Every 4-connected {claw,hourglass}-free graph is 2-Hamiltonian

A graph $G$ is $k$-hamiltonian if the graph $G-X$ is hamiltonian for any set $X$ of $k$ vertices of $G$. The claw is the complete bipartite graph $\claw$, and the hourglass is the unique graph with degree sequence $4,2,2,2,2$. We show that every 4-connected \{claw,hourglass\}-free graph is 2-Hamiltonian. The result can be easily extended to $k$-hamiltonicity for $k\geq 2$, and implies that $k$-hamiltonicity is for $k\geq 2$ polynomial-time decidable in the class of \{claw,hourglass\}-free graphs.