# Can two non-equivalent polytopes of same dimension have the same graph?

By a polytope I mean the convex hull of finitely many points. The graph of a polytope is the graph isomorphic to its 1-skeleton. By equivalence of polytopes I mean combinatorial equivalence, i.e. their face lattices are isomorphic.

I know that two polytopes can have isomorphic graphs while being non-equivalent, e.g. neighborly polytopes. However, all examples I know of are polytopes of different dimension. So I wonder:

Question: Can there be two non-equivalent polytopes of the same dimension with the same graph?

Especially, are all $$k$$-neighborly polytopes of the same dimension equivalent?

The 1-skeleton is usually not enough to recover the face lattice, but under some conditions it is. I did a quick google search, and read the abstract in this paper.

• I knew the paper but I failed to remember it and to draw the conclusion from the abstract. Thank you. Sep 4, 2018 at 12:16

There are many non-equivalent neighborly polytopes, already in dimension $$d=4$$. See for example