Hakimi, Schmeichel, and Thomassen (1979) [10] 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. In this paper, we show that every 4-connected planar triangulation on n vertices has $\mathrm{\Omega }(n^2)$ 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 $2^{\mathrm{\Omega }(n^{1/4})}$ Hamiltonian cycles.

中文翻译：

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在平面三角剖分中计算哈密顿循环

Hakimi、Schmeichel 和 Thomassen (1979) [10] 推测，在n个顶点上的每个 4 连通平面三角剖分G至少有$2(n-2)(n-4)$哈密​​顿循环，当且仅当G是双轮时相等。在本文中，我们证明了在n个顶点上的每个 4 连通平面三角剖分都有$\mathrm{\Omega }(n^2)$哈密​​顿循环。此外，我们证明如果G是n个顶点上的 4 连通平面三角剖分，并且G中任意两个 4 次顶点之间的距离至少为 3，则G有$2^{\mathrm{\Omega }(n^{1/4})}$哈密​​顿循环。