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: https://doi.org/10.1002/jgt.20570

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.

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About

• Item ORO ID
• 32160
• Item Type
• Journal Item
• ISSN
• 0364-9024
• Project Funding Details

• Funded Project Name	Project ID	Funding Body
  Not Set	Not Set	NSERC
  Not Set	Not Set	VEGA Research Grant [1/0489/08]
  Not Set	Not Set	APVV Research Grant [0040-06 and 0104-07]
  Not Set	Not Set	APVV LPP Research Grant [0145-06 and 0203-06]

• Keywords
• regular map; triangle group; self-dual map; self-Petrie-dual map
• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM)
  Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
• Copyright Holders
• © 2011 Wiley Periodicals, Inc
• Depositing User
• Jozef Širáň