What is the largest graph of positive curvature?

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Consider a simple connected planar graph embedded in the sphere, with minimum vertex degree three. For a vertex $v$, let $\deg(v)$ be its degree and let $|f|$ be the boundary length of a face $f$. Its combinatorial curvature is\[ \kappa(v)=1-\frac{\deg(v)}2+\sum_{f\ni v}\frac1{|f|}, \] where face incidences are counted around $v$. The prism graph on $2r$ vertices consists of two $r$-cycles joined by corresponding edges, for $r\geq3$. The antiprism graph has two $r$-cycles, with each vertex of one cycle joined to the two consecutive corresponding vertices of the other.

Statement recorded in the supplied source.

What is the greatest possible number of vertices of such a graph satisfying $\kappa(v)>0$ at every vertex, excluding the prism and antiprism families? The later primary source gives the exact maximum $208$. [S2] [S3]

Short English statement

How many vertices can a connected planar graph of minimum degree three have if every vertex has positive combinatorial curvature and the graph is neither a prism nor an antiprism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3328 (OPG-157). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, What is the largest graph of positive curvature?, node 157, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/what_is_the_largest_graph_of_positive_curvature.
• S3 Luca Ghidelli, On the largest planar graphs with everywhere positive combinatorial curvature (2017), abstract. https://arxiv.org/abs/1708.08502.