Hamiltonian cycles (seen as **spanning polygons**) are interesting for several reasons (only a few of which I am aware of), but especially because *not* every connected graph has a Hamiltonian cycle (is Hamiltonian), so the characterization of Hamiltonian graphs becomes interesting (see wikipedia article on [Hamiltonian paths][1]). 

Side note: Each platonic and archimedian solid is Hamiltonian. 

What about **spanning polytopes**, as
one possible generalization of
Hamiltonian cycles = spanning polygons?

(By "spanning polytope" I mean a spanning subgraph that is the 1-skeleton of a polytope of arbitrary dimension.)

There are connected graphs without spanning polytopes (trees obviously), but there are non-Hamiltonian graphs that have a spanning polytope of dimension d>2, e.g. the [Herschel graph][2].

A google search for "spanning polytope" yields only unrelated results, so my question is:

> Is there research on this or a related
> topic, only under another name?

If not so, does this have an obvious - or not so obvious - reason?


  [1]: http://en.wikipedia.org/wiki/Hamiltonian_path#Bondy.E2.80.93Chv.C3.A1tal_theorem
  [2]: http://mathworld.wolfram.com/HerschelGraph.html