Counterexamples to conjectures on strong maximality and minimality

TL;DR

This paper constructs counterexamples in infinite graphs and hypergraphs that disprove several conjectures related to strong maximality and minimality, resolving multiple open questions in the field.

Contribution

It provides explicit counterexamples to conjectures on strongly maximal matchings and minimal covers in infinite hypergraphs and graphs, advancing understanding of these structures.

Findings

Constructed 3-uniform hypergraphs without strongly maximal matchings

Built graphs with no strongly minimal colourings

Resolved multiple conjectures and open questions in the field

Abstract

We provide counterexamples to several conjectures concerning strongly maximal and strongly minimal structures in infinite graphs and hypergraphs. In particular, we construct 3-uniform hypergraphs without strongly maximal matchings and without strongly minimal covers, and from our construction for covers we build a graph with no strongly minimal colouring. We also consider several refinements of these problems. Our results resolve conjectures and questions of Aharoni; Aharoni and Berger; Aharoni, Berger, Georgakopoulos, and Spr\"{u}ssel; Aharoni and Korman; and Tardos.