  A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph
containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that
among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of
Hamiltonian subsets. We prove a near optimal lower bound that takes also the
order and the structure of a graph into account. For many natural graph
classes, it provides a much better bound than the extremal one ($\approx
2^{d+1}$). Among others, our bound implies that an $n$-vertex $C_4$-free graphs
with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian
subsets.
