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A unified Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groupsJ. Pascal Gollin,
Kevin Hendrey,
O-joung Kwon,
Sang-il Oum,
Youngho Yoo, 2025, original scientific article
Abstract: In 1965, Erdős and Pósa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs (l, z) of integers where such a duality holds for the family of cycles of length l modulo z. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.
Keywords: Erdős-Pósa property, cycle packing, group-labelled graph
Published in RUP: 17.11.2025; Views: 355; Downloads: 8
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