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A Fan-Type Theorem for Hamiltonian Berge Cycles in 3-Uniform Hypergraphs — arXiv:2610.41577

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.41577
License (as archived): CC0-1.0
Archived: 2026-11-02T05:00:38.235Z
SHA-256 of the archived text: 2153c2b5774436c1796f955152b4450b725857caa58df47e95bc2dc0d349166d
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How to cite

A Fan-Type Theorem for Hamiltonian Berge Cycles in 3-Uniform Hypergraphs — arXiv:2610.41577 — arXiv (en), https://arxiv.org/abs/2610.41577 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ba2ca3b9cebd (Metraipolis archive dc_ba2ca3b9cebd, sha256 2153c2b5774436c1796f955152b4450b725857caa58df47e95bc2dc0d349166d)

BibTeX

@misc{dcba2ca3b9cebd,
  title = {A Fan-Type Theorem for Hamiltonian Berge Cycles in 3-Uniform Hypergraphs — arXiv:2610.41577},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.41577},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ba2ca3b9cebd; sha256 2153c2b5774436c1796f955152b4450b725857caa58df47e95bc2dc0d349166d},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.41577. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $H$ be a simple $3$-uniform hypergraph on $n\ge7$ vertices, and put $\tau_n=\binom{\lfloor(n-1)/2\rfloor}{2}$. We prove that $H$ has a Hamiltonian Berge cycle if its $2$-shadow is $2$-connected, every pair at Berge distance two has an endpoint of hypergraph degree at least $\tau_n+1$, and every pair of codegree one has both endpoints above $\tau_n$. The distance-two threshold is sharp, and the order range cannot be extended to $n=4,5,6$. The proof separates two structural arguments. A low-degree endpoint of an extremal longest Berge path forces Hamiltonicity, independently of the order and the degree threshold. For high-degree endpoints, injective charging and its equality cases force pairs of codegree one that permit further rotations. The cases $n=7,8$ are settled by exhaustive verification, with a completeness proof and a second implementation using direct vertex--hyperedge cycle search. Keywords: Hamiltonian Berge cycle; Fan-type condition; 3-uniform hypergraph; longest-path rotation; exhaustive verification.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.41577 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-02T04:47:57.689Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).