A unified Erd\H{o}s-P\'{o}sa theorem for cycles in graphs labelled by multiple abelian groups

TL;DR

This paper characterizes when a duality between maximum cycle packings and minimum hitting sets holds in graphs labeled by multiple abelian groups, unifying and extending known results and providing new insights.

Contribution

It generalizes the Erdős-Pósa duality to cycles in labeled graphs with multiple abelian groups, characterizing all such cases and identifying obstructions.

Findings

Characterization of all pairs (ℓ, z) where duality holds for cycles of length ℓ modulo z.

Extension of duality results to graphs labeled with multiple abelian groups avoiding certain elements.

Identification of obstructions to duality and application to graphs on fixed surfaces.

Abstract

In 1965, Erd\H{o}s and P\'{o}sa 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 ${(\ell, z)}$ of integers where such a duality holds for the family of cycles of length $\ell$ 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.