Counterexamples to conjectures on strong maximality and minimality
Combinatorics
2025-11-18 v1
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.
Cite
@article{arxiv.2511.13709,
title = {Counterexamples to conjectures on strong maximality and minimality},
author = {Lawrence Hollom and Benedict Randall Shaw},
journal= {arXiv preprint arXiv:2511.13709},
year = {2025}
}
Comments
12 pages, 2 figures