Steinitz theorem says that combinatorial types of convex polyhedra is identified with 3-connected planar graph, called by a polyhedral graph.

A symmetry of polyhedral graph means that a vertex permutation( where the corresponding edges/faces permutations are also induced from vertices) preserving the combinatorial type. We can easily check that this is realized as a planar isotopy and reflection on 2-sphere. I want to say this is combinatorial symmetry and consider the group of combinatorial symmetries for a polyhedral graph.

Now, my question is the following.

Let us consider a polyhedral graph and the combinatorial symmetry group. Is there a Euclidean convex polyhedron of the polyhedral graph whose combinatorial symmetry group can be realized as Euclidean isometries?

For example, $K_4$ has the tetrahedral symmetry group. If one consider regular tetrahedra in Euclidean space, the all tetrahedral symmetries can be realized as Euclidean isometries.

Is this true for all polyhedral graph? .


Yes, this is true for all polyhedral graphs. See the following reference:

P. Mani, Automorphismen von polyedrischen Graphen, Mathematische Annalen Volume 192, Issue 4, pp 279-303, August 1971

Satz von Mani

My translation:

Theorem. For every graph $\mathfrak{Q}$ there is a three dimensional convex polyhedron $P$ with the properties:

(a) $\mathfrak{Q}$ is isomorphic to vertex-edge skeleton $\mathfrak{P}^1$ of $P$

(b) Every automorphism of $\mathfrak{P}^1$ is induced by a symmetry of the polyhedron $P$.

See this mathscinet link for an english summary of the result.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.