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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.13709 |
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| _version_ | 1866918206394859520 |
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| author | Hollom, Lawrence Shaw, Benedict Randall |
| author_facet | Hollom, Lawrence Shaw, Benedict Randall |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13709 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counterexamples to conjectures on strong maximality and minimality Hollom, Lawrence Shaw, Benedict Randall Combinatorics 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. |
| title | Counterexamples to conjectures on strong maximality and minimality |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.13709 |