  Koml\'os conjectured in 1981 that among all graphs with minimum degree at
least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian
subsets, where a subset of vertices is Hamiltonian if it contains a spanning
cycle. We prove this conjecture when $d$ is sufficiently large. In fact we
prove a stronger result: for large $d$, any graph $G$ with average degree at
least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$,
unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we
specify.
