Erdos-Posa theorems for undirected group-labelled graphs

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Yoo, Youngho
Yu, Xingxing
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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.
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