Skip to main content
SpringerLink
Log in

On the Strong Chromatic Number of Random Hypergraphs

  • MATHEMATICS
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

We study the probability threshold for the property of strong colorability with a given number of colors of a random \(k\)-uniform hypergraph in the binomial model \(H(n,k,p)\). A vertex coloring of a hypergraph is said to be strong if any edge does not have two vertices of the same color under it. The problem of finding a sharp probability threshold for the existence of a strong coloring with q colors for \(H(n,k,p)\) is studied. By using the second moment method, we obtain fairly tight bounds for this quantity, provided that q is large enough in comparison with k.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. J. Schmidt, “Probabilistic analysis of strong hypergraph coloring algorithms and the strong chromatic number,” Discrete Math. 66, 258–277 (1987).

    Article  MathSciNet  Google Scholar 

  2. E. Shamir, “Chromatic number of random hypergraphs and associated graphs,” Adv. Comput. Res 5, 127–142 (1989).

    Google Scholar 

  3. M. Krivelevich and B. Sudakov, “The chromatic numbers of random hypergraphs,” Random Struct. Algorithms 12 (4), 381–403 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Łuczak, “The chromatic number of random graphs,” Combinatorica 11 (1), 45–54 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Hatami and M. Molloy, “Sharp thresholds for constraint satisfaction problems and homomorphisms,” Random Struct. Algorithms 33 (3), 310–332 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Achlioptas and A. Naor, “The two possible values of the chromatic number of a random graph,” Ann. Math. 162 (3), 1335–1351 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Coja-Oghlan, “Upper-bounding the k-colorability threshold by counting cover,” Electron. J. Combinatorics 20 (3), Research Paper No. 32 (2013).

  8. A. Coja-Oghlan and D. Vilenchik, “The chromatic number of random graphs for most average degrees,” Int. Math. Res. Notices 2016 (19), 5801–5859 (2015).

    Article  MATH  Google Scholar 

  9. A. E. Balobanov and D. A. Shabanov, “On the strong chromatic number of a random 3-uniform hypergraph,” Discrete Math. 344 (3), 112231 (2021).

  10. A. E. Khuzieva, “On strong colorings of 4-uniform random hypergraphs,” Trudy Mosk. Fiz.-Tekh. Inst. 11 (2), 91–107 (2019).

    Google Scholar 

  11. M. Dyer, A. Frieze, and C. Greenhill, “On the chromatic number of a random hypergraph,” J. Combinatorial Theory, Ser. B 113, 68–122 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Ayre, A. Coja-Oghlan, and C. Greenhill, “Hypergraph coloring up to condensation,” Random Struct. Algorithms 54 (4), 615–652 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  13. D. A. Shabanov, “Estimating the r-colorability threshold for a random hypergraph,” Discrete Appl. Math. 282, 168–183 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Semenov and D. Shabanov, “On the weak chromatic number of random hypergraphs,” Discrete Appl. Math. 276, 134–154 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. S. Semenov, “Two-colorings of a random hypergraph,” Theory Probab. Appl. 64 (1), 59–77 (2019).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The second and third authors acknowledge the support of the Russian Foundation for Basic Research, project no. 20-31-70039. The research of the third author was supported by the Russian Federation President Grant, project no. MD-1562.2020.1.

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991, Moscow, Russia

    T. G. Matveeva & D. A. Shabanov

  2. HSE University, 101000, Moscow, Russia

    A. E. Khuzieva & D. A. Shabanov

  3. Moscow Institute of Physics and Technology (National Research University), 141701, Dolgoprudnyi, Moscow oblast, Russia

    D. A. Shabanov

Authors
  1. T. G. Matveeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. E. Khuzieva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. A. Shabanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. A. Shabanov.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matveeva, T.G., Khuzieva, A.E. & Shabanov, D.A. On the Strong Chromatic Number of Random Hypergraphs. Dokl. Math. 105, 31–34 (2022). https://doi.org/10.1134/S1064562422010094

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562422010094

Keywords:

Search

Navigation