Trinity symmetry and kaleidoscopic regular maps

Archdeacon, Dan; Conder, Marston and Siran, Jozef (2014). Trinity symmetry and kaleidoscopic regular maps. Transactions of the American Mathematical Society, 366(8) pp. 4491–4512.

DOI: https://doi.org/10.1090/S0002-9947-2013-06079-5

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 Me 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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