Zero $A$-paths and the Erd\H{o}s-P\'osa property

TL;DR

This paper characterizes $A$-paths with zero weight in Abelian groups that have the Erdős-Pósa property, extending to long paths and zero cycles, with structural results on zero walls and linkages.

Contribution

It provides a complete characterization of zero-weight $A$-paths with the Erdős-Pósa property in Abelian groups, including long paths and cycles, using structural graph theory techniques.

Findings

Zero $A$-paths with weight zero have the Erdős-Pósa property.

Characterization extends to paths of minimum length and zero cycles.

Structural results on zero walls and linkages support main theorems.

Abstract

Let $\Gamma$ be an Abelian group. In this paper I characterize the $A$-paths of weight $0\in\Gamma$ that have the Erd\H{o}s-P\'osa property. Using this in an auxiliary graph, one can also easily characterize the $A$-paths of weight $\gamma\in\Gamma$ that have the Erd\H{o}s-P\'osa property. These results also extend to long paths, that is paths of some minimum length. A structural result on zero walls with non-zero linkages will be proven as a means to prove the main result of this paper. This immediately implies that zero cycles with respect to an Abelian group $\Gamma$ have the Erd\H{o}s-P\'osa property.