Counting (and Randomly Generating) Hamiltonian Cycles in Rectangular Grids

TL;DR

This paper implements a method to count and generate Hamiltonian cycles in rectangular grid graphs, providing explicit generating functions for small widths and methods for random cycle generation.

Contribution

It fully implements and extends a 1995 method to derive generating functions for Hamiltonian cycles in grid graphs, including random cycle generation.

Findings

Generated explicit formulas for grids up to width 10

Developed a method for uniform random generation of Hamiltonian cycles

Derived generating functions for additional parameters

Abstract

We first fully implement, in Maple, the ingenious method of Robert Stoyan and Volker Strehl from 1995 to automatically derive generating functions for the number of Hamiltonian cycles in an m by n grid graph ,for a fixed width m, but general length n, and actually compute these generating functions for all m up to ten. We also show how to generate a uniformly-at-random such Hamiltonian cycle, and also derive more informative generating functions for other parameters besides the length of the grid graph.