Let $G$ be an $n$-node graph that can be decomposed into two disjoint union of spanning trees. In particular, $G$ has $2n-2$ edges.
It is not hard to show that the girth of $G$ is at most $O(\log n)$. However, I don't have an example with girth bigger than $4$.
Do we have an example of such graph whose girth is $\Theta(\log n)$, or is there a way to prove an even better upper bound?
I wonder if there are constant degree expanders that, after removing an appropriate number of edges, can be decomposed into two disjoint union of spanning trees.