Fast Exact Algorithms for Hamiltonicity in Claw-Free Graphs

The Hamiltonian Cycle problem asks if an n-vertex graph G has a cycle passing through all vertices of G. This problem is a classic NP-complete problem. So far, finding an exact algorithm that solves it in O∗(αn) time for some constant α < 2 is a notorious open problem. For a claw-free graph G, finding a hamiltonian cycle is equivalent to finding a closed trail (eulerian subgraph) that dominates the edges of some associated graph H. Using this translation we obtain two exact algorithms that solve the Hamiltonian Cycle problem for the class of claw-free graphs: one algorithm that uses O∗(1.6818n) time and exponential space and one algorithm that uses O∗(1.8878n) time and polynomial space. The latter algorithm also finds a hamiltonian cycle if it exists.