For a reason non-related with graph theory, I need to construct a family of graphs (acyclic, without loops) $\langle G_n:n\in \mathbb{N}\rangle$ satisfying the following:
- Each $G_n$ is regular (say $r_n$-regular)
- $\lim_{n\to\infty}r_n=\infty$
- $\lim_{n\to\infty}\operatorname{girth}(G_n)=\infty$
Here, the girth of a graph is the length of the smallest cycle in the graph (I am not sure if this notation is standard).
I am already familiar with the procedure of lifting, that for an $r$-regular graph $G$ of girth $n$ produces an $r$-regular graph $L[G]$ of girth 2n. So, I can put $G_n$ to be the graph obtained by applying $n$-times the lifting to the graph $K_{n+1}$, which is $n$-regular. The only issue is that the size of $L[G]$ is $|V(G)|\cdot 2^{|E(G)|}$, and so the size of the graphs grows as an exponential tower.
Is there a more effective way to obtain a family with a slow growth in the number of vertices and with the desire properties?
Even more generally (but possibly more difficult), what is the size (if known) of the smallest $r$-regular graph of girth $n$?
If $G$ has girth $g$, then for a given vertex $x$ the induced subgraph consisting of vertices of distance $< (g-1)/2$ from $x$ forms a tree. If the graph is $r$-regular, the number of vertices in this subgraph is greater than $(r-1)^{(g-3)/2}$. So it's unavoidable to have the number of vertices grow exponentially.
EDIT: It seems a smallest $r$-regular graph of girth $g$ is called an $(r,g)$ cage. See OEIS sequence A054760 and links there.