We say the rook graph, $R_n$, is the cartesian product of $K_n \times K_n$. Let $S$ be the set of graphs that are an induced subgraph of $R_n$ for some $n$.
Does there exist some constant $c$ such that if $G \in S$ is c-connected, it follows that $G$ is Hamiltonian? I know that if $c$ does exist, $c\geq 3$.
If possible, can we get values $c_d$ for induced subgraphs of d-dimensional rook graphs that are the product of $d$ complete graphs?