Hamiltonian Cycles in 4-Connected Planar and Projective Planar Triangulations with Few 4-Separators

Abstract

Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later, Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles. Along this direction, Brinkmann, Souffriau, and Van Cleemput in 2018 established a linear lower bound on the number of hamiltonian cycles in 4-connected planar triangulations. In stark contrast, Alahmadi, Aldred, and Thomassen in 2020 showed that every 5-connected triangulation of the plane or the projective plane has exponentially many hamiltonian cycles. This gives the motivation to study the number of hamiltonian cycles of 4-connected triangulations with few 4-separators. Recently, Liu and Yu in 2021 showed that every 4-connected planar triangulation with $O(n / \log n)$ 4-separators has a quadratic number of hamiltonian cycles. By adapting the framework of Alahmadi, Aldred, and Thomassen, we strengthen the last two aforementioned results. We prove that every 4-connected planar or projective planar triangulation with $O(n)$ 4-separators has exponentially many hamiltonian cycles.