Skip to content
DraconiticResearch
Search Sources API
197,761 studies31 sourcesUpdated 46 min ago
PreprintCC BY 4.0arXiv

Tuza's Ryser-conjecture claim for four-partite hypergraphs with matching number two

  • Patrick White
Open at sourcePDFDOICite

Abstract

We prove that every $4$-partite $4$-uniform hypergraph H with matching number ν(H)=2 satisfies τ(H)≤ 6, where τ denotes the vertex-cover number. This confirms a claim made by Tuza in his 1979 manuscript but never published with a proof, and closes the case (r,ν)=(4,2) of Ryser's conjecture. The best previous bound was τ≤ 7, an integrality consequence of the theorem of Haxell and Scott (2012). The proof uses Gyárfás's intersecting-case theorem (τ≤ 3 for intersecting $4$-partite $4$-uniform families), a short projection lemma (four base-disjoint edges in an intersecting family force a two-element cover), and Kőnig's matching theorem.

Related

  1. Perfect Matching in k-Partite k-Uniform Hypergraphs
    Jie Han, Hongliang Lu, Bin Wang +1
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Towards the Erdős matching conjecture for 4‐uniform hypergraphs: Stability and applications
    Péter Frankl, Hongliang Lu, Jie Ma +1
    MathematikaPeer-reviewedLicence not statedFull text11 citedCombinatorics & discrete mathematicsOpenAlex
  3. The Erdős Matching Conjecture for 4-uniform hypergraphs
    Jianfeng Hou, Caiyun Hu, Xizhi Liu
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Turán problems with bounded matching number in k-uniform hypergraphs
    Jialin Liu, Mingyang Guo, Xiumei Wang
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. On the chromatic number of pseudohemisphere hypergraphs
    Balázs István Szabó
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{white2026tuzas,
  title = {{Tuza's Ryser-conjecture claim for four-partite hypergraphs with matching number two}},
  author = {Patrick White},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14281},
  eprint = {2609.14281},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14281},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Tuza's Ryser-conjecture claim for four-partite hypergraphs with matching number two
AU  - Patrick White
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14281
UR  - https://arxiv.org/abs/2609.14281
AB  - We prove that every $4$-partite $4$-uniform hypergraph H with matching number ν(H)=2 satisfies τ(H)≤ 6, where τ denotes the vertex-cover number. This confirms a claim made by Tuza in his 1979 manuscript but never published with a proof, and closes the case (r,ν)=(4,2) of Ryser's conjecture. The best previous bound was τ≤ 7, an integrality consequence of the theorem of Haxell and Scott (2012). The proof uses Gyárfás's intersecting-case theorem (τ≤ 3 for intersecting $4$-partite $4$-uniform families), a short projection lemma (four base-disjoint edges in an intersecting family force a two-element cover), and Kőnig's matching theorem.
ER  -