Let $md_r^k(n)$ be the minimum diameter over all $r$-regular, $k$-connected graphs on at least $n$ vertices. (Let us assume $r, k \geq 2$).

Problem:Find lower and upper asymptotic bounds on $md_r^k(n)$.

There are fewer than $r^{(d+1)}$ vertices within distance $d$ of any given vertex in an $r$-regular graph; this implies the following trivial lower bound.

Fact:$\log_rn \ll_n md_r^k(n)$.

Jackson & Parson [1982] showed that for every $\varepsilon > 0$ there exist arbitrarily large $r$-regular, $r$-connected graphs with circumference at most $\varepsilon n$. When a graph is $2$-connected, the diameter is at most half the circumference, so we have the following upper bound when $r=k$.

Corollary:$md_r^r(n) = o(n)$.

Can we improve either of these bounds?