  The rook graph is a graph whose edges represent all the possible legal moves
of the rook chess piece on a chessboard. The problem we consider is the
following. Given any set $M$ containing pairs of cells such that each cell of
the $m_1 \times m_2$ chessboard is in exactly one pair, we determine the values
of the positive integers $m_1$ and $m_2$ for which it is possible to construct
a closed tour of all the cells of the chessboard which uses all the pairs of
cells in $M$ and some edges of the rook graph. This is an alternative
formulation of a graph-theoretical problem presented in [Electron. J. Combin.
28(1) (2021), #P1.7] involving the Cartesian product $G$ of two complete graphs
$K_{m_1}$ and $K_{m_2}$, which is, in fact, isomorphic to the $m_{1}\times
m_{2}$ rook graph. The problem revolves around determining the values of the
parameters $m_1$ and $m_2$ that would allow any perfect matching of the
complete graph on the same vertex set of $G$ to be extended to a Hamiltonian
cycle by using only edges in $G$.
