—   graph class (a class of finite simple planar graphs)

—   size statistic

—   labelling convention

If $c_n$ counts the members of $\mathcal C$ on the labelled vertex set $\{1,\ldots,n\}$, then $F_{\mathcal C}(x)=\sum_{n\geq0}c_nx^n/n!$ and $\gamma_{\mathcal C}=\lim_{n\to\infty}(c_n/n!)^{1/n}=1/\rho_{\mathcal C}$, where $\rho_{\mathcal C}$ is the radius of convergence of $F_{\mathcal C}$ [1].

Let $t_0$ be the unique solution in $(0,1)$ of $Y(t_0)=1$, where $Y(t)=\frac{1+2t}{(1+3t)(1-t)}\exp\!\left(-\frac{t^2(1-t)(18+36t+5t^2)}{2(3+t)(1+2t)(1+3t)^2}\right)-1$ [1].

For 2-connected planar graphs, $\gamma_{\mathrm{2-connected}}=1/R(t_0)$, where $R(t)=\frac{(1+3t)(1-t)^3}{16t^3}$ [1].

For all planar graphs and for connected planar graphs, $\gamma_{\mathrm{all}}=\gamma_{\mathrm{connected}}=1/\rho(t_0)$, where $\rho(t)=\frac{\sqrt{1+3t}(1-t)^3}{16t^3}\exp(A(t))$ and $A(t)=\frac{(3t-1)(1+t)^3\log(1+t)}{16t^3}+\frac{(1+3t)(1-t)^3\log(1+2t)}{32t^3}+\frac{(1-t)(185t^4+698t^3-217t^2-160t+6)}{64t(1+3t)^2(3+t)}$. All square roots are positive and all logarithms are real natural logarithms [1].

For 3-connected planar graphs, $\gamma_{\mathrm{3-connected}}=\frac{272+112\sqrt7}{27}$ [3].

For maximal planar graphs, $\gamma_{\mathrm{maximal}}=256/27$. The rooted simple triangulation count with $n+2$ vertices is $q_n=\frac{2(4n-3)!}{n!(3n-1)!}$, and $\lim_{n\to\infty}q_{n+1}/q_n=256/27$ [2].

An embedding is not part of the object being counted. Graphs have no loops or multiple edges. All entries use labelled enumeration by vertices; these constants are not the constants for unlabelled planar graphs [1].

Connectivity means vertex connectivity. For $k=2,3$, a $k$-connected graph has more than $k$ vertices and remains connected after deleting fewer than $k$ vertices. Including the single edge among the 2-connected graphs, as in the block decomposition used by [1], does not affect the growth constant.

A graph is maximal planar if adding any edge between nonadjacent vertices destroys planarity. For at least four vertices these are precisely the simple planar triangulations, and they are 3-connected. Tutte's rooted triangulation counts give the same exponential growth constant after removing the root and passing to labelled graphs [2].

All planar graphs and connected planar graphs have the same growth constant [1]. This equality concerns exponential growth; their counts are different.

Sage

import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from sage.rings.rational_field import QQ
R = RealBallField(400)
print((272 + 112*R(7).sqrt())/27)
print(QQ(256)/27)

O. Giménez and M. Noy, Asymptotic enumeration and limit laws of planar graphs, Journal of the American Mathematical Society 22 (2009), 309–329. Theorem 1, Lemma 4, Section 4 and Appendix. (arXiv) (doi)

W. T. Tutte, A census of planar triangulations, Canadian Journal of Mathematics 14 (1962), 21–38. (doi)

Giménez and Noy: Counting planar graphs and related families of graphs, Section 5

Growth_constants_of_the_classes_of_trees —   growth constants for tree classes; the labelled tree entries also use exponential generating functions counted by vertices

Entries are of type: real number

How well the digits are known: proven

How they were obtained:

The four decimal entries are computed to 100 significant decimal digits. The unique real root $t_0$ is enclosed by bisection with exact rational endpoints; Arb ball arithmetic certifies the signs and propagates the root enclosure through the radius formulas. Working precision is the bit equivalent of the requested digits plus 96 bits, and bisection uses 40 extra bits. At 100 digits every output ball has radius below $10^{-108}$. The maximal planar entry is the exact rational $256/27$.

more

Independent computations at 160 and 200 decimal digits use the planar-map characteristic system and the network equation, followed by the Appendix coefficient $B_2$ and $\rho=R\exp(B_2/R)$. They agree within $10^{-150}$. The attached checks also verify the 3-connected algebraic equation, the all/connected equality, the ordering implied by inclusion, and 50 exact successive ratios of Tutte's triangulation counts. The numerical cross-checks supplement the certified enclosures; they are not their error bounds.

Table is complete: no (it holds the labelled classes of all, connected, 2-connected, 3-connected and maximal planar graphs, counted by vertices, as the basic connectivity and triangulation cases; other restrictions and unlabelled enumeration are not included)