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

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

2007-09-14

arxiv.org/abs/0709.2322v1

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

10.1088/1751-8113/40/49/003

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.

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