Counting Self-avoiding Walks in Some Regular Graphs

Figure 1: A length-18 in the complete 2D grid. lems in different areas of science, such as combinatorics, statistical physics, theoretical chemistry, and computer science. By “twodimensional grid” we mean the two-dimensional rectangular lattice Z2 with origin (0, 0). One of the most prominent applications of counting SAWs is the modeling of spatial arrangement of linear polymer molecules in a solution. Here a SAW represents a molecule composed of monomers linked together in a chain by chemical bonds. Other application areas include the percolation model, the Ising model, and the network reliability model. Valiant [Val79b] is the first to find connections between the problem of counting SAWs and computational complexity theory. He showed that the problem of counting SAWs between two given points, the problem of counting Hamiltonian cycles, and the problem of counting Hamiltonian paths between two given points are all #P-complete under polynomial parsimonious reductions (that is, polynomial-time reductions of functions not requiring post-computation) both for directed graphs and for undirected graphs. One might ask for which types of graphs these counting problems remain #P-complete. The goal of this paper is to present some natural regular graphs for which the corresponding counting problems are #P-complete and discuss some open issues.