Many Hamiltonian subsets in large graphs with given density

Abstract 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 for large $d$ , 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 graph with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.