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Komlós conjectured in 1981 that among all graphs with minimum degree at least   d  , the complete graph    K   d  +  1minimises 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  +  1or a certain other graph which we specify.

Proof of Komlós's conjecture on Hamiltonian subsets