Herschel graph with different planar embeddings

Another example from a paper that I have mentioned before. This time the Herschel graph which is another of these crazy graphs thought up to prove or disprove some conjecture. The spring-layout paper gives two different layouts:

These really are the same graph :) One way to see this is to look at the vertex colors that I have applied. Each face is given a label (A or B) based on the key shown below the two graphs. The 'A' face is {Black, White, Grey, White} for example. The left-hand embedding (or layout) has such an A-face for its border, while the right-hand one has a B-face for a border.

Presumably, it is possible to convert a vertex partition (into equivalence classes) into a partition of faces. It seems easy for examples - like this - that have a planar embedding. More difficult for graphs that don't have one. On the other hand, not all planar layouts look very informative:

This is twistane (again) but not looking as symmetric as it can. However, the faces show the regularity - they are all the same, even the boundary. The colors used are the same as in the previous post.

Comments