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?



Does 2-connectivity imply Hamiltoniancy for subgraphs of the rook graph
It's known that $3\leq c \leq7$.


$S$ is the class of line graphs of bipartite graphs, by [ISGCI](http://graphclasses.org/classes/gc_736.html). 


The line graph of [this graph](https://math.stackexchange.com/questions/955285/2-connected-graphs-with-a-line-graph-containing-no-hamilton-cycle/955298#955298) is 2-connected and non-Hamiltonian, so $c\geq3$.


Siming Zhan proved that 7-connected line graphs are Hamiltonian, so $c\leq 7$. See [Zhan, Siming. "On hamiltonian line graphs and connectivity." Discrete Mathematics 89.1 (1991): 89-95.](https://core.ac.uk/download/pdf/82029817.pdf)


