Final doctoral examination and defense of dissertation of Youngho Yoo, June 10, 2022
Title: Erdos-Posa theorems for undirected group-labelled graphs
Thesis draft: https://people.math.gatech.edu/~yyoo41/thesis.pdf
Date: June 10, 2022
Time: 11:00am EST
Location: Skiles 006
Zoom link: https://gatech.zoom.us/j/96860495360?pwd=cktMRVVqMDRtVnJsb3ZLRll1bFRJQT09
Meeting ID: 968 6049 5360
Advisor: Dr. Xingxing Yu, School of Mathematics, Georgia Institute of Technology
Dr. Anton Bernshteyn, School of Mathematics, Georgia Institute of Technology
Dr. Grigoriy Blekherman, School of Mathematics, Georgia Institute of Technology
Dr. Chun-Hung Liu, Department of Mathematics, Texas A&M University
Dr. Mohit Singh, School of Industrial and Systems Engineering, Georgia Institute of Technology
Reader: Dr. Chun-Hung Liu, Department of Mathematics, Texas A&M University
Erdos and Posa 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 Erdos-Posa 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 Erdos-Posa 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 G and elements g for which A-paths of weight g satisfy the Erdos-Posa 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 Erdos-Posa 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 Erdos-Posa property of such cycles. This is a far-reaching generalization of a theorem of Reed that Escher walls are the only obstructions to the Erdos-Posa property of odd cycles. Consequently, we obtain a characterization of the sets of allowable weights in this setting for which the Erdos-Posa 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,M) for which cycles of length L mod M satisfy the Erdos-Posa 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 Erdos-Posa 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.