Enumerating all Hamilton Cycles and Bounding the Number of Hamilton Cycles in 3-Regular Graphs

We describe an algorithm which enumerates all Hamilton cycles of a given 3regular n-vertex graph in time O(1.276n), improving on Eppstein’s previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). Our method differs from Eppstein’s in that he considers in each step a new graph and modifies it, while we fix (at the very beginning) one Hamilton cycle C and then proceed around C, successively producing partial Hamilton cycles.