Title:
Erdos-Posa theorems for undirected group-labelled graphs
Erdos-Posa theorems for undirected group-labelled graphs
| dc.contributor.advisor | Yu, Xingxing | |
| dc.contributor.author | Yoo, Youngho | |
| dc.contributor.committeeMember | Bernshteyn, Anton | |
| dc.contributor.committeeMember | Blekherman, Grigoriy | |
| dc.contributor.committeeMember | Liu, Chun-Hung | |
| dc.contributor.committeeMember | Singh, Mohit | |
| dc.contributor.department | Mathematics | |
| dc.date.accessioned | 2022-08-25T13:34:03Z | |
| dc.date.available | 2022-08-25T13:34:03Z | |
| dc.date.created | 2022-08 | |
| dc.date.issued | 2022-06-14 | |
| dc.date.submitted | August 2022 | |
| dc.date.updated | 2022-08-25T13:34:03Z | |
| dc.description.abstract | Erdős and Pósa proved in 1965 that cycles satisfy an approximate packing-covering duality. Finding analogous approximate dualities for other families of graphs has since become a highly active area of research due in part to its algorithmic applications. In this thesis we investigate the Erdős-Pósa property of various families of constrained cycles and paths by developing new structural tools for undirected group-labelled graphs. Our first result is a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs. This structure theorem is then used to prove the Erdős-Pósa property of A-paths of length 0 modulo p for a fixed odd prime p, answering a question of Bruhn and Ulmer. Further, we obtain a characterization of the abelian groups Γ and elements l ∈ Γ for which A-paths of weight l satisfy the Erdős-Pósa property. These results are from joint work with Robin Thomas. We extend our structural tools to graphs labelled by multiple abelian groups and consider the Erdős-Pósa property of cycles whose weights avoid a fixed finite subset in each group. We find three types of topological obstructions and show that they are the only obstructions to the Erdős-Pósa property of such cycles. This is a far-reaching generalization of a theorem of Reed that Escher walls are the only obstructions to the Erdős-Pósa property of odd cycles. Consequently, we obtain a characterization of the sets of allowable weights in this setting for which the Erdős-Pósa property holds for such cycles, unifying a large number of results in this area into a general framework. As a special case, we characterize the integer pairs (l, z) for which cycles of length l mod z satisfy the Erdős-Pósa property. This resolves a question of Dejter and Neumann-Lara from 1987. Further, our description of the obstructions allows us to obtain an analogous characterization of the Erdős-Pósa property of cycles in graphs embeddable on a fixed compact orientable surface. This is joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum. | |
| dc.description.degree | Ph.D. | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | http://hdl.handle.net/1853/67225 | |
| dc.language.iso | en_US | |
| dc.publisher | Georgia Institute of Technology | |
| dc.subject | Graph theory, combinatorics | |
| dc.title | Erdos-Posa theorems for undirected group-labelled graphs | |
| 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 |