How big can the girth of the a graph $G$ consisting of two $C_k$ + some edges to make $G$ cubic?

Question

Let $G$ be a simple graph with vertex set $\{u_1, \dots, u_n\} \cup \{v_1, \dots v_n\}$, with $G_{\uparrow U}$ (subgraph induced by $U$) and $G_{\uparrow V}$ being cycles of size $n$. Now we want to add edges between each vertex of $U$ to a unique vertex of $V$ such that the length of the minimum cycle (girth) is as big as possible.

Question: Is it possible to have a girth of $\Omega(\log \log n)$ for infinitely values of $n$? If not, was girth lower bound can asymptotically achieve on such graphs?

An easy edge set would be $\{u_i, v_{i+k} \, $mod $ n\}$ but it gives cycles of size 4 almost everywhere.

I have a feeling the answer could be related to $\mathbb{Z}/p \mathbb{Z}$ groups as other simple construction could be the edge set "{$u_x, v_y$ | $y \equiv px$ mod $n\}$" with probably $p$ and $n$ being a prime numbers (I talk with 8+ years memory from my arithmetic classes).

However I can see how studying these groups would help to find cycles whose intersection with $G_{\uparrow U}$ is a single path, but not for other type of cycles.

Answer 1

this paper shows that for all girth $g$ there such a graph. Moreover it gives a recursive construction on $g$: if one can build such a graph with $n$ nodes of girth $g$, then there is a graph on $O(n 2^n)$ nodes with girth $g+1$.