A simple sphere theorem for graphs

Abstract

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.