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 $O^*(\aplha^n)$ time for some constant $\alpha < 2$ 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 $O^*(1.6818^n)$ time and exponential space, and one algorithm that uses $O^*(1.8878^n)$ time and polynomial space.