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arXiv · 2511.13709

Counterexamples to conjectures on strong maximality and minimality

Lawrence Hollom·Benedict Randall Shaw
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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 Sprüssel; Aharoni and Korman; and Tardos.

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Lawrence Hollom, Benedict Randall Shaw. 2025-11-17. Counterexamples to conjectures on strong maximality and minimality. https://arxiv.org/abs/2511.13709

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