Reconfiguration of Hamiltonian Paths and Cycles in Rectangular Grid Graphs
Albi Kazazi
TL;DR
This work studies reconfiguration of Hamiltonian cycles and paths on rectangular grid graphs via local moves (switch, double-switch, backbite). It develops a structural framework based on polyomino graphs, H-decomposition, and follow-the-wall construction, culminating in canonical forms and a Reconfiguration to Canonical Form (RtCF) algorithm for cycles, supported by MLC and 1LC subroutines. The main results show that any two Hamiltonian cycles on an $m\times n$ grid (with $n\ge m$) can be connected by at most $n^2m$ valid double-switch moves, and any two Hamiltonian paths can be connected using a mix of switch, double-switch, and backbite moves within a similar bound, with detailed complexity analyses. The fat-path framework (turns, weakenings, and sector analyses) underpins the move design and guarantees progress, enabling efficient reconfiguration and offering implications for Monte Carlo sampling of Hamiltonian configurations in lattice models. The work also discusses boundary effects, resistance phenomena, and extensions to higher dimensions and alternative lattices, framing future directions for reconfiguration theory and sampling algorithms.
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.