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Self‐dual and self‐petrie‐dual regular maps
Author(s) -
Richter R. Bruce,
Širáň Jozef,
Wang Yan
Publication year - 2012
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.20570
Subject(s) - combinatorics , mathematics , vertex (graph theory) , dual (grammatical number) , degree (music) , graph , generator (circuit theory) , dual graph , automorphism , discrete mathematics , planar graph , physics , art , power (physics) , literature , quantum mechanics , acoustics
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