A unified Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups
J. Pascal Gollin, Kevin Hendrey, O‐joung Kwon, Sang‐il Oum, Youngho Yoo · Mathematische Annalen · 2025
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 $${(\ell , z)}$$ ( ℓ , z ) of integers where such a duality holds for the family of cycles of length $$\ell $$ 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.