Reconfiguration of Hamiltonian Cycles in Rectangular Grid Graphs

An \textit{$m \times n$ grid graph} is the induced subgraph of the square lattice whose vertex set consists of all integer grid points $\{(i,j) : 0 \leq i < m,\ 0 \leq j < n\}$. Let $H$ and $K$ be Hamiltonian cycles in an $m \times n$ grid graph $G$. We study the problem of reconfiguring $H$ into $K$, \textcolor{blue}{\textbullet} where the Hamiltonian cycles are viewed as vertices of a reconfiguration graph \textcolor{blue}{\textbullet}, using a sequence of local transformations called \textit{moves}. A \textit{box} of $G$ is a unit square face. A box with vertices $a, b, c, d$ is \textit{switchable} in $H$ if exactly two of its edges belong to $H$, and these edges are parallel. Given such a box with edges $ab$ and $cd$ in $H$, a \textit{switch move} removes $ab$ and $cd$, and adds $bc$ and $ad$. A \textit{double-switch move} consists of performing two consecutive switch moves. If, after a double-switch move, we obtain a Hamiltonian cycle, we say that the double-switch move is \textit{valid}. We prove that any Hamiltonian cycle $H$ can be transformed into any other Hamiltonian cycle $K$ via a sequence of valid double-switch moves, such that every intermediate graph remains a Hamiltonian cycle. Moreover, assuming $n \geq m$, the number of required moves is bounded by $mn^2$.