What is the largest graph of positive curvature?

Discussion

Definition: For a graph $ G $ embedded in the sphere, the combinatorial curvature of a vertex $ v $ is defined to be $ 1 - \frac{ {\mathit deg}(v)}{2} + \sum_{f \sim v} \frac{1}{ {\mathit size}(f) } $ (here the summation is over all faces $ f $ incident with $ v $ ). Let $ G $ be a graph embedded in the sphere, and consider the polygonal surface formed by treating each face of size $ n $ as a regular $ n $ -gon of side length $ 1 $ . The gaussian curvature at a vertex $ v $ is defined to be $ 2 \pi $ minus the sum of the angles incident with $ v $ . So, our vertex $ v $ has positive curvature if the sum of the incident angles is less than $ 2 \pi $ . In fact, the combinatorial curvature at $ v $ is exactly $ 2 \pi $ times the gaussian curvature, so these quantities will always have the same sign. Let us call a convex polyhedron regular-faced if each face is a regular polygon. Based on the previous discussion, we know that every convex regular-faced polyhedron gives us a graph with everywhere positive combinatorial curvature. Indeed, we may view planar graphs with everywhere positive curvature as a kind of generalization of these polyhedra. The polyhedra in this class have been studied and classified. The Platonic solids and Archimedean solids are all convex and regular faced, and there are two infinite families: prisms and antiprisms . In addition to this, there are 92 other exceptional ones, known as Johnson Solids . Euler's formula tells us that the sum of the combinatorial curvatures over all of the vertices is equal to 2. Indeed, the combinatorial curvature is exactly what we get when we assign $ 1 $ to each vertex and face, $ -1 $ to each edge, and then "discharge" evenly onto the vertices. So, if we wish to construct large planar graphs where every vertex has positive curvature, we will need to make the curvature arbitrarily small. This can be achieved with prisms and antiprisms, but apart from these two families, all other graphs with everywhere positive curvature have a bounded number of vertices. Improving upon [DM], Zhang [Z] has shown this upper bound to be at most 580. The great rhombicosidodecahedron has 120 vertices and everywhere positive curvature (this is the largest regular-faced convex polyhedron which is not a prism or antiprism). Reti, Bitay, and Kosztolanyi [RBK] have improved upon this lower bound by constructing a graph with everywhere positive curvature and 138 vertices. These are the best bounds I (M. DeVos) know of.