Suppose we have a plane with several closed regions contained in it, under the requirement that these regions may only be tangent, but not intersect each other. For example, if we had 4 regions, a valid graph might be a 2x2 square grid.
In this case, we see that each region touches two other faces (singular points of contact don't count), and thus each region touches an average of two other regions.
In fact, the maximum such average for 4 regions is 3. This maximum is attained when every region touches every other region, which makes sense. But with 5 regions, a similar graph is not possible, as demonstrated here:
Proof that there can be no planar graph with 5 faces, such that any two of them share an edge.
My question is whether a general function $f(n)$ exists that takes the number of regions $n$ and returns the maximum average of sides shared in such a graph (the maximum "closeness" between regions).
We know that
$f(1)=0$
$f(2)=1$
$f(3)=2$
$f(4)=3$,
and trivially $f(n)\leq n-1$. From the link above we know that
$f(5)=\frac{18}{5}$.
If no explicit form for $f(n)$ exists, then I am also curious about what
$$\lim_{n\to\infty}f(n)$$
is, if it exists.
A few things I've tried:
If we consider an infinitely large square grid, then for side length $l$ there are $4$ corners, $4(l-2)$ side squares, and $(l-2)^2$ center squares. Thus, the limit using this method is
$$\lim_{l\to\infty}\frac{2\cdot 4+3\cdot 4(l-2)+4\cdot (l-2)^2}{l^2}=4\text{,}$$
so
$$\lim_{n\to\infty}f(n) \geq 4\text{,}$$
and similar logic with a hexagonal tesselation gives
$$\lim_{n\to\infty}f(n) \geq 6\text{.}$$
We can also consider fractals such as the Koch snowflake. Instead of removing previous triangle boundaries let every new triangle be a new region. At iteration $i$ where $i=0$ is one triangle and $i=1$ resembles a hexagram, we have
$$4^{i}\text{ regions}$$
and
$$3\cdot \left(2^i-1+\sum _{n=1}^i\left(2^i-1\right)\cdot 4^{i-n}\right)=2^{2i+1}-2\text{ shared faces.}$$
Interestingly enough, this average converges:
$$\lim_{i\to\infty}\frac{2^{2i+1}-2}{4^i}=2\text{.}$$
Thus, even though theoretically it seems like a self-similar Koch snowflake would have a divergent average, the limit converges to a nice number. Other fractals that I tried follow similar behavior....
I feel that this question is quite simple, but I couldn't find any source where something similar was considered. Apologies if this is a duplicate or a theorem of some sort.
I've no clue what type/area of mathematics this question could fall under, but any insight would be appreciated.