# The Half-integral Erd\"os-P\'osa Property for Non-null Cycles

## TL;DR

This paper proves that non-null cycles in group labeled graphs satisfy a half-integral Erdős-Pósa property, meaning either many such cycles can be packed with limited vertex overlap or a small vertex set intersects all of them.

## Contribution

The paper establishes the half-integral Erdős-Pósa property for non-null cycles in group labeled graphs, extending understanding of cycle packing and covering in this generalized setting.

## Key findings

- Non-null cycles have the half-integral Erdős-Pósa property.
- Existence of a function f(k) bounding the vertex set size for cycle intersection.
- Generalization of well-known cycle structures like odd cycles.

## Abstract

A Group Labeled Graph is a pair $(G,\Lambda)$ where $G$ is an oriented graph and $\Lambda$ is a mapping from the arcs of $G$ to elements of a group. A (not necessarily directed) cycle $C$ is called non-null if for any cyclic ordering of the arcs in $C$, the group element obtained by `adding' the labels on forward arcs and `subtracting' the labels on reverse arcs is not the identity element of the group. Non-null cycles in group labeled graphs generalize several well-known graph structures, including odd cycles. In this paper, we prove that non-null cycles on Group Labeled Graphs have the half-integral Erd\"os-P\'osa property. That is, there is a function $f:{\mathbb N}\to {\mathbb N}$ such that for any $k\in {\mathbb N}$, any group labeled graph $(G,\Lambda)$ has a set of $k$ non-null cycles such that each vertex of $G$ appears in at most two of these cycles or there is a set of at most $f(k)$ vertices that intersects every non-null cycle. Since it is known that non-null cycles do not have the integeral Erd\"os-P\'osa property in general, a half-integral Erd\"os-P\'osa result is the best one could hope for.