Reconfiguration of Hamiltonian Cycles in Rectangular Grid Graphs

TL;DR

The paper proves that any two Hamiltonian cycles on an $m\times n$ grid graph (with $n\ge m$) can be reconfigured into one another through a sequence of valid double-switch moves, while preserving the Hamiltonian property at every step. The approach builds a robust framework based on an $H$-decomposition, a follow-the-wall construction, and a detailed analysis of $H$-components, cookies, and necks, enabling controlled local moves. Canonical forms and two dedicated subroutines, MLC and 1LC, drive the reconfiguration by progressively simplifying the cookie structure across nested subgraphs, culminating in a canonical representation from which the target form can be reached. The work leverages Jordan’s curve theorem to manage interior/exterior regions and develops a comprehensive set of structural results for looping fat paths, turns, and weakenings to guarantee that progress can always be made toward canonical forms, yielding an explicit upper bound of $n^2m$ moves for the full reconfiguration (and $mn^2$ in the stated corollaries). The framework also extends the ideas to Hamiltonian e-cycles and discusses potential higher-dimensional generalizations, highlighting both the power and limitations of the double-switch approach in broader graph classes.

Abstract

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 $mn^2$.

Figures (78)

• Figure :
• Figure :
• Figure :
• Figure :
• Figure :