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Self-dual and self-Petrie-dual regular maps

Richter, Bruce R.; Siran, Jozef and Wang, Yan (2012). Self-dual and self-Petrie-dual regular maps. Journal of Graph Theory, 69(2) pp. 152–159.

DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1002/jgt.20570
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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|a2 =b2=c2=(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.

Item Type: Journal Article
Copyright Holders: 2011 Wiley Periodicals, Inc
ISSN: 0364-9024
Project Funding Details:
Funded Project NameProject IDFunding Body
Not SetNot SetNSERC
Not SetNot SetVEGA Research Grant [1/0489/08]
Not SetNot SetAPVV Research Grant [0040-06 and 0104-07]
Not SetNot SetAPVV LPP Research Grant [0145-06 and 0203-06]
Keywords: regular map; triangle group; self-dual map; self-Petrie-dual map
Academic Unit/Department: Mathematics, Computing and Technology
Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 32160
Depositing User: Jozef Širáň
Date Deposited: 03 Feb 2012 11:54
Last Modified: 30 Nov 2012 10:40
URI: http://oro.open.ac.uk/id/eprint/32160
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