A unified half-integral Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groups

Abstract

Erdős and Pósa proved in 1965 that there is a 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 if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example, cycles of length at least (Formula presented.), cycles of length (Formula presented.) modulo (Formula presented.), cycles intersecting a prescribed set of vertices at least (Formula presented.) times and cycles contained in given (Formula presented.) -homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.