Asymptotic Distribution of Form of Polyhedra

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.

Let $\mathcal P_k$ be the set of isomorphism classes of finite simple $3$-connected planar graphs with exactly $k$ edges, equivalently the topological polyhedral types in the source. Choose $P_k$ uniformly from this finite set and write \[\beta_k=\frac{|V(P_k)|}{k+2}.\] Euler's formula sends $|V|$ to $k+2-|V|$ under planar duality, so this counting distribution is symmetric about $1/2$.

Statement recorded in the supplied source.

Determine the asymptotic distribution of $\beta_k$ as $k\to\infty$: does $\beta_k$ converge in distribution, and is there a nonconstant limit distribution after a suitable centering and rescaling? In particular, does the distribution concentrate at $1/2$? [S2]

Short English statement

Determine the limiting distribution of the vertex-to-edge form parameter of unlabelled polyhedral graphs, including any nontrivial rescaled limit.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3154 (OPG-36933). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Asymptotic Distribution of Form of Polyhedra, original node 36933, statement and mathematical discussion. https://www.openproblemgarden.org/op/asymptotic_distribution_of_form_of_polyhedra.