Petrie Duality and the Anstee–Robertson Graph

Symmetry

Gareth A. Jones¹ 

[1] School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK

关键词: regular map;    Petrie polygon;    Petrie dual;    cage;    Anstee–Robertson graph;   

DOI  :  10.3390/sym7042206

来源: mdpi

【 摘 要 】

We define the operation of Petrie duality for maps, describing its general properties both geometrically and algebraically. We give a number of examples and applications, including the construction of a pair of regular maps, one orientable of genus 17, the other non-orientable of genus 52, which embed the 40-vertex cage of valency 6 and girth 5 discovered independently by Robertson and Anstee. We prove that this map (discovered by Evans) and its Petrie dual are the only regular embeddings of this graph, together with a similar result for a graph of order 40, valency 6 and girth 3 with the same automorphism group.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.