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

@inproceedings{Gollin2021AUH, title={A unified half-integral Erd\H\{o\}s-P\'\{o\}sa theorem for cycles in graphs labelled by multiple abelian groups}, author={Jochen Pascal Gollin and Kevin Hendrey and Ken-ichi Kawarabayashi and O-joung Kwon and Sang-il Oum}, year={2021} }

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…

## References

SHOWING 1-10 OF 36 REFERENCES

Half-integral packing of odd cycles through prescribed vertices

- MathematicsComb.
- 2013

This paper discusses packing disjoint S-cycles, i.e., cycles that are required to go through a set S of vertices, and shows the Erdős-Pósa-type result for the half-integral packing of odd S- cycles, which is a generalization of Reed (1999) when S=V.

Packing cycles in undirected group-labelled graphs

- Mathematics
- 2020

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs $(G,\gamma)$ where $\gamma$ assigns to each edge of an undirected graph $G$ an element of…

On the presence of disjoint subgraphs of a specified type

- MathematicsJ. Graph Theory
- 1988

A general sufficient condition for a family of graphs to have the Erdos-Posa property is derived and a polynomially bounded algorithm for finding a cycle of length divisible by m is obtained.

Unboundedness for Generalized Odd Cyclic Transversality

- Mathematics
- 2009

All graphs considered in this note are finite and without multiple edges and, unless the contrary is clear from the context, without loops. In 1962 (1984) P. Erdos and L. P6sa (V. Neumann-Lara)…

Any arithmetic progressions covering the first 2ⁿ integers cover all integers

- Mathematics
- 1970

In 1958 S. Stein [7] defined a system of n congruences x = a< (mod bi), 1^-i^n, to be disjoint if no x satisfies more than one of them. He conjectured that for every disjoint system of n congruences…

Packing cycles with modularity constraints

- MathematicsComb.
- 2011

We prove that for all positive integers k, there exists an integer N =N(k) such that the following holds. Let G be a graph and let Γ an abelian group with no element of order two. Let γ: E(G)→Γ be a…

Parity Linkage and the Erdős–Pósa Property of Odd Cycles through Prescribed Vertices in Highly Connected Graphs

- MathematicsJ. Graph Theory
- 2017

This result is proved via an extension of Kawarabayashi and Reed's result about parity-k-linked graphs and it is easy to deduce several other well known results about the Erd\H{o}s-P\'osa property of odd cycles in highly connected graphs.

The Erdős–Pósa Property for Odd Cycles in Graphs of Large Connectivity

- MathematicsComb.
- 2001

It is proved that every -connected graph contains a totally odd -subdivision, that is, a subdivision of in which each edge of corresponds to an odd path, if and only if the deletion of any vertex leaves a nonbipartite graph.

A Unified Erdős–Pósa Theorem for Constrained Cycles

- MathematicsComb.
- 2019

The main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to (Γ1,Γ2)-labeled graphs and shows that the half-integral Erdős–Pósa property holds for cycles not homologous to zero, odd cycles nothomologousto zero, and S-cycles not homologueous to 0 on an orientable surface.