Timeline for Which graphs embedded in surfaces have symmetries acting transitively on vertex-edge flags?
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| May 2, 2016 at 9:57 | comment | added | Noam Zeilberger | @JohnBaez On the other hand, this definition of "oriented regular map" would still rule out the cuboctahedron and the icosidodecahedron, since, for example, any automorphism of the cuboctahedron-as-combinatorial-map could not send a dart bordering a square to its left to a dart bordering a triangle to its left. | |
| May 2, 2016 at 9:56 | comment | added | Noam Zeilberger | @JohnBaez Glad you found it useful! On v-e-f flags vs v-e flags, Siran's definition of regular map allows for nonorientable regular maps, but in the oriented case it seems natural to me to define regular maps as combinatorial maps $M = (D,v,e)$ such that $Aut(M)$ (respectively $Mon(M) = \langle v,e\rangle$) acts transitively (respectively, freely) on the set of darts $D$. Note that a dart is essentially the same thing as a v-e flag, except in the case of loops. I see that this definition of "oriented regular map" is considered by Roman Nedela here: savbb.sk/~nedela/CMbook.pdf. | |
| Apr 30, 2016 at 20:57 | comment | added | John Baez | Thanks, that's extremely helpful. It's a very readable paper. It only gets at a special case of my question. He does consider orientation-reversing symmetries, and even maps on nonorientable surfaces, but he only studies maps whose symmetries act transitively on vertex-edge-face flags, not more general ones where the symmetries act transitively on vertex-edge flags (or 'arcs'). Nonetheless, this special case is so mathematically natural that it's very fun to read about, and it gives a big pile of examples of what I want. | |
| Apr 29, 2016 at 17:04 | history | answered | Noam Zeilberger | CC BY-SA 3.0 |