The minrank of random graphs over arbitrary fields

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


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.


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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.


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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

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