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Self-dual and self-petrie-dual regular maps

✍ Scribed by R. Bruce Richter; Jozef Širáň; Yan Wang

Publisher: John Wiley and Sons
Year: 2011
Tongue: English
Weight: 91 KB
Volume: 69
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20570

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

Regular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈a, b, c|a^2^ = b^2^ = c^2^ = (ab)^k^ = (bc)^m^ = (ca)^2^ = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map (G;a, b, c) is self‐dual if the assignment b↦b, c↦a and a↦c extends to an automorphism of G, and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca. In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k. We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012

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