  A finite simple graph G is declared to have positive curvature if every in G
embedded wheel graph has five or six vertices. A d-graph is a finite simple
graph G for which every unit sphere is a (d-1)-sphere. A d-sphere is a d-graph
G for which there exists a vertex x such that G-x is contractible. A graph G is
contractible if there is a vertex x such that S(x) and G-x are contractible.
The empty graph 0 is the (-1)-sphere. The 1-point graph 1 is contractible. The
theorem is that for d bigger than 1, every connected positive curvature d-graph
is a d-sphere. A discrete Synge result follows: a positive curvature graph is
simply connected and orientable. For every d larger than 1, there are only
finitely many positive curvature graphs. There are six for d=2 and all have
diameter less or equal to 3. To prove the theorem, we use a "geomag lemma"
which shows that every geodesic in G can be extended to an immersed 2-graph S
of positive curvature and must so be a 2-sphere with positive curvature. As
none of these has diameter larger than 3, also G has a diameter 3 or less. This
can be used to show that G-x is contractible and so must be a sphere.
