We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2 n/3 ) ≈ 1.260 n and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycles in a degree three graph in time O(2 3n/8 ) ≈ 1.297 n . We also solve the traveling salesman problem in graphs of degree at most four, by randomized and deterministic algorithms with runtime O((27/4)

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... n/3 ) ≈ 1.890 n and O((27/4 + ǫ) n/3 ) respectively. Our algorithms allow the input to specify a set of forced edges which must be part of any generated cycle. Our cycle listing algorithm shows that every degree three graph has O(2 3n/8 ) Hamiltonian cycles; we also exhibit a family of graphs with 2 n/3 Hamiltonian cycles per graph.