By "Steinitz's theorem" I mean his theorem that a graph is the graph of a three-dimensional convex polyhedron iff it is of genus 0 and has a vertex connectivity of at least three. Can a similar characterization be made for genus 1, or beyond?
$\begingroup$
$\endgroup$
3
-
$\begingroup$ The paper "The toroidal analogue to Eberhard's theorem" by Peter Gritzmann may be relevant, but I cannot access it. Cambridge Link. $\endgroup$– Joseph O'RourkeCommented Aug 18, 2013 at 1:47
-
$\begingroup$ What kind of generalization are you looking for? You are not going to find a convex embedding... $\endgroup$– Igor RivinCommented Aug 18, 2013 at 2:33
-
$\begingroup$ Obviously a replacement for convexity would be required. The point is to define a geometrically defined object, which would of course have a donut hole, whose graph is precisely genus 1 with vertex connectivity at least 3 (ie, 3-connected.) $\endgroup$– Gene Ward SmithCommented Aug 18, 2013 at 8:56
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Well, no direct generalization is known. One related result (which seems to have come out of an attempt to generalize Steinitz) is this paper by D. Eppstein and E. Mumford. However, since Steinitz' theorem is a simple consequence of the circle packing theorem, one can reasonably argue that the circle packing and its more general versions are the correct generalization -- unfortunately (or fortunately, for some of us) they move you into hyperbolic space. For recent work on this subject see the work of Francois Fillastre.
answered Aug 18, 2013 at 19:23