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Trinity symmetry and kaleidoscopic regular maps

Authors: Dan Archdeacon, Marston Conder and Jozef Širáň
Journal: Trans. Amer. Math. Soc. 366 (2014), 4491-4512
MSC (2010): Primary 05E18, 20B25, 57M15
Published electronically: December 6, 2013
MathSciNet review: 3206467
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Abstract: A cellular embedding of a connected graph (also known as a map) on an orientable surface has trinity symmetry if it is isomorphic to both its dual and its Petrie dual. A map is regular if for any two incident vertex-edge pairs there is an automorphism of the map sending the first pair onto the second. Given a map $ M$ with all vertices of the same degree $ d$, for any $ e$ relatively prime to $ d$ the power map $ M^e$ is formed from $ M$ by replacing the cyclic rotation of edges at each vertex on the surface with the $ e\,$th power of the rotation. A map is kaleidoscopic if all of its power maps are pairwise isomorphic. In this paper, we present a covering construction that gives infinite families of kaleidoscopic regular maps with trinity symmetry.

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  • [1] Dan Archdeacon, Pavol Gvozdjak, and Jozef Širáň, Constructing and forbidding automorphisms in lifted maps, Math. Slovaca 47 (1997), no. 2, 113-129. MR 1476862 (98k:05050)
  • [2] Dan Archdeacon and R. Bruce Richter, The construction and classification of self-dual spherical polyhedra, J. Combin. Theory Ser. B 54 (1992), no. 1, 37-63. MR 1142263 (92j:05058),
  • [3] Dan Archdeacon, R. Bruce Richter, Jozef Širáň, and Martin Škoviera, Branched coverings of maps and lifts of map homomorphisms, Australas. J. Combin. 9 (1994), 109-121. MR 1271195 (95c:05053)
  • [4] Norman Biggs, Algebraic graph theory, Cambridge University Press, London, 1974. Cambridge Tracts in Mathematics, No. 67. MR 0347649 (50 #151)
  • [5] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Computational algebra and number theory (London, 1993). MR 1484478,
  • [6] Robin P. Bryant and David Singerman, Foundations of the theory of maps on surfaces with boundary, Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 17-41. MR 780347 (86f:57008),
  • [7] Domenico A. Catalano, Marston D. E. Conder, Shao Fei Du, Young Soo Kwon, Roman Nedela, and Steve Wilson, Classification of regular embeddings of $ n$-dimensional cubes, J. Algebraic Combin. 33 (2011), no. 2, 215-238. MR 2765323 (2012b:57005),
  • [8] Marston D. E. Conder, Regular maps and hypermaps of Euler characteristic $ -1$ to $ -200$, J. Combin. Theory Ser. B 99 (2009), no. 2, 455-459. MR 2482963 (2010b:05084),
  • [9] Marston Conder and Peter Dobcsányi, Determination of all regular maps of small genus, J. Combin. Theory Ser. B 81 (2001), no. 2, 224-242. MR 1814906 (2002f:05088),
  • [10] Marston Conder, Primož Potočnik, and Jozef Širáň, Regular maps with almost Sylow-cyclic automorphism groups, and classification of regular maps with Euler characteristic $ -p^2$, J. Algebra 324 (2010), no. 10, 2620-2635. MR 2725192 (2012b:20100),
  • [11] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin, 1980. MR 562913 (81a:20001)
  • [12] Jonathan L. Gross and Thomas W. Tucker, Topological graph theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New York, 1987. A Wiley-Interscience Publication. MR 898434 (88h:05034)
  • [13] Lynne D. James and Gareth A. Jones, Regular orientable imbeddings of complete graphs, J. Combin. Theory Ser. B 39 (1985), no. 3, 353-367. MR 815402 (87a:05060),
  • [14] Gareth A. Jones, Regular embeddings of complete bipartite graphs: classification and enumeration, Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 427-453. MR 2679697 (2011g:05072),
  • [15] Gareth A. Jones and David Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273-307. MR 0505721 (58 #21744)
  • [16] G. A. Jones and J. S. Thornton, Operations on maps, and outer automorphisms, J. Combin. Theory Ser. B 35 (1983), no. 2, 93-103. MR 733017 (85m:05036),
  • [17] Aleksander Malnič, Roman Nedela, and Martin Škoviera, Lifting graph automorphisms by voltage assignments, European J. Combin. 21 (2000), no. 7, 927-947. MR 1787907 (2001i:05086),
  • [18] Roman Nedela, Regular maps--combinatorial objects relating different fields of mathematics, J. Korean Math. Soc. 38 (2001), no. 5, 1069-1105. Mathematics in the new millennium (Seoul, 2000). MR 1849340 (2002k:05071)
  • [19] Roman Nedela and Martin Škoviera, Exponents of orientable maps, Proc. London Math. Soc. (3) 75 (1997), no. 1, 1-31. MR 1444311 (98i:05059),
  • [20] R. Bruce Richter, Jozef Širáň, and Yan Wang, Self-dual and self-Petrie-dual regular maps, J. Graph Theory 69 (2012), no. 2, 152-159. MR 2864455 (2012k:20080),
  • [21] Jozef Širáň, The ``walk calculus'' of regular lifts of graph and map automorphisms, Proceedings of the 10th Workshop on Topological Graph Theory (Yokohama, 1998), 1999, pp. 113-128. MR 1732771 (2001j:05052)
  • [22] Jozef Širáň, Regular maps on a given surface: a survey, Topics in discrete mathematics, Algorithms Combin., vol. 26, Springer, Berlin, 2006, pp. 591-609. MR 2249288 (2007c:05063),
  • [23] Jozef Širáň and Yan Wang, Maps with highest level of symmetry that are even more symmetric than other such maps: regular maps with largest exponent groups, Combinatorics and graphs, Contemp. Math., vol. 531, Amer. Math. Soc., Providence, RI, 2010, pp. 95-102. MR 2757790 (2012d:20073),
  • [24] David B. Surowski, Lifting map automorphisms and Macbeath's theorem, J. Combin. Theory Ser. B 50 (1990), no. 2, 135-149. MR 1081218 (91j:05040),
  • [25] Stephen Edwin Wilson, New Techniques for the construction of regular maps, ProQuest LLC, Ann Arbor, MI, 1976. Thesis (Ph.D.)-University of Washington. MR 2626405
  • [26] Stephen E. Wilson, Operators over regular maps, Pacific J. Math. 81 (1979), no. 2, 559-568. MR 547621 (81a:57004)
  • [27] S. Wilson, personal communication, 2010.

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Additional Information

Dan Archdeacon
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05405

Marston Conder
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

Jozef Širáň
Affiliation: Department of Mathematics and Statistics, Open University, Milton Keynes, MK7 6AA, United Kingdom – and – Slovak University of Technology, 81368 Bratislava, Slovakia

Received by editor(s): September 16, 2012
Received by editor(s) in revised form: January 14, 2013
Published electronically: December 6, 2013
Additional Notes: The first author thanks the Open University which hosted him while much of this research was conducted.
The second author’s work was supported by the Marsden Fund of New Zealand, project UOA-1015, and assisted by use of the Magma system.
The third author acknowledges support from the VEGA Research Grant 1/0781/11, the APVV Research Grant 0223-10, and the APVV support as part of the EUROCORES Programme EUROGIGA, project GREGAS, ESF-EC-0009-10, financed by the European Science Foundation.
Article copyright: © Copyright 2013 American Mathematical Society

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