Abstract 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) ≈ 1.260ⁿ 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) ≈ 1.297ⁿ. We also solve the traveling salesman problem in graphs of degree at most four, by randomized and deterministic algorithms with runtime O((27/4)^n/3) ≈ 1.890ⁿ and O((27/4+ϵ)^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.