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

Jesper Lykke Jacobsen (1, 2)

1 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
2 SPhT - Service de Physique Théorique

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.

Mots clés: hamiltonian walk; hamiltonian circuit; exact enumeration; Lee-Yang zeros

Domaines: Mécanique statistique [cond-mat.stat-mech]

Jesper Lykke Jacobsen. Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions. 2007. ⟨hal-00172308⟩