Trinity symmetry and kaleidoscopic regular maps

Archdeacon, Dan; Conder, Marston; Širáň, Jozef

doi:10.1090/S0002-9947-2013-06079-5

Trinity symmetry and kaleidoscopic regular maps
HTML articles powered by AMS MathViewer

by Dan Archdeacon, Marston Conder and Jozef Širáň PDF

Trans. Amer. Math. Soc. 366 (2014), 4491-4512 Request permission

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.

References

Dan Archdeacon, Pavol Gvozdjak, and Jozef Širáň, Constructing and forbidding automorphisms in lifted maps, Math. Slovaca 47 (1997), no. 2, 113–129. MR 1476862
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, DOI 10.1016/0095-8956(92)90065-6
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
Norman Biggs, Algebraic graph theory, Cambridge Tracts in Mathematics, No. 67, Cambridge University Press, London, 1974. MR 0347649
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, DOI 10.1006/jsco.1996.0125
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, DOI 10.1093/qmath/36.1.17
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, DOI 10.1007/s10801-010-0242-8
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, DOI 10.1016/j.jctb.2008.09.003
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, DOI 10.1006/jctb.2000.2008
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, DOI 10.1016/j.jalgebra.2010.07.047
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-New York, 1980. MR 562913
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
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, DOI 10.1016/0095-8956(85)90060-7
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, DOI 10.1112/plms/pdp061
Gareth A. Jones and David Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307. MR 505721, DOI 10.1112/plms/s3-37.2.273
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, DOI 10.1016/0095-8956(83)90065-5
Aleksander Malnič, Roman Nedela, and Martin Škoviera, Lifting graph automorphisms by voltage assignments, European J. Combin. 21 (2000), no. 7, 927–947. MR 1787907, DOI 10.1006/eujc.2000.0390
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
Roman Nedela and Martin Škoviera, Exponents of orientable maps, Proc. London Math. Soc. (3) 75 (1997), no. 1, 1–31. MR 1444311, DOI 10.1112/S0024611597000245
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, DOI 10.1002/jgt.20570
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
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, DOI 10.1007/3-540-33700-8_{2}9
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, DOI 10.1090/conm/531/10458
David B. Surowski, Lifting map automorphisms and Macbeath’s theorem, J. Combin. Theory Ser. B 50 (1990), no. 2, 135–149. MR 1081218, DOI 10.1016/0095-8956(90)90069-C
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
Stephen E. Wilson, Operators over regular maps, Pacific J. Math. 81 (1979), no. 2, 559–568. MR 547621
S. Wilson, personal communication, 2010.