Hakimi–Schmeichel–Thomassen conjecture on Hamiltonian cycles in planar triangulations

Let $G$ be a $4$-connected planar triangulation on $n$ vertices. The Hakimi–Schmeichel–Thomassen conjecture. $G$ has at least

$$2(n-2)(n-4)$$

hamiltonian cycles, with equality if and only if $G$ is the double-wheel graph on $n$ vertices, namely, the join of a cycle of length $n-2$ and an empty graph on two vertices. This conjecture gives a sharp proposed lower bound for the number of Hamiltonian cycles in $4$-connected planar triangulations. The supplied paper proves exponential lower bounds for broad classes of such triangulations with few $4$-separators, but does not establish the conjectured exact quadratic bound and equality characterization in general.