Timeline for Which graphs embedded in surfaces have symmetries acting transitively on vertex-edge flags?
Current License: CC BY-SA 3.0
7 events
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| May 16, 2017 at 15:23 | comment | added | draks ... | Isn't the vertex-edge flag graph somehow related to the line digraph? | |
| Apr 29, 2016 at 20:42 | comment | added | verret | The only other option is that the arc-stabiliser is trivial and hence the group acts regularly on arcs. In that case, the vertex-stabiliser acts regularly on the neighbours, so it must be either cyclic of order $k$, or dihedral of order $k$ (in which case $k$ is even). These are all quite well studied objects, although not as much as the regular ones. In particular, they can be defined group-theoretically and, using this, can be enumerated up to a few thousand vertices and quite high genus, 100 say. (See math.auckland.ac.nz/~conder for example) | |
| Apr 29, 2016 at 20:42 | comment | added | verret | A very common term for a vertex-edge flag is an arc. Let me use this for simplicity. You are asking about maps with an arc-transitive group of automorphisms. It's not hard to see that the maximal amount of symmetry a map can have is to be arc-transitive with the arc-stabiliser having order $2$. This is the so-called "regular" case that Noam mentioned. In that case, the vertex-stabiliser is dihedral, of order $2k$, where $k$ is the valency. | |
| Apr 29, 2016 at 17:04 | answer | added | Noam Zeilberger | timeline score: 4 | |
| Apr 28, 2016 at 15:57 | history | edited | John Baez | CC BY-SA 3.0 |
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| Apr 28, 2016 at 15:23 | history | edited | John Baez | CC BY-SA 3.0 |
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| Apr 28, 2016 at 15:18 | history | asked | John Baez | CC BY-SA 3.0 |