Reconfiguration of Hamiltonian Paths and Cycles in Rectangular Grid Graphs

TL;DR

This paper demonstrates that any Hamiltonian cycle in a rectangular grid graph can be transformed into any other through a sequence of valid double-switch moves, preserving Hamiltonicity at each step, with extensions to Hamiltonian paths.

Contribution

It introduces a method for reconfiguring Hamiltonian cycles in grid graphs using local switch moves, ensuring all intermediate states are Hamiltonian, and extends the approach to Hamiltonian paths.

Findings

Any Hamiltonian cycle can be transformed into any other via valid double-switch moves.

The method preserves Hamiltonicity at every intermediate step.

Extensions include transformations of Hamiltonian paths with additional moves.

Abstract

\noindent 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 $K$ 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. This result extends to Hamiltonian paths. In that case, we also use single-switch moves and a third operation, the \textit{backbite move}, which enables the relocation of the path endpoints.