps Möbius regular maps. Möbius maps were earlier investigated by Wilson [27] under the name cantankerous maps. By using geometric arguments Wilson showed that the multiplicity of such a map must be 2 (see [27, p. 265] and also [16, Lemma 6]). For the sake of completeness we include a proof of this fact based on the determination of all non-orientable regular embeddings of dipoles. Observe that the dipole D2 has exactly one non-orientable embedding, which is regular and its supporting surface is the projective plane. This embedding is isomorphic to the map (H; l, r, t) where H is the dihedral group of order 8 with presentation H = (l, r, t | l2 = r2 = t2 = (lt)2 = 1, (rt)2 = 1, (lrt)2 = t>. (3.2) The next lemma shows that there is no other non-orientable regular embedding of any dipole. Lemma 3.1. There are no non-orientable regular embeddings of the dipole Dm except the unique embedding of D2 in the projective plane isomorphic to the map defined by the presentation (3.2). Proof. It is clear that the dipole D2 has a unique embedding in the projective plane and that the embedding is regular. Now let M = (H; l, r, t) be a non-orientable regular embedding of Dm with m > 2. Since Dm has just two vertices, the subgroup D = (r, t> has index two in H. Clearly, D is dihedral of order 2m. If we set a = rt, then (a> < D and D = (a, t>. Since D < H and D is dihedral, we get lal = a®tj, where i G Zm and j G Z2. (3.3) Suppose that j = 0. From (3.3) we then deduce that a = l2al2 = a®2, proving that i2 = 1 (mod m). It follows that H has presentation (a, t, l | am = l2 = t2 = (lt)2 = 1, tat = a-1, lal = a®>. However, it is straightforward to verify that the even-word subgroup H + = (rt, tl> = (a, lt> has index two in H, contradicting the assumption that M is non-orientable. Therefore j = 1. It follows that the element a®tj = a®t is an involution, and hence a is an involution as well. In particular, m = 2. Now suppose that i = 0. Then (3.3) reduces to lrlt lt=tl lrtl a=rt lal (3=3) t VI Vis - VI OV - VHiV — V ^ implying that r = 1, which is impossible. Therefore i = 1 and (3.3) reduces to the relation lal = at. Thus M is isomorphic to the previously defined embedding of D2 into the projective plane, and the proof is complete. □ The following result appears in [16] and [27]. We provide a purely algebraic proof. Theorem 3.2. Let X be a simple graph of order at least 2 and of valency d. Then a regular embedding M = (G; l, r, t) of the graph X(m) with multiplicity m is a Möbius regular map if and only if m = 2 and the generators l, r and t satisfy the identity (l(rt)d)2 = t. (3.4) Proof. Note that the action of the automorphism group G on the flags of M induces an action on the vertices and an action on the edges of M. We may therefore assume that the subgroup (r, t) of G fixes a vertex u, and the subgroup (l, t) fixes an edge joining the vertex u and an adjacent vertex v. Let H be the subgroup of G fixing the set {u, v}. Then H may be regarded as the automorphism group of a regular embedding H of the dipole Dm whose vertices are u and v and whose edges are the edges between u and v. The underlying graph structure implies that d is the smallest positive integer k such that the element (rt)k fixes both u and v. Therefore, H = (H; (rt)d, l, t). If M is a Möbius regular map, then the regularity of M implies that H is also a Möbius regular map. By Lemma 3.1, m = 2 and the identity (3.4) holds. Conversely, if m = 2 and M satisfies the identity (3.4), then H is a non-orientable embedding of the 2-cycle C2. The definition of H implies that M contains an orientation reversing cycle, so M is Mobius regular map. □ To formulate our classification theorem for non-orientable multicyclic regular maps we define two families of maps. First, let C (m, n) denote the non-orientable regular map (H (m, n); l, r, t) with H (m, n) being the group with presentation (l,r,t | l2 = r2 = t2 = (tl)2 = (rt)2m = (rl)2n = (rtrl)2 = 1, (rt)m(rl)n = r), (3.5) where n > 3 and m > 1 are odd integers. It is not difficult to see that the antipodal double cover of C (m, n) is the multicyclic orientably regular map C (2n, m, —1,0). Second, let M(n) denote the non-orientable regular map (H(n); l,r,t) with H(n) being the group with presentation (l,r,t | l2 = r2 = t2 = (tl)2 = (rt)4 = (rl)n = 1, (l(tr)2)2 = t), (3.6) where n = 0 (mod 3). By Theorem 3.2, the relation (l(tr)2)2 = t in the above definition forces M (n) to be a Mobius map. The following theorem is, except for the enumeration part, due to Li and Siran [15, Proposition 8]. Our proof is based on the classification of orientably regular embeddings of C.m) presented in the previous section and on Theorem 3.2 about Mobius maps proved above. The original proof of Li and Siran employed the analysis of regular maps with an unfaithful action of the map automorphism group on vertices. Theorem 3.3. Let M be a non-orientable regular embedding of an m-fold n-cycle C.m). Then either (i) m and n are both odd, and M is isomorphic to the map C(m, n), or (ii) m = 2, n = 0 (mod 3), and M is isomorphic to the Mobius map M(n). Moreover, for each pair (n, m) of admissible integers there is a unique non-orientable regular embedding of сПГ\ Proof. Let M = (G; l, r, t) be a non-orientable regular embedding of C„ \ and let M = (G; x, y) be the antipodal double cover of M, where x = rt and y = tl. We distinguish two cases. Case (i). Every 2-cycle of M is orientation-preserving. Since the antipodal cover is a smooth cover, the valency of M is preserved, and each set of m parallel edges in M lifts to a set of m parallel edges in M .So M is an orientably regular embedding of C^. By Theorem 2.2, M = (G; x, y), where G = G(2n, m, e, f ) = (x, y | x2m = y2 = 1, (x2)y = (x2)e, (xy)2n = (x2)f >, (3.7) where the parameters 2n, m, e, and f satisfy the numerical conditions stated in Theorem 2.2. Let K = (x2>. Then K < G and M/K = C2n, where C2n denotes the dihedral map of type {2n, 2} on the sphere. The inner reflection t of M projects onto the inner reflection t = (xy)n of C2n. It follows that t = x2k(xy)n for some k e Zm. Using the commuting rule (x2)y = x2e (3.8) we get x-1 (3=b) (x2k(xy)n)-1x(x2k(xy)n) (3=8) (xy)-nx(xy)n, (3.9) which implies x-2 (3=9) (xy)-nx2(xy)n (3=8) / x2 if n is even (3.10a) I x2e if n is odd. (3.10b) Suppose that n is even. From (3.10a) we deduce that x4 = 1, so m = 2. Theorem 2.2 now implies that e = 1 and therefore x-1 (3=9) (xy)-nx(xy)n = (yxx-2)nx(xy)n (3=8) x-2n+2(yx)2nx-1. (3.11) Since (xy)2n = x2f, we have (yx)2n = y-1((xy)2n)y = (x2f )y = x2ef (= x2f. If we combine this with (3.11), we get x-2n+2f +1 = 1. Consequently, —2n + 2f + 1 = 0 (mod 4), which cannot hold. It follows that n must be odd. By (3.10b), e = m — 1. We have -, (3-1a) ,2 2k/ \n 2k/ \n (3-8) / \2n (3-7) 2f 1 = t2 = x2k(xy)'x (xy)' = (xy) = x2f, so f = 0. Moreover, 1 (3=c)(x2k (xy)n)-1y(x2k (xy)n)y = (yx-1)nx-2k y(x2k (xy)n)y =(yxx-2)nx-4ky(xy)ny (3=8) x4k (yxx-2)n(yx)n (3=8) x4k+2(yx)2n (3=)x4k+2x2f = x4k+2. Since the order of x2 is m, we get 2k +1 = 0 (mod m), and hence k = (m — 1)/2. It can easily be verified that if both m and n are odd, then xm-1(xy)n is the unique inner reflection of the map C (2n, m; m — 1,0) that gives rise to a non-orientable regular embedding of C4m). Let t = xm-1(xy)n, r = xt and i = yt. Then, upon substitution, the group G(2n, m; m — 1,0) receives the presentation (3.5), and hence M = C (m, n). Case (ii). There exists an orientation-reversing 2-cycle in M. By Theorem 3.2, m = 2, every 2-cycle is orientation-reversing, and M = (G; i, r, t) is a Möbius regular map. It follows that the antipodal double cover M = (G; rt, ti) of M is an orientably regular embedding of the lexicographic product Cn[K2], where each vertex u of M lifts to two vertices u0 and u1 which are antipodal points of M. Without loss of generality we may assume that the generator x = rt fixes a certain vertex u0 and that the generator y fixes an edge u0v0 incident with v0. Set a = x2, b = ay, and K = (a, b). Then K acts regularly on the darts (u4, v j ) where i, j G Z2 .By regularity, K = Z2 ф Z2. We first show that K < G. It is evident that ax = a, ay = b and by = a. (3.12) Notice that since u0 and u1 are antipodal points, x fixes both u0 and u1. Similarly, yxy fixes both v0 and v1. Using regularity again we see that yx2y interchanges u0 and u1. Therefore, for any dart of the form (u4, v3 ) we get bx(ui, Vj ) = x-1yx2yx(u, v j ) = x-1yx2y(ui, Wk ) = x-1 (ui+1, Wh) = (ui+1, vs ), where k, h, s g Z2 and w0 and w1 are vertices adjacent to u0 but distinct from v0 and v1. Thus bx g K and hence K < G. Next we show that bx = ab. Since M is a halved quotient of M in which every pair of antipodal vertices u0 and u1 is identified to a single vertex u, there exists an inner reflection g which identifies the antipodal pairs. Notice that M /K = Cn, where Cn is the dihedral map of type {n, 2}. By regularity, g G K. From (3.1b) we see that g = 1 and g = a. If we had g = b, then from (3.1c) we would derive that by = b, which is a contradiction with the assumption by = a. Therefore g = ab = x2yx2y and consequently xg = x2(yx2y)x(x2)(yx2y) (3=b) x-1. Since [x2, yx2y] = 1, we get bx = x-1(yx2y)x = x-2yx2y = x2 yx2y = ab. (3.13) For convenience, set z = xy. Then zn G K, which implies that zn = a®bj for some i, j G Z2. From (3.12) and (3.13) we derive that az = b, bz = ab, and (ab)z = a, (3.14) so a4b3 = zn = (zn)z = (aibj)z (3^4) bi(ab)3 = a3bi+j, and hence ai-j = bž. Since (a) n (b) = 1, we see that i = 0 and j = 0. Therefore, zn = 1. Observe that the action of z on K defined by (3.14) induces the permutation (1)(a, b, ab), which implies that n = 0 (mod 3). Moreover, since zy = (xy)y = yx = yx 1x2 = (xy) 1x2 = z 1a, (3.15) we get G = (a, b, z, y) = (K x (z)) x (y). Therefore G is defined by the presentation (x, y | a = x2, b = ay, x4 = y2 = [a, b] = (xy)n = 1, bx = ab). (3.16) Conversely, It is straightforward to verify the group given by (3.16) with n = 0 (mod 3) gives rise to an orientably regular embedding of the graph Cn [K2]. To complete the proof it remains to show that t = ab is a unique inner reflection giving rise to a halved quotient with a multicyclic underlying graph. Let Ic(G) denote the subgroup of G generated by all its central involutions. From the previous part of the proof we know that G/K is isomorphic to the dihedral group of order 2n. If g G Ic(G), then gK G Z (G/K ). Since Z (G/K) 1 , n is odd, (žn/2), n is even, there exist elements i, j, k g Z2 such that g = a®^ if n is odd, and g = a®bj (zn/2)k if n is even. If n is odd, then aibj = g = gy = (a®bj )y = aj b®. Since (a) П (b) = 1, we have i = j. Moreover, we have a®b® = g = gz = (a®b®)z = a®b2i = a®, so i = 0 and hence Ic(G) = 1. Now we assume n is even. Recall that n = 0 (mod 3). We deduce from (3.14) that [a, zn/2] = [b, zn/2] = 1, and from (3.15) that [y, zn/2] = 1. So zn/2 G Ic(G). Applying similar techniques we may deduce that if an element h = a®bj belongs to Ic(G), then h =1. Therefore, Ic(G) = (zn/2). Summing up, we have proved that if n is odd, there is a unique inner reflection ab, and if n is even, there are two inner reflections ab and abzn/2. However, the latter inner reflection does not produce an embedding of CÌm). If we set t = ab, r = xt, and l = yt, then the presentation (3.16) transforms to the presentation (3.5). In either case, the antipodal regular covering that projects onto a non-orientable regular embedding of C(m) is unique, so for each pair (n, m) of admissible integers there is a unique non-orientable regular embedding of C(m), as claimed. □ As a corollary to the main theorem of this section we present a strengthening of a result due to Wilson [26] about non-orientable regular maps whose number of edges is a power of 2. Our result features two infinite classes of projective-planar regular maps arising as halved quotients of dihedral spherical maps: a unique embedding of the cycle Cn of length n > 1 in the projective plane which is a halved quotient of the map C2n, and its dual, the balanced embedding of the bouquet of n loops in the projective plane which is the halved quotient of the dipole map D(2n, —1). Theorem 3.4. If M is a regular map with nilpotent automorphism group, then Aut(M) is a 2-group. Furthermore, if M is non-orientable, then M is either the balanced embedding of the bouquet of 2n loops into the projective plane for some n > 0 or its dual, a projective-planar embedding of the cycle C2n. Proof. Let M = (G; l, r, t) be a regular map where G is nilpotent. Then G can be expressed as a direct product H x K where H is a 2-group and K has odd order. The elements l, r, and t belong to H, because they are involutions. It follows that H = G and K = 1. In other words, G is a 2-group. Now assume that M = (G; l, r, t) is non-orientable. Since G is a 2-group, M has 2n edges for some n > 0 and hence |G| = 2n+2. We proceed by induction on n to show that M is either the balanced embedding of the bouquet of 2n loops into the projective plane with n > 0 or its dual. An easy check of non-orientable regular maps with at most four edges shows that the claim is true for n < 2. For the induction step assume that the statement holds for some n > 2 and let M = (G; l, r, t) be a non-orientable regular map with 2n+1 edges, so that |G| = 2n+3. Since G isa 2-group, it has a non-trivial centre, and hence it contains a central involution z = 1. Clearly, (z) < G, so we may construct the quotient map M = (G; I, r, t) where G = G/(z) and I, r, and t are the images of r, l, and t, respectively. By the induction hypothesis, M is either an embedding of the cycle C2n into the projective plane or its dual. We may clearly assume that M is an embedded cycle. Then G has a presentation G = (i, r, i | i2 = r2 = t2 = (iii)2 = 1, (rt)2 = (ri)2n = 1). Since G is a cyclic central extension of G by (z), we get G = (l,r,t | l2 = r2 = t2 = (lt)2 = 1, (rt)2 = zi, (rl)2n = zj), for some i, j G Z2. If i = 0, then j = 1, and M is an embedding of C2n+i into the projective plane. If i = 1, then since lt(rt)2lt = z = (rt)2, the underlying graph of M is a cycle with multiplicity 2, which violates Theorem 3.3. This establishes the induction step, and the proof is complete. □ Remark 3.5. Breda d'Azevedo, Nedela, and Siran [3] showed that for any integer p = 7 (mod 12) with p > 7 the groups Gj,i = (r,s | x2j = s21 = (rs)2 = (rs-1)2 = 1), where j > l > 3 and (j - 1)(l - 1) = p + 1 give rise to infinitely many non-orientable regular maps of Euler characteristic -p. If p is a prime, this family forms a complete set of regular maps on the non-orientable surface of Euler characteristic -p. After setting l = m and j = n it becomes clear that these maps are identical with regular embeddings of CnT^ defined by (3.5). Remark 3.6. Malnic, Nedela, and Skoviera [18] proved that if the automorphism group of an orientably regular map M is nilpotent, then M can be decomposed into a direct product of two orientably regular maps, an orientably regular map whose automorphism group is a 2-group and a semistar of odd valency [18, Theorem 3.2]. Since the automorphism group of every non-orientable regular map is also the automorphism group of its antipodal double cover, it follows from Theorem 3.4 that no orientably regular maps with nilpotent automorphism groups are antipodal, except the dihedral maps {2n, 2} on the sphere and their duals. Acknowledgements Our research received support from the following grants: "Mobility - Enhancing Research, Science and Education" at Matej Bel University in Banska Bystrica, ITMS code 26110230082, under the Operational Programme Education co-financed by the European Social Fund, APVV-0223-10, VEGA 1/1085/11, VEGA 1/1005/12, by the EUROCORES Programme EUROGIGA (Project GReGAS) of the European Science Foundation under the contract APVV-ESF-EC-0009-10, and by the Slovak-Chinese bilateral grant APVV-SK-CN-0009-12. References [1] R.D.M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398-408. [2] H.R. Brahana, Regular maps and their groups, Amer. J. Math. 41 (1927), 268-284. [3] A. Breda d'Azevedo, R. Nedela and J. Siran, Classification of regular maps of negative prime Euler characteristic, Trans. Amer. Math. Soc. 357 (2005), 4175-4190. [4] D.A. Catalano, M. D. E. Conder, S.-F. Du, Y. S. Kwon, R. Nedela and S. 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Soc. 101 (2010), 427-453. [12] G.A. Jones and J.S. Thornton, Operations on maps, and outer automorphisms, J. Combin. Theory Ser. B 35 (1983), 93-103 . [13] Y.S. Kwon, New regular embeddings of n-cubes Qn, J. Graph Theory 46 (2004), 297-312. [14] Y.S. Kwon and R. Nedela, Non-existence of nonorientable regular embeddings of n-dimen-sional cubes, Discrete Math. 307 (3-5) (2007), 511-516. [15] C.H. Li and J. Siran, Regular maps whose groups do not act faithfully on vertices, edges, or faces, European J. Combin. 26 (2005), 521-541. [16] C.H. Li and J. Siran, Möbius regular maps, J. Combin. Theory Ser. B 97 (2007), 57-73. [17] C. MacLachlan, A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44 (1969), 265-272. [18] A. Malnic, R. Nedela, and M. Skoviera, Regular maps with nilpotent automorphism groups, European J. Combin. 33 (2012), 1974-1986. [19] R. Nedela and M. Skoviera, Regular maps of canonical double coverings of graphs. J. Combin. Theory Ser. B 67 (1996), 249-277. [20] R. Nedela and M. Skoviera, Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997), 807-823. [21] R. Nedela and M. Skoviera, Exponents of orientable maps, Proc. London Math. Soc. 75 (1997), 1-31. [22] R. Nedela, M. Skoviera and A. Zlatos, Regular embeddings of cycles with multiple edges, manuscript (2001). [23] M. Skoviera and A. Zlatos, Regular maps with multiple edges on orientable surfaces, manuscript (2000). [24] W. Threlfall, Gruppenbilder, Abh. sächs. Akad. Wiss. Math.-phys. Kl. 41 (1932), 1-59. [25] S.E. Wilson, Non-orientable regular maps, Ars Combin. 5 (1978), 213-218. [26] S.E. Wilson, Bicontactual regular maps, Pacific J. Math. 120 (1985), 437-451. [27] S.E. Wilson, Cantankerous maps and rotary embeddings of Kn, J. Combin. Theory Ser. B 47 (3)(1989), 262-279. [28] H. Zassenhaus, The Theory of Groups, Chelsea Publishing Co., New York, 1949. [29] J.-Y. Zhang and S.-F. Du, On the orientable regular embeddings of complete multipartite graphs, European J. Combin. 33 (2012), 1303-1312. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 195-213 Strongly regular m-Cayley circulant graphs and digraphs Luis Martinez * University of the Basque Country UPV/EHU, Department of Mathematics, 48080 Bilbao, Spain Received 16 September 2013, accepted 21 September 2014, published online 17 December 2014 The first part of this paper is a survey about strongly regular graphs and digraphs admitting a semiregular cyclic group of automorphisms. In the second part, some new types of such digraphs, called uniform and almost uniform, are studied. By using partial sum families, the form of the parameters is determined and some directed strongly regular graphs derived from these partial sum families with previously unknown parameters are obtained. Keywords: m-Cayley, circulant, strongly regular graphs, strongly regular digraphs, uniform partial sum families, almost-uniform partial sum families. Math. Subj. Class.: 05Cxx, 05C20, 05C25, 05C50, 05E30 1 Introduction Strongly regular graphs, which we define below, were introduced by Bose [4] in 1963. They constitute a very important class of graphs; in fact, they are one of the most basic association schemes, more specifically, they are the ones with two classes. Definition 1.1. A graph X without loops of valency k and order v is called a strongly regular graph with parameters v, k, A, ^ (for short, (v, k, A, -SRG) if any two adjacent vertices have exactly A common neighbours and any two distinct non-adjacent vertices have exactly ^ common neighbours. A SRG graph X is said to be trivial if X or its complement is a disjoint union of complete graphs. * Supported by the Spanish Government, grant MTM2011-28229-C02-02 and by the Basque Government, grant IT753-13. Technical and human support provided by IZO-SGIker (UPV/EHU, MICINN, GV/EJ, ESF) is gratefully acknowledged. Dedicated to the memory of my mother. E-mail address: luis.martinez@ehu.es (Luis Martinez) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ It is well known that if a non-trivial SRG is not a conference graph, that is, if the parameters are not of the form k = (v - 1)/2, p = (v - 1)/4, A = (v - 5)/4, then the eigenvalues of an adjacency matrix are integer numbers, and consequently Д = ß2 + 4y, where ß = A - p and y = k- p, is also an integer. The problem of studying what SRGs have nice cyclic automorphism groups has received a considerable attention over the past decades. A group of automorphisms of a graph is said to be regular if it acts transitively on the set of vertices and all the stabilizers are trivial. A graph is called circulant if it admits a cyclic regular group of automorphisms. By a classical result of W.G. Bridges and R.A. Mena [5] it is known that the Paley graphs are the only non-trivial circulant strongly regular graphs. Given integers m > 1 and n > 2, an automorphism group of a graph is called (m, n)-semiregular if it has m orbits of length n and no other orbit, and the action is regular on each orbit. An m-Cayley graph X is a graph admitting an (m,n)-semiregular group H of automorphisms. When H is abelian, we say that X is m-Abelian. If H is generated by an automorphism p (that is to say, when H is a cyclic group) and m = 1 (respectively, m = 2) we say that X is n-circulant (respectively, n-bicirculant). Sometimes, when a graph is m-Cayley over a cyclic group, we will just say that the graph is 'm-Cayley circulant', although that this terminology does not mean that the graph admits a regular group of automorphisms and should not be confused with the definition of circulant graph. Every m-Cayley graph X can be represented, following the terminology established by A. Malnic et al. in [18], by an m x m array of subsets of H in the following way. Let U0,..., Um_i be the m orbits of H, and for each i let щ z Uj. For each i and j, let Si j be defined by Sjj = {p e H I щ -»• p(uj)}. The family (Sj is called the symbol of Q relative to (H; uo,.. . ,um_1). The notion of strongly regular graph was generalized for directed graphs by Duval in [6]. A directed graph X without loops, of order v in which every vertex has both in-degree and out-degree k is called a directed strongly regular graph with parameters (v, k, p, A, t) (for short, (v, k, p, A,t) -DSRG or simply DSRG if we do not specify the parameters) whenever for any vertex u of X there are t undirected edges having u as an endvertex and for every two different vertices u and w of X the number of paths p(u, w) of length 2 starting at u and ending at w depends only on whether uw is an arc of X or not. In particular, (where A(X) denotes the arc set of X). A directed strongly regular graph will also be refereed to as a strongly regular digraph. DSRGs have received a considerable attention in the literature (see, for instance, [7], [8], [9], [11], [12], [13], [14]). The following relation between the parameters of a DSRG is obvious: k(k — ß) = pv + y, where ß = A - p and y = t - p. It is well known that ß2 + 4y is a square, unless (1.1) k = t= (v- 1)/2,p= (v- 1)/4,A= (v- 5)/4, (1.2) in which case the graph is undirected and is a conference graph, or k = (v- l)/2, p = (v + 1)/4, \=(v- 3)14, t = 0. (1.3) We define _ Д = sjß2 + 4y , which is an integer if the parameters of the digraph are not as indicated above. A DSRG X is called trivial if k = t and X is trivial as an undirected SRG. Next, we will review the main results in this paper. In the first part of this paper we will make a review of the study of strongly regular graphs and digraphs which are m-Cayley over a cyclic group (m-Cayley circulant SRGs and DSRGs). In Section 2 we will focus on the undirected graphs, and in Section 3 on the more general case of directed graphs. In the second part we will present some new results on two structures that produce circulant m-Cayley DSRGs. We study in Sections 4,5 and 6 the first structure, uniform partial sum families, which was proposed in the last section of [2], and we obtain there the general form of the parameters for the circulant case and give an sporadic uniform partial sum family which originates a DSRG with parameters previously undecided. In Section 7, we study almost uniform partial sum families, which are a generalization of uniform partial sum families, and obtain again the form of the parameters for the circulant case. Finally, we use almost uniform partial sum families to obtain three DSRG with parameters previously undecided. 2 Semiregular SRGs over cyclic groups In this section we will focuss on undirected graphs. In particular, we will review some results on strongly regular graphs admitting cyclic semiregular groups of automorphisms. Probably, the systematic study of circulant SRGs was began by D. Marušic in his seminal paper [21]. There, he obtained the form of the parameters of such graphs for the bicirculant case. Proposition 2.1. If a non-trivial (2 n, k, A, p) bicirculant strongly regular graph exists with prime n then, up to complementation, the parameters of the graph are v = 2n = 4s2 + 4s + 2, k = 2s2 + s, A = s2 - l, p = s2. He determined also some properties of the elements of the symbol. He denoted S0,0, Si i and S0,i by R, S and T, respectively, and he proved that the non-zero elements of the cyclic group Cn are the disjoint union of R and S, and that |R| = |S| = s2 + s and |T| = s2. He found also examples of such graphs for s = l and s = 2. He considered also non-trivial tricirculant SRGs over a cyclic group Cn of prime order and proved that, for such a graph or its complement, the elements of the symbol S0 0, S11 and S2 2 form a partition of the set of non-zero elements of Cn. His results were generalized by De Resmini and Jungnickel in [22]. They proved that the form of the parameters of a non-trivial bicirculant SRG over a cyclic group of order n was the one indicated in Proposition 2.1 if n is odd or not divisible by Д. They found also one such graph for s = 4. Unfortunately, the technique that they used to construct the graph, which is based on the existence of certain difference families over cyclic groups, was not fruitful to get examples with s > 5. This was further generalized by K.H. Leung and S.L. Ma in [17]. They found the general form of the parameters of non-trivial bicirculant SRGs, as stated in the following proposition, where с = |S0,i|, d = |S0,01: Proposition 2.2. Up to complementation, the parameters for any non-trivial bicirculant SRG over a cyclic group of order n are the following: 1. (n; с, d; Л, = (2m2 + 2 m + 1; m2, m2 + m; m2 - 1, m2) where m > 1. 2. (n; с, d; Л, = (2m2; m2, m2 — m; m2 — m; m2 — m) where m > 2. 3. (n; с, d; Л, = (2m2; m2, m2 + m; m2 + m; m2 + m) where m > 3. 4. (n; с, d; Л, = (2m2; m2 ± m, m2; m2 ± m; m2 ± m) where m > 2. Besides the graphs found by Marusic and by De Resmini and Jungnickel, Leung and Ma found one example for m = 2 in family 2 and one example for m = 2 in family 4 of the previous proposition. They also proved the non-existence of bicirculants SRG over cyclic groups of order 2m2 where m = pru,p is a prime congruent to 3 modulo 4, p and u are relatively prime and u2 < pr. More bicirculant SRGs were found by A. Malnic et al. in [19]. Concretely, they obtained bicirculant SRGs with parameters of the same form as in Proposition 2.1 for s = 3,4 and 5. The graphs that they found were the first known pairs of complementary bicirculant SRGs which are vertex-transitive but not edge-transitive. K. Kutnar et al. studied tricirculant SRGs in [16]. They proved that, under certain appropriate conditions, the elements on the symbol of a tricirculant SRG over a cyclic group Cn satisfy that S0,0, S1,1, S2,2 form a partition of Cn - {0} and the parameters of the graph can be determined. They denoted S0,0, S1,1, S2,2, S0,1, S1,2 and S2,0 by A, B, C, Д, S and T, respectively (the other elements in the symbol can be easily determined from these ones), and they denoted by TCay(C„; A, B, C ; Д, S, T) the associated tricirculant. In the next proposition, for a subset D £ Cn, we set D = Cn \ (Su {0} ). Proposition 2.3. Let X = TCay(C„;( A, B, C ; Д, S, T) be a non-trivial (3n, k, Л, -strongly regular tricirculant, where |A| + |B| + |C| < | A| + |B| + |C|. Assume A и B и C f 0. If n is a prime or n is coprime to 6Д then there exists an integer s such that the following two statements hold. 1. If the cardinalities of A, B and C are not all equal, then (3n, k, Л, = (3(12s2 + 9s + 2),( 4s + 1)(3s + 1), s( 4s + 3), s( 4s + 1)). If in addition |A| = \C\f \B\ (which is equivalent to |Д| = |S| ^ |T| ), then |A| = 2s(1 + 2s), |B| = (4s + 1)(s + 1), Д = s(4s + 1) and |T| = (1 + 2s)2. 2. If A B C then (3n, k, Л, = (3(3s2 - 3s + 1), s(3s ^ 1), s2 + s ^ 1, s2). In this case A s s 1 and Д S T s2. When A = \C\f |B| they proved that, for s = -1, exactly one tricirculant SRG exists up to isomorphisms, and for s = 1 no such graph exists. When A B C they proved that for s 2 exactly one tricirculant SRG exists up to isomorphisms, and for s 3 exactly four exist, and that for s 2 no such graph exists. 3 Semiregular DSRGs over cyclic groups DSRGs admitting a semiregular group of automorphisms were studied by Martinez and Araluze in [20] by using the concept of partial sum family. In the next definition, the third identity holds in the group ring Z[H]. As usual, we will identify a subset of H with the sum in Z[ H] of its elements. Definition 3.1. Let H be a group of order n and let m be an integer with m > 1. A family S = {Sij}, with 0 < i, j < m, of subsets of H is a (m,n, k,p, A, t) -partial sum family (for short, (m, n, k, p, A, t)-PSF, or simply PSF if we do not specify the parameters) if it satisfies: 1. for every i it holds that 0 j. Si,i, where 0 is the identity of H. 2. for every i it holds that E ^ |Sij = jO \Sj4 = k 3. for every i and j it holds that £mO1 Si,jSi,l = Si,jy{0} + ßSij + pH, where Si,j is the Kronecker delta, and where y = t - p and ß = A - p. The symbol notation, presented in the introduction for undirected graphs, is valid also for directed graphs. Martinez and Araluze proved that the existence of a (m, n, k, p, A, t)-partial sum family over a group H is equivalent to the existence of a (mn, k, p, A, t) -DSRG which admits a group of automorphisms isomorphic to H acting semiregularly and with m orbits (in fact, the elements of the PSF form the symbol of the digraph). They found 17 new DSRGs by using partial sum families. 14 of them had cyclic groups as groups of automorphisms, which are the kind of digraphs that we are considering in this paper. More specifically, they found: Four PSFs with parameters (2,15,13,6,5,8), (2,17,14,6,5,12), (2,17,15,7,6,9), (2,20,17,8,6,11) which produce bicirculant digraphs. Five PSFs with parameters (3,9,10,4,3,6), (3,11,11,4,3,4), (3,13,10,3,1,6), (3,14,14,5,4,5), (3,15,14,4,5,6) which produce tricirculant digraphs. Four PSFs with parameters (4,7,7,2,1,2), (4,8,9,3,1,6), (4,8,10,3,3,7), (4,11,10,2,3,4) which produce tetracirculant digraphs. One PSF with parameters (5,7,8,2,1,4) which produce a pentacirculant digraph. Finally, Araluze et al. found in [2] the form of the possible parameters of bicirculant DSRGs. They called the partial sum families partial sum quadruples in this case when m = 2. Proposition 3.2. Let S be a non-trivial PSQ in a cyclic group H. Then, up to complementation, the parameters of S are of the following form, where U = k - t and s, f are positive integers, and where q = |S1,0 and r = |S0,01: 1. n=s( 2f + 1), q= sf, r = sf,k = 2sf,p=sf,A=s(f- 1) ,t = sf. 2. n=s( 2f + 1), q= s(f + 1), r = sf,k = s( 2f + 1) ,p = s(f + 1) ,A = sf,t = s(f + 1). 3. n = 4s, q = 2s, r = 2s - 1, k = 4s - 1, p = s, A = 3s - 2, t = 3s - 1. 4. n = 2s2+ 2s+1+2U, q = s2 +U, r = s2 +s+U,k = 2s2 + ^ 2U,^ s4 U, ^ s^ U U and ^ 2s2 + ^ U. 5. ^ 2s4 2U, ^ s4 U, r = s4 ^U,^ 2s4 ^2U,^ ^U,^ s4 ^U and t 2s2 s U, where 2s divides s2 U. 6. n = 2 s2 + 2 U, q = s2 + U, r = s2^ s + U,k = 2s2- s + 2U,p=s2- s + U,A = s2 - s + U and t = 2s2 - s + U, where 2s divides s2 + U. 7. n = 2s2 + 2U, q = s2± s + U, r = s4 U, k = 2s2± s + 2U,^= s2± s+ U,\ = s2 ± s + U and t = 2s2 ± s + U, where s divides U. 8. n = 4s2, q = 2s2± 2s, r = 2 s2± s,k = 4s2± 3s,^ = (s± l)(2s± 1) ,A = (s± 1)(2s± 1) and t = s2 + (s ± 1)(2s ± 1) Families 4,5,6 and 7 correspond with the ones found by Leung and Ma in 2.2 (in fact, they correspond to the particular case U = 0). Araluze et al. proved that PSQs with parameters as in families 1,2 and 3 exists for every s and f. They found several sporadic examples for the other families, two of which produced DSRGs not isomorphic to any known ones with that parameters. Finally, we will mention a result about bicirculant association schemes. Let us recall first the definition of association scheme. An association scheme of class s is a pair X = (X, R, where R = {До, ...,RS} is a partition of X2 that satisfies: 1. Ro = {(x, ж) I ж e X}. 2. For every i = 0,..., n, we have {(y, x) | (x, y) € Rj} € R. 3. For every i,j,k = 0,..., n, there exists a non-negative integer pkj such that | {z e X | (x,z) € Rj, (z,y) € Rj}| = p^j whenever (x,y) € Rfc. The cardinality of X is called the order of the association scheme. The Rj are called the basic relations of the association scheme, and the (X, Rj) are called the basic digraphs. It is an easy consequence of part 3 of the definition that all the basic digraphs X, Rj are regular. The degree of Rj is defined as the degree of X, Rj . The association schemes of class one are called trivial. The linear span CR of the adjacency matrices of the Rj is called the Bose-Mesner algebra of the association scheme. The vector space CX is a left CR -module in a natural way. Since CR is a semisimple algebra, the mentioned vector space CX decomposes as a direct sum of irreducible CR-submodules, which are called the irreducible representations of X. An association scheme X = (X,{R0,..., Rs}) is said to be primitive if all the Ri are strongly connected. An automorphism of an association scheme X = (X, {R0,..., Rs}) is a bijection of X that is an automorphism of all the digraphs corresponding to the relations Ri. An association scheme X = (X, {R0,..., Rs}) is said to be bicirculant if there exists a cyclic group of automorphisms of X which acts semiregularly on X and has 2 orbits. I. Kovacs et al. obtained in [15] the following result: Proposition 3.3. Let X be a primitive bicirculant scheme on 2pe points, p > 2 a prime. Then 1. X is of class at most two; and 2. if the class is exactly 2, then 2pe = (2s + 1) 2 + 1 for some natural number s, and the degrees of basic digraphs of X are 1, s( 2s +1), (s + 1)(2s +1), and the multiplicities of the irreducible representations of X in its standard module are 1,pe,pe - 1. Although this result of I. Kovacs et al. is mainly applied to the study of primitive permutation groups, it has an interpretation in terms of the kind of objects in which we are interested in this paper, because association schemes of class two are DSRGs, so that they proved that non-trivial primitive bicirculant schemes of certain orders are in fact DSRGs, and they obtained additional restrictions on that orders. 4 Uniform partial sum families Uniform partial sum families were introduced in [2] as a kind of digraphs which generalizes what happens in family 4 of Proposition 3.2 and in part 2 of Proposition 2.3. Definition 4.1. Let H be a group of order n > 2 and let m > 1 be an integer. Then an (m,n,k,p,A,t)-partial sum family S = {Sj, with 0 < i,j < m, of subsets of H is uniform if it satisfies the following conditions: 1. The cardinalities of the 'diagonal' blocks Si,i are all equal. 2. The cardinalities of the 'non-diagonal' blocks Si,j, it j, are all equal. 3. The 'diagonal' blocks {Si,i : i e Zm} form a partition of H - {0}, where 0 is the identity element of the group. The following two propositions give infinite families, found in [2], of uniform PSFs: Proposition 4.2. If H is a group of order ef +1 and A0,..., Ae_ i is a partition of H- {0} and = f for every i, then S = {Sij}, where S^- = A j, is a uniform (e, ef +1 ,ef,f,f-1,/)-PSF in H. Proposition 4.3. If H is a group of order ef +1 and A0,..., Ae_ 1 is a partition of H- {0} with lA^ = f for every i, then S = {Si,^, where Si,j — Aj if i=j, i,j 0 Aj otherwise, is a uniform (e, ef + 1, e(f + 1) - 1, f + 1, e + f - 2, e + f - 1)-PSF in H. We let A = |S0,01, and B = |S0,11. Thus, IS^I = A for every i, and |Sj = B for every distinct i and j. Lemma 4.4. B = (2 k-ß± A)/(2m) and A = k + m)B. Proof. Using the second part of Definition 3.1 we obtain A + (m- ^^ = k, and applying the trivial character in the third part of the same definition with i 0, j 1 we obtain 2AB + (m- 2)B2 = ßB + pn. Using (1.1) with the DSRG associated to the PSF we get k(k — ß) = pmn + 7. Now, the result follows from the three previous identities and the definition of Д. □ 5 Form of the parameters Now we will derive the parameters of uniform PSFs when the group H is cyclic. We will need first a lemma. Lemma 5.1. Let & be a PSF in an abelian group H, and let x be a non-trivial character of H. Then, m— 1 XI x(S;,;) = m(ß - Д)/2 + гД for some i € {0,..., m}. i=о Proof. By using Definition 3.1, we have that the matrix Ax = (x(&ij)) satisfies AX = ßAX + yIm, and therefore its trace Y.mo1 x($i,i) is the sum of m roots of the polynomial x1 - ßx- y. Since those roots are 2 (ß± Д), the result follows. □ Proposition 5.2. If an m-circulant (m, n, k, p, A,t)-uniform partial sum family exists with m > 3, then the parameters have one of the following forms: 1. n = sm — rm + r2m + 1, k = sm + r2m - r,t = r^m — r + s. A = rm + s + T2 — 2r - l,p = s + r2 A = s - r + r2 ,B = s + r2 with r an integer and s a non-negative integer. 2. / ч О 0 0. 00 .0 2i(i— l) mr , -2 r2m2i + 2 m2r2i — sri — s + smri , k = n -2 sr2i2 - 4 sr i + 2 sr2'mi- 2 s + 4 m2r2i2 - 4 m2 r2^ 4 smr i + s2 2m2 s ' О О .0 0 0. о о 4 m2 r2i- 4 m2r2i — 2 smr + s2 + 2 rm2 s - 4 sm 2m2s s 2 r m 2 r mi s 2 r mi 2m2s 2 r2 mi 2 mr2 i2 s r i s 2 r m 2 r mi A =-,B = —--- with r an integer, s a non-negative integer and i € {2,..., m, and where 2m divides s. s sm sm sm Proof. By the previous lemma, for every non-trivial caracter x of H it holds that m—1 XI x(S;,;) = m(ß - A)/2 + iA for some i € {0,..., m}. i=0 By part 3 of Definition 4.1 we have that the previous sum equals -1, and hence we deduce that ß=((m- 2i)A-2)/m, (5.1) where A is a non-negative integer. Now, from the definition of A we obtain Ai 1 Ai 1 Am 7 = -----. (5.2) m2 Putting U = k-1, we deduce from (5.2) that Ai 1 Ai 1 Am k=U+u^--——-->-. (5.3) m2 From (1.1),(5.1),(5.2) and (5.3) we get i 2 2 2 2 2 2 \ n = [Um + m — 1 + um - 2 A i + mA i + A m - A m + A im — Ail (Um2 - 1 + m + um2 - 2 A i + A m + mA i + A2 im- A2 i2) j (m5 u) (5.4) Let us suppose that the plus sign holds in Lemma 4.4. Since A = (n - 1)/m, we obtain from that lemma that m A i + m-m2 A i + Um2 - 1 + um2 - 2A i + A m + A2 im- A2 i2 n=-^-. (5.5) m2 From (5.4) and (5.5) we find u as u=-ttt--г (-m2 A i + 2 A2 mi- 2A2i2 + 4 mA i- 4 A i- 2^ m2 + 3 m 2m2 1 m - 2 m2 A - Um3 + 2 mA + 2 Um4 m3 A i + A2i2m- A2m2i ± m((Um2 + 3 mA i + A2 im- 2 A i- A2i2 + m- 1- m2 A i)2 + 4im2A2 (m-1)(-1 + i))1/2) (5.6) Since ß is an integer, i = 0 is not possible (see (5.1) and use that m > 3). Let us suppose that i 1 . Then, Um2 - 1 + m- 2 A + 2 A m ^ A m2 ^ A2 + A2m U =--2--(5.7) m2 or Um^ 1 + m ^ 2 A + 2 A m- A2 + A2 m U =----. (5.8) m2 1 m Let us suppose that (5.7) holds. Then we get from (5.4) that n = 0, which gives a contradiction. Therefore, (5.8) must hold, and using this and (5.1),(5.2),(5.3) and (5.4) we obtain that the parameters are mU (Д + 1)2 -- Д+--—, (5.9) m1 , Um^ Д-Д2+ Д2^ + Д m k =-7-ч-, m m 1 —Д2 + Д + Д m + Um t =- m m1 A = Um2 - 1 + 3 m ^ 2 Д + 4 Д m ^ Д2 + Д^ - 3 Д m2 + Д m3 ^ 2 m2 m2 m 1 U (Д + 1)2 M=-7 + --jT-, (5.10) m 1 m2 „ Um^ Д m2 + 3 Д m + 2m + Д^- 2Д^1^Д2^ m2 A=- m2 m 1 Um2 1 m B = Um2 1 m 2 Д 2 Д m Д2 Д2 m m2 m 1 From (5.2), we have that m| (Д + 1)2, and then we deduce from (5.9) that m - 1 |mU. Since m - 1 and m are coprime, then m - 1 |U. Now, from (5.10) we deduce that m| Д + 1. Putting U = s(m -1 ) and Д = rm -1 we obtain parameters as in part 1 of the proposition. Now, let us suppose that i > 1. Since p is an integer, the square root in (5.6) is also an integer, and p =-TT—1-- (-m^ i + 2Д2ш^ 2Д2^ + 4 mД i- 4Д i- 2- m2 + 3 m 2m2 1 m - 2 m2 Д ^ U m3 + 2 m-Д + 2 Um4 m3 Д i + Д2i2m - Д2m2i ± ЦУшЧ 3 mД i + Д2^ - 2 Д i - Д2^ + m - 1 - m2 Д i + s)) (5.11) where s is a non-negative integer and U = (^4 m^2i + 4Д2^2 ^ 4 m^i2 + 4 m^2i2 - s2 ^ 6 smД i - 2 sД2im + 4 sД i + 2 sД2i2 ^ 2 sm + 2 s + 2 sm2Д i)(2sm2) Hence, (-s - 2 mA i + 2 m2 Д i) (-s + 2 mA ^ 2 mД i - 2 m2Д + 2 m2Д i) p = "1/2"---2, ч-- (5.12) sm2 1 m or (s- 2 m^ + 2 mД i)(s + 2 mД i) p = 1/2-2--(5.13) Let us suppose that (5.12) holds. Then we get from (5.4) that n = -s/(2m (- 1 + m) ), and this contradicts that n must be positive. Therefore, (5.13) must hold, and using this and (5.1),(5.2),(5.3) and (5.4) we obtain, by putting Д = r, that the parameters are as in part 2 of the statement. From the form of A we deduce that m divides s, and from the form of p we conclude that 2 divides s. From this two facts and the expression for A we obtain that 2m divides s. Finally, if the minus sign holds in Lemma 4.4, the parameters have the same form as when the plus sign holds by putting Д = -r, so that r can have both positive and negative values. □ By Proposition 4.2, m-circulant uniform PSFs as stated in part 1 of Proposition 5.2 always exist for r = 0 and, by Proposition 4.3, m-circulant uniform PSFs as stated in part 1 of Proposition 5.2 always exist for r = 1. 6 The tricirculant case Now we will consider the case when m = 3. In this case, we will call the PSF tricirculant. Proposition 6.1. If a tri-circulant (3, n, k, p, A,t)-uniform partial sum family exists, then either 1. n = 3r2 — 3r + 1 + 3s, k = 3r2 — r + 3s, p = r2 + s, A = r2 + r — 1 + s, t = 3r2 - r + s, A = r2 - r + s and B = r2 + s. (6.1) 2. n = 9r2 + 6r + 1, k = 9r2 + 10r + 2, p = 3r2 + 5r + 2, A = 3r2 + 4r + 1, t = 5r2 + 6r + 2, A = 3r2 + 2r and B = 3r2 + 4r + 1. (6.2) 3. n = 90r2 ^ 60r + 10, k = 90r2 ^ 40r + 3, p = 30r2 ^ 5r, A = 30r2 ^ 10r + 1, t = 80r2 - 40r + 6, A = 30r2 ^ 20r + 3 and B = 30r2 ^ 10r, (6.3) with r an integer and s a non-negative integer. Proof. Using the same notations as in the proof of Proposition 5.2, we have 2 + 2^ = 0 (mod 3), and hence i = 1 or i = 2. If i = 1, by following the proof of that proposition, we can see that the parameters are as in (6.1). Let us suppose that i = 2. We will assume that the plus sign holds in Lemma 4.4, because when the minus sign holds, it is easy to prove that the parameters have the same form, changing the sign of r. We have from (5.11) that (9U + 2(Д - 1)2)2 + 144Д2 = ((9U + 2(Д - 1)2) + s)2 for some positive integer s, and hence 2(9U + 2(Д ^ 1)2)s + s2 = 144Д2. Since U > 0, we deduce that s < 39 and, since 6 must divide s, then the possible values that s can take are 6,12,18,24,30,36. Now, having into account that Д = 1 (mod 3) and the expression for n in Proposition 5.2 we have that for s = 18,36 we obtain a contradiction with the fact that n is an integer, and for s = 6,24 we obtain a contradiction with the fact that 3 divides n - 1. For s = 12, by putting Д = 1 + 3r we obtain parameters as in part (6.2), and for s = 30, by putting Д = 1 + 3u we have U = 2" +58"^7 and hence u г -2 (mod 5). Putting now u = -2 + 5r we obtain parameters as in (6.3). □ We can eliminate some values of r in family (6.3). In the next proposition, Grobner bases are used (for definitions and results on Grobner bases, we refer the reader to [3]). Proposition 6.2. If a tricirculant uniform PSF exists with parameters as in family (6.3) of the previous proposition, then r 1 mod 2 . Proof. If X is any non-trivial character of H then we obtain from Definition 3.1, by applying the character x, that 2 Ex(Si,0 x(Sij) - Sij ßx(Si,j) = 0 for every i, j. (6.4) i=о By putting the indeterminate Ti,j instead of xC^ij) and calculating a Grobner basis for the polynomials obtained from the lefthandsides in (6.4) with respect to the lexicographic order with To,o > Ti,i> T2,2 > To,i > Ti,o> To,2 > T2,o > Tu> T2,i > r, we conclude that x(sM)2 ^ x(Si,i) - 6 + 35r + x(Su) XS2,i) - 50r2 + x(So,i)x(Si,o) + 5x(SM)r = 0. (6.5) Now we take the character x that takes the generator of the cyclic group H to вп2, where в is a primitive n-th root of unity. Since A is an odd number and B is an even number, we have that the x^j ) are integer numbers such that x(Si,j) is odd if i = j and even in other case. Now, we obtain from (6.5) that (x(Si,i) - 1)(x(Si,i) +r) = 2{r4 1) (mod 4). Suppose that r is even. Then the above implies that xCSm) = 3 (mod 4). We show below that this leads to a contradiction, hence r must be indeed odd. By the choice of the character x we get for every i e {0,1,2}, x(Si,i) = -|Si,i| + 2|SM n ker(x)l = -30r4 20r - 3 + 2|SM n ker(x)|. The group ker(x) is of order n/2 = 45r2 - 30r + 5, which is odd. Using this and that u?=o(S^ n ker(x)) = ker(x) ~ {0}, we see that at least one of the numbers n ker(x)l must be even. We may set the indices at the beginning so that |Si,i n ker(x)| is even, and hence above we find x(S^ ) s 1 (mod 4), a contradiction. □ Apart from the infinite families of uniform PSFs showed in Section 4, we will analyze some sporadic examples for the tricirculant case. When s = 0 in the first family in Proposition 6.1, that is, when the graph is undirected, Kutnar et al. found in [16] PSFs for r = 1,2 and 3. For r = -1, s = 2, a PSF with the parameters and structure indicated in Proposition 6.1 was found by the author and A. Araluze in [20]. We have found now for r = 2, s = 2, by using techniques of combinatorial optimization, a uniform tricirculant PSF which generates a DSRG with new parameters, which appear as an undecided case in Hobart and Brouwer's table [10]. This PSF is S0,0 = (3,4,7, 9), S^ = (0,4, 7, 8,11,12), S0,2 = (0,1,2,4,6,10), S^0 = (1, 2,5, 6, 8, 9), SM = (1, 5,6,12), S1,2 = (2, 3,5, 6,7, 8), S2,0 = (0,1, 3, 7, 9,11), S^ = (0, 5, 6,7, 8,11), S2,2 = (2,8,10,11). 7 Almost-uniform partial sum families In this section we will present a kind of partial sum families that generalizes the uniform ones. Definition 7.1. Let H be a group of order n > 2, H a subgroup of H of order n' and let m > 1 be an integer. Then an (m, n, k, p, A, t) -partial sum family S = {Sj, with 0 < i, j < m, with the Si,j subsets of H, is almost-uniform with respect to H if it satisfies the following conditions: 1. The cardinalities of the 'diagonal' blocks Si i are all equal. 2. The cardinalities of the 'non-diagonal' blocks Sij, it j, are all equal. 3. The'diagonal'blocks {Si,i : i e Zm} form a partition of H-H'. Remark 7.2. Observe that in the particular case when H e almost uniform PSFs are just uniform PSFs. In what follows, we will study the case when H is a proper subgroup of H. Let us analyze the form of the parameters of m-circulant (m,n, k,p, A, t)-almost-uniform partial sum families over a cyclic group H with respect to a proper and non-trivial subgroup H : Proposition 7.3. fan (m, n, k, p, A, t)-almost-uniform partial sum family exists with m > 3 with respect to a proper and non-trivial subgroup H' of the cyclic group H, then the parameters have one of the following forms: 1. n = (im — m + i) imr2 + (1 — i) mr + ms, k = (im — m + i) imr2 + (^ + m — im) r + ms, p = (im — m + i) ir2 - ri + s, A = (im — m + i) ir2 + (m — 3 i) r + s, t = —ri + i2r2m + s, A = (im — m + i) ir2 + (1 _ 2 i) r + s, and B = (im — m + i) ir2 + (1 — i) r + s. r2j2 = (j "" j r + ( —j — 1) r + sr ' m 2 2 rj 1 rj k = (j - jr ^rj + smH--, m rj rj m rm rj m ^ = s ■ , r j (—3 m — rm + rjm + rj) Л = s + r H-------, t = jl^j rj (rm^ 2 m — mr + rj ) A — s + -r-, rj (—m — mr + rmj + rj) where m divides rj. 3. 2p (p — 1) mr2 r (2r m2p — 2 rm2p4 si + sp — smpl -, k =--, / 2 • 2'2 2' 2 22 22 222 t = —(^2 sr mi + 2 sr i + 4 sr ip — 2 sr mp + 2 sr p + 4 m r p — 4 m r p + 2 srm — 4 srmp — s2)/(2m2s), 0 0 000 o o 4 m2r2p — 4 r2m2p2 + 2 srm — s— 2 4 srmi Л = ^v2--, (s + 2rmp)(s^ 2rm + 2rmp) м = V2-2-, r(^2 mrp2 + 2 mrp + si) rp(s^ 2 mr + 2 mrp) A =--, B =-. m2 m n s sm ms ms Proof. We have U i Si,i = H - H'. If x is a character of H which is non-trivial on W, then Si x(Si,i) = 0. If X is trivial on W but not on H, then £i x'(Si,i) = -|H'|. Since, by Lemma 5.1, the difference of both sums of characters must be divisible by Д, we have that WI = гД with i€ {1,..., m}. Since, again by Lemma 5.1, £; x(Si,i) = m(ß - Д)/2 + ]Д with je {0, ...m, we have ß=(m- 2j) Д/m, (7.1) where Д is a non-negative integer. We have 1=(jm-j 2 Д2 / m2. (7.2) Now, j=t-p= (jm-j 2) Д2/ m2, k = U + p + (jm-j 2) Д2 / m2(where U = k-t), ß=(m- 2j)Д/m. (7.3) From these equalities and (1.1) we get 2 2 2 22\/2 2 n=(Um + pm + Дmj + Д mj — Д j )(Um + pm + Дmj - Дш2 + jД2m- j2Д2)/(m5 p) (7.4) Let us suppose that the plus sign holds in Lemma 4.4 Since the PSF is almost-uniform we have n - гД = mA, and therefore Um2 + pm2 + Д mj - j2 Д2 + jД2m- Д m2j + гД m2 n =-2--(7.5) m2 From (7.4) and (7.5) we deduce / 23 223 2223 2 M = (—Д m — m U + mj Д - m гД + 2 Um - m jД + m Д j — Д m j + 2Д mj — 2 j2 Д2 + 2 jД2m±m((Um2 - Д m - 2Д mi + гД m2 + 3Д mj ^'2Д2 + j Д2ш - Д m2 j 2 + 4 ш2Д2( i-j + 1) (i-j)(m- 1))1/2) 2m2 m 1 . Let us suppose that j i. Then, Um2 + Д mi + i^2m- i2 Д2 M =--2--(7.6) or Um2 + Д mi- Д m2 + iT-m- i2 Д2 M =-27-n--(7.7) m2 m 1 If (7.6) holds, then we get from (7.4) that n = 0, which is a contradiction. Thus, (7.7) must hold, and we obtain Um2 + Д mi- Д m + iД2m-i2 Д2 (78) m1 Um2+ Д i- Д m + iA2m-i2A2 k=-д-гт--(7.9) m m 1 iA2m - i2 A2 + Um + A i - A m t =-,-т---(7.10) m m 1 л Um4 3 A mi - 2 A m2 + iA2m - i2 A2 + A m3 - 2 A m2i A =-27-n- (7.11) m2 m 1 Um2 A mi A m2 iA2m i2A2 p=-~s-n--(7.12) m2 m 1 Um4 2 A mi- A m + iA2m-i2A2 - A m2i ^-m-П--(7.13) m2 m 1 „ Um4 Am^ Am + iA2m-i2A2 B =-27-n--(7.14) m2 m 1 From (7.12) we have i2 A2 = 0 (mod m) and, from (7.9), - i? A2 + (mod m). Therefore, (mod m). From (7.13), (im - i2)A2 + (2i - 1)mA г 0 (mod m2). Using this and the previous congruence, we obtain mA 0 mod m2 , and hence A = 0 (mod m). (7.15) From (7.12), ue(ì2- 0 A2 + (1- 0 A (mod m- 1). (7.16) Now, using the two previous congruences and putting A = rm and U = i(i — 1)r2m2 + (1 - i)rm + s(m — 1), we obtain parameters as in part 1 of the proposition. Now, let us suppose that j = i + 1. In this case, by reasoning in a similar way as before, we obtain Um2- A m2 + j A2m + A m + A mj-j2 A2 , , n =-?—T\--(7.17) m m 1 Um2 jA jA2m j2A2 k =-j , j 14 j--(7.18) m m 1 U m2 + A mj + jA2m - j 2 A2 p=-n-n--(7.19) m2 m 1 3A mj + jA2m - j2A2 - A m2 + A m3- 2 A m2 j A =-^—T\- (7.20) m2 m 1 jA2m-j2A2 + Um + jA , t = j-^---— (7.21) m m 1 Um2+ + 2Д mj — j2Д2 - Д m2j A = ^П 7\ (7.22) m2 m 1 Um4 Д mj+jД2m^j2Д2 B =-^—V\--(7.23) m2 m 1 From (7.19) we have Д2/2 s 0 (mod m), and from (7.18), Д / - Д2j2 s 0 (mod m), and hence Д j s 0 (mod m). From (7.19), we have U + Дj + Д2/ - Д2?2 = 0 (mod m - 1). Putting U = -Л / - Д2/ + Д2.2 + s(m - 1) and Д = r, we obtain parameters as in part 2 of the proposition. When j ^ i and j f i + 1 we find the parameters following the line of the proof of Proposition 5.2 and putting p = j - i and r = Д. If the minus sign holds in Lemma 4.4, we find analogous families of parameters by putting r = — Д, so that r can be positive or negative. □ We have found, by using methods of combinatorial optimization, 14 examples of PSFs corresponding to family 1, which are listed in the appendix. For three of the examples, corresponding to m = 3, i = 1, r = -1, s = 3, m = 3, i = 1, r = -1, s = 4 and to m = 5, i = 1,r = 1, s = 2, the parameters appear as undecided cases in Hobart and Brouwer's table [10]. We wonder if such PSFs exist for m = 3, i = 1, r = 1 or r = and every positive integer s. 8 Appendix Family 1: For m = 3: 1. i 1, r 1, s 1: (((1), (0,1, 5), (2, 3,5)),(( 0, 3,5),( 3), (0,1,4)), ((1, 3,4),( 0,1, 2),( 5))) 2. i = 1,r = -1,s = 2: (((5,8), (0,4, 6, 7), (0,1,3, 7)), (( 1,2, 3, 5), (2,4), (0, 5, 6, 7)), ((2, 3,6, 8), (0, 3,4,7),( 1,7))) 3. i 1, r 1, s 3: (((5,9,11) ,(1,2,4,9,10),(0,1, 3, 8,9)), ((3, 5,8, 9,11), (1, 2,10), (0,1, 3, 7, 9)), ((0, 2,3,4,11), (4, 5,7, 8, 9), (3, 6,7))) 4. i 1 , r 1 , s 4: (((6,9,12,13), (0,1, 3,4,11,12), (6, 7,10,11,13,14)), ((1,4,5, 8,10,14),(1, 7,11,14), (0, 2, 5, 6,9,11)), ((1, 2,3,4,7, 9), (0,7, 8,9,10,13), (2,3,4,8))) 5. i 1 , r 1 , s 1 : (((5), (3,4), (2,3)),((2, 3), (1),(0,5)),((0, 3), (1,4),(3))) 6. i = 1, r = 1, s = 2: (((7, 8), (0,1, 2), (2,3, 4)), ((3, 7, 8), (1, 5), (2, 3, 7)), ((0, 5, 7),( 0,2, 7), (2,4))) 7. i 1 , r 1, s 3: (((3, 9,10), (0,5, 6,7), ( 6, 7, 8, 9) ),( (0, 3, 9,10), ( 5, 6,7) ,( 6,7, 8, 9) ), ((2, 3, 5, 8), (0, 5,10,11), (1, 2,11))) 8. i = 1,r = 1, s = 4: (((6, 8,12,14), (3, 5, 9,11,12), (5,7,11,13,14)), ((1, 3,4, 5,7),( 1, 2,4,13), (0, 2, 3,4, 6)), ((1,4, 8,10,12), (1, 5, 7, 9,13), (3,7, 9,11))) For m 4: 1. i 1, r 1, s 1: (((4, 6, 7), (4, 5), (4,7), (0,1)), ((3,4), (1, 2,4), (4, 5), (1, 6)), ((1,4), (2, 3), (4, 5, 6), (2, 3)), ((0,1), (2,7), (5, 6), (2, 3,4))) 2. i 1 , r 1 , s 1 : (((5), (4, 7), (3, 6), (4,7)), (( 1, 2), а), (0, 7), (0, 1)), ((1, 2), (0,1), (7), (0,1)), ( (1,4), ( 0,3), ( 2, 7), (3))) 3. i= 1,r = 1,s = 2: (((1, 2) ,(3,4, 5), (4,5, 6), (2, 6,10)), ((2,4, 9),( 5,7),( 1, 6,8),( 2,6,10) ), ((6, 7, 8),( 9,10,11) Д10,Д 0,4, 8) ), ((0, 2, ^ Д 3,5,Д 4,6,11),( 4, 8))) 4. i 1 , r 1, s 3: (((3,13,Д8,9,10,11), ( 1,2, 3,, (0, 9,10,11)), ((4, 5, 7,, (1,2,, (3,8, 9,10), (0,1,2, 7)), ((1, 2,4, 7), (12,13,14,15), (5,6, 7), (4,13,14,15)), ((0, 3,13,,( 8,9,10,11),( 1,2, 3,, (9,10,11))) For m 5: 1. i 1, r 1, s 1: (((^, (0, 7), (1, 2),( 6,7), ( 3, 6)),( (3, 8), , (4, 5), (0, 9), ( 6,9) ), ((4, 9), (6, 9), а), (5, 6), (2, 5)), ((3,8), (0, 3), (4, 5), (9), (6, 9)), ((4, 9), (1,4), (5, 6),( 0,1),( 7))) 2. i= 1,^ 1,^ 2: (((2,Д0, 2,4), (7,9,Д0,2, ^ Д10,12,14)), ((3,13,14), (1, 5), (7, 8,12), (0,1, 5), (0,10,11)), ((1, 5, 6), (3,7, 8), (10,14), (3, 7, 8), (2, 3,13)), ((2, 6,13), (0,4,8), (0,7,11), (4, 8), (3,10,14)), ((3,10,14), (1, 5,12), (4, 8,12), (1, 5,12), (7,11))) Acknowledgments: The author thanks the anonymous reviewers for their valuable suggestions and comments. 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Marušic, Strongly regular bicirculants and tricirculants, Ars Combin. 25 C (1988), 11-15. [22] M. J. de Resmini and D. Jungnickel, Strongly regular semi-Cayley graphs, J. Algebraic Combin. 1 (1992), 171-195. /^creative , ars mathematica ^commons contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 215-223 On automorphism groups of graph truncations Brian Alspach School of Mathematical and Physical Sciences University of Newcastle Callaghan, NSW 2308 Australia Edward Dobson Departmentment ofMathematics and Statisitics Mississippi State University Mississippi State, MS 39762 USA and UP IAM University of Primorska Muzejeska trg 2, 6000 Koper, Slovenia Received 11 May 2014, accepted 13 November 2014, published online 17 December 2014 Abstract It is well known that the Petersen graph, the Coxeter graph, as well as the graphs obtained from these two graphs by replacing each vertex with a triangle, are trivalent vertex-transitive graphs without Hamilton cycles, and are indeed the only known connected vertex-transitive graphs of valency at least two without Hamilton cycles. It is known by many that the replacement of a vertex with a triangle in a trivalent vertex-transitive graph results in a vertex-transitive graph if and only if the original graph is also arc-transitive. In this paper, we generalize this notion to t-regular graphs Г and replace each vertex with a complete graph Kt on t vertices. We determine necessary and sufficient conditions for T (Г) to be hamiltonian, show Aut(T(Г)) = Aut(r), as well as show that if Г is vertex-transitive, then T (Г) is vertex-transitive if and only if Г is arc-transitive. Finally, in the case where t is prime we determine necessary and sufficient conditions for T (Г) to be isomorphic to a Cayley graph as well as an additional necessary and sufficient condition for T (Г) to be vertex-transitive. Keywords: Truncation, automorphism group, Cayley graph, Hamiltonian. Math. Subj. Class.: 05C25 E-mail addresses: brian.alspach@newcastle.edu.au (Brian Alspach), dobson@math.msstate.edu (Edward Dobson) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 1 Introduction In 1969 Lovasz posed the problem below (this statement is written exactly as Lovasz wrote it). Problem 1.1 (Lovasz, 1969). Let us construct a finite, connected, undirected graph which is symmetric and has no simple path containing all elements. A graph is called symmetric, if for any two vertices x, y it has an automorphism mapping x into y. Usually this problem is stated as the conjecture that every vertex-transitive graph contains a Hamilton path (here "vertex-transitive" and "symmetric" are synonyms). Typically though it is usually the case that this conjecture is verified by showing that a a particular vertex-transitive graph contains a Hamilton cycle. Much work has been done in attempting to verify this conjecture — see [5] for some recent information regarding progress on this conjecture. The Petersen graph is vertex-transitive but does not contain a Hamilton cycle (see for example [3, Theorem 1.5.1]), while Tutte [9] first showed that the Coxeter graph is not hamiltonian, with an additional proof by Biggs [1]. The graphs obtained from the Petersen graph and the Coxeter graph by replacing each vertex with a triangle — called the truncation — are also vertex-transitive graphs that do not contain a Hamilton cycle. These four graphs are the only known connected vertex-transitive graphs, other than K2, that do not have a Hamilton cycle. The truncation of the Petersen graph is shown in Figure 1. It turns out that the truncation of the truncation of the Petersen and Coxeter graphs are not vertex-transitive. It is known by many that the truncation of a trivalent graph Г is vertex-transitive if and only if Aut(T) is also transitive on the edges of Г, or edge-transitive, although neither of the previously stated facts are proven in the literature. We will generalize the notion of truncation to vertex-transitive graphs of valencies other than 3. Note that as a triangle can be viewed as either a cycle of length 3 or a complete graph K3, there are two natural generalizations of the idea of truncation. Namely, one can "replace" each vertex with a cycle or with a complete graph, or even with an arbitrary graph. These have been studied for example in [2, 6, 11] We note that in [11], the graph obtained by replacing each vertex with a complete graph is called a clique-inserted graph, and that replacing each vertex with a cycle is motivated by map truncation. For a vertex v g V(Г), we denote the valency of v in Г by val(v). Definition 1.2. Let Г be a graph. The truncation T (Г) of Г is obtained from Г by replacing each vertex v of Г with a clique on val(v) vertices, denoted Tv, and whenever uv g E (Г), then one vertex of Tv is adjacent to one vertex of Tu and no vertex of Tv is adjacent to more than one vertex outside of Tv. Note that if uv g E (Г), we do not specify which vertex of Tu is adjacent to Tv. Obviously, different choices of such vertices will result in different graphs, but all such choices result in isomorphic graphs as each Tu and Tv is a complete graph and no vertex of Tv is adjacent to more than one vertex outside of Tv. Also, as each vertex in Tv is adjacent to val(v) - 1 vertices inside Tv and exactly one vertex outside Tv, a vertex u g Tv has valency val(v) in the truncation T (Г) . In particular, the truncation of a t-regular graph is still t-regular. We should point out that there is an equivalent definition of graph truncation introduced in [7]. For a graph Г, let £(Г) be the graph obtained from Г by subdividing each edge via the insertion of a single vertex, and Ь(Г) be the line graph of Г. Then T (Г) = £(£(Г)). In this paper we show T (Г) contains a Hamilton cycle if and only if Г has a spanning eulerian subgraph (Theorem 2.1). We then show that for a graph Г with minimal valency at least 3 the automorphism group of its truncation is isomorphic to Аи^Г) (Theorem 3.6), and subsequently that a connected vertex-transitive graph with minimal valency at least 3 has a vertex-transitive truncation if and only if it is arc-transitive (Theorem 3.7), or transitive on the set of directed edges or arcs of Г. We remark that this is consistent with the statements earlier that a trivalent vertex-transitive graph has vertex-transitive truncation if and only if it is edge-transitive as a vertex- and edge-transitive graph of odd valency is necessarily arc-transitive [10, 7.53]. Finally, for a vertex-transitive graph Г of prime valency, we also determine necessary and sufficient conditions for T (Г) be a Cayley graph provided that T (Г) is vertex-transitive (Theorem 3.8), as well as provide an alternative characterization of when T (Г) is vertex-transitive (Theorem 3.10). We begin with necessary and sufficient conditions for T (Г) to be Hamiltonian. 2 Hamiltonicity of Graph Truncations Theorem 2.1. If Г is a graph, then T (Г) contains a Hamilton cycle if and only if Г contains a connected spanning eulerian subgraph. Proof. First suppose that Г has a connected spanning eulerian subgraph Д, and let v0vi • • • vrv0 be an Euler tour of Д, where we traverse the tour so that the edge vjvi+1 is traversed from vi to vi+1. Given that the edge vjvi+1 is traversed from vi to vi+1, let uij0 and ui+1j1 denote the vertices of Tv. and Tvi+1, respectively, that are adjacent. For each x g V(Г), we let xm be the largest nonnegative integer for which vXm = x, set Yx = {i < xm : vi = x}, and Zx = {ui,o, ui,l : i e Yx}. For each 1 < i < r construct a path Pi as follows: Setting x = vi, we let Pi = ui,0 if i < xm, while if i = xm, we let Qi be a Hamilton path from ui, i to ui, о in T (r)[Tx - Zx], the subgraph of T (Г) induced by Tx - Zx. Let Pi be the path obtained from Qi by removing the initial vertex ui,l of Qi. We observe that Qi and consequently Pi certainly exist as T(r)[Tx - Zx] is a clique. Then uo,oui,iPiui,ou2,iP2u2,o ... ur,iPrur,ouo,iPouo,o is a Hamilton cycle in T (Г). Intuitively, we travel along the Euler tour until the last time we visit a Tv, at which point we traverse all the previously unvisited vertices of Tv. Conversely, suppose that H = vovl... vn is a Hamilton cycle in T (Г) (so vn = vo). For each 0 < i < n, let vi e Txi, xi e V(Г). Let E' = {xixi+l : Txi = Txi+1}, so that E' is simply the set of edges of H that connect different inserted cliques. Then the edges of E' form a spanning connected subgraph of Г as H is a Hamilton cycle in T (Г). Additionally, with the exception of Tx0 and Txn = Tx0, each time one traverses H and enters a Tx, one must exit that Tx. We conclude that every vertex of the graph formed by the edges of E' has even valency and so this graph is eulerian. □ In the case of trivalent graphs, as the only spanning eulerian subgraph of a trivalent graph is necessarily a Hamilton cycle, we have the following result. Corollary 2.2. The truncation of a trivalent graph Г is hamiltonian if and only if Г is hamiltonian. As the Petersen graph and the Coxeter graph are both trivalent graphs that are not hamiltonian, the following result is evident. Corollary 2.3. The truncations of the Petersen graph and the Coxeter graph are not hamil-tonian. 3 Vertex-transitive Graph Truncations While the truncation of any vertex-transitive trivalent graph that is not hamiltonian will give a trivalent graph that is not hamiltonian, the truncation of such a graph need not be vertex-transitive. Indeed, the truncations of the truncations of the Petersen and Coexeter graphs are not hamiltonian, but it turns out that they are not vertex-transitive. We now investigate when the truncation of a vertex-transitive graph is vertex-transitive. We begin by studying the relationship between Aut^) and Aut(T(Г)) for any graph Г. Definition 3.1. Let Г be a graph. We call the set T = {Tv : v e V(Г)} the fundamental vertex partition of V(T(Г). There is also a fundamental edge partition of E(T(Г)) with two cells, where one cell consists of those edges within a Tv, v e V(Г) (the clique edges), and the other cell consisting of those edges between two inserted cliques (the original edges). Lemma 3.2. Let Г be a graph with each vertex of valency at least 3. Then the fundamental vertex and edge partitions of T (Г) are invariant under Aut(T (Г)). Proof. We need only show that the fundamental vertex partition of T (Г) is invariant under Aut(T(Г)), as this implies that the fundamental edge partition is invariant under Aut(T(Г)). An edge xy with x G V(Tv) and y G V(Tu), u = v, cannot belong to a triangle because y is the only neighbor of x not in V(Tv ) and x is the only neighbor of y not in V(Tu). On the other hand, every edge xy with x, y in the same V(Tv) belongs to a triangle because Tv is a clique and t > 3. This then implies that Aut(T(Г)) permutes the sets in the partition T = {V (Tv ) : v G V (Г)}. □ We now introduce standard permutation group theoretic terms related to the fundamental vertex partition. Definition 3.3. Let G < Sn be transitive and act on Zn. Let B C Zn. If g(B) = B or g(B) n B = 0 for every g G G, then B is a block of G. If B is a block, then g(B) is also a block for all g G G, and {g(B) : g G G} is a G-invariant partition. Of course, singleton sets and Zn are blocks for every transitive group G < Sn, and the corresponding G-invariant partitions are trivial. If G has a nontrivial G-invariant partition, then G is imprimitive, and is primitive otherwise. Finally, for a G-invariant partition B, we denote by fixG(B) the subgroup of G which fixes each block of B setwise. That is, fixG(B) = {g G G : g(B) = B for all B G B}. We remark that if Aut(T(Г)) is transitive, then the fundamental vertex partition is an Aut(T (r))-invariant partition. Now observe that if Г is a cycle of length n, then T (Г) is a cycle of length 2n. Hence Aut(T (Г)) = D2n, the dihedral group of order 4n. Henceforth, we will assume that every vertex has valency at least 3, in which case T (Г) always contains a triangle. Theorem 3.4. Let Г be a graph where each vertex of Г has valency at least 3. Then Aut(T (Г)) acts faithfully on the fundamental vertex partition T and is isomorphic to a subgroup of Aut^). Proof. That Aut^) acts on T was established in Lemma 3.2. Let K be the kernel of the action of Aut(T(Г)) on T (if Aut(T(Г)) is transitive, then K = fixAut(T(r))(T)). We claim that K = 1. Indeed, if K = 1 with 1 = 7 g K, then let Tv G T such that K |Tv = 1. Then there exist distinct x, y G Tv such that 7(x) = y. Now, x is adjacent to some vertex z G Tu, u = v, and so y(x)y(z) is also an edge from Tv to Tu. However, there is only one edge from a vertex of Tv to a vertex of Tu in T (Г), a contradiction. Thus K =1. To finish the result, we need only show that if 7 G Aut(T(Г)), then 7 g Aut^), where 7 is the induced action of 7 on T. So suppose that uv g Е(Г). Then some vertex of Tu is adjacent to some vertex of Tv ,andas 7 G Aut(T (Г)), some vertex of y(Tu) = Ty(u) is adjacent to some vertex of y(Tv) = T^(v). But this occurs if and only if 7(u)7(v) G E (Г). □ Corollary 3.5. The truncations of the truncations of the Petersen and Coxeter graphs are not vertex-transitive. Proof. Let Г be the Petersen or Coexeter graph. If T (T (Г)) is vertex-transitive, then 9 divides |Aut(T(T(Г)))| as |V(T(T(Г)))| = 9|V(Г)| and the size of an orbit or a group divides the order of the group. By Theorem 3.4 applied twice, we see that 9 divides |Aut^)|. However, the automorphism group of the Petersen graph has order 120 as it is isomorphic to S5 (see for example [3, Theorem 1.4.6]) while the automorphism group of the Coexeter graph has order 336 as it is PGL(2,7) (see for example [1]), neither of which are divisible by 9. □ Theorem 3.6. If Г is a graph with each vertex of valency at least 3, then Aut(T (Г)) = Aut(r). Proof. In view of Theorem 3.4, it suffices to show that each element of Aut(r) induces an element of Aut(T(Г)), and that different elements of Aut(r) induce different elements of Aut(T (Г)). Let y G Aut(Г). Let x G V (T (Г)), and v G V (Г) with x G Tv. Then there exists a unique y G V (T ) not contained in Tv with xy G E(T (Г)). Let u G V (Г) such that y G Tu. Then y(u)y(v) g E (Г), and so there exists vertices xX g TY(v) and y' G TY(u) such that x'y' G E (T (Г)). Define Y : V (T (Г)) ^ (T (Г)) by Y(x) = x'. Note that as the original edges of T (Г) form a perfect matching, y is a well-defined bijection. Additionally, by definition, y maps the original edges of T (Г) to the original edges of T (Г). As y also map the fundamental vertex partition of T (Г) to itself, it maps the clique edges of T (Г) to the clique edges of T (Г). Thus Y G Aut(T (Г). Finally, as the induced action of y on T is y and Aut(T(Г) is faithful on T by Lemma 3.2, different elements of Aut^) induce different elements of Aut(T(Г). □ For a transitive group G acting on Zn, we denote the stabilizer in G of v by StabG (v). Then StabG(v) = {g G G : g(v) = v}. Concerning the statement of the following result, a vertex- and edge-transitive graph of odd valency is necessarily arc-transitive [10, 7.53]. Theorem 3.7. If Г is a connected vertex-transitive graph with each vertex of valency t > 3, then T (Г) is vertex-transitive if and only if Г is arc-transitive. Additionally, T (T (Г)) is not vertex-transitive. Proof. Before proceeding, some general observations about T (Г) are in order. As T (Г) is regular of valency t, T(Г)р^] is regular of valency t - 1. As | V(Tv) | = t, we see that there are exactly t edges with one end in Tv and the other end not in Tv, and of course each vertex of Tv is incident with exactly one such edge. Additionally, between some vertex of Tv and some vertex of Tu there is either exactly one edge if uv g Е(Г) or no edges if uv G Е(Г). Thus, each edge with one endpoint in Tv and the other endpoint outside of Tv uniquely determines a vertex in Tv and uniquely determines a Tu in which the other endpoint of the edge is a vertex. Conversely, each vertex x of Tv uniquely determines an edge with x as an endpoint and one endpoint not in Tv. Suppose that Г is arc-transitive, with v g V(Г). Then StabAut(-r) (v) is transitive on the neighbors Nr(v) of v. Set Nr(v) = {ub ..., ut} and let Yi,j G StabAut(r)(v) such that Yi(ui) = uj. As Aut^) = Aut(T(Г)) by Theorem 3.6 and Aut(T(Г)) acts faithfully on T = {Tv : v G V(Г)} by Theorem 3.4, there exists a unique Y^j G Aut(T(Г)) such that the action of Yi,j on T is Yi,j. As the action of Aut(T(Г)) on T is Aut^) which is transitive, in order to show that Aut(T(Г)) is transitive it suffices to show that {ó g Aut(T(Г)) : ó(Tv) = Tv} is transitive on Tv. Let x, y G Tv. Then there exist 1 < i, j < t suchthat xvui ,yvuj G E (T (Г)), where v^ G T^ and vu3- G T^. Then Yi,j (T^ ) = T^ and Yi,j (Tv ) = Tv. As each vertex of Tv is incident with exactly one edge whose other endpoint is not in Tv and Yi,j G Aut(T(Г)), we have that Yi,j(xvui) is the unique edge of T (Г) with one endpoint in Tv and the other in Tuj. That is, Yi,j (xvu. ) = yvuj. As Yi,j(Tv) = Tv, we conclude that (x) = y. Thus {J G Aut(T(Г)) : S(TV) = Tv} is transitive on Tv and the result follows. Conversely, suppose that T (Г) is vertex-transitive. It suffices to show that the stabilizer in Aut(r) of v G V(Г) is transitive on its neighbors. Observe that T = {Tv : v G V(Г)} is an Aut(T(^-invariant partition and Hv = {h g Aut(T(Г)) : h(Tv) = Tv}, v G V(Г), is transitive on Tv. Hence, if x, y G V(Tv), then there exists Yx,y G Aut(T(Г)) such that Yx,y (x) = y. For each vertex x of Tv, we denote the uniquely determined edge with x as an endpoint and with the other endpoint not in Tv by ^x — xZx. We let vx g V(Г) be such that zx G V(Tvx ), and observe that if x, y G V(Tv) with x = y, then Tvx = Tvy. Each edge xzx with x g V(Tv) then induces an edge vvx g Е(Г), and such edges are pairwise distinct. More specifically, there are exactly t edges vvx g Е(Г) induced by edges of the form xzx with x G V(Tv ). This then implies that the neighbors in Г of v are {vx : x G V(Tv)}. Finally, observe that Yx,y (xzx) is an edge with one endpoint y G V(Tv) and Yx,y (zx ) G V(Tv ). Denoting by Yx,y the automorphism of V(Г) induced by the action of Yx,y on T (with each Ta identified with the vertex a), we see that Yx,y (vvx) = vvy, and the stabilizer of v g V(Г) in Aut^) is transitive on the neighbors of v and so Г is arc-transitive. It now only remains to show that T (T (Г)) is not vertex-transitive. In view of our earlier arguments, it suffices to show that if Г is edge-transitive, then T (Г) is not edge-transitive. If Г is edge-transitive, then Aut(T(Г)) is transitive, and Aut(T(Г)) admits T as an Aut^-invariant partition. But T (Г) contains edges with both endpoints in Tv and edges with one endpoint in Tv and one endpoint outside of Tv. As T is an Aut^)-invariant partition, no automorphism will map an edge of the former type to an edge of the latter type. □ We now restrict our attention to graphs with prime valency. We remark that in the following result, the restriction to prime valency is only used to establish sufficiency. Theorem 3.8. If Г is a connected arc-transitive graph of prime valency t > 3 and order n, then T (Г) is isomorphic to a Cayley graph if and only if Aut^) contains a transitive group of order nt. Proof. As a vertex-transitive graph is isomorphic to a Cayley graph if and only if its automorphism group contains a regular subgroup [8], T (Г) being a Cayley graph implies that Aut(T(Г)) contains a transitive group R of order nt which is a isomorphic to a transitive subgroup of Aut^) of order nt by Theorem 3.4. Conversely suppose there exists R < Aut^) such that R is transitive on V(Г) and has order nt. It suffices to show that for fixed v g V(Г), the subgroup H of R fixing the set V(Tv) is transitive on V(Tv), as then R is transitive and as |R| = |V(T(Г))|, we have that R is regular. Now, T is an Aut(T(^-invariant partition, and the action of R on T, which we denote by R/T, is transitive. Then H/T fixes the vertex v g V(Г), and so |H/T| = t. Let t g H be of prime order t. Then every orbit of т has prime order t or has order 1. If т in its action on Tv is a t-cycle, then H is transitive on V(Tv) and the result follows. Otherwise, т is the identity in its action on V(Tv). As Г is connected and т has prime order t = 1, there exists u g V(Г) such that the action of т on T„ is the identity, and some vertex y of T„ is adjacent to a vertex z g V(Tw), w = v, and z is not fixed by т. Applying т to the edge yz, we see that y is adjacent to t vertices not in Tv and to t - 1 vertices contained in Tv. Then the valency of y is 2t - 1, a contradiction. □ Corollary 3.9. The truncations of neither the Petersen graph nor the Coxeter graph are isomorphic to Cayley graphs. Proof. As the automorphism groups of the Petersen graph and the Coxeter graph are S5 and PGL(2,7) of orders 120 and 336, respectively, according to Theorem 3.8 the truncations of these graphs are Cayley if and only if their automorphism groups contain subgroups of index 4. Each of these automorphism groups though contain simple groups of index 2 (A5 and PSL(2,7) respectively), and as neither of them are direct products, their only normal proper nontrivial subgroups are A5 and PSL(2,7), respectively. Any subgroup of either of index 4 would then give an embedding into S4 by [4, Corollary 4.9], which is solvable. □ Theorem 3.10. If Г is a connected vertex-transitive graph of prime valency p, then T (Г) is vertex-transitive if and only if Aut^) contains an element of order p with a fixed point. Proof. Suppose that T (Г) is vertex-transitive. Then T is an Aut(T (^-invariant partition, and so StabAut(T(r))(Tv)|Tv is transitive on Tv. As Tv has order p, it follows that StabAut(T(r))(Tv)|Tv contains an element of order p. Let 7 G StabAut(T(Г)) (Tv) such that y|Tv has order p. Without loss of generality, we assume that 7 has order p (although we will no longer necessarily have that 7 |Tv has order p — for our purposes we only need an element of order p that fixes some Tv). By Theorem 3.4, Aut(T(Г)) acts faithfully on T, and so 7/T = 1. Then 7/T has order p, fixes the point v, and by Theorem 3.4 the permutation 7/T is contained in Aut^). Suppose that Aut^) contains an element 7 of order p with a fixed point. As Г is connected, some fixed point u of 7 is adjacent in Г to some point u that is not fixed by 7. This follows as there is certainly vertices x, y G V(Г) with x fixed by 7 and y not fixed by 7. We then let u be the first vertex of an xy-path in Г that is not fixed by 7, and v its predecessor on the chosen xy-path. As 7 preserves adjacency, all elements in the (nontrivial) orbit of u are neighbors of v. Since 7 is of prime order p, the orbit of u is of length p, and since p is the valency of the graph, the orbit of u contains all the neighbors of v. Thus, Aut^) acts transitively on the arcs emanating from v and transitively on the vertices of Г, and is therefore arc-transitive. The result then follows by Theorem 3.7. □ There are a few questions which remain unanswered. First, is it true that Theorem 3.8 holds when the valency t is not prime? Similarly, does Theorem 3.10 hold when the valency is not prime? More specifically, if Г has valency t is it the case that T (Г) is vertex-transitive if and only if Aut^) contains a subgroup of order t with a fixed point (i.e. every element fixes the same point). Finally, what exactly is the group StabAut(t(r))(Tv) in its action on Tv ? Of course, as an abstract group it is isomorphic to StabAut(r)(v), but what is the action? Acknowledgement: The authors are indebted to the anonymous referees whose careful reading of this manuscript resulted in suggestions that improved the exposition, results, and proofs in this paper. References [1] N. Biggs, Three remarkable graphs, Canad. J. Math. 25 (1973), 397-411. [2] G. Exoo and R. Jajcay, Recursive constructions of small regular graphs of given degree and girth, Discrete Math. 312 (2012), 2612-2619, doi:10.1016/j.disc.2011.10.021. [3] D. A. Holton and J. Sheehan, The Petersen graph, volume 7 of Australian Mathematical Society Lecture Series, Cambridge University Press, Cambridge, 1993, doi:10.1017/ CBO9780511662058. [4] T. W. Hungerford, Algebra, volume 73 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1980, reprint of the 1974 original. [5] K. Kutnar and D. Marusic, Hamilton cycles and paths in vertex-transitive graphs—current directions, Discrete Math. 309 (2009), 5491-5500, doi:10.1016/j.disc.2009.02.017. [6] A. Malnic, T. Pisanski and A. Žitnik, The clone cover, Ars Math. Contemp. 8 (2015), 95-113. [7] T. Pisanski and T. W. Tucker, Growth in repeated truncations of maps, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), 167-176, dedicated to the memory of Professor M. Pezzana (Italian). [8] G. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426-438. [9] W. T. Tutte, A non-Hamiltonian graph, Canad. Math. Bull. 3 (1960), 1-5. [10] W. T. Tutte, Connectivity in graphs, Mathematical Expositions, No. 15, University of Toronto Press, Toronto, Ont., 1966. [11] F. Žhang, Y.-C. Chen and Ž. Chen, Clique-inserted-graphs and spectral dynamics of clique-inserting, J. Math. Anal. Appl. 349 (2009), 211-225, doi:10.1016/j.jmaa.2008.08.036. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 225-234 The equalization scheme of the residual voluntary health insurance in Slovenia Boris ZgrabliC University of Primorska, Faculty of Mathematics, Natural Sciences and Information Technologies, Glagoljaška ulica 8, SI-6000 Koper, Slovenia TRIGLAV, Health insurance company, Ltd., Pristaniška ulica 10, SI-6000 Koper, Slovenia Received 5 December 2014, accepted 29 December 2014, published online 10 January 2015 Residual voluntary health insurance in Slovenia covers the difference between the (recognised) value of the health service and the part of this value that is payed by the compulsory health insurance. From the inception of compulsory health insurance in 1992, residual voluntary health insurance has open enrolment. From 2006 community rating applied, as well as an equalization scheme with which the differences in health services expenses, arising from the different structures of the insurees of the single insurance undertakings regarding age and gender, shall be equalized. The equalization scheme of the residual voluntary health insurance in Slovenia is presented, along with a detailed explanation of the formulae required for the computation. Keywords: residual voluntary health insurance, equalization scheme, claims equalization, risk equalization. Math. Subj. Class.: 91B30 1 Introduction In Slovenia, healthcare financing from public sources is organized through compulsory health insurance as a healthcare system of Bismarkian type. The Health Care and Health Insurance Act (HCHIA) [7] regulates both the public compulsory health insurance as well as the private voluntary health insurance. Although the basket of health benefits for an insuree of the compulsory health insurance is extensive, for a particular health service, the extent of its financing in charge of the compulsory health insurance may, in line with Article 23 of HCHIA, for the majority of E-mail address: boris.zgrablic@guest.arnes.si (Boris Zgrablic) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ adult insurees vary from 100% to as low as 10% of the (recognized) value of the health service; the payment of the difference or the residual amount up to 100% of the health service value being the obligation of the insuree that received the health service. Covering the difference or the residual mentioned above is possible through voluntary health insurance of residual type, called residual voluntary health insurance. (The direct translation from Slovenian would be "complementary", but since the term "complementary health insurance" has many different meanings we rather use the word "residual".) Residual health insurance is the most expanded voluntary health insurance in Slovenia, with a dissemination of over 95% among the insurees of the compulsory health insurance who are liable for residual payments. Nowadays, residual health insurance in Slovenia is offered by three insurance undertakings. Among all types of voluntary health insurances determined in HCHIA, only the residual health insurance is fully regulated. From its introduction in 1992, open enrollment is its main characteristic. Since 2006, when the reform of residual health insurance applied, community rating is prescribed, with discounts limited to 3%, and late entry loadings of 3% per year, the total premium rise amounting to at most 80%. To restrict risk selection in residual insurance, easily achieved through contracting mainly young persons, the legislator decided to introduce the equalization scheme of the residual health insurance, with which the differences in health services expenses, arising from the different structures of the insurees of the single insurance undertakings regarding age and gender, shall be equalized. It is a fact that the purpose of the introduction of the equalization scheme is not given at a legislative level. The equalization scheme is implemented as a system of transfer of equalization amounts among insurance undertakings. It is calculated ex-post from data included in the reports on the performance of residual insurance for the past equalization period that the insurance undertakings transmit to the ministry, competent for health. Each report contains, for each of the seven age groups (up to 25 years, 25-35, 35-45, 45-55, 55-65, 65-75, above 75 years), for each gender and for each of the three months of the referential equalization period, the number of insurees of the individual insurance undertaking and the health services expenses for the referential period that where accounted up to one month after the referential period, where the health services expenses include the amounts of account claims arising from residual insurance coverage and the amounts accounted as compensations to the health service provider for the transmition of data needed for the functioning of residual health insurance and determined in HCHIA. Also, the report includes, for each age group and for each gender, for each of the three months of each of the three preceding equalization periods, the health services expenses for that preceding equalization period, that where accounted in the three months after the first month of the referential equalization period. Armstrong, Paolucci and Van de Ven [2] emphasized that a distinction needs to be made between the term "risk equalization" and the term "claims equalization", since besides the ex-ante nature of the first and ex-post of the latter, a major difference is that with the latter insurers have only limited incentives for efficiency as they retain only a limited financial responsibility because both the risk profiles and the claims costs per risk group are equalised, while the former is a mechanism that is used to ensure that risk solidarity principles apply, to prevent competition from occurring on the basis of risk selection, and to foster competition based on costs and quality of care. In this sense the equalization scheme of residual voluntary health insurance in Slovenia belongs to the class of claims equalization. A glossary of terms can be found in [1]. The benefits of risk equalization in health insurance markets are discussed in [5, 1]. Stam gives in [4] an insight to testing the effectiveness of risk equalization models in health insurance. The exact wording of the legislative regulation regarding the equalization scheme of residual voluntary health insurance in Slovenia, that is, a selection of the respective articles of HCHIA, can be found in the Appendix. The equalizations of differences shall be settled for each equalization period. In Section 2 we give a detailed presentation of all the needed calculations for the ministry, competent for health, to bring a decision about the equalization. The introduction of an equalization scheme in the specific extremely regulated market of residual voluntary health insurance in Slovenia should be necessarily accompanied by explicit provisions for protection of competition. The lack of such provisions and implications are discussed in [6]. 2 Computation of the equalization amount The computation of the equalization amount is described below. We use the following notation: i ... serial number of the insurance undertaking, included in the equalization in the referential equalization period; j ... serial number of the referential equalization period; k ... relative number of the preceding equalization period with respect to the referential equalization period (k = 0,1, 2, 3); k = 0 stands for the referential period; r ... age group of insurees; Nr j ... number of insurees of the insurance undertaking i, that in the referential equalization period j belong to the age group r; this number is calculated as the average of the numbers of insurees on the first day of every month of the referential equalization period j ; Nlr'j'k ... number of insurees of the insurance undertaking i that, according to the data on the day of reporting for the referential equalization period j, in the preceding equalization period (j - k) belonged to the the age group r. The following equalities hold, Ni,j'k = Nij-k'0 = Nij-k since the insurance undertakings ought to have up-to-date data of its insurance contracts. For further computations the amounts of health service expenses transmitted from insurance undertakings to the ministry, competent for health, in the reports on the performance of residual insurance for the past equalization period, are essential. We denote AErj'fc ... amount of health service expenses of the insurance undertaking i within the referential equalization period j relative to the preceding equalization period (j - k) and for the age group r. For k = 0 we denote АЕ^'0 as AE^'. The average amount of health service expenses of each insurance undertaking i is calculated for every age group j according to the formula Nr where AEi'j = AEt'j'° = AEj r -Ti'j'0 TVji'j : r AE^ ... the average amount of health service expenses of the insurance undertaking i within the referential equalization period j and for the age group r. The average amount of health service expenses of all insurance undertakings, included in the computation, is calculated for each age group according to the formula E AE: r AE*'j = N mj r where AE^'j ... the average amount of health service expenses of all insurance undertakings included in the computation, within the referential equalization period j and for the age group r. A common rounding policy should be applied. The standardized number of insurees in an age group of an individual insurance undertaking is calculated for every age group using the formula E N SNrj = E Nm'j s rr *—' » " Tmj N where SNr j ... the standardized number of insurees for the insurance undertaking i within the referential equalization period j and for the age group r. The standardized amount of health service expenses of an individual insurance undertaking is calculated within an age group r having more than 2000 persons for every equalization period (referential and preceding) according to the formula SAErj = SNrj • AEr,j, where SAEr,j ... the standardized amount of health service expenses for the insurance undertaking i within the referential equalization period j and for the age group r. In case the age group r of an individual insurance undertaking contains less than 2000 persons the standardized amount of health service expenses for this age group of this insurance undertaking is given by the formula SAErj = SNrj • AE'r,j. The basic equalization amount for the insurance undertaking for the referential equalization period j shall be calculated as the difference between the sum of amounts of expenses for health services in residual insurance of the insurance undertaking in all age groups, and the sum of standardized amounts of expenses for health services in residual insurance of the same insurance undertaking for all age groups. Hence BEAij = ^ AErj - ^ SAErj, rr where BEAi,j ... the basic equalization amount for the insurance undertaking i for the equalization period j . Let us determine the equalization amount of an insurance undertaking before taking in consideration the (possible) transfered amount (in case in the preceding equalization period the treshold was not attained). The calculation of the equalization amount of an individual insurance undertaking is based on the comparison of the sum of the positive basic equalization amounts, BEA-,j = ^ BEAij BEAi,j >0 where BEA-,j ... the sum of nonnegative basic equalization amounts for the referential equalization period j , and the sum of negative basic equalization amounts, BEA<'j = ^ BEAij where BEA< j ... the sum of negative basic equalization amounts for the referential equalization period j. If the conditions of Article 62.h of HCHIA are met, that is, if the equalization treshold is not attained, a transfer of equalization amounts in the subsequent equalization periods is carried. So for the insurance undertaking i and the equalization period j, the computing procedure determines first the equalization amount before taking into consideration the transfer, denoted EABi j, where EABij if (BEAij > 0) and (BEA-'j < -BEA —BEA<'j); (—BEA 0) and (BEA-'j > —BEA<'j); BEAij, BEA®'5, BEA®'5 • BEA; BEA- (—BEA THR^j. EA'J >0 In the referential equalization period the insurance undertaking is a payer in the equalization if its equalization amount for this equalization period is negative, otherwise the insurance undertaking is a receiver in the equalization. In this case the transfered equalization amount of the insurance undertaking i from the referential equalization period j to the subsequent referential equalization period (j + 1) equals zero, that is, TEA®'j-1 = 0. If the condition for performing equalization from the preceding paragraph is not met, then the equalization is not performed, the transfered equalization amount of the insurance undertaking i from the referential equalization period j to the subsequent referential equalization period (j+1) is equal to the equalization amount of the insurance undertaking i for the referential equalization period j, that is, TEAi j = EAi j. 3 Appendix - The legislative wording The equalization scheme of the residual voluntary health insurance as a method of risk equalization is regulated within the Health Care and Health Insurance Act (HCHIA) [7]. A selected list of provisions follows (translation from Slovenian by Liliane Strmcnik). Article 62.d The insurance undertakings providing residual insurance shall be included in the equalization scheme of residual insurance, with which the differences in health services expenses, arising from the different structures of the insurees of the single insurance undertakings regarding age and gender, shall be equalized. Health services expenses shall include the amounts of account claims arising from residual insurance coverage and the amounts accounted as compensations for the ensured data from point 7 of the second paragraph of Article 62 of this Act. The amount of these compensations shall be agreed between the insurance undertakings and the providers of health services as a percentage of the amounts of the gross account of claims, at the most up to 0,75 percent. The insurance undertakings providing residual insurance shall keep account records on the health services expenses for each insuree. Health services providers shall be bound to transmit the insurance undertakings all the necessary data. The insurance undertakings providing residual insurance shall participate in the calculation of equalization amounts and in the equalization of differences. The insurance undertakings starting to perform residual insurance shall be exempted from participating in the calculation of equalization amounts and in the equalization of differences for the period of the first twelve months of operation in the field of residual insurances. Article 62.e The basic equalization amount for the insurance undertaking shall be calculated as the difference between the amount of expenses for health services in residual insurance of the insurance undertaking and the standard amount of expenses for health services in residual insurance of the same insurance undertaking. Should the first amount of expenses be lower than the second, the insurance undertaking shall be payer in the equalization otherwise it shall be receiver in the equalization. The equalization amount shall be calculated for the insurance undertaking by preserving its basic equalization amount or proportionally reducing it in such a way that the sum of the equalization amounts of the payers as well as the sum of the equalization amounts of the receivers shall be equal to the lower of the sum of the basic equalization amounts of the receivers and the sum of the basic equalization amounts of the payers. The amount of health services expenses in residual insurance of the insurance undertaking shall be the sum of health services expenses arising from the coverage of residual insurance, increased for the amount of compensations for the ensured data from the preceding article. Into account shall be taken the health services expenses that shall have been accounted up to the last day inclusive of the month after the close of the equalization period. The standardized amount of health services expenses in residual insurance shall be the sum of standardized amounts of expenses according to age groups in the insurance undertaking. The standardized amount of health services expenses in residual insurance for an age class in the insurance undertaking equals the product of: 1. the average amount of health services expenses for an insuree in the insurance undertaking within this age class or, if the age class in the insurance undertaking comprehends less than 2.000 persons, the average amount of health services expenses for an insuree in all insurance undertakings within this age class; 2. the number of insurees in residual insurance within this age class in all insurance undertakings providing residual insurance and participating in the calculation of equalization amounts, and 3. the quotient of the number of insurees in residual insurance in the insurance undertaking and the number of all insurees in residual insurance in all insurance undertakings providing residual insurance and participating in the calculation of the equalization amounts. The average amount of health services expenses for an insuree in an insurance undertaking within an age class from point 1 of the preceding paragraph shall be determined so that the sum of all health services expenses of the insurance undertaking within the age class is divided by the number of insurees in the insurance undertaking within the same age class. The number of insurees in residual insurance in the insurance undertaking shall be the average number of insurees on the first day of each month within the equalization period, for which on that same day the insurance undertaking shall bear the responsibility of payment for the health services expenses from residual insurance. Age groups by gender and age shall start, separately for men and women, at the age of 15 and follow one another within the span of ten years up to the age of 75. The last age groups, separated by gender and age, shall be represented by the insurees aged 75 and more. The insurees who have not yet attained the age of 15 belong to the first two groups. The age of the insuree in residual insurance shall be reckoned as the age attained within the calendar year. The equalization period shall be a period of three successive calendar months. The first equalization period within a calendar year shall start with the first day of the calendar year. Article 62.f The payer insurance undertaking shall be obliged to pay the equalization amount to the receiver insurance undertakings. The ministry competent for health shall bring a decision about the equalization. Unless otherwise stipulated by this Act, for the decision procedure of the ministry, competent for health, the provisions of the Act regulating the general administrative procedure shall be applied. The equalizations of differences shall be settled for each equalization period. For the calculation of the equalization of differences in health services expenses between insurance undertakings, the latter shall be bound to transmit to the ministry, competent for health, the reports on the performance of residual insurance for the past equalization period within 20 days after the end of the month following an individual equalization period. Upon receipt of the reports from the preceding paragraph, the ministry, competent for health, shall hand over these reports to each of the insurance undertakings within 8 days. The insurance undertaking may give a statement on these reports within 8 days upon receipt of the reports. Should the insurance undertaking refer to documentary evidence in its statement, it should enclose these documents to the statement. Should the insurance undertaking not enclose the documentary evidence to its statement, the ministry competent for health shall take into account only the evidence enclosed to the statement for making its decision. The ministry competent for health shall decide about the equalization in a written order on equalization without an appointed day within at most 15 days upon expiry of the terms from the preceding paragraph. In its written order the Ministry, competent for health, shall state which insurance undertakings are the payers and which are the receivers in the equalization scheme and the amounts that the payer insurance undertakings shall pay on the accounts of the receiver insurance undertakings. The accounts of the insurance undertakings shall be stated in the written order. The payer insurance undertaking shall pay the equalization amount within 8 days upon receipt of the written order on equalization. In case of delay in payment, the insurance undertaking shall be obliged to pay legal interests on arrears according to the Legal Penalty Rate Act (Official Gazette RS, no. 56/03). For the time from the expiry of the equalization period until the issue of the written order on equalization, interest shall accrue in the amount of average pondered interest rate on the inter-bank money market. The ministry competent for health shall decide about the obligation of interest payment in the written order on equalization. The ministry, competent for health, may call for a debate if it estimate it necessary for the clarification or for the ascertainment of the decisive facts. In this case the insurance undertakings may give their statements in oral form as well, at the debate. The insurance undertaking in a state member of the European Union, acting within the frame of the European Union and being authorized, in accordance with the Insurance Act, to perform insurance business on the territory of the Republic of Slovenia either through a branch office or directly, shall appoint, for the duration of providing residual insurance business, a mandatory for the delivery of the written orders of the ministry competent for health. The insurance undertaking shall notify the ministry, competent for health, on the data about its authorized mandatory for delivery. For the time during which the insurance undertaking has no authorized mandatory for delivery or before notification of his data to the ministry competent for health, the written orders shall be delivered by publication on the notice board of the ministry, competent for health. The detailed instructions on the contents of the report shall be stipulated by the minister, competent for health. The detailed instructions for accounting monitoring and statement of business events regarding the performance of equalization shall be stipulated by the Insurance Supervision Agency. Article 62.h Equalization of differences in expenses for health services shall not be performed for the equalization periods in which the sum of the amounts for equalization that the payer insurance undertakings are obliged to pay is lower than one and a half percent of the amounts of gross calculated claims, increased for the amount of compensation for ensuring the data from point 7 of the second paragraph of Article 62 of this Act. The ministry, competent for health shall state, in a written order, that equalization shall not be performed. Should equalization not be performed in a particular accounting period, the equalization amount shall be transferred to the next accounting period, added to the sum from the prepreceding sentence and taken into account for equalization. Ackowledgement. The author is grateful to Ivan Gracar for all the fruitful conversations [3] and common work that supported this paper, in the year 2005 and on. References [1] J. Armstrong, F. Paolucci, H. McLeod, W.P.M.M van de Ven, Risk equalization in voluntary health insurance markets: A three country comparison, Health Policy 98 (2010), 39-49. [2] J. Armstrong, F. Paolucci, W.P.M.M van de Ven, Editorial: Risk equalization in voluntary health insurance markets, Health Policy 98 (2010), 1-2. [3] Ivan Gracar, personal communication from 2005 on. [4] P. J. A. Stam, Testing the effectiveness of risk equalization models in health insurance - A new method and its application, PhD Thesis, Erasmus University, Rotterdam, 2007. [5] W. P. M. M Van de Ven, R.P. Ellis, Risk adjustment in competitive health plan markets, in: Handbook of health economics, vol. 1, ed. A.J. Culyer and J.P. Newhouse, Elsevier Science BV, Amsterdam, 2000, 755-845. [6] B. Zgrablic, The equalization scheme of residual voluntary health insurance in Slovenia and protection of competition, in preparation. [7] Health Care and Health Insurance Act, Official Gazette of the Republic of Slovenia no. 9/1992 and further (in Slovenian). ars mathematica contemporanea Author Guidelines Papers should be prepared in ШЕХ and submitted as a PDF file. Articles which are accepted for publication have to be prepared in LTeX using class file amcjou.cls and bst file amcjou.bst (if you use BibTEX). These files and an example of how to use the class file can be found at http://amc-journal.eu/index.php/amc/about/submissions#authorGuidelines If this is not possible, please use the default LTeX article style, but note that this may delay the publication process. Title page. The title page of the submissions must contain: • Title. The title must be concise and informative. • Author names and affiliations. 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