Richter, Bruce R.; Siran, Jozef and Wang, Yan
Self-dual and self-Petrie-dual regular maps.
Journal of Graph Theory, 69(2) pp. 152–159.
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.
||2011 Wiley Periodicals, Inc
|External 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]|
||regular map; triangle group; self-dual map; self-Petrie-dual map
||Mathematics, Computing and Technology
Mathematics, Computing and Technology > Mathematics and Statistics
||03 Feb 2012 11:54
||30 Nov 2012 10:40
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