Annals of Combinatorics

, Volume 19, Issue 2, pp 243–268 | Cite as

Symmetry Type Graphs of Polytopes and Maniplexes

  • Gabe Cunningham
  • María Del Río-Francos
  • Isabel HubardEmail author
  • Micael Toledo


We extend the notion of symmetry type graphs of maps to include maniplexes and (abstract) polytopes, using them to study k-orbit maniplexes (where the automorphism group has k orbits on flags). In particular, we show that there are no fully-transitive k-orbit 3-maniplexes with k > 1 an odd number. We classify 3-orbit maniplexes and determine all face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators of the automorphism group of a maniplex, given its symmetry type graph. Finally, we extend these notions to oriented maniplexes, and we provide a classification of oriented 2-orbit maniplexes and a generating set for their orientation-preserving automorphism group.

Mathematics Subject Classification

52B15 05C25 51M20 


abstract polytope regular graph edge-colouring maniplex 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Gabe Cunningham
    • 1
  • María Del Río-Francos
    • 2
  • Isabel Hubard
    • 3
    • 4
    Email author
  • Micael Toledo
    • 3
    • 4
  1. 1.University of Massachusetts BostonBostonUSA
  2. 2.Institute of Mathematics Physics and MechanicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Instituto de Matemáticas, Universidad Nacional Autónoma de MéxicoCuernavacaMéxico
  4. 4.CuernavacaMéxico

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