The weak acyclic matching property in abelian groups

Abstract

A matching from a finite subset $A\subset {\mathbb{Z}}^n$ to another subset $B\subset {\mathbb{Z}}^n$ is a bijection $f:A\to B$ with the property that $a+f(a)$ never lies in $A$. A matching is called acyclic if it is uniquely determined by its multiplicity function. Alon et al. established the acyclic matching property for ${\mathbb{Z}}^n$, which was later extended to all abelian torsion-free groups. In a prior work, the authors of this paper settled the acyclic matching property for all abelian groups. The objective of this note is to explore a related concept, known as the weak acyclic matching property, within the context of abelian groups.

Remark 1.7 has been added. Several typos have been corrected. To appear in Séminaire Lotharingien de Combinatoire