Counting Hamiltonian cycles in planar triangulations

Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every 4-connected planar triangulation G on n vertices has at least 2(n-2)(n-4) Hamiltonian cycles, with equality if and only if G is a double wheel. We show that every 4-connected planar triangulation on n vertices has quadratic many Hamiltonian cycles. Moreover, we show that if G is a 4-connected planar triangulation on n vertices and the distance between any two vertices of degree 4 in G is at least 3, then G has exponentially many Hamiltonian cycles. Joint work with Zhiyu Wang and Xingxing Yu.