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

TL;DR

This paper generalizes Erdős-Pósa duality to cycles in graphs labeled by multiple abelian groups, establishing a relationship between half-integral packings avoiding certain values and hitting sets for cycles with specified properties.

Contribution

It extends Reed's odd cycle analogue to a broad class of cycle properties encoded by abelian group labels, unifying various cycle duality results.

Findings

Established a duality between half-integral packings and hitting sets for cycles with group-label constraints.

Unified multiple cycle properties, including length, homology, and intersection conditions, under a common framework.

Proved a general Erdős-Pósa type theorem for cycles satisfying finitely many properties in labeled graphs.

Abstract

Erd\H{o}s and P\'{o}sa 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 $\ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times, and cycles contained in given $\mathbb{Z}_2$-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.