报告题目: Counting Hamiltonian cycles in planar triangulations

报告人：刘肖男， 范德堡大学

地点：管理科研楼1318

时间：7月4号下午3：00-4：00

摘要：

Whitney showed that every planar triangulation without separating triangles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $\log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and, in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. We show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ 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 $2^{\Omega(n^{1/4})}$ Hamiltonian cycles. Joint work with Zhiyu Wang and Xingxing Yu.