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

2011 Wiley Periodicals, Inc

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]