Advertisement

The minrank of random graphs over arbitrary fields

  • Noga Alon
  • Igor Balla
  • Lior GishbolinerEmail author
  • Adva Mond
  • Frank Mousset
Article
  • 5 Downloads

Abstract

The minrank of a graph G on the set of vertices [n] over a field \(\mathbb{F}\) is the minimum possible rank of a matrix \(M \in \mathbb{F}^{{n}\times{n}}\) with nonzero diagonal entries such that Mi,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field \(\mathbb{F}=\mathbb{F}(n)\) and every p = p(n) satisfying n-1p ≤ 1 - n-0.99, the minrank of G = G(n, p) over \(\mathbb{F}\) is \(\Theta(\frac{n {\rm{log}}(1/p)}{{\rm{log}} n})\) with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev and Weinstein, who proved the above result for finite fields of size at most nO(1), with tools from linear algebra, including an estimate of Rónyai, Babai and Ganapathy for the number of zero-patterns of a sequence of polynomials.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We thank Peter Nelson for telling us about his paper [16]. The research on this project was initiated during a joint research workshop of Tel Aviv University and the Free University of Berlin on Graph and Hypergraph Coloring Problems, held in Berlin in August 2018, and supported by a GIF grant number G-1347-304.6/2016. We would like to thank the German–Israeli Foundation (GIF) and both institutions for their support.

References

  1. [1]
    N. Alon, The Shannon capacity of a union, Combinatorica 18 (1998), 1998–301.MathSciNetCrossRefGoogle Scholar
  2. [2]
    N. Alon, Lovász, vectors, graphs and codes, manuscript https://www.tau.ac.il/~nogaa/PDFS/ll70.pdf.
  3. [3]
    N. Alon and A. Kupavskii, Two notions of unit-distance graphs, Journal of Combinatorual Theory. Series A 125 (2014), 2014–1.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Z. Bar-Yossef, Y. Birk, T. S. Jayram and T. Kol, Index coding with side information, IEEE Transactions on Information Theory 57 (2011), 2011–1479.MathSciNetCrossRefGoogle Scholar
  5. [5]
    B. Bollobás, The chromatic number of random graphs, Combinatorica 8 (1988), 1988–49.MathSciNetCrossRefGoogle Scholar
  6. [6]
    A. Golovnev, O. Regev and O. Weinstein, The minrank of random graphs, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Leibniz International Proceedings in Informatics, Vol. 81, Schloss Dagstuhl–Leibniz- Zentrum fuer Informatik, Dagstuhl, 2017, Art. no. 46.Google Scholar
  7. [7]
    C. Grosu, Fp is locally like C, Journal of the London Mathematical Society 89 (2014), 2014–724.CrossRefGoogle Scholar
  8. [8]
    W. Haemers, An upper bound for the Shannon capacity of a graph, in Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978), Colloquia Mathematica Societatis János Bolyai, Vol. 25, North-Holland, Amsterdam–New York, 1981, pp. 267–272.Google Scholar
  9. [9]
    I. Haviv, On minrank and forbidden subgraphs, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Leibniz International Proceedings in Informatics, Vol. 116, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, 2018, Art. no. 42.Google Scholar
  10. [10]
    I. Haviv and M. Langberg, On linear index coding for random graphs, in 2012 IEEE International Symposium on Information Theory Proceedings, Institute of Electrical and Electronics Engineers, New York, 2012, pp. 2231–2235.CrossRefGoogle Scholar
  11. [11]
    S. Janson, T. Łuczak and A. Ruciński, Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000.Google Scholar
  12. [12]
    F. Juhász, The asymptotic behaviour of Lovász’ v function for random graphs, Combinatorica 2 (1982), 1982–153.CrossRefGoogle Scholar
  13. [13]
    D. E. Knuth, The sandwich theorem, Electronic Journal of Combinatorics 1 (1994), Art. no. 1.Google Scholar
  14. [14]
    L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory 25 (1979), 1979–1.MathSciNetCrossRefGoogle Scholar
  15. [15]
    E. Lubetzky and U. Stav, Non-linear index coding outperforming the linear optimum, IEEE Transactions on Information Theory 55 (2009), 2009–3544.CrossRefGoogle Scholar
  16. [16]
    P. Nelson, Almost all matroids are nonrepresentable, Bulletin of the London Mathematical Society 50 (2018), 2018–245.MathSciNetCrossRefGoogle Scholar
  17. [17]
    L. Rónyai, L. Babai and M. K. Ganapathy, On the number of zero-patterns of a sequence of polynomials, Journal of the American Mathematical Society 14 (2001), 2001–717.MathSciNetCrossRefGoogle Scholar
  18. [18]
    T. Tao, Rectification and the Lefschetz principle, 14 March 2013, http://terrytao.wordpress.com/2013/03/14/rectification-and-the-lefschetz-principle. Google Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Noga Alon
    • 1
    • 2
  • Igor Balla
    • 3
  • Lior Gishboliner
    • 4
    Email author
  • Adva Mond
    • 4
  • Frank Mousset
    • 4
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Schools of Mathematics and Computer ScienceTel Aviv UniversityRamat AvivIsrael
  3. 3.Department of MathematicsETH ZürichZurichSwitzerland
  4. 4.School of Mathematical SciencesTel Aviv UniversityRamat Aviv, Tel AvivIsrael

Personalised recommendations