Title:
Counting Hamiltonian cycles in planar triangulations
Counting Hamiltonian cycles in planar triangulations
| dc.contributor.advisor | Yu, Xingxing | |
| dc.contributor.author | Liu, Xiaonan | |
| dc.contributor.committeeMember | Bernshteyn, Anton | |
| dc.contributor.committeeMember | Blekherman, Greg | |
| dc.contributor.committeeMember | Kelly, Tom | |
| dc.contributor.committeeMember | Perkins, Will | |
| dc.contributor.department | Mathematics | |
| dc.date.accessioned | 2023-05-18T17:54:08Z | |
| dc.date.available | 2023-05-18T17:54:08Z | |
| dc.date.created | 2023-05 | |
| dc.date.issued | 2023-04-25 | |
| dc.date.submitted | May 2023 | |
| dc.date.updated | 2023-05-18T17:54:08Z | |
| dc.description.abstract | Whitney showed that every planar triangulation without separating $3$-cycles 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. | |
| dc.description.degree | Ph.D. | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | https://hdl.handle.net/1853/72035 | |
| dc.language.iso | en_US | |
| dc.publisher | Georgia Institute of Technology | |
| dc.subject | Hamiltonian cycles | |
| dc.subject | Planar triangulations | |
| dc.subject | Tutte paths | |
| dc.title | Counting Hamiltonian cycles in planar triangulations | |
| dc.type | Text | |
| dc.type.genre | Dissertation | |
| dspace.entity.type | Publication | |
| local.contributor.advisor | Yu, Xingxing | |
| local.contributor.corporatename | College of Sciences | |
| local.contributor.corporatename | School of Mathematics | |
| relation.isAdvisorOfPublication | 3b32a3b5-5417-4c47-8a35-79346368e87f | |
| relation.isOrgUnitOfPublication | 85042be6-2d68-4e07-b384-e1f908fae48a | |
| relation.isOrgUnitOfPublication | 84e5d930-8c17-4e24-96cc-63f5ab63da69 | |
| thesis.degree.level | Doctoral |