Although polyhedra can have much fewer edges than triangulations, many results about hamiltonicity proven for triangulations also hold for polyhedra. The most famous of these results is surely Whitney’s result from 1931 that 4-connected triangulations are hamiltonian, which was 25 years later generalised to 4-connected polyhedra by Tutte. Nevertheless the only known bounds for the number of hamiltonian cycles in 4-connected polyhedra are constant, though for triangulations a lower bound of $\left|V\right|∕\left({log}_2\left|V\right|\right)$ was already proven in 1979 and improved to a linear bound in 2018. In this article we prove linear lower bounds for 4-connected polyhedra and polyhedra with at most one 3-cut and sufficiently many edges.

尽管多面体可以比三角剖分具有更少的边，但许多关于三角剖分证明的半调性结果也适用于多面体。这些结果中最著名的肯定是惠特尼 1931 年的结果，即 4 连通三角剖分是汉密尔顿式的，25 年后，图特将其推广为 4 连通多面体。然而，4 连通多面体中哈密顿圈数的唯一已知界限是恒定的，尽管对于三角剖分，下界为$\left|伏\right|∕\left({日志}_2\left|伏\right|\right)$ 已经在 1979 年得到证明，并在 2018 年改进为线性边界。 在本文中，我们证明了 4 连通多面体和最多具有一个 3 切割和足够多边的多面体的线性下界。