Counterexamples on matchings in hypergraphs and full rainbow matchings in graphs

TL;DR

This paper disproves several conjectures related to matchings in hypergraphs and full rainbow matchings in graphs, advancing understanding of these combinatorial structures.

Contribution

It provides counterexamples to existing conjectures on matchings in hypergraphs and rainbow matchings in graphs, challenging previous assumptions.

Findings

Refutes multiple conjectures on hypergraph matchings

Disproves conjectures on full rainbow matchings in graphs

Advances understanding of combinatorial matching structures

Abstract

A graph $G$ whose edges are coloured (not necessarily properly) contains a full rainbow matching if there is a matching $M$ that contains exactly one edge of each colour. We refute several conjectures on matchings in hypergraphs and full rainbow matchings in graphs, made by Aharoni and Berger and others.