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Graphs and Combinatorics

, Volume 35, Issue 4, pp 933–939 | Cite as

A Note on Saturation for Berge-G Hypergraphs

  • Maria AxenovichEmail author
  • Christian Winter
Original Paper
  • 33 Downloads

Abstract

For a graph \(G=(V,E)\), a hypergraph H is called Berge-G if there is a hypergraph \(H'\), isomorphic to H, so that \(V(G)\subseteq V(H')\) and there is a bijection \(\phi : E(G) \rightarrow E(H')\) such that for each \(e\in E(G)\), \(e \subseteq \phi (e)\). The set of all Berge-G hypergraphs is denoted \(\mathcal {B}(G)\). A hypergraph H is called Berge-Gsaturated if it does not contain any subhypergraph from \(\mathcal {B}(G)\), but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Since each Berge-G hypergraph contains |E(G)| hypergedges, it follows that each Berge-G saturated hypergraph must have at least \(|E(G)|-1\) edges. We show that for each graph G that is not a certain star and for any \(n\ge |V(G)|\), there are Berge-G saturated hypergraphs on n vertices and exactly \(|E(G)|-1\) hyperedges. This solves a problem of finding a saturated hypergraph with the smallest number of edges exactly.

Keywords

Saturation Berge-hypergraph Non-uniform 

Notes

Acknowledgements

We thank Casey Tompkins for useful discussions and carefully reading the manuscript. Research of the second author was supported in part by Talenx stipendium.

References

  1. 1.
    Axenovich, M., Gyárfás, A.: A note on Ramsey numbers for Berge-G hypergraphs. Discrete Math. 342(5), 1245–1252 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    English, S., Graber, N., Kirkpatrick, P., Methuku, A., Sullivan, E.C.: Saturation of Berge hypergraphs (2017). arXiv:1710.03735
  3. 3.
    English, S., Gerbner, D., Methuku, A., Tait, M.: Linearity of saturation for Berge hypergraphs. European J. Combin. 78, 205–213 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Faudree, J., Faudree, R., Schmitt, J.: A survey of minimum saturated graphs. Electron. J. Combin. 1000, 19–29 (2011)zbMATHGoogle Scholar
  5. 5.
    Gerbner, D., Palmer, C.: Extremal results for Berge-hypergraphs. SIAM J. Discrete Math. 31(4), 2314–2327 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grósz, D., Methuku, A., Tompkins, C.: Uniformity thresholds for the asymptotic size of extremal Berge-\(F\)-free hypergraphs (2018). arXiv:1803.01953v1
  7. 7.
    Gyárfás, A., Lehel, J., Sárközy, G.N., Schelp, R.H.: Monochromatic Hamiltonian Berge cycles in colored complete uniform hypergraphs. J. Combin. Theory B 98, 342–358 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gyárfás, A., Sárközy, G.N.: The \(3\)-colour Ramsey number of a \(3\)-uniform Berge cycle. Combin. Probab. Comput. 20, 53–71 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Győri, E.: Triangle-free hypergraphs. Combin. Probab. Comput. 15, 185–191 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kászonyi, L., Tuza, Z.: Saturated graphs with minimal number of edges. J. Graph Theory 10(2), 203–210 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Palmer, C., Tait, M., Timmons. C., Wagner, A. Z.: Turán numbers for Berge-hypergraphs and related extremal problems. Discrete Math. 342(6), 1553–1563 (2019)Google Scholar
  12. 12.
    Pikhurko, O.: The minimum size of saturated hypergraphs. Combin. Probab. Comput. 8(5), 483–492 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pikhurko, O.: Results and open problems on minimum saturated hypergraphs. ARS Combin. 72, 111–127 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Winter, C.: Berge saturation of non-uniform hypergraphs, Bachelor Thesis. Karlsruhe Institute of Technology (2018)Google Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany

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