Talk:Regular map (graph theory)

Some suggestions

I think this article needs some more work, and I am not familiar enough with the field to do it myself.

• The first, topological, approach does not require that the graph be regular.

• I am having trouble with the second, group-theoretical approach. Let's try to construct an example. Let our group be S4 with flags a,b,c,d. Now we can have r₀=(ab), r₁=(bc), r₂=(cd). Thus they are all involutions, and (r₀r₂)²= I, as required. Now <r₀,r₁> generates D6, <r₀,r₂> generates C2×C2, <r₁,r₂> generates D6. So we seem to have a graph with 6 vertices, 4 edges, and 6 faces.

• The third, graph-theoretical, approach defines a "cubic map". I am unclear whether all "regular maps" are "cubic maps".

• The graph-theoretical approach refers to blue edges. The diagram has blue edges (but no red or yellow ones). Is this a coincidence?

• The three examples all have the projective plane as the 2-manifold (though this is only specified for the second one). The graph with seven hexagons on a torus would add some variety, if valid. So would something on a sphere, a dodecahedron maybe.

Maproom (talk) 11:43, 14 May 2009 (UTC)

Now that I understand what is meant by "flag", the second approach may make more sense. Maproom (talk) 09:28, 21 May 2009 (UTC)

Sorry, I have been using this article as a sandbox. In future I will do this on my userpage. Genusfour (talk) 13:50, 1 September 2009 (UTC)

Schläfli type

The terms "type" and "Schläfli type" are used in the article without being defined. (Or I have overlooked the definition?) I would try to fill in the details, but as I am not an expert in this area -- I have only read an easy popular article on this -- I am afraid I could write something which is not correct. --Kompik (talk) 11:51, 23 April 2010 (UTC)

Chirality

Some authors require a regular map to be flag-transitive, as specified in the second sentence of the article. But others, including Conder, allow chiral regular maps. I may get around to mentioning this in the article. Maproom (talk) 21:57, 27 December 2013 (UTC)

I see it is mentioned in the article, in the second paragraph of the Graph-theoretical approach section. And the paragraph is wrong. If Aut(M) acts regularly on the flags, the action is flag-transitive, and the map is not chiral. I shall get round to correcting this soon, unless someone else does first. Maproom (talk) 08:27, 23 July 2014 (UTC)

No, the article is correct. The statements are about the full automorphism group, not the rotational symmetry group as I had been thinking. Maproom (talk) 08:56, 11 January 2015 (UTC)

Table of properties

The table of regular maps in the article give the: Euler characteristic χ, Genus g, Schläfli symbol, Vertex Edge and Face counts, Group, Graph and name.

Not all of this is given in the cited source, Coxeter and Moser's book. Just how much is verifiable from the book, how much can be verified elsewhere and how much is original research to be deleted per WP:NOR? For example the two redlinked names, the "hemidihedron" and "hemihosohedron" appear to have no reliable source. Do all the other names? — Cheers, Steelpillow (Talk) 17:56, 2 February 2015 (UTC)

Toroidal polyhedra

I find the first sentence of the section Regular_map_(graph_theory)#Toroidal_polyhedra impossible to understand. It reads "Regular maps exist as torohedral polyhedra as finite portions of Euclidean tilings, wrapped onto the surface of a duocylinder as a flat torus."

1. Is the sentence about all regular maps, as the text suggests; or just about toroidal ones?
2. A duocylinder has two surfaces, not one.
3. Each surface of a duocylinder is a 3-manifold, but a regular map is an embedding in a 2-manifold.

Maproom (talk) 17:35, 20 February 2017 (UTC)

Explanation of subscripts (b,c) needed

The section Toroidal polyhedra begins as follows:

"Regular maps exist as torohedral polyhedra as finite portions of Euclidean tilings, wrapped onto the surface of a duocylinder as a flat torus. These are labeled {4,4}_b,c for those related to the square tiling, {4,4}. {3,6}_b,c are related to the triangular tiling, {3,6}, and {6,3}_b,c related to the hexagonal tiling, {6,3}. b and c are whole numbers. There are 2 special cases (b,0) and (b,b) with reflective symmetry, while the general cases exist in chiral pairs (b,c) and (c,b)."

At no point in the article is the meaning of the subscripts (b,c) ever explained.

Nor is the highly unusual word "torohedral" ever explained.

I hope that someone knowledgeable about this subject can fix these serious omissions.