Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions

TL;DR

This paper introduces an exact enumeration algorithm for Hamiltonian chains, walks, and circuits on low-dimensional lattices, providing detailed counts and physical insights for two and three dimensions.

Contribution

The paper presents a novel algorithm for exact enumeration of Hamiltonian structures on lattices, extending previous methods to low-dimensional cases with comprehensive data.

Findings

Enumerated Hamiltonian chains up to 12x12 in 2D

Enumerated Hamiltonian walks up to size 17 in 2D

Enumerated Hamiltonian circuits up to size 20 in 2D and some 3D results

Abstract

We present an algorithm for enumerating exactly the number of Hamiltonian chains on regular lattices in low dimensions. By definition, these are sets of k disjoint paths whose union visits each lattice vertex exactly once. The well-known Hamiltonian circuits and walks appear as the special cases k=0 and k=1 respectively. In two dimensions, we enumerate chains on L x L square lattices up to L=12, walks up to L=17, and circuits up to L=20. Some results for three dimensions are also given. Using our data we extract several quantities of physical interest.