The table below shows all maps with three edges.
A green border indidates that the map is a regular map.
Cyan lines join duals.
Red lines join Petrie duals.
A cyan S indicates a self-dual.
A red S indicates a self-Petrie dual.
Show duals and Petrie duals:
hexad 1
hexad 2
hexad 3
hexad 4
hexad 5
hexad 6
hexad 7
triad 1
triad 2
triad 3
triad 4
triad 5
triad 6
duad
monad
ALL
Some observations follow.
For a fixed number of edges (as in the table above), the more vertices a graph has, the fewer loops it has, and so the more limited it is in the manifolds which it can partition into disc-like regions. The graphs are ordered from top to bottom by numbers of vertices, so the ones at the top of the table, with few vertices, have entries further to the right, for manifolds with more negative Euler characteristic.
The rightmost entries, for the manifolds with the most negative Euler characteristic, must be the duals of maps also in the rightmost column. This restricts them to having duals in the same row. So many of them are self-dual.
The lowest entries, for the maps with the most vertices, must be the Petrie duals of maps in the same row. So they are self-Petrie dual.
This is a sub-page of Regular Maps and other Maps.
See a similar page on Regular Maps and other Maps with fewer than three edges.
See also Index to pages on Regular Maps.