# Two dimensional graphs with positive curvature

 There are exactly 6 two-dimensional boundary less graphs with positive combinatorial curvature. They have vertex cardinality v in {6,7,8,9,10,12}. Their edge and face cardinalities e=(v-2) 3, f=(v-2) 2 are determined by Gauss-Bonnet telling that their curvatures add up to 12. The octahedron and the icosahedron are the only two dimensional constant curvature graphs.

polyhedron curvatures 3D Plot Planar Plot
octahedron (2,2,2,2,2,2)
10 hedron (2,2,2,2,2,1,1)
12 hedron (2,2,2,2,1,1,1,1)
14 hedron (2,2,2,1,1,1,1,1,1)
16 hedron (2,2,1,1,1,1,1,1,1,1)
icosahedron (1,1,1,1,1,1,1,1,1,1,1,1)

They were found by hand and checking with the built in polyhedral graph libraries (of degree 6-12) in Mathematica 8. For vertex size smaller or equal to 9, I searched with brute force over 215 suitable adjacency matrices and checking the two dimensional ones for graph isomorphism.

One can see by hand that there is no positive curvature graph with 11 vertices. Proof: Such a graph has exactly one degree 4 vertex and otherwise only degree 5 vertices. The edge formula mentioned above shows that there are 27 edges. Look at the point with cardinality 4 and look at spheres around it: the next sphere S2 has 4 entries. Attach all necessary edges to get 24. We have only 3 edges left to attach two more vertices. This is not possible. Remark. There are more two -dimensional graphs with vertex cardinality smaller or equal to 12 and nonnegative curvature. While there are infinitely many fullerene type graphs with nonnegative curvature, there are finitely many graphs of nonnegative curvature which have no flat disc of fixed radius r. this page posted, July 2, 2011