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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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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Additional Information

Dan Archdeacon
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05405
Email: dan.archdeacon@uvm.edu

Marston Conder
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
Email: m.conder@auckland.ac.nz

Jozef Širáň
Affiliation: Department of Mathematics and Statistics, Open University, Milton Keynes, MK7 6AA, United Kingdom – and – Slovak University of Technology, 81368 Bratislava, Slovakia
Email: j.siran@open.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06079-5
PII: S 0002-9947(2013)06079-5
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