- Series
- Dissertation Defense
- Time
- Thursday, April 13, 2023 - 2:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Xiaonan Liu – Georgia Tech – xliu729@gatech.edu – https://xliu729.math.gatech.edu/
- Organizer
- Xiaonan Liu

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 $n/ \log _2 n$ 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.