The Minrank of Random Graphs over Arbitrary Fields

TL;DR

This paper establishes tight bounds on the minrank of random graphs over any field, resolving a long-standing problem and extending previous finite field results to arbitrary fields.

Contribution

It provides the first tight bounds for the minrank of Erdős–Rényi graphs over all fields, including the real numbers, for a wide range of edge probabilities.

Findings

Minrank of G(n,p) is Θ(n log(1/p)/log n) over any field.

Results hold for p between n^{-1} and 1 - n^{-0.99}.

Settles a problem posed by Knuth in 1994.

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 $M_{i,j}=0$ whenever $i$ and $j$ are distinct nonadjacent vertices of $G$. This notion, over the real field, arises in the study of the Lov\'asz 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 $n^{-1} \leq p \leq 1-n^{-0.99}$, the minrank of $G=G(n,p)$ over $\mathbb{F}$ is $\Theta(\frac{n \log (1/p)}{\log n})$ 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 $n^{O(1)}$, with tools from linear algebra, including an estimate of R\'onyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials.