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I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.

I am looking for polytopes which are edge- but not vertex-transitive. There are infinitely many of these for $d=2$, and exactly two for $d=3$ (rhombic dodecahedron and rhombic tricontrahedron, see below).

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I do not know a single example for $d\ge 4$.

I believe it is easy to see that the edge-graph of such a polytope must be bipartite, and thus, zonotopes might be a good place to start looking. But my constructions fail for $d\ge 4$.

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  • $\begingroup$ To echo your bipartite point: Wikipedia says "Every edge-transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular." $\endgroup$ – Joseph O'Rourke Aug 14 at 13:51
  • $\begingroup$ @JosephO'Rourke That was my thought. But one has to be careful: the symmetry group of the polytope might be smaller than the one of its edge-graph. "Bipartite" should still hold, but I do not immediately know about "semi-symmetric or biregular". Update: I have read the definition of these terms, and they should probably still hold. $\endgroup$ – M. Winter Aug 14 at 13:54
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    $\begingroup$ Both your examples in d=3 are the convex hull of the union of a regular polyhedron and its dual (appropriately scaled). Have you tried this construction in d=4? $\endgroup$ – Yoav Kallus Aug 14 at 14:02
  • $\begingroup$ @YoavKallus That's a great idea. Thank you. What else comes to my mind now is to let a vertex grow out of each facet of a (regular) polytope and at some point the original edges will vanish in the inside of a facet. This might give examples. This corresponds to your idea of looking for the "appropriate scaling". And no, have not tried this, but will do so now. $\endgroup$ – M. Winter Aug 14 at 14:08
  • $\begingroup$ @YoavKallus At least the construction using the 4-cube and the 4-crosspolytope does not work (no matter the scaling, their are at least two orbits of edges). $\endgroup$ – M. Winter 2 days ago

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