A fact from Herschel graph appeared on Wikipedia's Main Page in the Did you know column on 18 October 2009 (check views). The text of the entry was as follows:
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The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
Lead: a bit on the short side; will comment later whether I think anything major is missing
Definition and properties: the description of the graph is a bit confusing. Each of the three pairs of degree-four vertices makes it sound like there are six degree-four vertices. Maybe better to say "for any two distinct degree-four vertices"?
It would also be great to have a labeled illustration making it easier to follow the description. Or at least say that the three degree-four vertices are the blue ones in the middle, and the other two blue ones are the two degree-3 ones not belonging to the four-vertex cycles.
Polyhedron: I understand what you mean by all of the same symmetries as the underlying graph but in one case we have a permutation group and in the other case a subgroup of O(3); is it worth trying to explain this more?
I think it would be worth making the connection between the Lich's nemesis story and the fact that the graph supports a Hamiltonian path, but not cycle, more explicit.
Added more of an explanation for this story, including why it relates to the nonexistence of a Hamiltonian cycle.
Hamiltonicity: It is the smallest non-Hamiltonian polyhedral graph strictly speaking, you haven't said what a Hamiltonian graph is yet.
Also, "the smallest non-Hamiltonian polyhedral graph by vertices" seems to be contradicted by "other polyhedral graphs with 11 vertices and no Hamiltonian cycles". It has the minimal number of vertices, yes, but as it is not unique with this property, it is not "the smallest".
two vertices are adjacent in the medial graph whenever the corresponding edges of the Herschel graph are consecutive on one of its faces. here it might help to have defined the face of a planar graph, but perhaps the polyhedral picture is sufficient for it.
So far (and here) we're really only using faces of a polyhedral graph, for which I think it's ok to treat them in an intuitive way as being the same as the faces of the polyhedron. —David Eppstein (talk) 01:16, 18 August 2023 (UTC)[reply]
The article k-edge-connected graph has no information about being "essentially 6-edge-connected".
Moved "essentially" out of the wikilink and added a gloss.
History: In British English, "British astronomer Alexander Stewart Herschel" would like a "the". I think it is fine in American English though, but it is too long since I lived in the US so I am uncertain.
I think I am happy with your responses. However, there are a bit many "howevers" in the History section; could you try to cut down a bit? (These are WP:WTW). —Kusma (talk) 10:17, 18 August 2023 (UTC)[reply]
I'll be travelling from middle of next week; should we not finish by then, my responses might also be delayed a bit. —Kusma (talk) 10:24, 18 August 2023 (UTC)[reply]
1: Content checks out; this is an established blog by mathematicians so it barely passes I think (Christian Lawson-Perfect is a maths educator; I recently came across his name because he is the mathstodon.xyz admin).
3a: Citation should use location (to prevent people from thinking "Dover" is a location). I don't see content about Steinitz' theorem in Coxeter p. 8 in the 1947 edition; as Dover usually just does photomechanical reproductions, I would expect the Dover edition to be the same.
Swapped to London/Methuen/1948 since I have easier access to that edition (which is I expect the same). He doesn't mention Steinitz but he certainly strongly suggests that this graph forms a polyhedron. —David Eppstein (talk) 16:29, 18 August 2023 (UTC)[reply]
Well, for the Herschel graph this is true, but the general statement of Steinitz' theorem is currently in the article and not supported by a reference. You could just add refs 5 or 8 from Steinitz's theorem. —Kusma (talk) 21:22, 18 August 2023 (UTC)[reply]
Ok, added Grünbaum since I think we don't need both, and reworded to clarify that Grünbaum is sourcing the general statement of Steinitz's theorem and not its specific application to this graph. —David Eppstein (talk) 09:41, 19 August 2023 (UTC)[reply]
3b: Coxeter calls it something like the simplest example and isn't totally explicit about number of edges.
7: is this Tait's comment that ends in a two line proof of the four colour theorem?
The bogus proof of the four color theorem is paragraph (15) of the cited work; the conjectured (but not incorrectly stated as proven) Hamiltonicity of cubic polyhedra is paragraph (16). Added more specific pointer to the reference. —David Eppstein (talk) 09:37, 19 August 2023 (UTC)[reply]
9: this is a user generated site; also, it should say "remains open as of 2011" or something. Might be better to just rephrase "A refinement of Tait's conjecture is Barnett's conjecture, which ..."
The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
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