Petrie duality and the Anstee-Robertson graph

Gareth A. Jones, Matan Ziv-Av

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish
Pages (from-to)2206-2223
Number of pages18
Issue number4
StatePublished - 1 Jan 2015


  • Anstee-Robertson graph
  • Cage
  • Petrie dual
  • Petrie polygon
  • Regular map

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • General Mathematics
  • Physics and Astronomy (miscellaneous)


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