Hamiltonian Paths in Distance Graphs

The object of study is the graph
$$ G(n,r,s)=(V(n,r),E(n,r,s)) $$
with
\begin{align*} V(n,r)&=\{v : v \subset \{1,\dots,n\}, \, |v|=r\}, \\ E(n,r,s)&=\{\{v,u\} : v,u \in V(n,r), \, |v \cap u|=s\}; \end{align*}
i.e., the vertices of the graph are $r$-subsets of the set $\mathcal{R}_n=\{1,\dots,n\}$, and two vertices are connected by an edge if these vertices intersect in precisely $s$ elements. Two-sided estimates for the number of Hamiltonian paths in the graph $G(n,k,1)$ as $n \to \infty$ are obtained.